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Thread: Why is space infinite?

  1. #1 Why is space infinite? 
    Senior Member MaxPayne's Avatar
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    They say it all started just 13.7 billion years ago. Can't we calculate the size of the Universe from that data?

    Surely it could never be less than 13.7 billion lys, and due to accelerated expansion it could be even larger. But it still ends up falling short of infinity.

    So why is Universe infinite for you?... Any particular reason other than "coz Einstein said so"?did he really?
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    Quote Originally Posted by MaxPayne View Post
    They say it all started just 13.7 billion years ago. Can't we calculate the size of the Universe from that data?

    Surely it could never be less than 13.7 billion lys, and due to accelerated expansion it could be even larger. But it still ends up falling short of infinity.

    So why is Universe infinite for you?... Any particular reason other than "coz Einstein said so"?did he really?
    When you say the universe, are you referring to space generally or only the part of space that has matter in it? I can't see how space can be anything other than infinite otherwise we would have to ask the question "what lies beyond the end?" The material universe however I would think is finite.
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    No prefered frame of reference. Finite yet unbounded. Olber's paradox.
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    Quote Originally Posted by MaxPayne View Post
    They say it all started just 13.7 billion years ago. Can't we calculate the size of the Universe from that data? Surely it could never be less than 13.7 billion lys, and due to accelerated expansion it could be even larger.
    You are actually referring to the observable universe which, when the age of the universe and the expansion rates are taken into account, imply an observable universe of about 46 billion light-years in radius.

    But it still ends up falling short of infinity.

    So why is Universe infinite for you?... Any particular reason other than "coz Einstein said so"?did he really?
    We don't know if the universe is infinite, so your question is presuming something that isn't so. However, there is a class of solutions to Einstein's field equations which, when fed our best experimental data, yields an infinite size for the universe. That may be what you're thinking of.
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    Greetings again, MaxPayne.

    As far as I understand it, pikpobedy is correct in that Einstein's universe is "finite but unbounded", like a toy train on a circular track, the train circling the track endlessly and repetitively. He is also right in that there is no preferred reference frame. Hence Einstein does not teach an infinite universe - unless you redefine the meaning of 'infinite.'

    Newton's universe was spatially infinite but practically finite since Newton posited a finite amount of matter distributed in that ring we call the Milky Way.

    The point is that we cannot detect infinity observationally, hence philosophy is necessarily introduced into this question.

    Hence too I cannot discuss this further on the main forums or I will be banned yet again. You can start a post on the Personal & Alternative Theories forum and I can discuss it with you freely there!

    TFOLZO
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    Quote Originally Posted by TFOLZO View Post
    The point is that we cannot detect infinity observationally, hence philosophy is necessarily introduced into this question.
    You are too eager to inject philosophy, perhaps to admit less-developed ideas that you favor personally. But you should not overlook the scientific alternative, which is to state the current observational evidence. And that evidence has allowed us to determine the radius of curvature of the universe to exceed 70 or so billion light-years. The error bars in the measurements are large enough to permit a flat Euclidean universe, as well as one with positive or negative curvature. The maths are simplest for the Euclidean case, so that's a common, but not necessary, choice. As more observational evidence is gathered, the error bars will tighten up, as they have continually over the last half-century or thereabouts.
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    Ha! Yes! Indeed tk421!
    Quote Originally Posted by tk421 View Post
    You are too eager to inject philosophy, perhaps to admit less-developed ideas that you favor personally. But you should not overlook the scientific alternative, which is to state the current observational evidence. And that evidence has allowed us to determine the radius of curvature of the universe to exceed 70 or so billion light-years. The error bars in the measurements are large enough to permit a flat Euclidean universe, as well as one with positive or negative curvature. The maths are simplest for the Euclidean case, so that's a common, but not necessary, choice. As more observational evidence is gathered, the error bars will tighten up, as they have continually over the last half-century or thereabouts.
    In the earliest editions of Einstein's RSGT Einstein described the universe as 26,000,000 lightyears in circumference! It has been expanding very rapidly ever since - and I can't see any reason for that to change!

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    What does "space is infinte" mean?
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    If you accept the speed of light is the maximum speed limit, then according to Big Bang Theory, the maximum distance you can ever travel is the outer limit of a Hubble Sphere.

    Well.... kind of. You would never reach that point because the universe will keep expanding too fast for you to ever reach it.

    So, in that sense the universe's size is totally infinite. Big Bang Theory proposes that the universe's size is not infinite now, but after an infinite period of time it will be.

    Taking all these considerations together, we arrive at the conclusion that an individual observer must experience the universe to be infinite. For the simple reason that an observer cannot travel fast enough to hope to reach the outer edge before it expands away.
    A mathematician and an engineer were at a party. An older colleague of theirs was there with his daughter. The two each asked if they could speak to her. He said it was ok, but they had to approach her by going half way across the room, then stop, then half way again and stop and proceed in that manner. The mathematician realized that he would never reach her and gave up. The engineer determined that he could get close enough to talk. --Approximate retelling of a joke by my math teacher.
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    Quote Originally Posted by Kojax View Post
    If you accept the speed of light is the maximum speed limit, then according to Big Bang Theory, the maximum distance you can ever travel is the outer limit of a Hubble Sphere.

    Well.... kind of. You would never reach that point because the universe will keep expanding too fast for you to ever reach it.

    So, in that sense the universe's size is totally infinite. Big Bang Theory proposes that the universe's size is not infinite now, but after an infinite period of time it will be.

    Taking all these considerations together, we arrive at the conclusion that an individual observer must experience the universe to be infinite. For the simple reason that an observer cannot travel fast enough to hope to reach the outer edge before it expands away.
    Very good point. Size depends upon an imposed coordinate system that is not unique. So knowing this already, those clever mathematicians talk about other things like bounded, open and closed.
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    Quote Originally Posted by Kojax View Post
    ...Big Bang Theory proposes that the universe's size is not infinite now
    Sounds good to me.

    Quote Originally Posted by Kojax
    ...but after an infinite period of time it will be.
    Or in other words, it will never ever be infinite.
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    Yeah. On the one hand, it will never be infinite.

    But on the other hand, it is important to remember that it is only finite from the perspective of a "privileged observer". In physics, there is no such thing as a privileged observer. Only non-privileged observers.

    All the non-privileged observers agree that they can only see as far as the Hubble Sphere, and they don't know what may lie beyond it. Furthermore, they agree that it is impossible to reach beyond the edge of the Hubble Sphere by any means.
    A mathematician and an engineer were at a party. An older colleague of theirs was there with his daughter. The two each asked if they could speak to her. He said it was ok, but they had to approach her by going half way across the room, then stop, then half way again and stop and proceed in that manner. The mathematician realized that he would never reach her and gave up. The engineer determined that he could get close enough to talk. --Approximate retelling of a joke by my math teacher.
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    Why is space infinite?
    The pertinent question should be: Why would space be finite? To be finite, space must either stop somewhere or loop back to some remote part of itself. This would require an explanation beyond space simply not doing either of those thing and just continuing along. An analogy of the point I'm making would be computer code that contains a loop. If the code within the loop doesn't change the state of the system, then the code will never stop, which is the same as saying that it will run for an infinite amount of time. One thing to note is that saying the code "never stops" is a somewhat local way of saying that the code "runs for an infinite amount of time", which is the equivalent global notion. The essential point of the analogy is that the code within the loop need not do anything for the state of system not to change. In other words, if the code within the loop does nothing, the code will run forever. It requires an explicit change in the state of the system for the code to stop. That is, the code needs a reason to stop. It doesn't need a reason to continue running forever. Getting back to space, it needs a reason to be finite. It doesn't need a reason to be infinite. Thinking that the energy of the universe is somehow limited is a mundane way of thinking and one should not apply mundane notions to the universe.
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    Kojax: IMHO it's either finite or it's not, regardless of whether there's any observers.

    KJW: yes, to be finite space must either stop or loop back on itself. But note that your code analogy relates to time rather than space, and one could equally assert that space has to have a reason to be infinite. As for why I think it's finite, take a look at the stress-energy-momentum tensor. See the pressure diagonal? And the shear stress? You could liken space to a squeezed-down stress-ball which expands because it's under pressure. If space was infinite, the pressure would be counterbalanced at all locations. Space would not expand.

    I think the expansion of space is the evidence that it isn't finite. As for the end of space or the edge of space, I don't know. But I'd rather think about that than be told there's an infinite number of John Duffields tapping away on their laptops out there.
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    Quote Originally Posted by Farsight View Post
    KJW: yes, to be finite space must either stop or loop back on itself. But note that your code analogy relates to time rather than space, and one could equally assert that space has to have a reason to be infinite.
    My analogy was just an analogy. It really isn't about space or time as such, but about trivial vs non-trivial topologies (with the trivial topology being infinite). In other words, an infinite space is simpler to define. That is why the "Why?" question should be applied to a finite space rather than an infinite space.


    Quote Originally Posted by Farsight View Post
    As for why I think it's finite, take a look at the stress-energy-momentum tensor. See the pressure diagonal? And the shear stress? You could liken space to a squeezed-down stress-ball which expands because it's under pressure. If space was infinite, the pressure would be counterbalanced at all locations. Space would not expand.
    I think you need to have a look at the Friedmann equations as it appears you misunderstand the relationship between pressure and the expansion of the universe.
    A tensor equation that is valid in any coordinate system is valid in every coordinate system.
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    Quote Originally Posted by KJW View Post
    My analogy was just an analogy. It really isn't about space or time as such, but about trivial vs non-trivial topologies (with the trivial topology being infinite). In other words, an infinite space is simpler to define. That is why the "Why?" question should be applied to a finite space rather than an infinite space.
    You've missed the obvious here. One could define a trivial topology that is finite. Where space has an edge. There is no space beyond the edge of space. Light waves undergo total internal reflection or something. Ditto for matter waves.

    Quote Originally Posted by KJW
    I think you need to have a look at the Friedmann equations as it appears you misunderstand the relationship between pressure and the expansion of the universe.
    I've looked at this previously. Note that it starts with the simplifying assumption that the universe is spatially homogeneous. If you've seen this paper it may occur to you that a universe that is spatially homogeneous is a flat universe. It may also occur to you that two out of the three solutions were always going to be wrong.

    As far as we know from WMAP and then Planck space is "flat", and we are not privileged. We can reason that observers a billion years ago would have deemed that space was flat and that they were not privileged. And they can reason that observers a billion years ago would have deemed that space was flat and that they were not privileged. You can take this all the way back, winding everything back to a universe the size of a grapefruit. In recent texts the observable universe is wound back to the size of a grapefruit. But it used to be the universe. A grapefruit is a simple sphere, or a "ball" in mathematics. With an edge.

    Yes, it's unfamiliar to you. But for myself, I will not accept a universe that was already infinite when the big bang occurred.
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    Quote Originally Posted by Farsight View Post
    Kojax: IMHO it's either finite or it's not, regardless of whether there's any observers.


    Distance and time only exist in the presence of matter. It appears that all matter is incapable of reaching the edge of its own Hubble Sphere.

    It's not perfect logic, but I think it more or less can be inferred from that, that there may not be any fact of the matter at all. Matter tells space how big it is. Not the other way around.
    A mathematician and an engineer were at a party. An older colleague of theirs was there with his daughter. The two each asked if they could speak to her. He said it was ok, but they had to approach her by going half way across the room, then stop, then half way again and stop and proceed in that manner. The mathematician realized that he would never reach her and gave up. The engineer determined that he could get close enough to talk. --Approximate retelling of a joke by my math teacher.
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    Sorry Kojax, I don't think that helps much. IMHO the universe is either finite or not, regardless of whether there's any observers or matter.
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    Quote Originally Posted by Farsight View Post
    You've missed the obvious here. One could define a trivial topology that is finite. Where space has an edge.
    Sorry, my use of the term "trivial topology" was not technically correct. However, I did consider both the space that loops back to a remote part of itself and the space with an edge. Both of these involve a topological structure that is more complicated than .


    Quote Originally Posted by Farsight View Post
    a universe that is spatially homogeneous is a flat universe.
    No. All three solutions are both homogenous and isotropic.


    Quote Originally Posted by Farsight View Post
    In recent texts the observable universe is wound back to the size of a grapefruit. But it used to be the universe. A grapefruit is a simple sphere, or a "ball" in mathematics. With an edge.
    But it is the observable universe that is finite with an edge because we can't observe beyond the cosmic microwave background. But that doesn't mean that beyond the cosmic microwave background doesn't exist. Indeed, beyond the cosmic microwave background corresponds to a time before recombination which occurred when the universe was approximately 379,000 years old.


    Quote Originally Posted by Farsight View Post
    Yes, it's unfamiliar to you. But for myself, I will not accept a universe that was already infinite when the big bang occurred.
    I wouldn't say it's "unfamiliar" to me. However, while I do have a strong preference towards the spatial topology, I haven't actually been arguing about this preference. Instead, I've been arguing about the "naturalness" of the topology, noting that "naturalness" is not a substitute for empirical evidence.
    A tensor equation that is valid in any coordinate system is valid in every coordinate system.
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    Quote Originally Posted by KJW View Post
    Sorry, my use of the term "trivial topology" was not technically correct. However, I did consider both the space that loops back to a remote part of itself and the space with an edge. Both of these involve a topological structure that is more complicated than .
    OK noted. I guess it comes back to this: I will not accept a universe that was already infinite when the big bang occurred. It's turtles all the way down, a non-answer, not much better than God did it.

    Quote Originally Posted by KJW View Post
    No. All three solutions are both homogenous...
    IMHO the nub of the issue is that if space is homogeneous, light goes straight. Really straight. Then there is no intrinsic curvature. There is no looping back.

