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Thread: Why is space infinite?

  1. #101  
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    Quote Originally Posted by Kojax View Post
    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.
    There's a simpler mechanism in the guise of Compton scattering and inverse Compton scattering. A photon is accelerated in the vector sense, and loses or gains energy.

    Quote Originally Posted by Kojax View Post
    That's why photons shift toward blue when they descend into a gravitational field. It's the equivalent of a falling baseball gaining momentum.
    Only gravity is not a force in the Newtonian sense, and there's no energy being added to the photon or the baseball.

    Quote Originally Posted by Kojax View Post
    It is hard to "do work on" a photon. But it is not impossible.
    It isn't hard either. It's just the inverse Compton.

    Quote Originally Posted by Kojax View Post
    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.
    No problem. But when we're talking about gravity, it's just a photon or just a ball moving through space, and it isn't interacting with anything other particle or object. So it doesn't gain any energy or lose any.
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  2. #102  
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    Quote Originally Posted by KJW View Post
    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...
    No problem.

    Quote Originally Posted by KJW View Post
    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...
    All points noted, I take no issue with what you said.

    Quote Originally Posted by KJW View Post
    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?
    I think the interpretation of that mathematics is mistaught.
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  3. #103  
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    Quote Originally Posted by Kojax View Post
    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.
    The latter is something we need to talk about.

    Quote Originally Posted by Kojax
    An edge we can never reach. Which therefore doesn't exist in any practical sense.
    It either exists or it doesn't.

    Quote Originally Posted by Kojax
    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.
    Maybe it is a part of a particular observer's universe.

    Quote Originally Posted by KJW View Post
    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...
    Schrodinger proposed his cat to demonstrate the absurdity of the Copenhagen interpretation.
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  4. #104  
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    Quote Originally Posted by KJW View Post
    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.
    All points noted. When I said contradiction I was thinking of the photon energy, wherein observer A says it has an energy of x and observer B says it has an energy of y. I concur the your point about the all-frames "true" reality, and have referred to the "big picture" in an attempt to get this across.
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  5. #105  
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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.
    I understand what you're saying, but I disagree with what you're saying. I'd say there is a universal truth as to the energy of that photon. And I would add that at the root of all this is that we view energy differently. You see it as merely a measure, I see it as something real. Matter is made of it.

    Quote Originally Posted by Markus Hanke
    ...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.
    The Penrose process deserves a thread in its own right. And it's another black hole issue. It's interesting how important black holes are for all sorts of things.

    Quote Originally Posted by Markus Hanke
    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.
    No problem. I guess what I object to is people saying "energy is not conserved in GR". There's a difference between the lack of definition and breaking the laws of physics, if you know what I mean. Ex-nihilo and all that.

    Quote Originally Posted by Markus Hanke
    Explicitly, in Schwarzschild space-time, the sum total of kinetic and gravitational potential energy per unit mass of a particle is a conserved quantity...
    Good stuff.

    Quote Originally Posted by Markus Hanke
    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.
    I think the wrong nuance comes out of what KJW referred to as an obstruction.

    Quote Originally Posted by Markus Hanke
    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 think highly of Motl's physics. He pays no regard to conservation of energy. I like Philip Gibbs, and while I agree with his sentiment that energy is conserved in all established physical laws including general relativity, I actually don't like his explanation. And look who wrote this.


    Quote Originally Posted by Markus Hanke View Post
    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...
    Noted Markus. I'm not too concerned about the "spirit" of GR for this, because IMHO blueshifted photons gaining energy is even uglier when gravity is not a force in the Newtonian sense.
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  6. #106  
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    Quote Originally Posted by Farsight View Post
    I see it as something real. Matter is made of it.
    It's just a property of a system, same as mass.

    No problem. I guess what I object to is people saying "energy is not conserved in GR". There's a difference between the lack of definition and breaking the laws of physics, if you know what I mean. Ex-nihilo and all that.
    Yes, I know what you mean, and that was my point. The law is not really broken, because it isn't defined.

    I think the wrong nuance comes out of what KJW referred to as an obstruction.
    Yes. The obstruction is just the presence of curvature. This is quite analogous to the situation with the the derivative - you can't use the standard definition of the derivative on a curved manifold, because curvature will mess this up for you. Therefore you need to generalise, and introduce the notion of a covariant derivative, which carries extra terms to compensate for curvature ( not rigorous, but you get the idea ). The situation is quite similar with the energy-momentum tensor.

    I'm sorry Markus, I'm not buying it.
    It works fine mathematically. Other than that I see little point in arguing about it, because once we start accounting for quantum effects the singularity will more than likely vanish anyway. I don't believe in physical singularities any more than you do, for me they are merely a symptom of the fact that GR is purely classical, and hence inherently incomplete. However, even in the classical domain you can actually eliminate singularities, simply by allowing torsion onto your manifold ( this gives a model called Einstein-Cartan gravity ).

    Noted. Would you like to start a thread?
    Feel free to. I'll be rather busy over the next few days, but will reply when I have time.
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    Quote Originally Posted by Markus Hanke View Post
    It's just a property of a system, same as mass.
    I think this is a crucial difference between us, one that is worth pursuing. I mentioned Compton scattering yesterday. I've mentioned the wave nature of the photon plenty of times. When you put the photon through Compton scattering, some of its E=hf energy is converted into the kinetic energy of the electron. In theory you could do another Compton scatter on the residual photon, and another, and another. In the limit you take all the energy out of the wave, and it just doesn't exist any more. Because energy is "what it is", not just some measureable property. Hence Einstein said the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy. Energy is a thing. In fact I'd go so far as to say that at the fundamental level it's the only thing, but that's definitely one for another thread. Meanwhile, ask yourself if action is a property of a system called a photon.

    Quote Originally Posted by Markus Hanke View Post
    Yes, I know what you mean, and that was my point. The law is not really broken, because it isn't defined.
    Good stuff.

    Quote Originally Posted by Markus Hanke View Post
    Yes. The obstruction is just the presence of curvature. This is quite analogous to the situation with the the derivative - you can't use the standard definition of the derivative on a curved manifold, because curvature will mess this up for you. Therefore you need to generalise, and introduce the notion of a covariant derivative, which carries extra terms to compensate for curvature ( not rigorous, but you get the idea ). The situation is quite similar with the energy-momentum tensor.
    Again good stuff. Though I'd say it's a mathematical obstruction.

    Quote Originally Posted by Markus Hanke View Post
    It works fine mathematically. Other than that I see little point in arguing about it, because once we start accounting for quantum effects the singularity will more than likely vanish anyway. I don't believe in physical singularities any more than you do...
    We seem to be agreeing a lot, Markus.

    Quote Originally Posted by Markus Hanke View Post
    ...for me they are merely a symptom of the fact that GR is purely classical, and hence inherently incomplete.
    IMHO they're a symptom of misinterpretation.

    Quote Originally Posted by Markus Hanke View Post
    However, even in the classical domain you can actually eliminate singularities, simply by allowing torsion onto your manifold ( this gives a model called Einstein-Cartan gravity ).
    Which is a step towards combining electromagnetism and gravity.

