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Thread: Special Relativity Primer

  1. #1 Special Relativity Primer 
    Member Janus's Avatar
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    I've noticed that there is a lot of variation in some of the "explanations" of Special Relativity I've seen and it can be confusing to some. Therefore, I thought I'd try to clear up some of the confusion with a step by step walk through some of the main concepts, concluding with how they all fit together to explain everybody's favorite punching bag: "The Twin Paradox".

    Since some people's eyes glaze over when they see a bunch of equations, I'm going to try do it without the math.


    1. Inertial frames of reference

    Essentially, this is any frame of reference which is not accelerating. Imagine An astronaut(A) floating in space far away from any gravity source. His can be said to be at rest according to a inertial reference frame (We'll call this "A's frame"). If we add another astronaut(B) with a relative motion with respect to A, He is moving in A's frame, But is motionless according to another frame(B's frame).

    So in A's frame, A is stationary and B is moving, and in B's frame A is moving and B is stationary.

    We can choose from an infinite number of inertial reference frames, all of which A
    and B would have different velocities, What frame of reference we use for any situation depends on which is more convenient.

    One point about reference frames is that they are not the same thing as "point of view". A point of view is attached to an object or observer, a frame of reference is not. If A accelerates, his "point of view" goes with him, However, the frame of reference we originally called A's frame does not. Instead, after the acceleration, A is now motionless with respect to a different frame (A's new frame) and is in motion with respect to his old "A's frame".

    (While accelerating, A is in what is known as a non-inertial reference frame. All you need to know about this right now is that that rules for dealing with non-inertial frames differ from those dealing with inertial frames.)


    2. The postulates of Relativity

    SR is built on two postulates:

    1. The laws of physics is the same for all inertial reference frames.
    2. The speed of light is a constant for all inertial frame.

    The first simply means that there is no physical experiment that can distinguish between two reference frames. Any experiment you perform at rest with respect to one frame gives the same results as a those done at rest in a frame moving with respect to the first.
    The result is that there is no "preferred" inertial reference frame. You can't say who is "really" moving and who is "really" at rest.

    The second state that the speed of light in one of these physical laws that is the same for all inertial reference frames. And when we mean that the speed of light is "constant" we mean relative to the inertial frame from which it is being measured.

    Example, going back to our astronauts A and B. Assume that their relative velocity is towards each other (A see's B approaching him and B see's A approaching him).

    A flash of light occurs at a point of light between them.

    The following animation shows how events occur in A's frame with the expanding circle representing the flash of light.



    Note that the flash expands in a circle from a point that maintains a constant distance from A, so that the speed of the light flash traveling in all directions is the same relative to A.

    Now here is the same light flash as it occurs in B's frame.





    Note that for the speed of light to be a constant in B's frame, the light has to expand in a circle from a point that remains a constant distance from B.

    Note that this is not what A and B "see", it is what they determine has happened.
    (For example, A would "see" the light from the flash, before he "see's" the light reach B, because of the time it would take for the light to reflect back to A from B. , but once A takes this into account, he will conclude that the light reached B first. )

    This last paragraph brings up our next point. Note that in the animations, according to A's frame, the light hits B first, but according to B's frame, the light Hits A first.

    This takes us to our next topic which I'll discuss in my next post.
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  2. #2  
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    Okay, we'll pick things up where we left off:

    3. The Relativity of Simultaneity


    As noted in the last post, our flash either hits A or B first depending on whether determined by A's or B's frame of reference. This leads us to the conclusion that the order of events can be dependent upon the frame in which they are determined. It can also mean that events that are simultaneous in one frame, may not be so in another.

    To demonstrate, imagine you have two observers, each at rest in his own frame. Frame 1, which we'll call the "ground frame", has an observer which is standing next to a train track. Frame 2, which we'll call the "railway car frame" has an observer which is riding a car which is moving relative to the ground frame.

    Two lightning strikes hit the tracks at points an equal distance from our ground observer. (He is at the midpoint between the strikes). He will obvious see the flashes from the strikes at the same time.

    Now imagine that the railway car observer is passing the ground observer at the very instant the light from the flashes reach him. he will see the flashes at the same time also, as shown in the following animation which shows things from the ground frame:



    But what of the railway car frame? We've already established that he will "see" the flashes at the same time. But since light has a finite speed, and he himself is moving with respect to the tracks, he can't be at the midpoint when the strikes occur. He will be closer to one strike than the other. Since, as we showed in the earlier animations, the light in his frame will expand in a circle from a point a constant distance from himself, if the lightning strike occurred simultaneously in his frame, his being closer to one strike than the other would result in the flashes reaching him at different times. Therefore the only way that the light from the flashes can reach him at the same time is for the strike to occur at different times as shown in the following animation which shows the same situation as the last animation, but shown from the railway car frame.