    Quote Originally Posted by KJW
    ...and isotropic
    I've seen this used to justify a universe which looks the same in every direction to all observers. Even though the cosmological principle is only a principle. Even though we don't know that there aren't some observers out there for whom half the sky is totally dark.

    Quote Originally Posted by KJW
    But it is the observable universe that is finite with an edge because we can't observe beyond the cosmic microwave background. But that doesn't mean that beyond the cosmic microwave background doesn't exist.
    Agreed. And nor does it mean that beyond our horizon, space continues on and on forever.

    Quote Originally Posted by KJW
    Indeed, beyond the cosmic microwave background corresponds to a time before recombination which occurred when the universe was approximately 379,000 years old.
    No problem.

    Quote Originally Posted by KJW
    I wouldn't say it's "unfamiliar" to me. However, while I do have a strong preference towards the spatial topology, I haven't actually been arguing about this preference. Instead, I've been arguing about the "naturalness" of the topology, noting that "naturalness" is not a substitute for empirical evidence.
    OK noted. I'm not a fan of naturalness myself.

    By the by, I think there's a clue to it all in dark energy and conservation of energy. You'll have seen the balloon analogy on the Discovery Channel. It isn't ideal, but there's something about it that is worth noting: the pressure of the air within the balloon is counterbalanced by the tension within the skin. If you have an inflated balloon (in a vacuum of course), there are two ways to make it bigger. One is to blow more air into it, which is akin to adding more energy. Another way is to make the skin weaker. Tension is of course negative pressure. And if you make the skin weaker the balloon gets bigger, but then the skin gets even weaker. Then it's not so much a balloon analogy, as this. Maybe that's unfamiliar, but see [0912.2678] New Physics at Low Accelerations (MOND): an Alternative to Dark Matter where Milgrom says this on page 5:

    "We see that the modification of GR entailed by MOND does not enter here by modifying the ‘elasticity’ of spacetime (except perhaps its strength), as is done in f(R) theories and the like".

    I'm not a fan of MOND either. But I am a fan of conservation of energy.
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    Quote Originally Posted by Farsight View Post
    Sorry Kojax, I don't think that helps much. IMHO the universe is either finite or not, regardless of whether there's any observers or matter.

    What happened to "matter tells space time how to curve"?

    If there were no matter, space would have a distance equal to .... well.... it simply wouldn't have a "distance" attribute.



    If we re-ask the question as being "how far apart are the two furthest apart bits of matter in the universe"? (As we ought to ask it) Then we're still left with the problem of how to measure that. Since bits of matter a Hubble Sphere Radius apart cannot communicate with each other by any means whatsoever.
    A mathematician and an engineer were at a party. An older colleague of theirs was there with his daughter. The two each asked if they could speak to her. He said it was ok, but they had to approach her by going half way across the room, then stop, then half way again and stop and proceed in that manner. The mathematician realized that he would never reach her and gave up. The engineer determined that he could get close enough to talk. --Approximate retelling of a joke by my math teacher.
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    Quote Originally Posted by Kojax View Post
    What happened to "matter tells space time how to curve"?
    Wheeler actually said "matter tells space how to curve", which is wrong. Replacing space with spacetime is better, but it's still not right. It ought to be a concentration of energy tells spacetime how to curve. If you skim a light beam past a star it curves. But if you have two stars close to one another and you throw your light beam between them, it doesn't curve.

    Quote Originally Posted by Kojax View Post
    If there were no matter, space would have a distance equal to .... well.... it simply wouldn't have a "distance" attribute.
    I think you could claim that if there were no motion. We define the second and the metre using the motion of electromagnetic phenomena. Or light if you prefer. We can conceive of time and distance in a universe that consists of light and energy even if there is no matter present.

    Quote Originally Posted by Kojax
    I we re-ask the question as being "how far apart are the two furthest apart bits of matter in the universe"? (As we ought to ask it) Then we're still left with the problem of how to measure that. Since bits of matter a Hubble Sphere Radius apart cannot communicate with each other by any means whatsoever.
    Yes, it's a problem, and we don't know the answer. But if we knew all the answers we wouldn't have anything to talk about, would we?
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    Quote Originally Posted by Farsight View Post
    But if you have two stars close to one another and you throw your light beam between them, it doesn't curve.
    So you are claiming that the null geodesics ( light ) between two massive object are the exact same as null geodesics in the complete absence of any massive bodies ?
    This is probably a good place to remind you that this forum is mostly dedicated to mainstream science, so please do not answer the above with your "variable speed of light & aether" ideas; these will be moved and confined to the "Personal Theories" section, because that is what they are.
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    Quote Originally Posted by Markus Hanke View Post
    So you are claiming that the null geodesics ( light ) between two massive object are the exact same as null geodesics in the complete absence of any massive bodies?
    No. I said the light goes straight. As you know massive bodies would result in something akin to the Shapiro delay.

    Quote Originally Posted by Markus Hanke View Post
    This is probably a good place to remind you that this forum is mostly dedicated to mainstream science, so please do not answer the above with your "variable speed of light & aether" ideas; these will be moved and confined to the "Personal Theories" section, because that is what they are.
    Markus, please cease and desist with this notion that something unfamiliar to you is some personal theory. Do your own research. Look on arXiv for example.
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    Quote Originally Posted by Farsight View Post
    Sorry Kojax, I don't think that helps much. IMHO the universe is either finite or not, regardless of whether there's any observers or matter.

    I don't think that follows quantum physics though. The observer may be actually part of the Universe , and consciousness may effect and even create matter.
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    Quote Originally Posted by Markus Hanke View Post
    So you are claiming that the null geodesics ( light ) between two massive object are the exact same as null geodesics in the complete absence of any massive bodies ?
    This is probably a good place to remind you that this forum is mostly dedicated to mainstream science, so please do not answer the above with your "variable speed of light & aether" ideas; these will be moved and confined to the "Personal Theories" section, because that is what they are.
    Cool, I like talking with people with personal theories. I might also add that I do think it is pretty mainstream that photons do not all have the same lengths or wavelengths or frequencies? Also that light does not travel through all mediums of transport at the same speed?
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    Quote Originally Posted by MaxPayne View Post
    They say it all started just 13.7 billion years ago. Can't we calculate the size of the Universe from that data?

    Surely it could never be less than 13.7 billion lys, and due to accelerated expansion it could be even larger. But it still ends up falling short of infinity.

    So why is Universe infinite for you?... Any particular reason other than "coz Einstein said so"?did he really?
    I am not so sure he said so.
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    Quote Originally Posted by mayflow View Post
    Cool, I like talking with people with personal theories...
    I'm not some guy with personal theories, mayflow, despite what you may have heard. It's just that I'm very well read, and I know about things that go back to the likes of Einstein Minkowski and Maxwell which don't feature in modern popscience. These can be a source of conflict.

    Quote Originally Posted by mayflow
    I don't think that follows quantum physics though
    It does.

    Quote Originally Posted by mayflow
    The observer may be actually part of the Universe
    He certainly is.

    Quote Originally Posted by mayflow
    and consciousness may effect and even create matter.
    Who told you that? It isn't true I'm afraid.
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    So, Farsight. Everyone is going to create their own personal ideas and theories. It was you who chose to read about what you read about. You process it in your own ways. Just as I and or anyone else does. We base them surely on what we have read and researched and studied, and it was we each that chose to read and research and study whatever we have. Relativity you know.

    Two things you said to me in your answers seem contradictory to my own ideas, which is fine because my ideas are just my ideas.

    I don't think if an observer is part of the Universe, he or she or it does not have an effect on the Universe.

    Observer effect (physics) - Wikipedia, the free encyclopedia
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    Quote Originally Posted by MaxPayne View Post
    They say it all started just 13.7 billion years ago. Can't we calculate the size of the Universe from that data?

    Surely it could never be less than 13.7 billion lys, and due to accelerated expansion it could be even larger. But it still ends up falling short of infinity.

    So why is Universe infinite for you?... Any particular reason other than "coz Einstein said so"?did he really?

    I dont know whether it is right or wrong...
    as v know stars or constellation always try to expand so cant v just find the distance between 2 constellations or ratio in which they expand??
    it would be something like arithmetic progression or geometric so that v could find how much compact it was 13.7 billion years ago...
    pls correct me if i am wrong...
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    Quote Originally Posted by mayflow
    So, Farsight. Everyone is going to create their own personal ideas and theories.
    Maybe. But I'm not some "my theory" guy. I'm a "relativist".

    Quote Originally Posted by mayflow
    It was you who chose to read about what you read about. You process it in your own ways. Just as I and or anyone else does. We base them surely on what we have read and researched and studied, and it was we each that chose to read and research and study whatever we have. Relativity you know.
    All I can say is that when I tell you what Einstein said, follow it up and read it for yourself. Don't accuse me of being some "my theory" guy because I've told you something you didn't know.

    Quote Originally Posted by mayflow
    Two things you said to me in your answers seem contradictory to my own ideas, which is fine because my ideas are just my ideas. I don't think if an observer is part of the Universe, he or she or it does not have an effect on the Universe.
    Sure. But if you observe a table, your eye intercepts light from the table. That doesn't change the table one bit. If however you're in the dark and you turn on the light in order to see, then you do affect the table. But then you aren't being a passive observer. You are interacting with your surroundings. You are being more than just an observer.
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    Quote Originally Posted by Farsight View Post
    Markus, please cease and desist with this notion that something unfamiliar to you is some personal theory. Do your own research. Look on arXiv for example.
    Farsight, whether you like it or not, if you choose to participate here then you need to come to terms with the fact that it is the mods and admins who decide what goes into Personal Theories, and what doesn't. I am sure you are aware that I know about your ideas, and so far as I am concerned they are a Personal Theory, and nothing more. You are welcome to participate in discussions, but once you bring up things like variable light speeds in lieu of curvature etc, then I will categorize these into Personal Theories, despite your protests. The basis on which I decide what is accepted science and what isn't will be established textbooks on the subject matter, and whenever I make a decision I will always be able to back those with textbook references.

    Btw, arXiv is a pre-print server, so while these articles are good sources to support a point, they are not yet established science due to the lack of peer-review.

    These are the parameters of your participation here. It's your choice now whether you wish to go with them or not; your participation here is voluntary, but subject to the above terms.

    No. I said the light goes straight.
    You are contradicting yourself. Light traces out null geodesics; if those are "straight", then that would mean we are dealing with Minkowski space-time. Is that what you are claiming, that null geodesics between two massive bodies are the same as in Minkowski space-time ?
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    Quote Originally Posted by Markus Hanke View Post
    Farsight, whether you like it or not, if you choose to participate here then you need to come to terms with the fact that it is the mods and admins who decide what goes into Personal Theories, and what doesn't. I am sure you are aware that I know about your ideas, and so far as I am concerned they are a Personal Theory, and nothing more...
    Markus, you are still in denial about Einstein referring to the aether of general relativity. That isn't some personal theory, that's a matter of historical accuracy, and the references on arXiv bear me out. But if you'd like to throw that into some personal-theories trashcan, you are free to do so.

    Quote Originally Posted by Markus Hanke
    You are welcome to participate in discussions, but once you bring up things like variable light speeds in lieu of curvature etc, then I will categorize these into Personal Theories, despite your protests. The basis on which I decide what is accepted science and what isn't will be established textbooks on the subject matter, and whenever I make a decision I will always be able to back those with textbook references.
    And I am always able to back up my posts with references to Einstein papers. Which trump your references to some textbook bible written by some non-entity who doubtless believes in time travel and the multiverse. Now can we stop the carping and get back to the physics please?

    Quote Originally Posted by Markus Hanke View Post
    You are contradicting yourself. Light traces out null geodesics; if those are "straight", then that would mean we are dealing with Minkowski space-time. Is that what you are claiming, that null geodesics between two massive bodies are the same as in Minkowski space-time?
    No. As Einstein said, the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. The space between those two nearby stars is affected by those two stars, affecting your measurements of space and time. In the rubber-sheet analogy, there's a saddle point between the two stars. But light does not curve when it passes between those stars. It doesn't curve towards one star or the other, it goes straight.
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    Quote Originally Posted by Farsight View Post
    But if you'd like to throw that into some personal-theories trashcan, you are free to do so.
    Good, so we have an understanding here.
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    Quote Originally Posted by Farsight View Post
    No. As Einstein said, the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration.
    Right, so Einstein said the metric of space-time differs from Minkowski space-time in the presence of energy-momentum, i.e. is affected by the stars. We also know that light traces out null geodesics in aforementioned space-time. Yet you claim that those null geodesics ( = light ) do not curve between two massive stars. I am merely asking you to clarify this, in light of the above quote by Einstein that you yourself have provided. To be more precise - as we all know, geodesics in a general are solutions to the system of equations



    The "metrical qualities" Einstein was referring to are encapsulated by the Christoffel symbols . These Christoffel symbols are linear combinations of derivatives of the metric; in order for the above to yield a straight line, the Christoffel symbols, and hence the derivatives of the metric, must identically vanish :



    the solutions of which are straight lines. Now tell me, if, as you claim, light propagates in a straight line between two massive stars, how can this be consistent with the above quote you have provided ? It would necessarily imply that the Christoffel symbols vanish ( otherwise no straight line solution ), which in turn means that the metric structure of space-time is not affected by the stars - in direct contradiction to the quote you yourself gave. Please explain in appropriate mathematical detail.
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    Quote Originally Posted by Markus Hanke View Post
    Right, so Einstein said the metric of space-time differs from Minkowski space-time in the presence of energy-momentum, i.e. is affected by the stars.
    Yes. The metric is "what you measure". So if you tracked along the light-beam with a clock and plotted your clock reading against some convenient pulsar, you would find you'd drawn a curve. The gravitational time dilation is at a maximum directly between the stars.