    Quote Originally Posted by Markus Hanke View Post
    Feel free to. I'll be rather busy over the next few days, but will reply when I have time.
    I'm been busy too. And there is so much to talk about. I could start a dozen spin-off threads from all this.
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  8. #108  
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    Quote Originally Posted by Farsight View Post
    Hence Einstein said the energy of the gravitational field shall act gravitatively in the same way as any other kind of energy.
    While the essence of what this means physically is certainly true, one needs to be careful to remember that gravitational self-energy does not form a part of the energy-momentum tensor in the field equations. Rather, this self-interaction is encoded in the mathematical structure ( non-linearity ) of the equations themselves. The reason is that "gravitational energy" is once again not a covariant quantity, but dependent on the observer and/or coordinate system. This is the physical meaning of the aforementioned Landau-Lifschitz pseudotensor - it represents the "gravitational self-energy". So, while energy-momentum in itself is not globally defined and conserved, the combination of energy-momentum and gravitational energy is defined and conserved, even globally.

    IMHO they're a symptom of misinterpretation.
    I don't agree. Singularities are mathematically inevitable in standard GR, they are not merely interpretation. It is important to recall the definition of what a gravitational singularity is - it arises if the curvature invariants formed from the Riemann tensor ( as well as the tensor itself ) cease to be well defined. For example, in Schwarzschild space-time this happens if you take the limit r -> 0, and it happens in all coordinate systems, so this cannot be mathematically eliminated or "interpreted away" without changing the geometry itself. However, I am of the opinion that no such mathematical singularities will arise once quantum effects are accounted for.

    Which is a step towards combining electromagnetism and gravity
    Actually, Einstein-Cartan gravity does not incorporate EM - you might be confusing this with Kaluza-Klein theory perhaps...?

    I'm been busy too. And there is so much to talk about.
    I think you might start with just one thread - Kruskal-Szekeres charts seem like a good topic to me.

    We seem to be agreeing a lot, Markus.
    We are agreeing on some things, but not on others; but so far as I am concerned that is now perfectly fine. It used to be important to me to convince everyone else of my own point of view, and I used to get terribly frustrated if I couldn't make that work. There were occasions where I was a bit of an arse, really, and I am not too proud to acknowledge that retrospectively. But to be honest, for me at least, that approach took a good-sized chunk of the enjoyment of being on Internet forums away from me, so now I am focussing on having meaningful discussions instead of allowing things to escalate into silly arguments. I work two jobs, look after my family, do some charity work as well, and try to self-study on top of everything else - as you can imagine, that makes me a busy man. As such I need to focus on what is important, instead of always pushing an ego trip; sometimes just agreeing to disagree and moving on goes a long way. I enjoy meaningful discussions ( even if no agreement is reached ), and in particular I enjoy helping others who have genuine questions, and I enjoy learning from people who know more than I do, such as KJW.

    I think you would agree also that we will always remain in disagreement over the fundamental truths of GR, and even the value of the "mainstream" in general; but at the same time that should not prevent us from talking and discussing. We can simply acknowledge our differences, and nonetheless have a meaningful debate. All I am asking as a forum administrator is that we are honest with ourselves, and label our personal opinions as what they are, given that this is a mainstream-oriented forum. But I did notice that you are making an effort to do so by using expressions such as "IMHO", and I try to do the same; so long as we keep that up there will be no problems whatsoever.

    I am taking a "breather" from self-studying for a few weeks right now; I am trying to prepare myself mentally for the next challenge which I have set for myself, and that is learning the mathematics and deeper physics of quantum field theory and the Standard Model. I must admit that I have taken a deep-sitting and really quite irrational dislike towards QFT, and I have no idea why. It is largely intuition I suppose - something about the entire concept of quantum fields just does not feel "right" to me at all, but I can't put my finger on why that is. I am not saying it is wrong, I am perfectly aware that much of it is in perfect accord with experiment and observation; it is just that the geometric approach of GR feels very right to me, but QFT as it is presented and used does not. Incidentally we know that somewhere in between the two there appears to sit a fundamental flaw somewhere, as is evidenced by the "vacuum catastrophe" ( which is evidently more than just an incompatibility ! ), so I want to know everything about it, mathematics and all. I'm just a naturally curious person when it comes to these things.

    But enough of my ramblings
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    Quote Originally Posted by Markus Hanke View Post
    While the essence of what this means physically is certainly true, one needs to be careful to remember that gravitational self-energy does not form a part of the energy-momentum tensor in the field equations...
    The point is that he's referring to energy as a thing in its own right. Like he did when he said the mass of a body is a measure of its energy-content.



    Quote Originally Posted by Markus Hanke
    I don't agree. Singularities are mathematically inevitable in standard GR, they are not merely interpretation. It is important to recall the definition of what a gravitational singularity is - it arises if the curvature invariants formed from the Riemann tensor ( as well as the tensor itself ) cease to be well defined. For example, in Schwarzschild space-time this happens if you take the limit r -> 0, and it happens in all coordinate systems, so this cannot be mathematically eliminated or "interpreted away" without changing the geometry itself. However, I am of the opinion that no such mathematical singularities will arise once quantum effects are accounted for.
    I take the view that GR stops at the event horizon. Along with everything else. I don't have an issue with the Schwarzschild singularity at r=rs. Just with the point singularity in the middle.

    Quote Originally Posted by Markus Hanke
    Actually, Einstein-Cartan gravity does not incorporate EM - you might be confusing this with Kaluza-Klein theory perhaps...?
    I really was thinking of electromagnetism. Einstein struggled for years trying to unify electromagnetism and gravity. You know how you mentioned torsion? In solid mechanics, "torsion is the twisting of an object due to an applied torque". Now take a look at the screw nature of electromagnetism. An electromagnetic field is a "twist" field. When you move through it you turn. If you don't know you're moving you think of it as a "turn" field. All totally unfamiliar to you I know, and absolutely not in the textbooks. But IMHO one day it will be.

    Quote Originally Posted by Markus Hanke
    I think you might start with just one thread - Kruskal-Szekeres charts seem like a good topic to me.
    The thing is I really need to talk about something else first to justify what I'd say about KS coordinates. And then something else to justify that, and so on.

    Quote Originally Posted by Markus Hanke
    We are agreeing on some things, but not on others; but so far as I am concerned that is now perfectly fine. It used to be important to me to convince everyone else of my own point of view, and I used to get terribly frustrated if I couldn't make that work. There were occasions where I was a bit of an arse, really, and I am not too proud to acknowledge that retrospectively. But to be honest, for me at least, that approach took a good-sized chunk of the enjoyment of being on Internet forums away from me, so now I am focussing on having meaningful discussions instead of allowing things to escalate into silly arguments. I work two jobs, look after my family, do some charity work as well, and try to self-study on top of everything else - as you can imagine, that makes me a busy man. As such I need to focus on what is important, instead of always pushing an ego trip; sometimes just agreeing to disagree and moving on goes a long way. I enjoy meaningful discussions ( even if no agreement is reached ), and in particular I enjoy helping others who have genuine questions, and I enjoy learning from people who know more than I do, such as KJW.
    All points noted Markus.

    Quote Originally Posted by Markus Hanke
    I think you would agree also that we will always remain in disagreement over the fundamental truths of GR, and even the value of the "mainstream" in general; but at the same time that should not prevent us from talking and discussing. We can simply acknowledge our differences, and nonetheless have a meaningful debate. All I am asking as a forum administrator is that we are honest with ourselves, and label our personal opinions as what they are, given that this is a mainstream-oriented forum. But I did notice that you are making an effort to do so by using expressions such as "IMHO", and I try to do the same; so long as we keep that up there will be no problems whatsoever.
    It's good to talk. And if we agreed about everything what would we talk about?


