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SR derived solely from one postulate

  1. Feb 6, 2010 #1
    The two postulates of SR are:

    1) the laws of physics are the same in every inertial frame
    2) light is measured travelling isotropically at c in every inertial frame

    I intend to derive SR by applying only the second postulate alone, that the speed of light is measured isotropically at c in every inertial frame. We will start with a reference frame where the light is always measured to travel isotropically at c. Whether this particular frame is considered absolute or just a preferred arbitrary frame is of no consequence since we will only be determining the results according to how the physics directly relates between observers, which is how SR would relate the physics since there is otherwise nothing else to relate it to within the philosophy of SR. All other frames are to be viewed from the perspective of this preferred frame. We will apply certain properties that might occur to observers as they move relative to that frame of reference, along with their clocks and rulers. The clocks of observers in motion to this frame may time dilate by a factor of z, lengths contract in the line of motion by a factor of Lx and perpendicularly to the line of motion by a factor of Ly. Each of these ratios are to be compared to that of the reference frame, so if any of these are found not to occur, then the value for that property will be 1.

    Alice will always be considered the reference frame observer that remains stationary to that particular frame of reference for the purposes of this demonstration. We will have Bob and Carl travel away from Alice with a speed of v with a distance of d between them, Carl before Bob, as these measurements are taken by Alice. Light always travels isotropically to the reference frame at c, and we want to find what must be true for Bob and Carl to measure the same isotropic speed. Let's say a pulse is sent from Bob to Carl. According to Alice, the light travels a distance of d plus the extra distance Carl has travelled away from the pulse in the time it took for the pulse to reach Carl, so c t_BC = d + v t_BC, t_BC = d / (c - v). The time Alice will measure for a light pulse to travel from Carl to Bob while Bob moves toward the pulse in that time is c t_CB = d - v t_CB, so t_CB = d / (c + v).

    If the clocks of Bob and Carl are synchronized to each other as Alice views them, then Bob and Carl will say that their own clocks are unsynchronized because the time it takes for the pulse to travel from Bob to Carl is different from the time it takes to travel from Carl to Bob. So Bob and Carl establish a new simultaneity convention between their own clocks. This is done by having Bob turn his own clock forward or having Carl turn his back by an amount which will produce a time lag between their readings of tl. Bob's clock now reads a greater time than Carl's according to Alice, but Bob and Carl say their clocks are synchronized. When the pulse travels from Bob to Carl, it travels from Bob when Bob's clock reads TB = tl and Carl's clock reads TC = 0, then reaches Carl when Carl's clock reads TC' = z t_BC, so with a difference in times as measured by Bob and Carl of t'_BC = TC' - TB = z t_BC - tl. When the pulse travels from Carl to Bob, Carl sends it when the clocks read TC = 0 and TB = tl, and Bob receives it when his clock reads TB' = tl + z t_CB, so in a time measured by them of t'_CB = TB' - TC = tl + z t_CB. Since they must measure these two times to be equal, then t'_BC = t'_CB, z t_BC - tl = tl + z t_CB, 2 tl = z t_BC - z t_CB = z d / (c - v) - z d / (c + v) = z d [(c + v) - (c - v)] / [(c - v) (c + v)] = 2 z d v / (c^2 - v^2), whereby tl = z d v / (c^2 - v^2). This is now the simultaneity difference between the clocks of Bob and Carl according to Alice. In the frame of Bob and Carl according to Alice, their rulers have been contracted in the line of motion by Lx, so if Alice measures a distance between them of d, then they will measure d' = d / Lx . So when the pulse is sent from Bob to Carl, it will be measured to have a speed of c' = d' / t'_BC = (d / Lx) / (z t_BC - tl) = (d / Lx) / [z d / (c - v) - z d v / (c^2 - v^2)] = [(c^2 - v^2) / (z Lx)] / [(c + v) - v] = c (1 - (v/c)^2) / (z Lx), whereby if c' = c, then z Lx = 1 - (v/c)^2 . Likewise, the speed measured from Carl to Bob gives the same result.