    Quote Originally Posted by Markus Hanke View Post
    We also know that light traces out null geodesics in aforementioned space-time. Yet you claim that those null geodesics ( = light ) do not curve between two massive stars. I am merely asking you to clarify this, in light of the above quote by Einstein that you yourself have provided.
    You have misunderstood something here. I referred to a saddle-like region in the curved spacetime depiction, which is curved. But the light moving through space goes through the middle of it and doesn't curve.

    Quote Originally Posted by Markus Hanke
    To be more precise - as we all know, geodesics in a general are solutions to the system of equations



    The "metrical qualities" Einstein was referring to are encapsulated by the Christoffel symbols . These Christoffel symbols are linear combinations of derivatives of the metric; in order for the above to yield a straight line, the Christoffel symbols, and hence the derivatives of the metric, must identically vanish :



    the solutions of which are straight lines. Now tell me, if, as you claim, light propagates in a straight line between two massive stars, how can this be consistent with the above quote you have provided ? It would necessarily imply that the Christoffel symbols vanish ( otherwise no straight line solution ), which in turn means that the metric structure of space-time is not affected by the stars - in direct contradiction to the quote you yourself gave. Please explain in appropriate mathematical detail.
    Along the centre line the curvature is restricted to the time aspect of spacetime. Apologies, I don't know how I can give any mathematical detail that says it any better.
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    Quote Originally Posted by Farsight View Post
    Wheeler actually said "matter tells space how to curve", which is wrong. Replacing space with spacetime is better, but it's still not right. It ought to be a concentration of energy tells spacetime how to curve. If you skim a light beam past a star it curves. But if you have two stars close to one another and you throw your light beam between them, it doesn't curve.
    The "time" aspect would still curve, though, wouldn't it? The beam of light would shift blue as it moves toward the center, and then red again as it goes past the center.

    I'm pretty sure that passing through the exact middle of a two star system (where the stars have identical mass, spin... etc) is basically approximately the same as the "hole through the center of the Earth" problem.

    Hole Through the Earth Example


    The only major difference I can see is that since the two stars are orbiting around each other, there should be a small amount of frame dragging or something like that in the direction the system is spinning toward. Or would there be?


    I think you could claim that if there were no motion. We define the second and the metre using the motion of electromagnetic phenomena. Or light if you prefer. We can conceive of time and distance in a universe that consists of light and energy even if there is no matter present.
    We could use a pair of photons instead of a pair of particles. The communication problem still persists, though. If they're a Hubble Sphere Diameter away from each other, then they will never be able to cross paths with each other, even if they were on a collision course.



    Yes, it's a problem, and we don't know the answer. But if we knew all the answers we wouldn't have anything to talk about, would we?
    Yeah. It's good to take the conversation in this direction, so we can have something to discuss. We can speculate a little without getting too far off course.

    So long as we remember that we don't really know the answer (unless an answer becomes apparent, which agrees fully with established science).
    A mathematician and an engineer were at a party. An older colleague of theirs was there with his daughter. The two each asked if they could speak to her. He said it was ok, but they had to approach her by going half way across the room, then stop, then half way again and stop and proceed in that manner. The mathematician realized that he would never reach her and gave up. The engineer determined that he could get close enough to talk. --Approximate retelling of a joke by my math teacher.
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    Quote Originally Posted by Farsight View Post
    Along the centre line the curvature is restricted to the time aspect of spacetime
    ...which means null geodesics in that region ( which are solutions to the geodesic equations ) are not straight lines. Thank you.
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    Quote Originally Posted by Kojax View Post
    The "time" aspect would still curve, though, wouldn't it?
    Yes, of course.

    I'm pretty sure that passing through the exact middle of a two star system (where the stars have identical mass, spin... etc) is basically approximately the same as the "hole through the center of the Earth" problem.
    It is not precisely the same, but conceptually similar. The full GR treatment of this is much more complicated, though.

    The only major difference I can see is that since the two stars are orbiting around each other, there should be a small amount of frame dragging or something like that in the direction the system is spinning toward. Or would there be?
    There would be quite a number of higher-order non-linear effects in the full GR treatment, which are not present in the Newtonian approximation. I am not actually aware of an exact analytical metric for this case, all the treatments I have seen are numerical ones.
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    Quote Originally Posted by Farsight View Post
    OK noted. I guess it comes back to this: I will not accept a universe that was already infinite when the big bang occurred. It's turtles all the way down, a non-answer, not much better than God did it.
    But you might have to accept that the big bang is not a single point, but a null space of the same shape as the universe at other times. I don't think that one can sustain the notion that the universe encloses the big bang as a single point at the centre in spacetime (the most simple form of the expanding balloon analogy). This would require that the current universe have a very specific curvature determined by its age. The Friedmann-Lemaître-Robertson-Walker metric, even if describing an expanding spherical universe, it not so constrained. Thus, one needs to be careful about taking analogies too far.


    Quote Originally Posted by Farsight View Post
    IMHO the nub of the issue is that if space is homogeneous, light goes straight. Really straight. Then there is no intrinsic curvature. There is no looping back.
    It is a geometrical fact that a sphere is homogenous and isotropic.

    Light is irrelevant because the shape of the universe is the shape of the three-dimensional universe at an instant in time. Light-like trajectories are in space and time.


    Quote Originally Posted by Farsight View Post
    Agreed. And nor does it mean that beyond our horizon, space continues on and on forever.
    That's true, but the universe is bigger than the observed universe.


    Quote Originally Posted by Farsight View Post
    I'm not a fan of naturalness myself.
    I'm not using the term "naturalness" in the same way as given in that article, but rather in a way that mathematicians might use that term.
    A tensor equation that is valid in any coordinate system is valid in every coordinate system.
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    Quote Originally Posted by KJW View Post
    Light-like trajectories are in space and time.
    Yes, and this is the crucial fact that also underlies the proper understanding of the Shapiro delay in conjunction with the geodesic equation.
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    Quote Originally Posted by Kojax View Post
    The "time" aspect would still curve, though, wouldn't it?
    Yes.

    Quote Originally Posted by Kojax
    The beam of light would shift blue as it moves toward the center, and then red again as it goes past the center.
    Actually, no. The frequency of light doesn't change. Think E=hf and imagine you threw a 511keV photon into a black hole. The black hole mass increases by 511keV/c². You measure a blueshift when you're lower because you and your measuring devices change.

    Quote Originally Posted by Kojax
    I'm pretty sure that passing through the exact middle of a two star system (where the stars have identical mass, spin... etc) is basically approximately the same as the "hole through the center of the Earth" problem.
    Agreed.

    Quote Originally Posted by Kojax
    The only major difference I can see is that since the two stars are orbiting around each other, there should be a small amount of frame dragging or something like that in the direction the system is spinning toward. Or would there be?
    In reality there would be some, but gravitomagnetism is very weak, and I was discounting it to be honest.

    Quote Originally Posted by Kojax
    We could use a pair of photons instead of a pair of particles. The communication problem still persists, though. If they're a Hubble Sphere Diameter away from each other, then they will never be able to cross paths with each other, even if they were on a collision course.
    I'm not quite sure about that. Have a look at [astro-ph/0310808] Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe and note the abstract: We show that we can observe galaxies that have, and always have had, recession velocities greater than the speed of light. I wrote something about a caterpillar marching along a piece of elastic towards your nose to illustrate this.

    Quote Originally Posted by Kojax
    Yeah. It's good to take the conversation in this direction, so we can have something to discuss. We can speculate a little without getting too far off course. So long as we remember that we don't really know the answer (unless an answer becomes apparent, which agrees fully with established science).
    Yep. But for myself I'm a bit unhappy about the way the multiverse seems to be turning into established science.
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    Quote Originally Posted by Markus Hanke View Post
    ...which means null geodesics in that region ( which are solutions to the geodesic equations ) are not straight lines. Thank you.
    The point is that the light goes straight and there's no gravity. And you can apply this to the early universe.
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    Quote Originally Posted by KJW View Post
    But you might have to accept that the big bang is not a single point...
    Actually, I don't. In similar vein I don't accept that black holes feature a point singularity in the centre. I think the original "frozen star" interpretation is the one that's right. Kevin Brown mentions it in the formation and growth of black holes, but doesn't favour it.

    Quote Originally Posted by KJW View Post
    but a null space of the same shape as the universe at other times.
    Fair enough. But what's a null space?

    Quote Originally Posted by KJW View Post
    I don't think that one can sustain the notion that the universe encloses the big bang as a single point at the centre in spacetime (the most simple form of the expanding balloon analogy). This would require that the current universe have a very specific curvature determined by its age. The Friedmann-Lemaître-Robertson-Walker metric, even if describing an expanding spherical universe, it not so constrained. Thus, one needs to be careful about taking analogies too far.
    Agreed. The balloon analogy has its problems. It comes with the suggestion that if you kept going thataway you end up coming back thisaway.

    Quote Originally Posted by KJW View Post
    It is a geometrical fact that a sphere is homogenous and isotropic.

    Light is irrelevant because the shape of the universe is the shape of the three-dimensional universe at an instant in time. Light-like trajectories are in space and time.
    I can't see how light is irrelevant. By the way, when I think of the shape of the universe, I think of the raisin-cake analogy. And I think of the universe as being shaped like a ball. You can draw lines within that ball to represent light-paths. The ball is expanding, causing some lines to curve. But if you set aside the expansion, the lines are straight. At any point in time, the universe is flat.

    Quote Originally Posted by KJW View Post
    That's true, but the universe is bigger than the observed universe.
    No probs.

    Quote Originally Posted by KJW View Post
    I'm not using the term "naturalness" in the same way as given in that article, but rather in a way that mathematicians might use that term.
    OK noted. But sorry, how might a mathematician use that term?
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    Quote Originally Posted by Farsight View Post
    Yes.

    Actually, no. The frequency of light doesn't change. Think E=hf and imagine you threw a 511keV photon into a black hole. The black hole mass increases by 511keV/c². You measure a blueshift when you're lower because you and your measuring devices change.
    Actually the frequency does change. The reason, in layman's terms, is because the front of the light wave is moving slower than the rear of the light wave.

    It was demonstrated by the Pound-Rebka Experiment.

    Pound



    Agreed.

    In reality there would be some, but gravitomagnetism is very weak, and I was discounting it to be honest.
    It's a good idea to point out that you are being approximate in your statement, so people don't get the wrong idea.


    I'm not quite sure about that. Have a look at [astro-ph/0310808] Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe and note the abstract: We show that we can observe galaxies that have, and always have had, recession velocities greater than the speed of light. I wrote something about a caterpillar marching along a piece of elastic towards your nose to illustrate this.
    Yeah. It depends on what version of Cosmology one is using. If it is an accelerating expansion (which I think is the established version) , then the Hubble Sphere shrinks over time, so an object outside the present Hubble Sphere might still be visible because it was inside the Hubble Sphere back when it emitted the light we are seeing today.

    In that case, using (using h as the Hubble Constant) isn't sufficient. We need to add a few terms to accommodate the acceleration also, and then calculating it gets nasty.

    However, if an object was already outside our Hubble Sphere back when the light was emitted, it still won't reach us now. There still does exist a distance that can serve as an event horizon. It's just not the "Hubble Sphere". I was incorrect in choosing that term.



    Yep. But for myself I'm a bit unhappy about the way the multiverse seems to be turning into established science.
    I'm unhappy with it too. But I'm not sure it's wrong. I hate having a speed limit of "C", but I have to admit that it may in fact be the speed limit. I really don't like the BBT at all, but it might be correct. So until I can actually refute it, I'll have to work with it.
    A mathematician and an engineer were at a party. An older colleague of theirs was there with his daughter. The two each asked if they could speak to her. He said it was ok, but they had to approach her by going half way across the room, then stop, then half way again and stop and proceed in that manner. The mathematician realized that he would never reach her and gave up. The engineer determined that he could get close enough to talk. --Approximate retelling of a joke by my math teacher.
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    Quote Originally Posted by Farsight View Post
    The point is that the light goes straight and there's no gravity.
    There is no gravity at the Lagrange point(s) between two massive bodies ? Surely I am misunderstanding you, because that is trivially false, as we know even from good old Newtonian physics. Likewise, I have already demonstrated ( post #35 ) that, precisely because there is gravity, the light geodesics cannot be straight.

    As mentioned before - remember that light traces out null geodesics in space-time, not space.
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    Quote Originally Posted by Kojax View Post
    Actually the frequency does change...
    Cross my heart and hope to die, but it doesn't. I know people say it does, but conservation of energy says it doesn't.

    Quote Originally Posted by Kojax
    Yeah. It depends on what version of Cosmology one is using. If it is an accelerating expansion (which I think is the established version) , then the Hubble Sphere shrinks over time, so an object outside the present Hubble Sphere might still be visible because it was inside the Hubble Sphere back when it emitted the light we are seeing today. In that case, using (using h as the Hubble Constant) isn't sufficient. We need to add a few terms to accommodate the acceleration also, and then calculating it gets nasty. However, if an object was already outside our Hubble Sphere back when the light was emitted, it still won't reach us now. There still does exist a distance that can serve as an event horizon. It's just not the "Hubble Sphere". I was incorrect in choosing that term.
    All points noted, and the one about being approximate.

    Quote Originally Posted by Kojax
    I'm unhappy with it too. But I'm not sure it's wrong.
    If we had solid evidence that it was, we could shoot it down. But we can't, just as we can't shoot down fairies.