    Quote Originally Posted by Markus Hanke
    I am taking a "breather" from self-studying for a few weeks right now; I am trying to prepare myself mentally for the next challenge which I have set for myself, and that is learning the mathematics and deeper physics of quantum field theory and the Standard Model. I must admit that I have taken a deep-sitting and really quite irrational dislike towards QFT, and I have no idea why. It is largely intuition I suppose - something about the entire concept of quantum fields just does not feel "right" to me at all, but I can't put my finger on why that is. I am not saying it is wrong, I am perfectly aware that much of it is in perfect accord with experiment and observation; it is just that the geometric approach of GR feels very right to me, but QFT as it is presented and used does not. Incidentally we know that somewhere in between the two there appears to sit a fundamental flaw somewhere, as is evidenced by the "vacuum catastrophe" ( which is evidently more than just an incompatibility ! ), so I want to know everything about it, mathematics and all. I'm just a naturally curious person when it comes to these things.
    Might I suggest you look at the geometry of electromagnetism and QED before you spend time on QCD.
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    Quote Originally Posted by Farsight View Post
    I don't have an issue with the Schwarzschild singularity at r=rs. Just with the point singularity in the middle.
    The two are not the same. The singularity at the event horizon is purely a coordinate effect, it is not physical; all curvature invariants are finite, regular, and well defined there. You can eliminate this artefact by a simple choice of different coordinates; for example, if you choose Kruskal-Szekeres coordinates, the singularity will disappear.

    The same is not true for the r->0 singularity, because the curvature invariants all diverge there, and they do so in all coordinate systems; so this is not just a coordinate artefact, but a curvature singularity where the theory breaks down.

    Trust me, everyone has an issue with this singularity, but then it arises only because GR can't account for quantum effects, so no one expects this to be physically real.

    I take the view that GR stops at the event horizon
    I think that is perhaps a bit too conservative. I would say that GR still works reasonably well for a substantial part of the interior region too, at least as a classical approximation.

    You know how you mentioned torsion? In solid mechanics, "torsion is the twisting of an object due to an applied torque".
    True, but in differential geometry it has a somewhat different meaning. It's hard to explain without getting technical, but the essence is that it is one of two principle properties of a connection, the other one being the usual curvature. In standard GR, a connection is chosen for which torsion always vanishes ( the Levi-Civita connection ), but it is in fact possible to write a theory which does the opposite and models gravity via torsion only, on an otherwise flat manifold with vanishing curvature. It's called teleparallel gravity, and was a bit of a pet project for Einstein in his later years.

    The problem with incorporating EM is that there aren't enough degrees of freedom in standard GR to do so ( not even with torsion incorporated ); I once saw a formal proof that such a geometric unification is only possible in five or more dimensions, but I can't immediately find the paper now.

    And if we agreed about everything what would we talk about?
    Yes, precisely. These forums would be very boring places

    Might I suggest you look at the geometry of electromagnetism and QED before you spend time on QCD.
    Actually I want to take a top-down approach and learn about the mathematics and underlying ideas of quantum field theory, gauge theory and Yang-Mills theory in general before I turn to the special cases of QED, QCD and all the rest of the Standard Model. This is exactly the opposite way of how it is usually taught, but I believe it will work better for me personally. I will also need to revise my quantum mechanics, since it has been a long time since I paid any serious attention to it - my main focus has been on GR for the last few years.

    The geometry of classical (!) electrodynamics is very clear and intuitive once written in terms of differential forms, but QED is quite a bit more complicated since it is formulated as a Lagrangian. It is just much harder to grasp intuitively.
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    Quote Originally Posted by Markus Hanke View Post
    The two are not the same. The singularity at the event horizon is purely a coordinate effect, it is not physical...
    The coordinate singularity demarks the boundary between two events that are finitely time displaced and those with infinite temporal displacement. I refer to an interior event and an exterior event. Without this particular distinction black holes would not black nor so interesting.

    From an information perspective, finite time-like connectivity is distinctive from infinite time-like connectivity. In the first, information is conserved. In the later it is not. This is what the big deal is about as of late. Even the cheddar headed twit occupying Newton's seat has discovered that this is a problem.

    You can invent any number of new coordinates to replace Schwarz. coords. to smooth-out the event horizon, but they don't change the causal distinction.
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    Quote Originally Posted by Useful Idiot View Post
    The coordinate singularity demarks the boundary between two events that are finitely time displaced and those with infinite temporal displacement. I refer to an interior event and an exterior event. Without this particular distinction black holes would not black nor so interesting.
    It does indeed mark a form of boundary, but the singularity isn't essential or physical. You can always eliminate it by an appropriate choice of coordinates - for example, Kruskal-Szekeres coordinates still mark out the event horizon, but the chart is not singular there.

    You can invent any number of new coordinates to replace Schwarz. coords. to smooth-out the event horizon, but they don't change the causal distinction.
    That's true, and it's exactly my point. The singularity is merely an artefact of your choice of coordinates, not a result of the actual geometry of spacetime.
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    Quote Originally Posted by Farsight View Post
    I think the wrong nuance comes out of what KJW referred to as an obstruction.
    The term "obstruction" is used to describe a quantity that prevents the existence of a solution of an equation. It arises because the necessary and sufficient conditions for the existence of a solution may not be satisfied, this being the obstruction. The "wrong nuance" as you put it was precisely the nuance I was intending. Curvature is an obstruction in that it does obstruct the ability of a space to satisfy the properties that are satisfied by flat spaces. Because I regard all physical quantities as having a mathematical basis for existence, the notion of curvature as an obstruction is more important to me than the geometrical notion of curvature.
    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
    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.
    Perhaps you could do a small mathematical demonstration to show us where all working cosmologists get this wrong?
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    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.
    This seems like you are admitting that you are putting forward a personally theory, not one that is accepted by the mainstream.
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    Quote Originally Posted by Markus Hanke View Post
    The two are not the same. The singularity at the event horizon is purely a coordinate effect, it is not physical; all curvature invariants are finite, regular, and well defined there. You can eliminate this artefact by a simple choice of different coordinates; for example, if you choose Kruskal-Szekeres coordinates, the singularity will disappear.

    Right, but is there any effect in GR or SR that you *couldn't* eliminate by choosing a different set of coordinates?

    Besides that, Relativity isn't really about coming up with a "global truth". It has nothing to say about what a privileged observer would see if privileged observers existed. It only tells us what non-privileged observers will see. (I'm thinking that those observers probably don't have a choice about what coordinate system they want to use, because their frame of reference is deciding that for them) .

    That said, this concept you mentioned of Kurska-Szekeres coordinates looks really interesting. I looked up the wiki page.

    Kruskal



    I get frustrated seeing people evoke "proper time" as a refutation of the idea that time stops at the event horizon. Even in science fiction, if you're wearing a wristwatch and the bad guy puts you in a "stasis field" (or some other sci-fi form of suspended animation) your wrist watch would be frozen just as much as your body is frozen. If the hero shows up and turns off the stasis field, and you then look down at your wristwatch, you will see that zero proper time has passed between the villain freezing you in place and the hero rescuing you.

    So.., according to you and your wristwatch, you were never in a stasis field. It didn't happen. All other nearby observers agree that you were in one, but you disagree. I suppose all the other observers are wrong?