    Now let's say that in the frame of Bob and Carl, an apparatus has been set up where light is allowed to travel across the lengths of two perpendicular arms of equal lengths d' and back. Since the pulses must have the isotropic speed of c along equal lengths of d', then the times to travel both arms are measured the same in the frame of the apparatus with c = d' / t' which is to be accepted as the usual definition of speed with distance measured over time measured, not a law that must be derived, but a given definition for speed, so since t' = d' / c where c and d' are measured the same along both arms, then so must be t' be the same along both arms. Also, because the pulses coincide in the same place upon separating and then coincide in the same place again when returning, then all observers in all frames must agree that the times to traverse both arms is the same for whatever time they measure between these two events. Let's say that one arm travels directly in the line of motion of the apparatus to the reference frame. From Alice's frame, the apparatus is contracted in the line of motion by Lx and perpendicularly by Ly, so the lengths of the arms are dx = Lx d' and dy = Ly d'. The time that Alice measures for the pulse to travel the arm in the line of motion and back is t_forward = dx / (c - v) and t_back = dx / (c + v), whereby tx = dx / (c - v) + dx / (c + v) = dx [(c + v) + (c - v)] / (c^2 - v^2) = 2 (Lx d') c / (c^2 - v^2). In the perpendicular direction, Alice measures a time of (c t_away)^2 = (v t_away)^2 + dy^2, so t_away = dy / sqrt(c^2 - v^2). The pulse travels in the same way along the same angle away and back, so t_perp = 2 dy / sqrt(1 - (v/c)^2) = 2 (Ly d') / sqrt(1 - (v/c)^2). In order for these times to be the same as Alice measures them, then 2 (Lx d') c / (c^2 - v^2) = 2 (Ly d') / sqrt(c^2 - v^2), and from this we gain Lx / Ly = sqrt(1 - (v/c)^2).

    Okay, here's where things get interesting. One might think that two observers measuring the same relative speed of each other would follow from the first postulate, since if the laws of physics is the same in all inertial frames, then with nothing else to relate the physics to except between the observers, then each must measure the same relative speed between them as the other does, but it actually follows from the second principle alone. Let's say that Bob and Carl pass Alice. Alice says that the time for Carl to pass and Bob to reach her is t = d / v, all as measured in her own frame. Now, from what is measured in the frame of Bob and Carl, when Carl passes Alice, his clock reads TC = 0 and Bob's reads TB = tl. When Bob passes Alice, his clock then reads TB' = tl + z t. Bob and Carl will read the difference in times that has passed between their clocks as t' = TB' - TC = tl + z t = z d v / (c^2 - v^2) + z d / v = [z d / (v (c^2 - v^2))] [v^2 + (c^2 - v^2)] = z d / (v (1 - (v/c)^2)) and the distance Alice has travelled of d' = d / Lx, giving a relative speed for Alice as Bob and Carl measure it of v' = d' / t' = (d / Lx) / [z d / (v (1 - (v/c)^2)] = v (1 - (v/c)^2) / (z Lx), but since we have already established earlier that z Lx = 1 - (v/c)^2, then we gain v' = v (1 - (v/c)^2) / (z Lx) = v, so the observers will measure the same relative speed of each other in both frames.