    Quote Originally Posted by Kojax
    I hate having a speed limit of "C", but I have to admit that it may in fact be the speed limit.
    It definitely is, on account of the wave nature of matter.

    Quote Originally Posted by Kojax
    I really don't like the BBT at all, but it might be correct. So until I can actually refute it, I'll have to work with it.
    I don't share you view there. I think space has to expand. It can't do anything but expand. So when you wind it back, there's the big bang. I like the BBT. But I don't like inflation. I think it's a solution to a problem that doesn't exist.
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    Quote Originally Posted by Farsight View Post
    but conservation of energy says it doesn't.
    I hate to disappoint you, but conservation of energy is a local law only. In flat space-times, this doesn't make any difference, since :



    so energy is conserved both locally and globally over some region M. In curved space-times, however, we need to generalise this to the covariant divergence once the integration is being done over some region. Watch closely what happens then :



    What this means is that in curved space-times, conservation of energy holds only locally, but not globally over some extended region. It is also possible to explicitly calculate the difference in energies of the in-falling light beam, and unsuprisingly the result matches exactly the shift in frequencies.

    So yes - the frequency changes, which is of course exactly what we measure.

    This is why the Pound-Rebka experiment is considered to be a direct measurement of space-time curvature, which hence becomes not just a theoretical concept, but an empirically measurable quantity.
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    Quote Originally Posted by Markus Hanke View Post
    There is no gravity at the Lagrange point(s) between two massive bodies? Surely I am misunderstanding you, because that is trivially false, as we know even from good old Newtonian physics.
    The point is that you don't fall towards either of those massive bodies.

    Quote Originally Posted by Markus Hanke View Post
    Likewise, I have already demonstrated ( post #35 ) that, precisely because there is gravity, the light geodesics cannot be straight. As mentioned before - remember that light traces out null geodesics in space-time, not space.
    Yes, but the light moves straight through space. And you don't fall towards either of those massive bodies. So there is no gravity where the two gravitational fields counterbalance one another.

    IMHO this is relevant to the universe as a whole. Forget about the expansion of space for a minute. Imagine I take all the matter in the stars and planets of the universe and turn it into dust. Then I spread this dust uniformly throughout the universe. Then I convert this dust into "the energy of space itself". Now I shine light beams through it. Those light beams go straight. There is no preferential direction in which they curve. There is no constant curvature. There is no gravity. And even if I were to magic some up, it would alter the motion of light and matter through space, but it wouldn't make space fall inwards. Despite the waterfall analogy, we do not live on a Chicken-Little planet - the sky is not falling in. The universe didn't collapse under its own gravity, and it never ever will.
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    Quote Originally Posted by Markus Hanke View Post
    I hate to disappoint you, but conservation of energy is a local law only...
    No, it's global. There is no creation of energy eh-nihilo. There are no perpetual motion machines getting energy from nowhere.

    Quote Originally Posted by Markus Hanke View Post
    In curved space-times, however, we need to generalise this to the covariant divergence once the integration is being done over some region. Watch closely what happens then :


    I see an expression that proves nothing, and no justification.

    Quote Originally Posted by Markus Hanke View Post
    What this means is that in curved space-times, conservation of energy holds only locally, but not globally over some extended region. It is also possible to explicitly calculate the difference in energies of the in-falling light beam, and unsuprisingly the result matches exactly the shift in frequencies.
    Then calculate the difference in energies for a 511keV photon falling from here into a black hole. And then tell me how much that black hole mass increases by.

    Quote Originally Posted by Markus Hanke View Post
    So yes - the frequency changes, which is of course exactly what we measure.
    We measure a change in frequency because we and our clocks go slower when we're lower. We changed, the photon didn't. It's akin to accelerating towards a photon in gravity-free space. We measure an increased frequency, but we changed, the photon didn't.
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    Quote Originally Posted by Farsight View Post
    The point is that you don't fall towards either of those massive bodies.
    Of course not, but that doesn't mean there is no gravity. Gravity isn't the net force at a point in space, but the gravitational potential present there. This is the same in both GR and Newtonian gravity.

    Yes, but the light moves straight through space
    We are doing GR, so we are dealing with space-time, not just space, as pointed out several times now. This may not be to your liking, but it is how GR works, and incidentally it is also in accord with the quote you have provided us earlier. If you consider space-time, you get a consistent and full explanation of everything that happens in the scenario, which the space-only perspective cannot do.

    So there is no gravity where the two gravitational fields counterbalance one another.
    See above. If there was no gravity, light would propagate in the exact same way as it does in free space without sources present, which is of course not the case, as we all know. There is no net force at the Lagrange point, but there is gravity.
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    Quote Originally Posted by Farsight View Post
    No, it's global. There is no creation of energy eh-nihilo. There are no perpetual motion machines getting energy from nowhere.
    No, it can only be local in general. In curved spacetime, there is no consistent notion of the global conservation of energy-momentum. Markus' mathematical expression basically says that attempts to define a notion of global conservation of energy-momentum will lead to inconsistencies between different frames of reference. It is the peculiar nature of spacetime curvature that energy-momentum is conserved everywhere (local) but not necessarily conserved overall (global). Ultimately, this is because the energy-momentum at different locations cannot be consistently added together to form a total.
    A tensor equation that is valid in any coordinate system is valid in every coordinate system.
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    Quote Originally Posted by Farsight View Post
    No, it's global.
    Energy is not even defined globally in GR, it is always a local property, hence there is no global conservation, nor does there need to be one :

    Is Energy Conserved in General Relativity?

    There are no perpetual motion machines getting energy from nowhere.
    The generalisation of the laws of thermodynamics to curved space-times still hold, so there are no perpetual motion machines here. I do not know why you would think otherwise. There is no energy being "created" with which one could perform any work, nor did I claim that there was. You need to have a more careful look at the mathematics I gave.

    I see an expression that proves nothing, and no justification.
    The expression formulates what I have explained in terms of the appropriate mathematics. It is a basic flux integral, and the equalities come in via the generalised Stokes theorem, which is valid in any coordinate system and on any manifold. You are welcome to point out any mathematical mistakes, if you can find them, otherwise this is just the integral of a divergence over a region M.

    What justification do you expect me to give you ? If I reference a textbook ( which of course I can, the above maths are elementary differential geometry ) you will reject it anyway because it deviates from your own ideas. But if you want the reference still, then let me know, and I'll provide it - for starters you might look at the references section of the Baez article I linked to above. Other than that, you should know by now that I can back up anything and everything I say with textbook references.

    Then calculate the difference in energies for a 511keV photon falling from here into a black hole. And then tell me how much that black hole mass increases by.
    With all due respect Farsight, but what does this have to do with the discussion at hand ? We were talking about the Pound-Rebka experiment and red/blueshift of light in the vicinity of Earth, not black holes, so please do not change the scenario now. The calculation for light falling towards Earth is just a straightforward application of the standard frequency shift formula in Schwarzschild space-times.

    We measure a change in frequency because we and our clocks go slower when we're lower
    Everyone's clocks continue to tick at the exact same rate of "1 second per second" locally. The only thing that changes is the geometry of the observers' world lines ( and hence recorded total proper times along some section of them ), due to the background curvature of space-time around the earth, i.e. the global relationships between observers. This is the basic idea of GR - you locally continue using Minkowskian laws, but globally the relationships become more complicated.

    We changed, the photon didn't.
    We didn't change either. What is different though are the geodesics in our region of space-time, as compared to some reference point very far away, which is why the photon arrives here at a different frequency. It's all just geometry.
    Last edited by Markus Hanke; 04-04-2014 at 11:37 AM.
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    Quote Originally Posted by KJW View Post
    No, it can only be local in general. In curved spacetime, there is no consistent notion of the global conservation of energy-momentum. Markus' mathematical expression basically says that attempts to define a notion of global conservation of energy-momentum will lead to inconsistencies between different frames of reference.
    Precisely, thank you KJW.

    It is worth noting though that, while in general the notion of "energy" can't be defined globally, the symmetries of a space-time in question give rise to certain constants of motion - for example, in Schwarzschild space-time we have





    as constants of motion. It needs to be stressed though that this is dependent on a particular metric, so in other space-times one would get different constants of motion.
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    It gets more complicated in GR, but even in classical physics, the light will be blue shifting, and then redshifting again.

    It's possible the red shift might cancel the blue shift, leaving the light in the same state upon departure as it had on entry. I guess it depends how the problem is set up, and how perfectly the stars match. If they're orbiting, then it would have to pass through them perpendicular to the orbiting plain in order to not have a course change. (Even doing that might not work in GR.)


    Quote Originally Posted by Farsight View Post
    Cross my heart and hope to die, but it doesn't. I know people say it does, but conservation of energy says it doesn't.
    If I dropped a rock between the two stars, the rock would accelerate (gain kinetic energy) as it approaches the center point between the stars, and then begin to lose kinetic energy is it passes the center point and moves away.

    By shifting toward blue, light is also gaining energy as it moves to the center point. Then by shifting red as it leaves the system, it loses that energy again.


    Anyway, clearly we must agree that the energy of falling objects is not conserved. Only the combination of the object's potential and kinetic energy is conserved. As it loses potential energy, it gains kinetic energy. With light, it's the same except the light is gaining a different kind of energy other than kinetic energy.

    Quote Originally Posted by Farsight View Post
    The point is that you don't fall towards either of those massive bodies.

    No. But you do fall toward the center of mass for the whole system. In this case, the center of mass doesn't actually have any mass in it. But that doesn't matter. It's still the center.

    It's just like that "hole in the center of the Earth" situation I linked above. When you fall through the donut hole in that thought experiment, the mass is actually on either side of you. It's not directly "below" you. But you're still pulled toward the center.

    Having mass on all sides of you is no different than having it on two sides of you. Until you reach the exact center point between the two stars, both of them are pulling on you at an angle.
    A mathematician and an engineer were at a party. An older colleague of theirs was there with his daughter. The two each asked if they could speak to her. He said it was ok, but they had to approach her by going half way across the room, then stop, then half way again and stop and proceed in that manner. The mathematician realized that he would never reach her and gave up. The engineer determined that he could get close enough to talk. --Approximate retelling of a joke by my math teacher.
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    Quote Originally Posted by Markus Hanke View Post
    Precisely, thank you KJW.

    It is worth noting though that, while in general the notion of "energy" can't be defined globally, the symmetries of a space-time in question give rise to certain constants of motion - for example, in Schwarzschild space-time we have





    as constants of motion. It needs to be stressed though that this is dependent on a particular metric, so in other space-times one would get different constants of motion.
    I would need a lot more explanation before I could really make sense of that.

    If energy cannot be defined globally, then mass cannot be defined globally. If mass cannot be defined globally, then observers may disagree about the gravitational field created by a given body of mass (because they disagree how much mass is present).

    Once observers are disagreeing about that, how do you carry out transformations between their perspectives?
    A mathematician and an engineer were at a party. An older colleague of theirs was there with his daughter. The two each asked if they could speak to her. He said it was ok, but they had to approach her by going half way across the room, then stop, then half way again and stop and proceed in that manner. The mathematician realized that he would never reach her and gave up. The engineer determined that he could get close enough to talk. --Approximate retelling of a joke by my math teacher.
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    Quote Originally Posted by Kojax View Post
    If energy cannot be defined globally, then mass cannot be defined globally. If mass cannot be defined globally, then observers may disagree about the gravitational field created by a given body of mass (because they disagree how much mass is present).
    But one can define scalars from the energy-momentum tensor such as or even and these can be totalled over some region of spacetime. The problem is that one can't form a conservation law from scalar quantities.

    I should remark that one can construct an energy-momentum vector for which one can obtain a global conservation law (Stokes theorem can be applied). However, the energy-momentum vector is somewhat ill-defined in the context of general relativity.

    In general relativity, the only fields that describe gravitation in a covariant form are the curvature tensor fields (in particular, the Weyl conformal tensor field), which describe the tidal effect but not the Newtonian gravitational field itself.

    For stationary spacetimes, one can define a scalar gravitational potential from which one can define the Newtonian gravitational field. Such a field is conservative. This means for example that one can shine a beam of light from high in space onto a mirror on the ground which reflects the beam back to the source in space, and the frequency of the reflected beam at the source will be the same as the original source beam.

    In the Schwarzschild metric, the parameter is interpreted as the mass of the blackhole but strictly speaking isn't the mass of anything, but is actually the arbitrary parameter of a one-parameter family of Schwarzschild spacetime metrics. In other words, the parameter isn't locally associated with the blackhole itself but associated globally with the entire spacetime.
    A tensor equation that is valid in any coordinate system is valid in every coordinate system.
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    Quote Originally Posted by Farsight View Post
    I'm a "relativist".
    In what way are you a "relativist"?
    A tensor equation that is valid in any coordinate system is valid in every coordinate system.
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    Quote Originally Posted by Kojax View Post
    It gets more complicated in GR, but even in classical physics, the light will be blue shifting, and then redshifting again.
    Yes, indeed.

    It's possible the red shift might cancel the blue shift
    Yes, this is possible, but only if the whole setup is perfectly stationary, which is kind of difficult for two massive stars

    Only the combination of the object's potential and kinetic energy is conserved.
    Yes, this is exactly correct ( this "combination" is the first one of the aforementioned constants of motion ) - but only in Schwarzschild space-time. If you consider a different geometry, such as Reissner-Nordstroem or something more exotic, you will also get different constants of motion. This is the point we were trying to make - there is no general law of global energy conservation in GR; you may get constants of motion, but those depend on the symmetries of the geometry considered.

    I would need a lot more explanation before I could really make sense of that.
    You can safely ignore the mathematical details so long as you can make sense of the fact that there are certain "constants of motion", i.e. quantities which are conserved at all points along a free-fall trajectory. In GR, those quantities are dependent on the metric, and hence do not generally correspond to the Newtonian concepts of kinetic and potential energies.