    That's why I say "proper time" is bullox. It's completely uninformative, un-useful information. Coordinate time is the only time that means anything.
    Farsight likes this.
    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
    It does indeed mark a form of boundary, but the singularity isn't essential or physical. You can always eliminate it by an appropriate choice of coordinates - for example, Kruskal-Szekeres coordinates still mark out the event horizon, but the chart is not singular there.
    I don't know what essential means, but the a physical distinction remains between the interior and the exterior for an external observers. The event horizon is simply the boundary. It is the regions that are of interest in my view.

    That's true, and it's exactly my point. The singularity is merely an artefact of your choice of coordinates, not a result of the actual geometry of spacetime.
    this is something I don't understand. Without a good theory of quantum gravity, or a few strong descriptive conjectures over the field of possible theories I don't see any way to advance further. however, I don't think space-time is quantized a dimension at a time, but quantized over 4 volumes. It really doesn't make sense otherwise, and should be what it takes to understand the event horizon with quantum character, as it should be. In other words, we should expect no definite boundary between regions even if a causally distinctive interior can be physically realizable under quantum considerations.

    I was focused on AdS/CFT for a while. It appears to be one of the strong descriptive conjectures of quantum gravity.

    I don't take discussions or most conclusions about black holes too seriously. How can we, without considering some strong rules obtained by those working in the quantum domain, and some known constraints on how how quantum gravity should be.
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    Quote Originally Posted by Kojax View Post
    Right, but is there any effect in GR or SR that you *couldn't* eliminate by choosing a different set of coordinates?
    Anything that is relativistically invariant cannot be "eliminated" or changed. This for example includes the presence of the central singularity, the area of the event horizon, or the length of world lines.

    Besides that, Relativity isn't really about coming up with a "global truth".
    I agree.

    That's why I say "proper time" is bullox. It's completely uninformative, un-useful information. Coordinate time is the only time that means anything.
    Proper time is the length of your world line, and it is what your wrist watch will physically measure; all observers will always agree on the length of your world line, no matter where they are and what their state of relative motion is. The same is not true for coordinate time - it explicitly depends on the chosen coordinate system and is hence a completely arbitrary concept. I can't really imagine anything more useful than an actual, physical measurement that everyone agrees on, as opposed to something completely arbitrary that is valid only for a specific observer in a specific set of coordinates.

    If you set aside all notions of watches and measurements and times for a minute, and just look at the world line in spacetime of an observer falling into a black hole, you will find that the world line is of a finite and well defined length; this is true in all coordinate systems and for all observers, hence there is no physical singularity at the event horizon, and time will not "stop" there for someone falling through ( which would mean that his world line is of infinite length ). You can see this most easily by drawing your world line onto a Kruskal-Szekeres diagram - it simply extends right through the event horizon, and meets the singularity, while remaining of finite length.

    What happens is just that light rays eminating from a freely falling emitter become increasingly redshifted and "bent" as the emitter approaches the event horizon, so they will become dimmer and dimmer and take longer and longer to arrive at some observer very far away; because of that, such an observer will see the falling object slowing down and fading as it gets close to the event horizon. Of course, that does not mean that the falling object never crosses the horizon as experienced by the object itself, it means only that it is never seen to be doing so by an observer far away.

    I get frustrated seeing people evoke "proper time" as a refutation of the idea that time stops at the event horizon.
    There is no reason to get frustrated, nor is there anything to be refuted. The thing is just this - GR gives us a way to relate measurements made by different observers at different places. All observers are right in what they measure, but only in their own local frames of reference. If a far-away observer measures an object to slow down and fade away as it falls towards the event horizon ( using his own system of assigning coordinates to events ), then he is right in his own frame ( which happens to be far away from the horizon ). Likewise, if an observer falling together with the object determines that he reaches the horizon in finite time and keeps falling thereafter, then he is also right, but once again only in his own local frame, which happens to be attached to the freely falling object. It isn't a question of who is right and who is wrong, but rather of how the two measurements are related; in GR - unlike in Newtonian physics - there is no longer a requirement for all observers to agree on all measurements, nor is that even possible or meaningful in curved space-times. Both coordinate and proper measurements have their own usefulness and place, but it is important to understand how they are related, and what they physically mean.

    Generally speaking, much unnecessary confusion is avoided if one abandons specific measurements and focusses on the geometry of the problem instead; for example, for an infalling observer I ask myself what geometric object all observers can universally agree on - the answer is of course the world line that it traces out in spacetime. Since the world line as a geometric object is of finite length and smooth and differentiable everywhere, then it is immediately obvious that nothing special happens at the event horizon; there is no singularity there, as is unambiguously clear from a Kruskal-Szekeres diagram of the spacetime.
    Last edited by Markus Hanke; 04-12-2014 at 09:07 AM.
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    Quote Originally Posted by Useful Idiot View Post
    I don't know what essential means, but the a physical distinction remains between the interior and the exterior for an external observers. The event horizon is simply the boundary. It is the regions that are of interest in my view.
    Yes, I agree.

    Without a good theory of quantum gravity, or a few strong descriptive conjectures over the field of possible theories I don't see any way to advance further.
    I'm not quite sure what you mean here. I was merely trying to point out that the Schwarzschild coordinate chart is singular at the event horizon, but that this singularity can be eliminated simply by choosing different coordinates - meaning that it is merely a coordinate artefact, but not a physical singularity. For example, Kruskal-Szekeres coordinates are well behaved everywhere outside r = 0, even at the event horizon. The curvature tensor remains regular at the event horizon too, as do all curvature invariants.

    I don't take discussions or most conclusions about black holes too seriously. How can we, without considering some strong rules obtained by those working in the quantum domain, and some known constraints on how how quantum gravity should be.
    Yes, you are right - fully accounting for all quantum effects might considerably change many of the details that we are so accustomed to from classical GR.
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    Quote Originally Posted by Markus Hanke View Post
    The two are not the same. The singularity at the event horizon is purely a coordinate effect, it is not physical; all curvature invariants are finite, regular, and well defined there. You can eliminate this artefact by a simple choice of different coordinates; for example, if you choose Kruskal-Szekeres coordinates, the singularity will disappear.
    That's what they say. See the formation and growth of black holes and note the frozen star interpretation. This is the original black hole interpretation, and I think it's right. Kevin Brown and most other people don't.

    Quote Originally Posted by Markus Hanke View Post
    The same is not true for the r->0 singularity, because the curvature invariants all diverge there, and they do so in all coordinate systems; so this is not just a coordinate artefact, but a curvature singularity where the theory breaks down.
    It's certainly a different animal.

    Quote Originally Posted by Markus Hanke View Post
    Trust me, everyone has an issue with this singularity, but then it arises only because GR can't account for quantum effects, so no one expects this to be physically real...
    Let's just say I think it isn't physically real for a GR reason rather than a QM reason.

    Quote Originally Posted by Markus Hanke View Post
    True, but in differential geometry it has a somewhat different meaning. It's hard to explain without getting technical, but the essence is that it is one of two principle properties of a connection, the other one being the usual curvature. In standard GR, a connection is chosen for which torsion always vanishes ( the Levi-Civita connection ), but it is in fact possible to write a theory which does the opposite and models gravity via torsion only, on an otherwise flat manifold with vanishing curvature. It's called teleparallel gravity, and was a bit of a pet project for Einstein in his later years.