    Now let's look at the addition of speeds. Let's say that according to Alice, who is stationary with the reference frame, Bob and Carl are travelling in one direction at v and Danielle is travelling past them in the other direction at u. According to Alice, it takes a time of t = d / (u + v) for Danielle to travel from Carl to Bob, so Bob and Carl measure their difference in times to be TC = 0 and TB' = tl + z t, so t' = TB' - TC = z t + tl = z d / (u + v) + z d v / (c^2 - v^2) = z d [(c^2 - v^2) + (u + v) v] / [(u + v) (c^2 - v^2)] = z (Lx d') [c^2 - v^2 + u v + v^2] / [(u + v) (c^2 - v^2)] = d' [c^2 + u v] / [(u + v) c^2] = d' [1 + u v / c^2] / (u + v). Therefore, the speed that Bob and Carl measure of Danielle is w = d' / t' = (u + v) / (1 + u v / c^2). If Danielle were to travel in the same direction as Bob and Carl at u, then Alice would measure a time for Danielle to travel from Bob to Carl of t = d / (u - v), whereby Bob and Carl would measure their difference in times to be TB = tl and TC' = z t, for a difference in times of t' = TC' - TB = z t - tl = z d / (u - v) - z d v / (c^2 - v^2) = [z d / ((u - v) (c^2 - v^2))] [(c^2 - v^2) - v (u - v)] = [z d / ((u - v) (c^2 - v^2))] [c^2 - u v] = [z (Lx d') / ((u - v) (1 - (v/c)^2)] [1 - u v / c^2] = [d' / (u - v)] [1 - u v / c^2]. The relative speed Bob and Carl measure for Danielle when travelling in the same direction, then, is w = d' / t' = (u - v) / (1 - u v / c^2).

    Now let's say Bob and Danielle are both travelling in ships that they measure of a length of d' in their own frames. Let's determine what the length contraction Bob measures of Danielle's ship will be. Bob cannot measure the length of Danielle's ship at a distance or even directly by using his ruler while Danielle's ship is in motion to his, so he has to find another way. What he does is to find the difference in time that it takes for the front of Alice's ship to pass an antenna on his ship and then the back of her ship to pass the same antenna. At T=0 on his clock, the front of Alice's ship passes the antenna. According to Alice, the time that it takes for Danielle's ship to pass Bob's antenna is t = (Lx(u) d') / (u + v), where Lx(u) is the length contraction Alice measures of Danielle's ship. If t passes in Alice's frame, then z(v) t, where z(v) = z from before but now we are adding more speeds so must become more specific, passes for Bob and all observers must agree that this is Bob's reading when Danielle passes since the events of the readings upon his clock coincide in the same place as the front and back of Danielle's ship with Bob's antenna when the clock is placed in the same place as the antenna also. The length of Danielle's ship as Bob measures it, then, is d" = w t' = [(u + v) / (1 + u v / c^2)] [z(v) (Lx(u) d') / (u + v)] = z(v) Lx(u) d' / (1 + u v / c^2). The observed length contraction, then, is Lx(w) = d" / d' = z(v) Lx(u) / (1 + u v / c^2).

    So now let's find out what Bob and Carl measure for the time dilation of Danielle's clock. We will place Carl in the front of the ship of proper length d' and Bob at the back. Alice says Danielle travels from Carl to Bob in a time of t = (Lx(v) d') / (u + v). When Danielle passes Carl, the readings upon the clocks according to Alice are TC=0 and TB = tl. When Danielle passes Bob, Bob's reading is TB' = tl + z(v) t, and again, all observers must agree since the events of the clock readings and Danielle directly passing the clocks coincide in the same places. Bob and Carl say the difference in times that has passed between their clocks is TB' - TC = tl + z(v) t = z(v) (Lx(v) d') v / (c^2 - v^2) + z(v) (Lx(v) d') / (u + v) = [z(v) Lx(v) d' / ((c^2 - v^2) (u + v))] [v (u + v) + (c^2 - v^2)] = d' [c^2 + u v] / (c^2 (u + v)) = d' (1 + u v / c^2) / (u + v) = d' / w. The amount of time that has passed upon Danielle's clock while travelling from Carl to Bob is t" = z(u) t = z(u) (Lx(v) d') / (u + v), so the time dilation Bob and Carl measure of Danielle's clock is z(w) = t" / t' = [z(u) (Lx(v) d') / (u + v)] / [d' / w] = [z(u) Lx(v)] [w / (u + v)] = z(u) Lx(v) / (1 + u v / c^2).