    The light still reaches both observers at the same time, but the right lightning strike occurs first and then the left strike.

    Events that are simultaneous in one frame are not always simultaneous in another.

    4.The Relativity of Simultaneity and Clocks

    Imagine you have two clocks separated by some distance and you want to synchronize them. A good way would be to sent a light speed signal to them from a point halfway between them. Each clock reads zero until the flash reaches them and then starts running (we'll assume identical clocks that will run at the same rate from then on)

    Something like this:



    This works for the frame in which the clocks are at rest, but what happens according to a frame in which the clocks are moving? Then, according to the postulates of Relativity, we get this:



    Note that the light hits the left clock first, starting it and then the right clock, starting it. Now even though both clocks run at the same rate after both have started, the rightmost clock will always run behind the left clock.

    Or put another way, while in the frame in which the clocks are at rest, it is always the same time at the position of each clock, In the frame where the clocks are moving from left to right, it is always a later time at the position of the right clock than it is at the position of the left clock.

    Time is relative between frames. Thus when you say things like "at the same time" or "at this time the right clock reads 12:00am", you have to say in which frame you are making this determination.

    At this point I would like to reiterate the concept of "no preferred frame". Even though the two frame come to different conclusions as whether or not the clocks show the same time, each is equally entitled to his conclusion, and there is no absolute way to say who is "really" right.

    This also takes us to the next aspect of the relativity of time; the fact that the rate of time flow can differ between frames, which I will discuss in my next post.
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    5. Time dilation


    Take two mirrors placed a certain vertical distance from each other and bounce a light pulse between them. This is called a "light clock". Every round trip of the pulse is one tick of the clock. Any other type of clock(mechanical, biological, atomic, etc) in at rest with respect to the light clock, will keep the same time.

    Now take two light clocks, with a relative motion to each other perpendicular to the travel of the pulses. Each clock in its own frame ticks normally.

    But how do the clocks tick according to each other?

    We'll start the two clocks at the same position and with the pulses staring at the same point, as per this animation:



    The red numbers represent a clock keeping time with its own light clock in its own frame. The expanding circles illustrate how the pulses travel at a constant speed in the frame being shown.

    Note that when the light pulse of the "stationary" clock reaches the mirror, the light pulse of the "moving" light clock has only completed a part of the trip the mirror. In fact, the pulse can only complete one round trip for the two made by the "stationary" clock.

    Of course, if we showed the "moving" clock as stationary, then we would see the "Stationary" clock as moving to the left, and ticking slower.

    Once again, which clock ticks slower depends on which frame you choose and neither is a better choice than the other.

    Time moves slower compared to your time for objects moving relative to you.

    Now let's add a second set of mirrors so that the pulses bounce back and forth in the same direction as the relative motion of the light clocks, like thus.



    Immediately, we see a problem. The two pulses for our "stationary" clock bounce back and forth twice, always meeting up where they started. The vertical pulse of the "moving" clock still makes its one round trip. But, the horizontal pulse of the "moving" clock hasn't even reached the mirror, let alone bounced back to its starting place by the time the vertical pulse has completed its round trip.

    Since in the "moving" clock's frame, the two pulse do meet up again, we have an apparent contradiction. This is resolved by the concept I'll discuss in the next post.
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    As we noted in the last post, something doesn't quite seem right with our last animation, to correct this we need:

    6. Length Contraction

    To discuss length contraction, let's go back to our discussion of the Relativity of Simultaneity. Remember that we said the events are not always simultaneous between frames. We will use a another example of this (this is a thought experiment Einstein used)

    We'll go back to the railroad tracks and our two observers, only this time the railway car observer is on a car that is halfway between the ends of a long train. Also instead the both observers see the flashes at the same time, the strikes will occur simultaneously according the the ground frame, At the instant the railway car observer passes him. and strike at points that are even with the ends of trains and leave char marks on the ground. The ground observer sees the flashes at the same time, but concludes that one flash reaches the train observer first. (the expanding domes are the light flashes)



    The fact that the light flashes reach the train observer at different time is true for both frames. But since the train observer is halfway between the ends of the train, and the lightning struck the ends of the train, the light should take an equal time to travel from either end to the observer, thus the front strike must have taken place first.