    The problem with incorporating EM is that there aren't enough degrees of freedom in standard GR to do so ( not even with torsion incorporated ); I once saw a formal proof that such a geometric unification is only possible in five or more dimensions, but I can't immediately find the paper now.
    All points noted. I find this sort of thing very interesting.

    Quote Originally Posted by Markus Hanke View Post
    Actually I want to take a top-down approach and learn about the mathematics and underlying ideas of quantum field theory, gauge theory and Yang-Mills theory in general before I turn to the special cases of QED, QCD and all the rest of the Standard Model. This is exactly the opposite way of how it is usually taught, but I believe it will work better for me personally. I will also need to revise my quantum mechanics, since it has been a long time since I paid any serious attention to it - my main focus has been on GR for the last few years.
    OK noted.

    Quote Originally Posted by Markus Hanke View Post
    The geometry of classical (!) electrodynamics is very clear and intuitive once written in terms of differential forms...
    We never seem to hear about it.
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    Quote Originally Posted by KJW View Post
    The term "obstruction" is used to describe a quantity that prevents the existence of a solution of an equation. It arises because the necessary and sufficient conditions for the existence of a solution may not be satisfied, this being the obstruction. The "wrong nuance" as you put it was precisely the nuance I was intending. Curvature is an obstruction in that it does obstruct the ability of a space to satisfy the properties that are satisfied by flat spaces. Because I regard all physical quantities as having a mathematical basis for existence, the notion of curvature as an obstruction is more important to me than the geometrical notion of curvature.
    Noted.
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    Quote Originally Posted by Markus Hanke
    ...Proper time is the length of your world line, and it is what your wrist watch will physically measure...
    This conjured up a vision of a little chalky tailor stretching out his tape measure to measure the length of a world line. IMHO the whole of physics hangs on what clocks do.
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    Quote Originally Posted by Farsight View Post
    This is the original black hole interpretation, and I think it's right. Kevin Brown and most other people don't.
    They/we don't agree with that older interpretation because it is easy to show that the geometry of space-time at the event horizon is perfectly regular, so there is no singularity there. I can do it here for you, if you like; in fact it is so straightforward that it can be done in just one line. There is a good case to be made that Einstein himself wasn't fully aware of these finer points of differential geometry.

    Let's just say I think it isn't physically real for a GR reason rather than a QM reason.
    You can eliminate the singularity even in the classical domain, simply by allowing torsion on your manifold, i.e. by a choice of a different connection. It turns out that this has far-reaching consequences, but that's not the topic of this thread.

    We never seem to hear about it.
    I'm busy today, but I might write a few words about this when I get a chance.
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    Quote Originally Posted by Farsight View Post
    This conjured up a vision of a little chalky tailor stretching out his tape measure to measure the length of a world line. IMHO the whole of physics hangs on what clocks do.
    The advantage of using the geometric length of world lines to define proper time is that it is entirely independent of the clock mechanism; it works for a digital clock just as well as for a decaying particle, or an ageing body, or any other type of clock.
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    Quote Originally Posted by Markus Hanke View Post
    Proper time is the length of your world line, and it is what your wrist watch will physically measure; all observers will always agree on the length of your world line, no matter where they are and what their state of relative motion is. The same is not true for coordinate time - it explicitly depends on the chosen coordinate system and is hence a completely arbitrary concept. I can't really imagine anything more useful than an actual, physical measurement that everyone agrees on, as opposed to something completely arbitrary that is valid only for a specific observer in a specific set of coordinates.

    It's possible I am misapplying the concept of gravitational time dilation. If an object were to fall very near, but remain hovering just outside the event horizon for 10 billion years, and then be retrieved to a location far from any big source of gravity - its internal clock would show that very little time had passed.

    But for it to hover there, it would need to be under constant acceleration. Perhaps the constant acceleration is what is really causing the time dilation effect?

    If instead, the object were allowed to fall in freely, I'm thinking then it would not be under acceleration, because from its perspective falling inward is an inertial state. And in that event it would make sense to say it does fall inward.

    So, is that the piece of the puzzle I have been missing?


    If you set aside all notions of watches and measurements and times for a minute, and just look at the world line in spacetime of an observer falling into a black hole, you will find that the world line is of a finite and well defined length; this is true in all coordinate systems and for all observers, hence there is no physical singularity at the event horizon, and time will not "stop" there for someone falling through ( which would mean that his world line is of infinite length ). You can see this most easily by drawing your world line onto a Kruskal-Szekeres diagram - it simply extends right through the event horizon, and meets the singularity, while remaining of finite length.
    Models are great, but actual events and/or scenarios are only things that can be experimentally verified. And in science, experimental verification is king.

    If the model can't leave us with a description of a series of events that could (in principle) be created and then observed, then it's just speculation.
    Last edited by Kojax; 04-13-2014 at 02:48 AM.
    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 an object were to fall very near, but remain hovering just outside the event horizon for 10 billion years, and then be retrieved to a location far from any big source of gravity - its internal clock would show that very little time had passed.
    Yes, there is indeed gravitational time dilation near the event horizon, as there is near any source of energy-momentum. The degree of time dilation depends on the mass of the black hole, but the crucial point to understand is that the time dilation is not infinite - someone falling through the event horizon will not see the entire history of the universe play out in front of him :

    general relativity - Does someone falling into a black hole see the end of the universe? - Physics Stack Exchange

    But for it to hover there, it would need to be under constant acceleration. Perhaps the constant acceleration is what is really causing the time dilation effect?
    The underlying source of gravitational time dilation is differences in gravitational potential; looking at it this way should make it obvious that the gravitational potential cannot be infinite at the event horizon, and hence neither is gravitational time dilation.

    So, is that the piece of the puzzle I have been missing?
    For simple Schwarzschild black holes, you can indeed look at it that way, so long as you consider only a small local region for the observers ( no tidal forces ).

    Models are great, but actual events and/or scenarios are only things that can be experimentally verified. And in science, experimental verification is king
    That's true. Unfortunately though we don't have a black hole to play around with.

    If the model can't leave us with a description of a series of events that could (in principle) be created and then observed, then it's just speculation.
    If we had a black hole to play with, we could let a clock fall freely towards it on a highly elliptical orbit, in such a way that it comes very close to the event horizon and then back out to the starting point. We could then compare what the two clocks ( freely falling & reference at starting point ) have recorded. Mathematically, and for a BH of a few solar masses, one would expect a difference in total recorded times of a few dozen percent, but not billions of years.
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    Quote Originally Posted by Kojax View Post
    But for it to hover there, it would need to be under constant acceleration. Perhaps the constant acceleration is what is really causing the time dilation effect?
    Yes. This is the equivalence principle between gravity and acceleration.
    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
    Yes. This is the equivalence principle between gravity and acceleration.
    Thanks. That helps me get the picture of this.

    Is there another aspect to gravitational time dilation beyond the acceleration required to keep an object from falling?


    Quote Originally Posted by Markus Hanke View Post
    The underlying source of gravitational time dilation is differences in gravitational potential; looking at it this way should make it obvious that the gravitational potential cannot be infinite at the event horizon, and hence neither is gravitational time dilation.
    Do you need infinite acceleration in order to bring time to a stop? I was under the impression that a finite amount of acceleration would be enough depending on the distance between the observer and the accelerating object.