    If energy cannot be defined globally, then mass cannot be defined globally
    Also correct. Defining the amount of "mass" in a region of space-time is highly non-trivial in GR, and there is in fact no agreed way to do this for the general case. There are a number of different concepts, such as Hawking mass and Bondi mass, but they apply only to certain special cases. See here also :

    https://today.duke.edu/2014/03/hawkingmass2

    Once observers are disagreeing about that, how do you carry out transformations between their perspectives?
    You can relate measurements of length, area, angles etc etc between observers using the metric, but things are a lot more complicated when it comes to mass and energy. You can define an energy-momentum tensor field at each point in space-time, on which all observers will agree; what they won't agree on in general is the sum total of energy-momentum across some region, as demonstrated earlier on in this thread. This is because there isn't any consistent, covariant way to integrate this over some region of curved space-time, so the problem becomes difficult. Here is a fairly recent paper which treats this issue in the context of one particular approach, the "Hawking Mass" :

    [1402.3287] Time flat surfaces and the monotonicity of the spacetime Hawking mass II

    All in all, many things in GR curved space-times which may seem clear and unambiguous at first glance turn out to be highly non-trivial and counterintuitive on closer inspection. That is just the nature of things, I suppose.
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    Quote Originally Posted by KJW View Post
    But one can define scalars from the energy-momentum tensor such as or even and these can be totalled over some region of spacetime.
    Interesting, I never considered that. But as you say in the following paragraph, this isn't much help to formulate a general conservation law.

    For stationary spacetimes, one can define a scalar gravitational potential from which one can define the Newtonian gravitational field. Such a field is conservative.
    Is it being stationary a sufficient condition to be able to define a conservative field ? Intuitively I would have though that this holds true only if the metric is also spherically symmetric in addition to being stationary, but I might well be wrong.

    In the Schwarzschild metric, the parameter is interpreted as the mass of the blackhole but strictly speaking isn't the mass of anything, but is actually the arbitrary parameter of a one-parameter family of Schwarzschild spacetime metrics. In other words, the parameter isn't locally associated with the blackhole itself but associated globally with the entire spacetime
    True. The M originally enters the metric only because we demand that the Schwarzschild solution asymptotically approaches Newtonian gravity.
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    Quote Originally Posted by Markus Hanke View Post
    Is it being stationary a sufficient condition to be able to define a conservative field ?
    It is sufficient because the gravity field (including an artificial gravity field) is derived from the definition of the Killing vector field, and the potential field is derived from that. The existence of the potential field in this case establishes the conservative nature of the gravity field. However, it should be remarked that while the gravity field and corresponding potential field are manifestly covariant, they are defined in terms of stationary trajectories only. The derivation establishes that the gravity field is due to the time dilation. This is also true for an artificial gravity field which is due to the time dilation associated with acceleration (the derivation makes no mention of spacetime curvature).


    Quote Originally Posted by Markus Hanke View Post
    The M originally enters the metric only because we demand that the Schwarzschild solution asymptotically approaches Newtonian gravity.
    One thing that I'm curious about is how the parameter of the Kerr metric was matched to angular momentum given that there is no frame-dragging in Newtonian gravity.
    A tensor equation that is valid in any coordinate system is valid in every coordinate system.
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    Quote Originally Posted by KJW View Post
    One thing that I'm curious about is how the parameter of the Kerr metric was matched to angular momentum given that there is no frame-dragging in Newtonian gravity.
    I'd say it's simply because the Kerr metric can be written in a form that is equivalent to a co-rotating reference frame with some angular velocity and colatitude; you can then simply match the free parameters. On the other hand, there isn't actually a requirement to do so - just as you pointed out that in the Schwarzschild metric you can consider M to be a parameter for a family of metrics, we can do the same for the Kerr metric, and make this a 2-parameter family, and keep everything abstract without mixing Newtonian notions into it.

    It is sufficient because the gravity field (including an artificial gravity field) is derived from the definition of the Killing vector field, and the potential field is derived from that. The existence of the potential field in this case establishes the conservative nature of the gravity field. However, it should be remarked that while the gravity field and corresponding potential field are manifestly covariant, they are defined in terms of stationary trajectories only. The derivation establishes that the gravity field is due to the time dilation. This is also true for an artificial gravity field which is due to the time dilation associated with acceleration (the derivation makes no mention of spacetime curvature).
    Ok, this makes sense
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    Does anyone know if there has been progress in finding a rotating perfect-fluid interior solution that can be matched to a Kerr exterior metric ?
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    Quote Originally Posted by Markus Hanke View Post
    I'd say it's simply because the Kerr metric can be written in a form that is equivalent to a co-rotating reference frame with some angular velocity and colatitude; you can then simply match the free parameters. On the other hand, there isn't actually a requirement to do so - just as you pointed out that in the Schwarzschild metric you can consider M to be a parameter for a family of metrics, we can do the same for the Kerr metric, and make this a 2-parameter family, and keep everything abstract without mixing Newtonian notions into it.
    But note that in the case of the Schwarzschild metric, can be expressed in kilograms because we can correlate it to Newton's law of gravity. But whereas we can say that is an angular momentum, can we express this angular momentum in terms of joule-seconds given that there is no correspondence in Newton's law of gravity?
    A tensor equation that is valid in any coordinate system is valid in every coordinate system.
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    Quote Originally Posted by Markus Hanke View Post
    Does anyone know if there has been progress in finding a rotating perfect-fluid interior solution that can be matched to a Kerr exterior metric ?
    It is my understanding that there is no equivalent to Birkhoff's theorem for the Kerr metric.
    A tensor equation that is valid in any coordinate system is valid in every coordinate system.
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    Quote Originally Posted by KJW View Post
    But note that in the case of the Schwarzschild metric, can be expressed in kilograms because we can correlate it to Newton's law of gravity. But whereas we can say that is an angular momentum, can we express this angular momentum in terms of joule-seconds given that there is no correspondence in Newton's law of gravity?
    Good point - I seem to remember that this was explained somewhere in MTW, but I am away until Monday and don't have the book with me. I'll check when I get back
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    Quote Originally Posted by KJW View Post
    It is my understanding that there is no equivalent to Birkhoff's theorem for the Kerr metric.
    Yes, that's exactly the issue. There has been a long-standing problem in trying to manually find an rotating interior solution that can be matched to an exterior Kerr metric; the Wiki article mentions a Wahlquist fluid ( which I had never heard of ), which was up until recently thought to allow this, but it turns out that in fact it doesn't. It can be done with a thin disk of rotating dust, though.

    I presume that this is hence still an open problem.
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    Quote Originally Posted by Markus Hanke View Post
    Of course not, but that doesn't mean there is no gravity. Gravity isn't the net force at a point in space, but the gravitational potential present there. This is the same in both GR and Newtonian gravity...
    The point being that if gravitational potential is the same everywhere, light goes straight and you don't fall down. And the early universe doesn't collapse under its own gravity.

    Quote Originally Posted by Markus Hanke View Post
    We are doing GR, so we are dealing with space-time, not just space, as pointed out several times now. This may not be to your liking, but it is how GR works, and incidentally it is also in accord with the quote you have provided us earlier. If you consider space-time, you get a consistent and full explanation of everything that happens in the scenario, which the space-only perspective cannot do.
    I do consider it. But like Einstein, I don't confuse it with space.
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    Quote Originally Posted by KJW View Post
    No, it can only be local in general. In curved spacetime, there is no consistent notion of the global conservation of energy-momentum. Markus' mathematical expression basically says that attempts to define a notion of global conservation of energy-momentum will lead to inconsistencies between different frames of reference. It is the peculiar nature of spacetime curvature that energy-momentum is conserved everywhere (local) but not necessarily conserved overall (global). Ultimately, this is because the energy-momentum at different locations cannot be consistently added together to form a total.
    There is no peculiar nature to spacetime curvature, frames of reference are abstract things, energy is not. Matter is made of it. Do not dismiss conservation of energy because you don't know how to add energy up.

    Quote Originally Posted by KJW
    In what way are you a "relativist"?
    I'm forever quoting Einstein. I root for relativity. I think it's much mistaught and much misunderstood. I think it's the Cinderella of contemporary physics.
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    Quote Originally Posted by Farsight View Post
    There is no peculiar nature to spacetime curvature, frames of reference are abstract things, energy is not.
    Do not dismiss conservation of energy because you don't know how to add energy up.
    Energy is a frame-dependent quantity; the only manner in which to formulate it covariantly is via the energy-momentum tensor, but then you cannot consistently integrate that over a region, as demonstrated, so its conservation holds everywhere locally, but not globally.

    The point being that if gravitational potential is the same everywhere, light goes straight and you don't fall down
    The gravitational potential at a Lagrange point is different than at a reference point far away.
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    Quote Originally Posted by Farsight View Post
    But like Einstein, I don't confuse it with space.


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    Quote Originally Posted by Markus Hanke View Post
    Energy is not even defined globally in GR, it is always a local property, hence there is no global conservation, nor does there need to be one :

    Is Energy Conserved in General Relativity?
    The lack of a definition does not mean energy can be magicked up out of fresh air. And the Baez article isn't great. See where it says this?

    but includes NO contribution from "gravitational energy". So one can argue that "gravitational energy" does NOT act as a source of gravity.

    I pointed out the other day that on page 185 of the Doc30 Foundation of the General Theory of Relativity Einstein said this:

    the energy of the gravitational field shall acti gravitaively in the same way as any other kind of energy.

    Quote Originally Posted by Markus Hanke View Post
    The generalisation of the laws of thermodynamics to curved space-times still hold, so there are no perpetual motion machines here. I do not know why you would think otherwise. There is no energy being "created" with which one could perform any work, nor did I claim that there was. You need to have a more careful look at the mathematics I gave.
    Then conservation of energy holds good.

    Quote Originally Posted by Markus Hanke View Post
    The expression formulates what I have explained in terms of the appropriate mathematics. It is a basic flux integral, and the equalities come in via the generalised Stokes theorem, which is valid in any coordinate system and on any manifold. You are welcome to point out any mathematical mistakes, if you can find them, otherwise this is just the integral of a divergence over a region M.
    Note Kelvin's involvement in Stokes theorem.

    Quote Originally Posted by Markus Hanke View Post
    What justification do you expect me to give you ? If I reference a textbook ( which of course I can, the above maths are elementary differential geometry ) you will reject it anyway because it deviates from your own ideas. But if you want the reference still, then let me know, and I'll provide it - for starters you might look at the references section of the Baez article I linked to above. Other than that, you should know by now that I can back up anything and everything I say with textbook references.
    A justification to support your argument, which appears to be non-conservation of energy.

    Quote Originally Posted by Markus Hanke View Post
    With all due respect Farsight, but what does this have to do with the discussion at hand?
    It proves my point. If the 511keV photon really did increase in frequency and really did gain energy, the black hole mass would increase by more than 511keV/c².

    Quote Originally Posted by Markus Hanke View Post
    We were talking about the Pound-Rebka experiment and red/blueshift of light in the vicinity of Earth, not black holes, so please do not change the scenario now. The calculation for light falling towards Earth is just a straightforward application of the standard frequency shift formula in Schwarzschild space-times.
    It's straightforward, but conservation of energy applies. There is no magical mysterious mechanism by which the photon gains energy from nowhere.

    Quote Originally Posted by Markus Hanke View Post
    Everyone's clocks continue to tick at the exact same rate of "1 second per second" locally. The only thing that changes is the geometry of the observers' world lines ( and hence recorded total proper times along some section of them ), due to the background curvature of space-time around the earth, i.e. the global relationships between observers. This is the basic idea of GR - you locally continue using Minkowskian laws, but globally the relationships become more complicated.
    Come on Markus. When you're lower than me your clock is going slower than mine. And you measure the photon to have a higher frequency than I do.

    Quote Originally Posted by Markus Hanke View Post
    We didn't change either. What is different though are the geodesics in our region of space-time, as compared to some reference point very far away, which is why the photon arrives here at a different frequency. It's all just geometry.
    Geometry is important, but you did change. When you fell to your lower location some of your mass-energy was converted into kinetic energy and dissipated. Your mass reduced. You know about the mass deficit, don't you?

    You know, maybe we need a new thread: Conservation of energy and gravity.
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    Quote Originally Posted by Kojax View Post
    If I dropped a rock between the two stars, the rock would accelerate (gain kinetic energy) as it approaches the center point between the stars, and then begin to lose kinetic energy is it passes the center point and moves away.
    Yes. But this kinetic energy doesn't come from nowhere. It comes from the mass-energy of the rock. Lift a rock and you do work on it. You expend energy. This energy is in the rock. Yes the Earth and the rock are a system, but the Earth is so big we ignore it and focus on the rock. That rock gained mass because you lifted it. When you drop it, some of that mass, some of that gravitational potential energy, is converted into kinetic energy.

    Quote Originally Posted by Kojax View Post
    By shifting toward blue, light is also gaining energy as it moves to the center point. Then by shifting red as it leaves the system, it loses that energy again.
    No. The photon is just kinetic energy. It doesn't gain any energy at all.

    Quote Originally Posted by Kojax View Post
    Anyway, clearly we must agree that the energy of falling objects is not conserved. Only the combination of the object's potential and kinetic energy is conserved. As it loses potential energy, it gains kinetic energy. With light, it's the same except the light is gaining a different kind of energy other than kinetic energy.
    The energy of falling things is conserved because gravity is not a force in the Newtonian sense. You do work on the rock when you lift it. But no work is done on the rock when it falls. All that happens is that potential energy in the rock is converted into the rock's kinetic energy. The photon doesn't have any potential energy, it's all kinetic energy.