    Now let's compare what we have gained for the time dilation and length contraction Bob and Carl measure of Danielle and dive into a little math logic for this part of the demonstration. We have found that Lx(w) = z(v) Lx(u) / (1 + u v / c^2) and z(w) = z(u) Lx(v) / (1 + u v / c^2), whereby (1 + u v / c^2) = z(v) Lx(u) / Lx(w) = z(u) Lx(v) / z(w), so by rearranging we gain [z(v) / Lx(v)] [Lx(u) / z(u)] [z(w) / Lx(w)] = 1. The values here that are represented by u and v are what Alice measures for the relative speeds of Danielle and of Bob and Carl, respectively. w represents the relative speed that is measured by Bob and Carl of Danielle. Now, from Alice's perspective, u and v can have any arbitrary values for the relative speeds to the reference frame and w will be determined by what those values are. u and v can have the same value and still be arbitrary, so let's say that u = v. In that case, [z(v) / Lx(v)] [Lx(u) / z(u)] = 1 so [z(w) / Lx(w)] = 1 also. Since w can still have any arbitrary value with any arbitrary values of u and v where u=v, then [z(w) / Lx(w)] = 1 for any arbitrary value whatsoever, therefore z(w) / Lx(w) always equals 1 for any relative speed of w. If that is the case, then [z(v) / Lx(v)] [Lx(u) / z(u)] [z(w) / Lx(w)] = [z(v) / Lx(v)] [Lx(u) / z(u)] = 1 for any arbitrary values of u and v even when the speeds are not equal, and the only way they can do that when changing the speed of u slightly while keeping the speed of v the same, for instance, is if z(v) / Lx(v) = 1 and z(u) / Lx(u) = 1 always also.

    So since we had z(v) Lx(v) = 1 - (v/c)^2 and Lx(v) / Ly(v) = sqrt(1 - (v/c)^2) as found at the beginning of the demonstration, and now we have z(v) / Lx(v) = 1, whereby z(v) = Lx(v), then z(v) Lx(v) = z(v)^2 = 1 - (v/c)^2, giving z(v) = Lx(v) = sqrt(1 - (v/c)^2), as well as Lx(v) / Ly(v) = sqrt(1 - (v/c)^2), giving Ly(v) = 1, so no contraction takes place perpendicularly to the line of motion. And there we have it. We have determined that all of the basic principles of SR can be determined from the second postulate alone.
     
    Last edited: Feb 6, 2010
  2. jcsd
  3. Feb 6, 2010 #2

    JesseM

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    Here's a simple demonstration that you are not uniquely led to the Lorentz transformation if you only assume the second postulate without assuming the first. Let's say that we have an unprimed frame with coordinates x,t where light always moves at c. Now define the coordinates x',t' of a primed frame as follows:

    x' = A*(x - vt)
    t' = A*(t - vx/c2)

    Here A can be any dimensionless number, it need not equal the gamma factor of 1/sqrt(1 - v2/c2). For example, we could set A=3v/c.

    OK, now consider two arbitrary events (x0, t0) and (x0+ct1, t0+t1) such that distance/time between these two events is equal to c in the unprimed frame. The coordinates of the first event in the primed frame will be:

    x' = A*(x0 - vt0)
    t' = A*(t0 - vx0/c2)

    And the coordinates of the second event in the primed frame will be:

    x' = A*((x0+ct1) - v*(t0+t1))
    t' = A*((t0+t1) - (v/c2)*(x0+ct1))

    So, subtracting the first x' coordinate from the second one, delta-x' will be:
    A*(ct1 - vt1) = A*(c-v)*t1
    and delta-t' will be:
    A*(t1 - (v/c)*t1) = A*(1 - (v/c))*t1

    So, delta-x'/delta-t' = (c-v)/(1 - (v/c)) = c*(c-v)/(c-v) = c

    ...which means that regardless of the value of A, this coordinate transformation ensures that two events which have distance/time=c in the unprimed frame will also have distance/time=c in the primed frame, satisfying the second postulate. You can't pin down the value of A unless you bring in the first postulate too.
     