    For simple Schwarzschild black holes, you can indeed look at it that way, so long as you consider only a small local region for the observers ( no tidal forces ).
    Yeah. I think I'll stick with those for now. Best to master the basics, before moving on.



    If we had a black hole to play with, we could let a clock fall freely towards it on a highly elliptical orbit, in such a way that it comes very close to the event horizon and then back out to the starting point. We could then compare what the two clocks ( freely falling & reference at starting point ) have recorded. Mathematically, and for a BH of a few solar masses, one would expect a difference in total recorded times of a few dozen percent, but not billions of years.

    During the nearest point of approach, when it is closest to the event horizon, the clock would be traveling at very nearly C.

    So even if it is not being accelerated (because it is in free fall) and General Relativistic acceleration doesn't affect its clocks, it is still moving very fast, which means Special Relativity has something to say.

    I guess that, technically, both clocks think the other one is the one moving fast (both the clock on the satellite far away from the event horizon and the one in elliptical orbit).

    But somehow they don't disagree about which one is accelerating. Even though technically neither of them is accelerating, because they're both just following their own inertial paths. Yet clearly they must perceive that they are accelerating relative to each other because their relative velocities are continuously changing.

    ... Ok so this last bit has me confused for now.
    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
    Do you need infinite acceleration in order to bring time to a stop? I was under the impression that a finite amount of acceleration would be enough depending on the distance between the observer and the accelerating object.
    In order to "stop time" you would need an infinite dilation factor; how would you propose to get that using only finite accelerations ?

    During the nearest point of approach, when it is closest to the event horizon, the clock would be traveling at very nearly C.
    Not necessarily. This depends on the mass of the black hole as well as the exact orbital parameters.

    So even if it is not being accelerated (because it is in free fall) and General Relativistic acceleration doesn't affect its clocks, it is still moving very fast, which means Special Relativity has something to say.
    Yes, but remember that we want to compare the total recorded times ( = proper times ) on both clocks once they are brought back together at rest after the experiment; we aren't interested in instantaneous measurements while the clocks are in relative motion. SR time dilation due to relative motion is purely a coordinate artefact, and has no effect on total recorded proper times - the world line of the freely falling clock is completely determined by the geodesics of spacetime.

    But somehow they don't disagree about which one is accelerating.
    They are both in free fall, so proper acceleration is zero for both of them, and they both trace out geodesics in spacetime. The reason why their trajectories are nonetheless different is because spacetime is not everywhere flat, but curved, so the geometry of their geodesic world lines cannot be the same. That is the whole essence of GR - observers locally experience the exact same physics, yet nonetheless their global relationship is non-trivial.

    You might find this interesting, just to play around with - most notably, it will tell you what the time dilation factor for these various orbits would be :

    http://demonstrations.wolfram.com/Or...ildBlackHoles/
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    Quote Originally Posted by Kojax View Post
    Is there another aspect to gravitational time dilation beyond the acceleration required to keep an object from falling?
    The aforementioned gravitational potential, but note that uniform acceleration is physically equivalent to differences in gravitational potential ( at least in Schwarzschild spacetime ), so these are just two different ways to look at the same physics.
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    Quote Originally Posted by Markus Hanke View Post
    In order to "stop time" you would need an infinite dilation factor; how would you propose to get that using only finite accelerations ?
    The effect is partially described in this:

    The Relativistic Rocket


    I'm having trouble finding the formula again. I had it once and I should have put it in a safe place. Basically for a uniformly accelerating object, like say a rocket, the time dilation observed on faraway objects in the direction of motion depends partly on the amount of acceleration and partly on their distance from you. And there is a distance that acts like a horizon, where the redshift is so great they simply vanish.

    Start reading at the bold text that says "Below the rocket, something strange is happening." By "below", I think they mean "behind". Or in the direction the rocket is accelerating away from.






    Yes, but remember that we want to compare the total recorded times ( = proper times ) on both clocks once they are brought back together at rest after the experiment; we aren't interested in instantaneous measurements while the clocks are in relative motion. SR time dilation due to relative motion is purely a coordinate artefact, and has no effect on total recorded proper times - the world line of the freely falling clock is completely determined by the geodesics of spacetime.
    That's the odd thing, isn't it? When we finally do bring them back together after they've been orbiting a long time, their clocks do disagree.

    As with the Global Positioning System., where there is a need to slow the clocks just slightly in order to accommodate the relativistic effect of them orbiting in lighter gravity.

    What the Global Positioning System Tells Us About Relativity


    However, I'm not sure if the reason the adjustment is necessary is because they are in lighter gravity, or if it's because we humans down on Earth are not in free fall. Maybe if we were in free fall just like the satellites, there wouldn't be any need for the adjustment?

    Instead, there is that N force acting upon us, accelerating us "upward", which might also be causing the time dilation we experience? I'm confused about which accelerations "count" and which ones don't count.
    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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    And there is a distance that acts like a horizon
    You probably mean the Rindler horizon, which is a feature of the Rindler coordinate chart. Note that this is once again a coordinate singularity, but not a physical ( gravitational ) one.

    That's the odd thing, isn't it? When we finally do bring them back together after they've been orbiting a long time, their clocks do disagree.
    Yes, it is just a consequence of the fact that space-time just isn't the same everywhere globally, and hence neither are observers' world lines.

    As with the Global Positioning System
    Yes, indeed.

    However, I'm not sure if the reason the adjustment is necessary is because they are in lighter gravity, or if it's because we humans down on Earth are not in free fall.
    The reason is both that there is relative motion between the satellite and the earth, and more importantly that there is a difference in gravitational potential. Both of these are accounted for in GPS satellites.

    Maybe if we were in free fall just like the satellites, there wouldn't be any need for the adjustment?
    There would still be a difference in gravitational potentials, and hence gravitational time dilation which needs to be adjusted for, because satellite and observer are at different radial coordinates in the Schwarzschild space-time.

    I'm confused about which accelerations "count" and which ones don't count.
    I think since you are restricting yourself to Schwarzschild geometry anyway, it might be much easier to think about differences in potential instead.
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    Quote Originally Posted by Kojax View Post
    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.
    Sure. The cause of the received EMR or sound can be energy from the object as primary source (a star, a motor roar) or from elsewhere "bouncing" (reflected light, radar, sonar, an echo) off the object. The wavelengths are doppler shifted and also any time signal will have its beat shortened or lengthened.

    As for a classical ball gaining energy (velocity) bouncing off an oncoming object or losing energy (velocity) bouncing off a receding object. Sure. An observer on that object sees the ball change velocity as does a ground based observer. The observation can be made by using light bounced off it or it can also be observed by catching the ball in the samemanner a radar echo or sound echo are "caught". In both cases a change in direction is observed. In one case a change in magnitude is observed. (am I correct).


    In an extreme case: the ball is launched toward a receding target object is going at a velocity equal to or faster than the ball. No bounce back, no echo, no velocity or direction change for that ball.
    The equivalent for EMR whether it is object source emitted or reflected (absorbed and re-emitted) is what? I can speculate and buzzword that, but I have no formal training in extreme cosmology and superluminal stuff. I do not bumble where I can easily fumble and stumble.


    A bit off the subject but similar in terms of energy transfer. I think that space probes use an approach to a massive object to change velocity and direction. Gravitational boosts when catching up to a "receding" planet. Conversely approaching a diffrent way can slow the probe.
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    Quote Originally Posted by Markus Hanke View Post
    You probably mean the Rindler horizon, which is a feature of the Rindler coordinate chart. Note that this is once again a coordinate singularity, but not a physical ( gravitational ) one.