    So far this just confirms what we've already said. But, remember, each end of the train was even with the strike point and subsequent char mark on the ground at the time of the strikes. If the distance between the strikes on the ground and the length of the train are the same, then the both ends would be at both strikes points at the same time leading to simultaneous strikes, which we already established doesn't happen in the train frame. The only way for for the front strike to occur before the rear strike is for the front of the train to be even with the front char mark before the rear of the train is even with the rear char mark. This means that the char marks have to be closer together than the ends of the train in the train frame.

    The ground frame is length contracted in the train frame and we get this result in the train frame:



    Note that the flashes of light still reach the ground observer at the same time.

    Of course this also means that in the ground frame, it is the train that is length contracted. (if we were to stop the train so that it was at rest with respect to the ground, it would still be longer than the distance between the char marks, though not as much as it measured in its frame when they were in relative motion.)

    Objects with relative motion to a frame are contracted in length as measured from the frame.

    So, going back to our light clocks, this means that the horizontal distance between the mirror of the "moving" clock must be contracted according to the "stationary" frame. Thus you end up with this:



    In my next post, I'll put it all together.
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  5. #5  
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    Now we'll apply all we've gone over to:

    7. The Twin Paradox

    We'll keep it simple and assume instantaneous acceleration up to speed. (this avoids dealing with those Non-inertial frame of references. While finite accelerations can be dealt with and don't change the basic outcome, they add an extra complexity to the problem. We'll also assume clocks at both ends of the trip, synchronized in the Earth frame.

    We'll break it down for each Twin:

    Earth twin

    The Space twin jumps to near light speed and according the Earth twin begins to under go time dilation and ages at a slower rate. He reaches the endpoint having aged less than the time that has past on both the Earth clock and the endpoint clock (we'll assume that 49 years have past on Earth and the endpoint clock, and the Space twin has aged 7 yrs.
    The Space twin does an instantaneous turn around and heads back to Earth. He is still undergoing time dilation and ages another 7yrs, while 49 yrs pass on Earth and at the End point.

    He returns to Earth 98 years later, having aged 14 yrs.

    Space Twin

    He jumps to near light speed. (or in his frame the Earth and endpoint jump to near light speed) Because of length contraction, the distance between Earth and the endpoint has contracted to 1/7 the distance it was as measured in the Earth frame. Because the Relativity of Simultaneity, the endpoint clock no longer reads the same time as the Earth clock but reads 48 yrs ahead. (the Earth clock does not change reading because it is still right next to the Space twin.) The distance is crossed, during which time the Earth and endpoint clocks run slow due to time dilation. Due to the length contracted distance, the distance passes in 7yrs by the space twin's clock. In the meantime, 1 yr pass on the Earth and endpoint clocks. Upon being even with the endpoint, the space twin clock reads 7yrs, the endpoint clock reads 49 yrs, and the Earth clock reads 1 yr.

    The return trip begins. As a result, the Space twin changes reference frames to one with a relative velocity towards Earth from one away. The distance between endpoint and Earth is still 1/7 that as measured in the Earth frame, but now, due to Relativity of Simultaneity, it is the Earth clock that is ahead of the Endpoint clock. Since the Space twin is still right next to the Endpoint clock, it is the Earth clock that changes to read 48 years ahead of the Endpoint clock and thus now reads 97 yrs. Both Earth and endpoint clock run slow during the second half of the trip, advancing 1 yr to the 7yrs passing on the Space twin clock. Upon reaching Earth, the Earth clock now reads 98 yrs, the space twin's clock 14 yrs and the endpoint clock 50 yrs. The space twin matches speed with the Earth entering the Earth frame and the clock at the endpoint jumps forward to 98 yrs to match the Earth clock.

    Remember this is not what each twins "sees", it is what he concludes. (for instance when the space twin is right next to the Endpoint clock, he "sees" the same time on Earth as the someone at the Endpoint would since the same light reaches his eyes, he does disagree as to how long ago that light left Earth.)

    I hope this helps. There are a lot of ways of trying to explain this and not all ways make sense to everyone. After I take a break from the keyboard for a while I might continue and take this on from a different angle
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    MODERATOR NOTE : I have taken the liberty to create a dedicated thread for the discussion of the contents of this sticky :

    Discussion of Special Relativity Primer

    Please continue on over there - some very good points being raised ! I have cross-referenced this on the other thread also, so that it is clear those posts were moved from here.
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