    Quote Originally Posted by Kojax
    No. But you do fall toward the center of mass for the whole system. In this case, the center of mass doesn't actually have any mass in it. But that doesn't matter...
    No problem with any of this.
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    Quote Originally Posted by Farsight View Post
    The lack of a definition does not mean energy can be magicked up out of fresh air.
    I did not say that it can. It is conserved every locally, after all, so there are no sources or sinks of energy-momentum. Yet observers do not agree on the sum total of energy-momentum in a region of space-time.

    Then conservation of energy holds good.
    Locally yes, not globally.

    It proves my point. If the 511keV photon really did increase in frequency and really did gain energy, the black hole mass would increase by more than 511keV/c².
    It neither gains nor looses energy; it is the energy itself which different observers don't necessarily agree on. All you can do if you want everyone to agree is formulate an energy-momentum tensor for the photon - but then you no longer have a Schwarzschild space-time !

    Come on Markus. When you're lower than me your clock is going slower than mine. And you measure the photon to have a higher frequency than I do.
    Yes, but note what you have written - "me vs you". It is about the comparison between observers; each observer in isolation, without an outside reference point, notices nothing special. Locally, space-time is everywhere Minkowskian, but globally it is not. That is the central point of GR.

    Geometry is important, but you did change.
    When I am far away I am in a local patch of Minkowskian space-time. When I am closer to the central mass ( in free fall ), I am also in a local patch of Minkowskian space-time. It is only the relationship between those frames in a globally curved space-time that allows me to deduce GR effects such as gravitational time dilation or frequency shift. You cannot see time dilation on just one clock, and neither can you see frequency shift if there is only an isolated receiver.

    Everything comes down to local vs global in GR.
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    Quote Originally Posted by Markus Hanke View Post
    Energy is a frame-dependent quantity; the only manner in which to formulate it covariantly is via the energy-momentum tensor, but then you cannot consistently integrate that over a region, as demonstrated, so its conservation holds everywhere locally, but not globally.
    Go back to the SR photon, and your measurement of energy is a frame-dependent quantity. But your frame is little more than your state of motion. And no matter how you move, the photon energy does not change. Now apply the principle of equivalence and think about your mass deficit and that 511keV photon falling into the black hole. What you measure isn't always a complete picture of reality.

    Quote Originally Posted by Markus Hanke View Post
    The gravitational potential at a Lagrange point is different than at a reference point far away.
    Yes it is. Just as the gravitational potential at the reference point 13.6 billion years ago is different to that at the reference point now. The gravitational potential changes over time as the universe expands, but when it's the same at all points in space you don't fall down. And the universe didn't collapse when it was small and dense, did it?
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    Quote Originally Posted by Markus Hanke View Post


    You can safely ignore the mathematical details so long as you can make sense of the fact that there are certain "constants of motion", i.e. quantities which are conserved at all points along a free-fall trajectory. In GR, those quantities are dependent on the metric, and hence do not generally correspond to the Newtonian concepts of kinetic and potential energies.
    That makes more sense. Because now we're not saying there is no conservation of energy. We're just saying the energy can take more forms.

    In Newtonian gravity we needed potential energy and kinetic energy to form a conservation law. Now I guess we just need more "kinds" of energy. New concepts to fill in the gaps.

    And I see what you mean, that it might make more sense to just scrap the "energy" concept altogether and go with "constants of motion" or some other concept that's more global, or requires fewer separate concepts in order to make a model out of.


    All in all, many things in GR curved space-times which may seem clear and unambiguous at first glance turn out to be highly non-trivial and counterintuitive on closer inspection. That is just the nature of things, I suppose.
    Or one might take it as evidence that GR is flawed. But it's the best we've got.

    The Ptolemaic system of the modern era. An almost unapproachably complicated system of loops within loops, but which gives accurate predictions. And if we want to navigate we've got to use it because there isn't anything better.

    And it could be another 1000 years before we figure out the better answer, so if one wants to practice physics within this lifetime, GR it is.
    A mathematician and an engineer were at a party. An older colleague of theirs was there with his daughter. The two each asked if they could speak to her. He said it was ok, but they had to approach her by going half way across the room, then stop, then half way again and stop and proceed in that manner. The mathematician realized that he would never reach her and gave up. The engineer determined that he could get close enough to talk. --Approximate retelling of a joke by my math teacher.
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    Quote Originally Posted by Farsight View Post
    And the Baez article isn't great. See where it says this?

    but includes NO contribution from "gravitational energy". So one can argue that "gravitational energy" does NOT act as a source of gravity.

    I pointed out the other day that on page 185 of the Doc30 Foundation of the General Theory of Relativity Einstein said this:

    the energy of the gravitational field shall acti gravitaively in the same way as any other kind of energy.
    Both of these are correct. The source of gravity in the Einstein equations is the energy-momentum tensor, which does not include contributions by gravitational self-energy, as Baez rightly points out; on the other hand the field equations themselves are highly non-linear in the sense that the differential equations for the components of the metric tensor are mutually interdependent - this is where gravitational self-interactions come in, as Einstein himself rightly recognised. Therefore, you can get topological constructs which are held together by their own gravitational self-interaction, such as geons, in the complete absence of any sources of energy-momentum - later ( 1960s ) proven with explicit examples by Wheeler.
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    Quote Originally Posted by Kojax View Post
    New concepts to fill in the gaps.
    Yes, in a way. The Newtonian notions of energy are replaced by constants of motion, which arise from the symmetries of the space-time in question. These conserved quantities are hence explicitly dependent on the metric itself, and therefore only valid in specific scenarios, but not in general.

    And I see what you mean, that it might make more sense to just scrap the "energy" concept altogether and go with "constants of motion" or some other concept that's more global, or requires fewer separate concepts in order to make a model out of.
    Exactly.

    Or one might take it as evidence that GR is flawed. But it's the best we've got.
    I don't agree. Just because it is counter-intuitive does not mean it must be flawed, because "being intuitive" is a very subjective notion. What is intuitive to me mightn't be intuitive to you at all, and vice versa. Do you know what I mean ?
    On the other hand, we know already that GR is incomplete, because it is a purely classical model.
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    Quote Originally Posted by Markus Hanke View Post
    I did not say that it can. It is conserved every locally, after all, so there are no sources or sinks of energy-momentum. Yet observers do not agree on the sum total of energy-momentum in a region of space-time.
    OK no problem.

    Quote Originally Posted by Markus Hanke View Post
    Locally yes, not globally.
    There's no sources or sinks, so even though observers don't agree, they know the problem is with their summation.

    Quote Originally Posted by Markus Hanke View Post
    It neither gains nor looses energy;
    Exactly.

    Quote Originally Posted by Markus Hanke View Post
    it is the energy itself which different observers don't necessarily agree on. All you can do if you want everyone to agree is formulate an energy-momentum tensor for the photon - but then you no longer have a Schwarzschild space-time !
    They aren't agreeing on their measurement of this energy. This is akin to you putting the pedal to the metal in your gedanken spaceship. The distance between here and Alpha Centauri doesn't change just because you decided to go fast.

    Quote Originally Posted by Markus Hanke View Post
    Yes, but note what you have written - "me vs you". It is about the comparison between observers; each observer in isolation, without an outside reference point, notices nothing special.
    But we compare notes to learn more. To see the big picture beyond our local parochial view.

    Quote Originally Posted by Markus Hanke View Post
    Locally, space-time is everywhere Minkowskian, but globally it is not. That is the central point of GR.
    I honestly don't share that view Markus. The principle of equivalence applies to an infinitesimal region. To a region of zero extent. Which means it doesn't apply at all. It was an "enabling principle" only. Hence the Synge quote about the midwife being laid to rest with full honours. You cannot transform away a real gravitational field. If spacetime really was locally Minkowskian, your pencil wouldn't fall down.

    Quote Originally Posted by Markus Hanke View Post
    When I am far away I am in a local patch of Minkowskian space-time. When I am closer to the central mass ( in free fall ), I am also in a local patch of Minkowskian space-time. It is only the relationship between those frames in a globally curved space-time that allows me to deduce GR effects such as gravitational time dilation or frequency shift. You cannot see time dilation on just one clock, and neither can you see frequency shift if there is only an isolated receiver.
    No problem. When you and your pencil are falling, you can claim that your pencil doesn't fall down.

    Quote Originally Posted by Markus Hanke View Post
    Everything comes down to local vs global in GR.
    If you say so. And I say this: to understand the world, we compare what you see with what I see, and we agree that whilst we see it different, the world is what it is. Is it a square? Is it a hexagon? No, it's a cube.
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    Quote Originally Posted by Farsight View Post
    Yes it is. Just as the gravitational potential at the reference point 13.6 billion years ago is different to that at the reference point now. The gravitational potential changes over time as the universe expands, but when it's the same at all points in space you don't fall down. And the universe didn't collapse when it was small and dense, did it?
    You cannot describe an expanding universe using the Schwarzschild metric. How do you define the notion of a global "gravitational potential" from the Killing vectors of the FLRW metric ?

    And the universe didn't collapse when it was small and dense, did it?
    Of course not. Remember that this is a FLRW space-time, not a Schwarzschild space-time.

    And no matter how you move, the photon energy does not change.
    When you are in relative motion wrt to the emitter, the photon will impact you at a different frequency as compared to the same scenario without relative motion. Likewise, if you encounter an in-falling photon close to the event horizon, you will observe it having a frequency very different from that at the far-away emitter, even though energy is locally conserved at every point of its trajectory.

    What you measure isn't always a complete picture of reality.
    It's a complete picture of your reality, but different observers can have different "realities". There is no universally valid frame in GR, globally speaking.
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    Quote Originally Posted by Farsight View Post
    OK no problem.
    There's no sources or sinks, so even though observers don't agree, they know the problem is with their summation.
    Actually, the point is that there is no problem in them disagreeing. Unlike in Newtonian physics, the notion of energy does not need to be globally defined or conserved, since it is replaced by other conserved quantities. I know that this isn't an easy concept to grasp, nor is it what you probably want to believe, but since you are on a mainstream forum I think it is reasonable to ask you to at least consider this idea, if nothing else. Newton is restricted to a very highly symmetric kind of space-time, but GR is much more general, so it is no surprise that many Newtonian concepts just simply no longer hold.

    But we compare notes to learn more. To see the big picture beyond our local parochial view.
    Yes, indeed. But we need to make sure that we actually do this, because without comparing notes we will not see any of the GR-related effects.

    The principle of equivalence applies to an infinitesimal region. To a region of zero extent.
    No, it actually applies to a region on the order of 1/g, wherein g is the magnitude of proper acceleration. In other words, it applies to regions of uniform fields, i.e. regions where no tidal forces are present, as is the case for a uniformly accelerating frame, for instance.

    No problem. When you and your pencil are falling, you can claim that your pencil doesn't fall down.
    Obviously. If I am in free fall, my proper acceleration is zero, and I experience no gravity in my local free-fall frame, so my pencil will float with me and not fall. If you do the same somewhere far away from me, you will experience the exact same in your own free-fall frame. However, if we somehow manage to compare our measurements of space and time and energy, we will find that strangely enough they might differ. This is both the beauty and the challenge of GR.

    to understand the world, we compare what you see with what I see
    I agree. You measure and I measure, and then we relate those measurements. Then we proceed to do this at every possible pair of points in a region of space-time, and relate all of these measurements to each other - the result of which is the metric.
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    Quote Originally Posted by Markus Hanke View Post
    Both of these are correct. The source of gravity in the Einstein equations is the energy-momentum tensor, which does not include contributions by gravitational self-energy, as Baez rightly points out; on the other hand the field equations themselves are highly non-linear in the sense that the differential equations for the components of the metric tensor are mutually interdependent - this is where gravitational self-interactions come in, as Einstein himself rightly recognised.
    That takes me back to what I was saying about the energy of space itself.

    Quote Originally Posted by Markus Hanke View Post
    Therefore, you can get topological constructs which are held together by their own gravitational self-interaction, such as geons, in the complete absence of any sources of energy-momentum - later (1960s) proven with explicit examples by Wheeler.
    Shame he didn't think about electromagnetic self-interaction.

    Quote Originally Posted by Markus Hanke
    You cannot describe an expanding universe using the Schwarzschild metric.
    I didn't say you could.

    Quote Originally Posted by Markus Hanke
    How do you define the notion of a global "gravitational potential" from the Killing vectors of the FLRW metric?
    I don't know how from the killing vectors. But the FLRW metric starts with the assumption of homogeneity and isotropy of space. We've all read Einstein referring to a gravitational field as inhomogeneous space. So the FLRW metric assumes no gravitational field. It assumes the same gravitational potential everywhere.

    Quote Originally Posted by Markus Hanke
    Of course not. Remember that this is a FLRW space-time, not a Schwarzschild space-time.
    And yet people talk about the Big Crunch, and others marvel that space is flat. Even though it's always been flat.

    Quote Originally Posted by Markus Hanke
    When you are in relative motion wrt to the emitter, the photon will impact you at a different frequency as compared to the same scenario without relative motion. Likewise, if you encounter an in-falling photon close to the event horizon, you will observe it having a frequency very different from that at the far-away emitter, even though energy is locally conserved at every point of its trajectory.
    No problem with that. But like you said, it neither gains nor loses energy.

    Quote Originally Posted by Markus Hanke
    What you measure isn't always a complete picture of reality. It's a complete picture of your reality, but different observers can have different "realities". There is no universally valid frame in GR, globally speaking.
    I know. But when two observers have two different "realities" and they contradict one another, we put our thinking caps on.
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    Quote Originally Posted by Farsight View Post
    That takes me back to what I was saying about the energy of space itself.
    I think it is really more accurate to say that gravity self-interacts, as opposed to saying it has energy, once again because energy is difficult to define.