    But again, it is what you would observe if you were on an accelerating space ship. Not because you "chose" the wrong coordinate system, but because that's what your eyes would actually be seeing.

    Your body/eyes are incapable of choosing any other coordinate system other than the one they are in, so this Rindler Horizon is the only reality you can possibly experience aboard that rocket.





    The reason is both that there is relative motion between the satellite and the earth, and more importantly that there is a difference in gravitational potential. Both of these are accounted for in GPS satellites.



    There would still be a difference in gravitational potentials, and hence gravitational time dilation which needs to be adjusted for, because satellite and observer are at different radial coordinates in the Schwarzschild space-time.



    I think since you are restricting yourself to Schwarzschild geometry anyway, it might be much easier to think about differences in potential instead.
    What is confusing me is comparing what happens if an object is allowed to freely fall, compared with what happens if it is made to hover. If it is made to hover at a specific distance from the center of gravity, it experiences the time dilation appropriate to that radius, right? If it freely falls, it experiences the time dilation appropriate to each distance it occupies as it falls for however long it occupies those distances.

    But that doesn't seem right. The one that is being made to hover is subject to an additional acceleration effect. Whatever is making it hover is accelerating it.

    It seems like that ought to change how much time dilation it experiences. But it doesn't, right? It's all just based on where it is in the gravitational field?
    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
    Your body/eyes are incapable of choosing any other coordinate system other than the one they are in, so this Rindler Horizon is the only reality you can possibly experience aboard that rocket.
    Yes, that's true, but it is also true for any other rocket observer, who will see a different Rindler horizon depending on his own state of motion and acceleration.
    That is why I was saying before - all observers are right, but only in their own frame of reference.

    If it is made to hover at a specific distance from the center of gravity, it experiences the time dilation appropriate to that radius, right?
    Time dilation as compared to what ?

    If it freely falls, it experiences the time dilation appropriate to each distance it occupies as it falls for however long it occupies those distances.
    Again - time dilation as compared to what ?

    It seems like that ought to change how much time dilation it experiences. But it doesn't, right? It's all just based on where it is in the gravitational field?
    To be honest I am not really sure what you are getting at. If you have a freely falling clock, and a clock that is hovering ( stationary ) somewhere, how do you propose to meaningfully compare them, and what exactly is it you are comparing ? Their accumulated proper times, or some instantaneous measurement ? And how do you synchronise them in the first place to perform such a comparison and give it any physical meaning ? In curved space times none of these issues are quite as trivial as they initially appear to be. Locally, in their own frames, both clocks tick at exactly "one second per second" - you can't detect time dilation in an isolated local frame. Only when you choose some outside reference point can you detect it.

    As for gravitational potential, it depends only on the metric ( specifically the term ), which in the case of the Schwarzschild metric depends on the radial coordinate. The very same is true for a hovering observer - he finds himself in a uniformly accelerating frame, the term for which also depends on the radial coordinate ( the closer you are to the central body, the harder you need to accelerate to remain stationary ); he has to uniformly accelerate by the exact equivalent of the gravitational attraction at his location to remain hovering. Hence, at a given instant, uniform acceleration and the presence of a gravitational potential are physically equivalent, and their magnitude depends on the radial coordinate. Likewise, the difference in gravitational potential between freely falling observers depends on the difference in their radial coordinates. All of this is due to the symmetries of the Schwarzschild space-time.
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    Quote Originally Posted by Markus Hanke View Post
    Yes, that's true, but it is also true for any other rocket observer, who will see a different Rindler horizon depending on his own state of motion and acceleration.
    That is why I was saying before - all observers are right, but only in their own frame of reference.
    But only privileged observers are free of a frame of reference. And there is no such thing as a privileged observer.

    I think instead of saying both observers are right, but only in their own frame of reference, it makes more sense to just plain say both observers are right.

    Their observations contradict, but they're both accurate.




    Time dilation as compared to what ?



    Again - time dilation as compared to what ?


    To keep things simple, I propose comparing their state of time dilation against that of an observer who is very far away from any massive objects. Someone who is experiencing virtually no gravity.

    What's confusing me is the idea that I can use the same formula for both cases. Of course I have to do an integral for the falling rock, because that rock's location is changing over time, and I don't have to worry about doing an integral for the hovering rock, since it is staying in one place.

    However, aside from that, it's the same formula. (For the simple case of a non-rotating, spherically symmetric body.)


    But... I would think it shouldn't be. I would expect that I'd also have to deal with the upward acceleration that is being experienced by the hovering rock.



    To be honest I am not really sure what you are getting at. If you have a freely falling clock, and a clock that is hovering ( stationary ) somewhere, how do you propose to meaningfully compare them, and what exactly is it you are comparing ? Their accumulated proper times, or some instantaneous measurement ? And how do you synchronise them in the first place to perform such a comparison and give it any physical meaning ? In curved space times none of these issues are quite as trivial as they initially appear to be. Locally, in their own frames, both clocks tick at exactly "one second per second" - you can't detect time dilation in an isolated local frame. Only when you choose some outside reference point can you detect it.
    Oh ok. I want to compare the two rocks' coordinate times, as observed by an observer very far away from any massive bodies (so an observer who is not in a gravitational field.)


    After that, maybe I'll try comparing their proper times in a scenario where they both start at a location very far from any massive bodies, and both end up at a location very far away from any massive bodies.

    When dealing with gravity, we always have the wonderful benefit of being able to use a frame of reference with no gravity (or very small gravity) as our "staging area" for the experiment, if we want to. We can synchronize the clocks in that environment at the outset, and then we can compare the clocks there when the experiment is over.


    As for gravitational potential, it depends only on the metric ( specifically the term ), which in the case of the Schwarzschild metric depends on the radial coordinate. The very same is true for a hovering observer - he finds himself in a uniformly accelerating frame, the term for which also depends on the radial coordinate ( the closer you are to the central body, the harder you need to accelerate to remain stationary ); he has to uniformly accelerate by the exact equivalent of the gravitational attraction at his location to remain hovering. Hence, at a given instant, uniform acceleration and the presence of a gravitational potential are physically equivalent, and their magnitude depends on the radial coordinate. Likewise, the difference in gravitational potential between freely falling observers depends on the difference in their radial coordinates. All of this is due to the symmetries of the Schwarzschild space-time.

    Depending on how we look at this - I'm trying to figure out which way is the convention. Is it:

    1) - Treat gravity as not being acceleration, so if you hover stationary you are subject to one and only one acceleration effect (in that you are being constantly accelerated upward.)

    or

    2) - Treat gravity as yes being acceleration. So if you hover stationary you are subject to two accelerations. You are being accelerated downward by gravity, and accelerated upward by whatever is preventing you from falling.


    I'm not sure which of those two perspectives is the one that is the correct one to use.
    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 only privileged observers are free of a frame of reference.
    Every observer is associated with a frame of reference.

    I think instead of saying both observers are right, but only in their own frame of reference, it makes more sense to just plain say both observers are right.
    I don't necessarily agree that this is a good way to look at things, but then again it is minor point; so long as you realise that there is no requirement for them to agree on everything ( unlike in Newtonian physics ), then all is good.