    Last edited: Feb 6, 2010
  4. Feb 6, 2010 #3
    Yes, that's what I thought too when I first applied z, Lx, and Ly to the formulas. I could only determine that z Lx = 1 - (v/c)^2 and Lx / Ly = sqrt(1 - (v/c)^2) originally. I could go no further until applying the Doppler shift as Alice and Bob send pulses to each other as they move away from each other with a relative speed of v with D = z / (c + v) as Alice observes and D = (c - v) / z as Bob observes, so relating the first postulate to that where the observations are the same, then z / (c + v) = (c - v) / z, so z^2 = 1 - (v/c)^2, whereas z = Lx = sqrt(1 - (v/c)^2) and Ly = 1. However, the second to last paragraph shows how the same values can be found by introducing Danielle in the two previous paragraphs and finding the time dilation and length contraction that Bob and Carl observe of Danielle and comparing the results.
     
  5. Feb 6, 2010 #4

    JesseM

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    But are you arguing that the *second* postulate would be violated in some way if we let A take some value other than the relativistic gamma, like my example of A=3v/c? How can it be, if you agree that even with A not equal to gamma, it would still be true that any pair of events with distance/time=c in one frame would also have distance/time=c in the other frame? Or do you disagree that this would always be true if we set A to some other value like A=3v/c?

    On the other hand, if you do agree that we can completely satisfy the second postulate with a coordinate transformation that's different from the Lorentz transformation because A is not equal to gamma, then logically that means that the second postulate does not uniquely imply the Lorentz transformation.
     
    Last edited: Feb 6, 2010
  6. Feb 6, 2010 #5
    At the risk of being old fashioned, I submit the following quote from Sect. 15-1 of The Feynman Lectures on Physics, V1:

    "We now know that the mass of a body increases with velocity. In Einstein's corrected formula m has the value m=m(0)/(1-v^2/c^2)^(1/2) where the 'rest mass' m(0) represents the mass of a body that is not moving and c is the speed of light... For those who want to learn just enough about it so they can solve problems, that is all there is to the theory of relativity --- it just changes Newton's laws by introducing a correction factor to the mass."

    I believe Feynman was suggesting that all of the rest of SRT can be "reverse-engineered" from the expression for m(v).
     
  7. Feb 6, 2010 #6

    atyy

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    I looked up Weinberg's relativity text. He says if you require the dtau to be invariant, then you get the Poincare group, but if you only require dtau to be invariant when dtau is zero then you get a larger group called the conformal group, in which case the statement that a massive particle moves at constant velocity would not be an invariant statement.
     
  8. Feb 6, 2010 #7

    bcrowell

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    The way I've often presented it in the past is that there is only one postulate: the laws of physics are the same in every inertial frame. Maxwell's equations are a law of physics, so #2 follows from #1. To me, the split between #1 and #2 is a historical artifact of the 19th-century picture of physics:

    (a) In the 19th century, mechanics and optics were thought of as completely separate branches of physics, and most people, including Einstein's professors in college, weren't bothered by the fact that they were fundamentally inconsistent with one another.

    (b) In the 19th century, the only known fundamental field was the electromagnetic field. Therefore it made sense to single out the electromagnetic field for special treatment. From the modern perspective, it's kind of silly to give the electromagnetic field preferential status compared to other massless fields, and there's clearly no reason to describe c as the speed of light. c is a geometrical property of spacetime, the maximum speed of causality.

    When you start playing around with alternative formulations of axiomatic systems, one of the things you find is that the game is uninteresting unless you hold yourself to a fairly high standard of rigor in the statement of the axioms. When Einstein stated the postulates of SR in 1905, he was presenting them as philosophical criteria to apply to candidate theories, not as a rigorous axiomatization like Euclid's postulates of plane geometry. For instance, in his 1905 paper he says at a certain point, "In the first place it is clear that the equations must be *linear* on account of the properties of homogeneity which we attribute to space and time." This is clearly an implicit assumption that would have to be listed as a postulate if you were really going to do a formal axiomatization. But that wasn't what he was trying to do in that paper, so he just brings homogeneity up when he needs it.
     