    Shame he didn't think about electromagnetic self-interaction.
    Actually, he did - there is such a thing as an electromagnetic geon.

    It assumes the same gravitational potential everywhere.
    Which answers your original question.

    And yet people talk about the Big Crunch, and others marvel that space is flat. Even though it's always been flat.
    I'm not sure what you mean by this now.

    But like you said, it neither gains nor loses energy.
    Yes. It's really just a matter of the emitter and the receiver not agreeing on what the energy/frequency actually is, hence the red/blueshift. Different energies are determined in different frames. Though it is counterintuitive, this does not imply creation or destruction of energy anywhere, so the conservation law can still hold locally at all points. I find this really fascinating.

    I know. But when two observers have two different "realities" and they contradict one another, we put our thinking caps on.
    We should never take our thinking hats off in the first place
    But jokes aside, it once again comes down to the same thing - there is no absolute reality in GR, and unlike in Newtonian physics, there is no requirement for there to be one. In GR, observers have more "freedom" in that they are permitted to disagree on certain things, whereas in Newtonian physics that is not possible. As a rule of thumb, all proper measurements are invariant; for example, if an arbitrary observer uses a stop watch to record a certain amount of elapsed time, then the maths of GR work out in such a way that all observers will agree on what the clock has physically recorded. Same for proper length and proper accelerations, and so on. This way, there are no physical contradictions, ever ( the hands of the clock, once stopped, will not suddenly jump to a different position just because you move to a different reference frame ). There are only disagreements about coordinate measurements, because those depend on the system of coordinates chosen, and the observer, both of which are purely arbitrary.
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    Quote Originally Posted by Markus Hanke View Post
    Actually, the point is that there is no problem in them disagreeing. Unlike in Newtonian physics, the notion of energy does not need to be globally defined or conserved, since it is replaced by other conserved quantities. I know that this isn't an easy concept to grasp, nor is it what you probably want to believe, but since you are on a mainstream forum...
    Whoa there. You said it neither gains nor loses energy. I've persuaded you that the 511keV photon neither gains nor loses energy. I've convinced you that conservation of energy does apply. I don't have an issue with other conserved quantities. Apart from invariant mass.

    Quote Originally Posted by Markus Hanke View Post
    I think it is reasonable to ask you to at least consider this idea, if nothing else. Newton is restricted to a very highly symmetric kind of space-time, but GR is much more general, so it is no surprise that many Newtonian concepts just simply no longer hold.
    I don't have an issue with other conserved quantities. If you'd like me to say something about them please start a thread.

    Quote Originally Posted by Markus Hanke View Post
    Yes, indeed. But we need to make sure that we actually do this, because without comparing notes we will not see any of the GR-related effects.
    Sounds good to me Markus. It's good to talk.

    Quote Originally Posted by Markus Hanke View Post
    No, it actually applies to a region on the order of 1/g, wherein g is the magnitude of proper acceleration. In other words, it applies to regions of uniform fields, i.e. regions where no tidal forces are present, as is the case for a uniformly accelerating frame, for instance.
    Not the uniform gravitational field again. Let's have a new thread on that.

    Quote Originally Posted by Markus Hanke View Post
    Obviously. If I am in free fall, my proper acceleration is zero, and I experience no gravity in my local free-fall frame, so my pencil will float with me and not fall.
    OK.

    Quote Originally Posted by Markus Hanke View Post
    I agree. You measure and I measure, and then we relate those measurements. Then we proceed to do this at every possible pair of points in a region of space-time, and relate all of these measurements to each other - the result of which is the metric.
    Good stuff.
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    Quote Originally Posted by Markus Hanke View Post
    I think it is really more accurate to say that gravity self-interacts, as opposed to saying it has energy, once again because energy is difficult to define.
    I beg to differ. But LOL, you'd say it wasn't mainstream.

    Quote Originally Posted by Markus Hanke
    Actually, he did - there is such a thing as an electromagnetic geon.
    That's a bit of a mouthful, Markus. Couldn't we shorten that to something else? Ah, now what could that be?

    Quote Originally Posted by Markus Hanke
    Which answers your original question.
    Only if the gravitational potential was very very low, the early universe would have been like a frozen-star black hole and we don't need inflation.

    Quote Originally Posted by Markus Hanke
    I'm not sure what you mean by this now.
    Space is flat and it's expanding so I can't see how it can be infinite. Which means it has to have some kind of edge.

    Quote Originally Posted by Markus Hanke
    Yes. It's really just a matter of the emitter and the receiver not agreeing on what the energy/frequency actually is, hence the red/blueshift. Different energies are determined in different frames. Though it is counterintuitive, this does not imply creation or destruction of energy anywhere, so the conservation law can still hold locally at all points. I find this really fascinating.
    What's even more fascinating is that energy isn't local. A photon takes many paths. An electron's field is what the electron is, and it doesn't have an edge.

    Quote Originally Posted by Markus Hanke
    We should never take our thinking hats off in the first place But jokes aside, it once again comes down to the same thing - there is no absolute reality in GR, and unlike in Newtonian physics, there is no requirement for there to be one. In GR, observers have more "freedom" in that they are permitted to disagree on certain things, whereas in Newtonian physics that is not possible. As a rule of thumb, all proper measurements are invariant; for example, if an arbitrary observer uses a stop watch to record a certain amount of elapsed time, then the maths of GR work out in such a way that all observers will agree on what the clock has physically recorded. Same for proper length and proper accelerations, and so on. This way, there are no physical contradictions, ever (the hands of the clock, once stopped, will not suddenly jump to a different position just because you move to a different reference frame ). There are only disagreements about coordinate measurements, because those depend on the system of coordinates chosen, and the observer, both of which are purely arbitrary.
    All that is pretty reasonable. But hmmn, maybe we should talk about black holes and Kruskal-Szekeres coordinates some more. But not on this thread.

    And not now. Because I have to go.
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    Quote Originally Posted by Farsight View Post
    Space is flat and it's expanding so I can't see how it can be infinite.
    Actually, we don't know for certain whether it is flat or not; we can only tell that the observable part of it appears to be flat within certain observational limits.
    But regardless - there isn't really a contradiction between it being flat and infinite and expanding. The trick is to realise that the BB was not necessarily a single point; think of it rather as a collection of infinitely many points, the separation of which was initially exactly zero. As the universe came into being, that separation became well defined and non-zero; you can now pick to arbitrary points and define the distance between them. This doesn't preclude the possibility that for every pair of points you choose, there exists another pair which is even further apart, ad infinitum. Again, this is not intuitive, but the mathematics are unambiguous and quite rigorous.

    But hmmn, maybe we should talk about black holes and Kruskal-Szekeres coordinates some more.
    Ok, I would be happy to. But I should say in advance that there are aspects to the KS chart which I haven't fully grasped myself yet; it is a very important system though, because it is a maximally extended chart, and hence covers the entire Schwarzschild space-time, which Schwarzschild coordinates themselves can't do ( you'd need two separate patches for this ).
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    Quote Originally Posted by Farsight View Post
    Yes. But this kinetic energy doesn't come from nowhere. It comes from the mass-energy of the rock. Lift a rock and you do work on it. You expend energy. This energy is in the rock. Yes the Earth and the rock are a system, but the Earth is so big we ignore it and focus on the rock. That rock gained mass because you lifted it. When you drop it, some of that mass, some of that gravitational potential energy, is converted into kinetic energy.

    No. The photon is just kinetic energy. It doesn't gain any energy at all.
    Photons have an equivalent to gaining and losing kinetic energy. For a photon, shifting toward blue or red does that. If I'm driving at 100 km/h and I throw a baseball ahead at 20 km/h, the baseball will be moving at 120 km/h, right? Or if I threw it behind me as I'm driving at 100 km/h, it would be going at 80 km/h forward.

    Ok. With light, the speed of light is invariant. But if I shine a flashlight forward as I'm driving at 100 km/h it gets slightly blue shifted. That's the equivalent of the baseball moving faster when thrown from a moving car. A blue photon has more energy than a red photon. The number of photons in the beam of light stays the same, but each photon has more energy because it has a shorter wavelength and higher frequency.

    That's why photons shift toward blue when they descend into a gravitational field. It's the equivalent of a falling baseball gaining momentum.



    The energy of falling things is conserved because gravity is not a force in the Newtonian sense. You do work on the rock when you lift it. But no work is done on the rock when it falls. All that happens is that potential energy in the rock is converted into the rock's kinetic energy. The photon doesn't have any potential energy, it's all kinetic energy.
    It is hard to "do work on" a photon. But it is not impossible.

    Whenever the photons from a police officers' radar gun bounces off a moving vehicle, some work is done on those photons. Either the car is moving away, and the photon comes back slightly red shifted, or the car is moving toward and the photon comes back slightly blue shifted.

    If the officer had instead thrown a bouncy ball at the car as it was moving away and waited for it to bounce back, the ball would return to him with less momentum than it had when he threw it. (This is true even ignoring air resistance.) If he threw the ball toward a car that was coming toward him and waited for it to bounce back, then it would return to him with more momentum than it had when he threw it.
    A mathematician and an engineer were at a party. An older colleague of theirs was there with his daughter. The two each asked if they could speak to her. He said it was ok, but they had to approach her by going half way across the room, then stop, then half way again and stop and proceed in that manner. The mathematician realized that he would never reach her and gave up. The engineer determined that he could get close enough to talk. --Approximate retelling of a joke by my math teacher.
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    Quote Originally Posted by Kojax View Post
    But if I shine a flashlight forward as I'm driving at 100 km/h it gets slightly blue shifted. That's the equivalent of the baseball moving faster when thrown from a moving car. A blue photon has more energy than a red photon. The number of photons in the beam of light stays the same, but each photon has more energy because it has a shorter wavelength and higher frequency.
    Change the flashlight to an omnidirectional source of EMR.

    The blue shift and red shift are not observed by you in the moving car. Those stationary, directly on the road in front of the car will see the maximum blue shift in the incident beam. Those stationary directly on the road behind will see the maximum red shift. Those directly off to the side will see no shift. Those that are upstream or downstream will see shift as a function of the relative angle of the car's movement (and the volicity of course).

    All this to say that the gain or loss of energy of the light beam is a perception of the observer relative to the source. To some the beam is bluer, to others redder. So does the light change gain or lose energy. The same effect occurs if the observer moves while the source is fixed. Moved and fixed relative to the earth or to any frame of reference.
    Last edited by pikpobedy; 04-06-2014 at 02:51 AM.
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    Quote Originally Posted by mayflow View Post
    and consciousness may effect and even create matter.
    I thought I had explained this to you already. We have no evidence that consciousness affects matter, let alone that it creates matter.
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    Quote Originally Posted by Farsight View Post
    There is no peculiar nature to spacetime curvature, frames of reference are abstract things, energy is not. Matter is made of it. Do not dismiss conservation of energy because you don't know how to add energy up.
    Spacetime curvature is peculiar in that properties that hold true in the familiar world of Euclidean geometry do not simply hold true in curved spacetime. Frames of reference are what the name indicates: They are the reference to which our measurements are relative. How would you specify a location on Earth without using the latitude and longitude coordinate system (a frame of reference)? I'm not dismissing the conservation of energy-momentum. What I said was that it can only be local. And it is not just the case that I don't know how to add energy-momentum, but that there is no consistent way to add energy-momentum. Curvature is an obstruction to doing this. For example, suppose I have an arrow pointing in some direction and I choose to move this such that it always remains parallel to itself at its immediate previous location. I move around a large loop back to my initial starting point. Then the arrow will not necessarily point in the same direction as it originally started from. The angle through which the arrow rotated depends on the curvature inside the loop. In other words, even though there was no rotation of the arrow at any point around the loop, the total rotation upon going all the way around the loop is non-zero (if the curvature is non-zero). That is, the total rotation is not the sum of the individual rotations. Furthermore, if starting from the same location, I choose different loops to go around, the rotation angle will also differ. So the total change in the rotation angle while going between two points depends on the path between them and not simply on the points themselves. But note that it is curvature that causes this apparent anomaly, and indeed curvature can be defined by this apparent anomaly.


    Quote Originally Posted by Farsight View Post
    I'm forever quoting Einstein. I root for relativity. I think it's much mistaught and much misunderstood. I think it's the Cinderella of contemporary physics.
    Einstein is not god and should not be taken as the final word on relativity. Ultimately, it is the mathematics that has the final word on the theory. In saying that you think it is mistaught, in what way do you think the mathematics of relativity is mistaught?
    A tensor equation that is valid in any coordinate system is valid in every coordinate system.
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    Quote Originally Posted by Farsight View Post
    Space is flat and it's expanding so I can't see how it can be infinite. Which means it has to have some kind of edge.
    The metric of a constantly expanding infinite flat space:

    A tensor equation that is valid in any coordinate system is valid in every coordinate system.
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    Quote Originally Posted by Markus Hanke View Post
    I don't agree. Just because it is counter-intuitive does not mean it must be flawed, because "being intuitive" is a very subjective notion. What is intuitive to me mightn't be intuitive to you at all, and vice versa. Do you know what I mean ?
    On the other hand, we know already that GR is incomplete, because it is a purely classical model.
    Oh yeah. I guess that's true. It doesn't match up with QM yet, so I guess nobody is expecting it to be entirely correct at this point. Either that, or nobody is expecting QM to be entirely correct. Or both.