    To keep things simple, I propose comparing their state of time dilation against that of an observer who is very far away from any massive objects.
    Ok, then you can compare instantaneous dilation factors, and you will find that they depend only on their radial coordinates.

    But... I would think it shouldn't be. I would expect that I'd also have to deal with the upward acceleration that is being experienced by the hovering rock.
    The upward acceleration is just a manifestation of the gravitational field, so these are physically equivalent. You don't count them twice.

    Oh ok. I want to compare the two rocks' coordinate times, as observed by an observer very far away from any massive bodies (so an observer who is not in a gravitational field.)
    Same as above - the dilation factors ( coordinate measurements ! ) depend on the radial distance from the event horizon. If the object is in free fall, you also need to account for the coordinate velocity, i.e. relative motion.

    After that, maybe I'll try comparing their proper times in a scenario where they both start at a location very far from any massive bodies, and both end up at a location very far away from any massive bodies.
    We can synchronize the clocks in that environment at the outset, and then we can compare the clocks there when the experiment is over.
    Yes, this is the only really meaningful way to do it, because you are comparing proper times between clocks that start and end up in the same frame of reference

    I'm not sure which of those two perspectives is the one that is the correct one to use.
    It's number 1 - a uniform acceleration is equivalent to a locally uniform gravitational field. This is just what the equivalence principle tells us.
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    Quote Originally Posted by Markus Hanke View Post
    Every observer is associated with a frame of reference.
    It is good that we agree on that.


    I don't necessarily agree that this is a good way to look at things, but then again it is minor point; so long as you realise that there is no requirement for them to agree on everything ( unlike in Newtonian physics ), then all is good.
    Right. Just like in Special Relativity. We can do transformations between their perspectives, but without the transformation there is likely going to be disagreement.

    The main difference is that in General Relativity, that transformation is quite often a big awful Tensor that you'd need several very fast computers working together and calculating for a month in order to resolve.



    Yes, this is the only really meaningful way to do it, because you are comparing proper times between clocks that start and end up in the same frame of reference

    When we do that, we find that gravitational time dilation was in fact, quite real, do we not?

    If I take two satellites out beyond the Oort cloud (far from any gravitational mass), synchronize their clocks, then send one into orbit around the Sun at the same distance as Earth (but far from Earth), and the other into orbit around the Sun at the same distance as Mercury (but far from Mercury) - then leave them there for a few decades. Then bring them back to their original starting location out beyond the Oort cloud (far from any gravitational mass)

    I would notice that their clocks had gotten out of sync.

    That leads me to believe that gravitational time dilation is more than a coordinate effect. That would be an example of taking two measurements of proper time and having them disagree. Or at least I think it would. Wouldn't 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
    The main difference is that in General Relativity, that transformation is quite often a big awful Tensor that you'd need several very fast computers working together and calculating for a month in order to resolve.
    Actually, tensor calculus makes a lot of things much easier. There is of course no requirement to use it though - you could in principle write down the Einstein equations are a system of 10 ( in general ) coupled partial differential equations without ever making reference to tensors. What you would get is an awful mess that completely obscures the physical meaning, so I don't see the point.

    Sometime though you are right - there are many scenarios that cannot be treated analytically, and require numerical methods with substantial computing resources.

    When we do that, we find that gravitational time dilation was in fact, quite real, do we not?
    Yes, the dilation of proper times - being physical clock readings - is quite real.

    I would notice that their clocks had gotten out of sync.
    Yes, absolutely.

    That leads me to believe that gravitational time dilation is more than a coordinate effect.
    Gravitational time dilation is not a coordinate effect - relative time dilation ( as described by SR for inertial frames in relative motion ) on the other hand, is. These two are physically distinct effects.
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    Is it possible that gravity is just an artifact of time dilation?
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    Quote Originally Posted by bird11dog View Post
    Is it possible that gravity is just an artifact of time dilation?
    One thing to bear in mind, is that a theory of gravitation based on a single, real, scalar function varying from point to point (e.g. a time dilation factor, or a refractive index, or a simple scalar potential of some form) does not have enough degrees of freedom at each point of spacetime to replicate General Relativity; in other words, sooner or later you are going to be making predictions which contradict General Relativity, and it is quite likely that as a result you will be contradicting observation. The exact details obviously depend on the exact details of your idea.
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    Quote Originally Posted by btr View Post
    One thing to bear in mind, is that a theory of gravitation based on a single, real, scalar function varying from point to point (e.g. a time dilation factor, or a refractive index, or a simple scalar potential of some form) does not have enough degrees of freedom at each point of spacetime to replicate General Relativity; in other words, sooner or later you are going to be making predictions which contradict General Relativity, and it is quite likely that as a result you will be contradicting observation. The exact details obviously depend on the exact details of your idea.
    Yes, indeed. This is something that a lot of people with "personal theories" do not realise; it is easy to write down theories of gravity that are based on scalar fields in flat space-time, but it is just as easy to show that those will be in conflict with GR in the strong field regime.

    As for bird11dog's post - what we in our everyday lives consider as "gravity" can in fact be understood as curvature along the time direction, i.e. as gravitational time dilation. However, this is of course not the whole picture, because gravity also comprises tidal effects, which are far more complex than just a single scalar value. It is rather trivial to show that, based on the fact that gravity arises from geodesic deviation, you need a rank-4 tensor to completely capture all aspects of that. The number of physical degrees of freedom at each point is then further constrained by the various symmetries present, but they cannot be captured by a single scalar value.
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    So if someone posts an idea you could show where it fails?
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    If you have an idea in mind, start up a thread in the Personal Theories and Alternative Hypotheses section, and see what transpires.
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    It's done, have at it.
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    I have read hundreds of times the relationship between space and time. And that they are actually a fabric, called spacetime. I have even come to the point where after thinking so much about it, I can actually understand space and time as a single entity, and it has come to make sense to me in my hours of amateur astronomy.

    However, in my mind this concept ceases to work when “empty space” enters the discussion. I mean, if space and time are part of one fabric, then I don’t see how space can exist without time, and vice versa. For me is hard to understand “the part of space that has matter in it”. And the other part, “the part of space without any matter in it” is even harder to understand. Just like the existence of an infinite empty space before the Big Bang.

    Of course talking about the relationship between space and time forces to introduce the concept of time, which for sure is very complex and my personal interpretation is probably not of the interest of anybody.

    Anyway, the main idea that I’m trying to pass at the same time that I’m trying to understand it myself is that space is as finite as time, because they both are one. And if time has a beginning, in this case the 13,7 thousand million years, so should space (with its matter). And they are both expanding, mass getting away from itself and time just passing by.

    This makes me come to the conclusion that spacetime is a growing entity. And if it has a beginning and it is also expanding, at the current moment it should have a current age/dimension.
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    I don’t see how space can exist without time, and vice versa.
    Within our current models, space does not exist independently of time, and vice versa as well.

    Just like the existence of an infinite empty space before the Big Bang.
    We do not currently have an established model of the Big Bang event, and what gave rise to it. There are a number of speculations, but they are just that - speculative.

    And if time has a beginning, in this case the 13,7 thousand million years, so should space (with its matter).
    Yes, they both have a "beginning" of sorts. Just be careful not to confuse the concepts of "boundedness" with "finity" - they are not the same notions. Something can be bounded, yet still be infinite - the universe could have originated at a BB event, but then continue to expand into all eternity. That still makes it infinite, even though space and time are bounded by the BB event.
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