    Quote Originally Posted by Farsight View Post

    Only if the gravitational potential was very very low, the early universe would have been like a frozen-star black hole and we don't need inflation.
    Maybe it did set there for an infinite length of time from the perspective of someone outside the system looking in.

    Sort of like how events transpiring inside a black hole appear to be frozen in time from the perspective of an observer outside looking in. But to an observer inside the event horizon time is continuing normally.


    Space is flat and it's expanding so I can't see how it can be infinite. Which means it has to have some kind of edge.
    An edge we can never reach. Which therefore doesn't exist in any practical sense.

    The mathematician can go crazy insisting it exists, but the engineer doesn't worry about it because it's not part of his/her universe. Maybe it's part of "the" universe, but not part of any particular observer's universe.

    I'm strongly inside the QM camp, in terms of how I view these matters. As far as I am concerned, unobservable reality is like Schrodinger's Cat. It has no definite state. Whatever lies beyond the (acceleration adjusted) Hubble Sphere may be a superposition of many possible states. It doesn't have to choose one for us, because we won't be observing it anyway.
    A mathematician and an engineer were at a party. An older colleague of theirs was there with his daughter. The two each asked if they could speak to her. He said it was ok, but they had to approach her by going half way across the room, then stop, then half way again and stop and proceed in that manner. The mathematician realized that he would never reach her and gave up. The engineer determined that he could get close enough to talk. --Approximate retelling of a joke by my math teacher.
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    Quote Originally Posted by Farsight View Post
    Quote Originally Posted by Markus Hanke View Post
    How do you define the notion of a global "gravitational potential" from the Killing vectors of the FLRW metric ?
    I don't know how from the killing vectors. But the FLRW metric starts with the assumption of homogeneity and isotropy of space. We've all read Einstein referring to a gravitational field as inhomogeneous space. So the FLRW metric assumes no gravitational field. It assumes the same gravitational potential everywhere.
    The FLRW metric is not (in general) stationary. It does have Killing vectors, but they are spacelike, whereas the definition of the gravitational potential I mentioned earlier involves timelike Killing vectors and stationary spacetime trajectories, which there are none because the FLRW metric is not stationary. So the notion of gravitational potential doesn't apply to the FLRW metric.

    The notion of gravitation in GR is more complicated than what you suggest. It is true that the FLRW metric with arbitrarily expanding flat space is conformally flat. This means that the Weyl conformal tensor field is zero everywhere at all times. The Weyl conformal tensor represents the gravitation external to an energy-momentum distribution, so there is no gravitation of this type in the FLRW metric. But there is also the local gravitation associated with the energy-momentum itself, described by the Einstein curvature tensor. For a flat space, it arises because this flat space is expanding over time.
    A tensor equation that is valid in any coordinate system is valid in every coordinate system.
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    Quote Originally Posted by Farsight View Post
    But when two observers have two different "realities" and they contradict one another, we put our thinking caps on.
    Although different frames of reference have difference descriptions of the reality, they do not contradict one another. One can transform the description in one frame of reference to the description in another frame of reference by applying the relationship between the two frames of reference. The "true" reality then becomes the set of all possible descriptions from all frames of reference with no frames of reference being preferred over any of the others.

    Suppose one has a map of the world and turns this upside down. The upside down map is a different description of the world, but it's still the same map. We can transform the upside down description to the right-side up description by a rotation of 180°, and the ability to do this tells us that the two descriptions are indeed of the same map.
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    Quote Originally Posted by Farsight View Post
    Whoa there. You said it neither gains nor loses energy. I've persuaded you that the 511keV photon neither gains nor loses energy. I've convinced you that conservation of energy does apply.
    I think you are still not grasping the point of what I am trying to say; maybe I am just not explaining it well enough. The first thing you must realise is that the energy you say the photon has is already an observer-dependent quantity - 511keV as measured by whom ? The determination that the photon has this particular amount of energy is true only in the frame of whoever makes this measurement ( probably the rest frame of the emitter ). If we consider another observer somewhere else ( past whom the photon falls on its way to the event horizon ), then that observer will generally disagree as to how much energy it has. At the same time, we know that energy is locally conserved at every point along the photon's in-fall trajectory, so what is the solution to the dilemma ? The solution is quite simply that all observers are right, but only in their own local frames of reference. Reality is what a particular observer measures - there is no "universal" truth as to the amount of energy the photon has. A global energy conservation law as in Newtonian physics cannot be consistent in such a set-up, nor is it needed.

    Now, as it so happens, Schwarzschild space-time is a highly symmetric type of geometry; the parameter "M", as KJW as already pointed out, is not actually the mass of the black hole in isolation, but the mass-energy of the entire space-time. This means that, if we add a photon into our space-time, anywhere at all, the "M" will go up by exactly the total amount of energy it has as determined by some observer - in reality it would also mean that the space-time ceases to be Schwarzschild.
    It is important though to realise and understand that this is generally true only in Schwarzschild space-time, due to the symmetries present there. It cannot be generalised to arbitrary geometries, so there is no corresponding global conservation law. A rather striking example of this is the Penrose process - one can actually reduce the total ( "irreducible" ) mass of a Kerr or Kerr-Newman black hole by adding energy ( in-falling matter or light ) in just the right manner, but only up to a maximum of 29% of initial mass. This process is not possible in Newtonian physics, since it would globally violate conservation laws, but it is allowed in GR.

    I must stress again one very crucial insight for all of this - in GR, conservation of energy is not violated, but rather it isn't generally defined in the first place. These are very different concepts - you cannot violate a law that isn't defined. The underlying reason is that the definition of "conservation of energy" is connected to symmetries via Noether's theorem; conservation of energy corresponds to space-time translation invariance, and this invariance does not hold for general curved space-times, but of course it will always hold in Euclidean space, and hence in classical mechanics.

    Explicitly, in Schwarzschild space-time, the sum total of kinetic and gravitational potential energy per unit mass of a particle is a conserved quantity ( which is why it adds linearly to the "M" parameter ), but you can't consistently split it into its "kinetic" and "potential" parts as we do in classical mechanics. This is most easily seen if we look at the Lagrangian of the falling particle; in classical mechanics, the Lagrangian is a linear combination of kinetic and potential energy :



    so this can be nicely "split" into the two forms of energy. The same is not true in GR, because in Schwarzschild space-time, the same Lagrangian becomes



    which cannot be consistently split into kinetic and potential parts. Since both kinetic and potential energy are observer-dependent quantities, and you can't consistently define them from the Lagrangian, it is meaningless to attempt to write a global conservation law. In the case of Schwarzschild, their sum total happens to be a constant of motion that corresponds to the classical total energy, but this is only because this space-time has all the symmetries it has, and isn't true in the case of more general geometries.

    What I am trying to say with all of this is - when we say that "in GR energy is not globally conserved", then we refer to the most general case of Einstein manifolds, i.e. all geometries which are valid solutions to the GR field equations. It isn't possible to write a general, global law of energy conservation that holds on all such manifolds, quite simply because space-time translation invariance does not generally hold for such manifolds ( via Noether's theorem ). What is possible is, given a metric such as Schwarzschild, to examine its symmetries and deduce a set of conserved quantities, but these will apply only to the specific metric in question, and they do not necessarily correspond to any Newtonian concepts. In the specific case of Schwarzschild, the two constants I gave earlier in the thread are globally conserved, but the integral of the energy-momentum tensor is not, unlike in Newtonian physics.

    It isn't really my aim anymore to try and convince you of things, Farsight, since ultimately you must arrive at your own conclusions, as do we all. I can hope only that the above shed some light on why KJW and myself ( and all textbooks on the matter ) say the things we say, and what the underlying thought processes and mathematics are. I will for now reference the corresponding question on PSE instead of textbooks to back up the above, but if you are looking for references from the latter I can provide them separately :

    general relativity - Is the law of conservation of energy still valid? - Physics Stack Exchange

    I don't have an issue with other conserved quantities.
    Excellent. Given this, all that is needed is to understand that those quantities do not necessarily correspond to any Newtonian concepts, but rather derive from the metric and hence the symmetries of the specific geometry in question.
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    Quote Originally Posted by Farsight View Post
    When you fell to your lower location some of your mass-energy was converted into kinetic energy and dissipated. Your mass reduced. You know about the mass deficit, don't you?
    Quote Originally Posted by Farsight View Post
    The photon is just kinetic energy. It doesn't gain any energy at all.
    Let's consider what you said here. High up, my mass is larger and the photon has some amount of energy. On the ground, my mass is lower and the photon has the same amount of energy. If one considers the ratio of the photon's energy to my mass, then on the ground this will be larger than high up. Because as far as I'm concerned, my mass has not changed, I will observe that the photon's energy has increased (increased relative to my mass). I will observe the photon as blueshifted. Ultimately, this is because of gravitational time dilation.
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    I would like to mention something else which is of importance in this context - it is actually possible to define an object called the Landau-Lifschitz pseudotensor, which is somewhat of an "add-on" to the normal energy-momentum tensor, to include gravitational self-energy as well. If you add this to the energy-momentum tensor and form the covariant divergence of the resulting object, you will find that this is always conserved, even globally; in that sense we can regard that construct as an extension of the energy conservation law to GR - be careful to realise though that this pseudotensor is not an exact equivalent of any Newtonian energy notion, but rather a mathematical trick to eliminate the extra terms appearing under the integral.

    Of course, the price you pay for this is that the Landau-Lifschitz pseudotensor is not a covariant object ( though its divergence is ), so it is once again dependent on the observer, as one would intuitively expect. But if you can live with this compromise, then we can both agree that it is indeed possible to form a global quantity which is conserved in general, and which roughly corresponds to a concept of energy. Personally I find this very ugly and not at all in the spirit of GR, but mathematically it works just fine.
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    Quote Originally Posted by Markus Hanke View Post
    I think you are still not grasping the point of what I am trying to say; maybe I am just not explaining it well enough. The first thing you must realise is that the energy you say the photon has is already an observer-dependent quantity - 511keV as measured by whom ? The determination that the photon has this particular amount of energy is true only in the frame of whoever makes this measurement ( probably the rest frame of the emitter ). If we consider another observer somewhere else ( past whom the photon falls on its way to the event horizon ), then that observer will generally disagree as to how much energy it has. At the same time, we know that energy is locally conserved at every point along the photon's in-fall trajectory, so what is the solution to the dilemma ? The solution is quite simply that all observers are right, but only in their own local frames of reference. Reality is what a particular observer measures - there is no "universal" truth as to the amount of energy the photon has. A global energy conservation law as in Newtonian physics cannot be consistent in such a set-up, nor is it needed.
    Good example.

    It makes sense that different observers would observe the same photon to have different energies, depending on their own direction of motion.


    Quote Originally Posted by pikpobedy View Post
    Change the flashlight to an omnidirectional source of EMR.

    The blue shift and red shift are not observed by you in the moving car. Those stationary, directly on the road in front of the car will see the maximum blue shift in the incident beam. Those stationary directly on the road behind will see the maximum red shift. Those directly off to the side will see no shift. Those that are upstream or downstream will see shift as a function of the relative angle of the car's movement (and the volicity of course).

    All this to say that the gain or loss of energy of the light beam is a perception of the observer relative to the source. To some the beam is bluer, to others redder. So does the light change gain or lose energy. The same effect occurs if the observer moves while the source is fixed. Moved and fixed relative to the earth or to any frame of reference.
    That is true of the baseball also. If I'm in the car traveling at 100 km/h and I throw the ball at 20 km/h I observe the ball to move at 20 km/h away from me.

    Also you didn't address the question of a police officer using a radar gun, and seeing the photons he emitted from his radar gun return blue shifted or red shifted. As compared to what would happen if he threw a bouncy ball at the cars, and noted that the ball gained or lost speed returning to him after it had bounced off the moving car. That second example is a better example.
    A mathematician and an engineer were at a party. An older colleague of theirs was there with his daughter. The two each asked if they could speak to her. He said it was ok, but they had to approach her by going half way across the room, then stop, then half way again and stop and proceed in that manner. The mathematician realized that he would never reach her and gave up. The engineer determined that he could get close enough to talk. --Approximate retelling of a joke by my math teacher.
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    Quote Originally Posted by KJW View Post
    Let's consider what you said here. High up, my mass is larger and the photon has some amount of energy. On the ground, my mass is lower and the photon has the same amount of energy. If one considers the ratio of the photon's energy to my mass, then on the ground this will be larger than high up. Because as far as I'm concerned, my mass has not changed, I will observe that the photon's energy has increased (increased relative to my mass). I will observe the photon as blueshifted. Ultimately, this is because of gravitational time dilation.
    Sounds good. Note thought that there's a subtlety associated with matter being "deflected" only half as much as light. The reduction in your mass is only half the apparent increase in the photon energy. So you might think you'd gained mass.
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    Sorry, I've been tied up for a few days, I'll backtrack to where we got up to.

    Quote Originally Posted by Markus Hanke View Post
    Actually, we don't know for certain whether it is flat or not...
    Fair enough. But I think it is, and always has been.

    Quote Originally Posted by Markus Hanke View Post
    But regardless - there isn't really a contradiction between it being flat and infinite and expanding. The trick is to realise that the BB was not necessarily a single point; think of it rather as a collection of infinitely many points, the separation of which was initially exactly zero...
    I'm sorry Markus, I'm not buying it. We should talk black holes.

    Quote Originally Posted by Markus Hanke View Post
    OK, I would be happy to. But I should say in advance that there are aspects to the KS chart which I haven't fully grasped myself yet; it is a very important system though, because it is a maximally extended chart, and hence covers the entire Schwarzschild space-time, which Schwarzschild coordinates themselves can't do ( you'd need two separate patches for this ).
    Noted. Would you like to start a thread?
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