  9. Feb 6, 2010 #8
    Yes, that is a very good way to put it. :) I am saying that the second postulate will not be fully satisfied with any other value for A other than A = y = 1/z = 1/sqrt(1 - (v/c)^2).
     
  10. Feb 6, 2010 #9
    That is an interesting way to look at it also. But then, how do we know that Maxwell's equations lead to the result that light always travels isotropically at c in every inertial frame until it has been tested, so then requiring that the second postulate be true and is to be included with SR? Until it has been tested, that law in itself might not be true for light as anything else such as rockets would not be measured isotropically in any direction for any frame, but then once it is proved, it must be included, or some form of Maxwell's overall result, in order to determine the precise values of z, Lx, and Ly.
     
    Last edited: Feb 6, 2010
  11. Feb 6, 2010 #10
    Well folks, looks like the show might be over already. Although I believe the mathematics for what I have demonstrated to be solid, an observant poster in this other thread has already shown me that unless the first postulate is also taken into account, there is nothing that says that different materials might not contract to varying degrees in different frames unless the physics is applied in the same way for all inertial frames, and I am inclined to agree. To that end, if this thread needs to be moved elsewhere in this forum or closed altogether, then that is fine. :)
     
  12. Feb 6, 2010 #11

    rbj

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    actually, we should be able to derive SR from the first postulate alone.

    the laws of nature have both form (the equations) and content (the parameters in those equations). the laws expressing interaction have parameters such as [itex]c[/itex] or [itex]\hbar[/itex] or [itex]G[/itex] in them. if the laws are exactly the same for two different inertial observers, then the parameters contained therein are also the same.
     
  13. Feb 6, 2010 #12
    Right, but we would still have to include one of those laws as the second postulate. For instance, using the first postulate alone that the physics is the same in every inertial frame does not lead directly to observers measuring light to travel isotropically at c, so we would still have to make that the second postulate in order to work through the calculations that derive SR.
     
  14. Feb 6, 2010 #13

    JesseM

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    But then do you disagree with my derivation that seems to show that regardless of the value of A, if two events have a coordinate distance/time of c in one frame, they will have a coordinate distance/time of c in the other frame too?

    We could look at a numerical example if it'd help. Say we set v=0.6c and A=2 (unlike in the Lorentz transformation, where v=0.6c would imply a gamma factor of 1.25). Then the transformation from unprimed to primed is:

    x' = A*(x - vt) = 2*(x - (0.6c)*t)
    t' = A*(t - vx/c^2) = 2*(t - 0.6x/c)

    And a little algebra shows that the inverse transformation from primed back to unprimed will be:

    x = (x' + vt')/[A*(1 - v^2/c^2)] = (x' + (0.6c)*t')/[2*(1 - 0.36)] = (x' + (0.6c)*t')/1.28
    t = (t' + vx'/c^2)/[A*(1 - v^2/c^2)] = (t' + 0.6x'/c)/1.28

    So, do you think it is possible to find a case where something is moving at c in one coordinate system but not moving at c in the other? If so it should be easy enough to find a numerical example.

    Let's try a random example. Suppose in the unprimed frame a light signal is emitted at x=2 light years, t=8 years, and the signal is received at x=7 light years, t=13 years. You can see that here the signal moved 5 light years in the course of 5 years, so it was moving at c in the unprimed frame.

    In the primed frame the signal was emitted at:
    x' = 2*(2 - 0.6*8) = -5.6
    t' = 2*(8 - 0.6*2) = 13.6

    And was received at:
    x' = 2*(7 - 0.6*13) = -1.6
    t' = 2*(13 - 0.6*7) = 17.6

    So, in the primed frame the light traveled a distance of -1.6 - (-5.6) = 4 light years, and took a time of 17.6 - 13.6 = 4 years to do it. So again, you can see that in this example the second postulate is satisfied: the light moved at a coordinate speed of c in both coordinate systems.
     
  15. Feb 6, 2010 #14
    Of course, the claim that only the second postulate is required to produce SR has already been shown false, but for a different reason then you stipulate, but simply that there is no reason that materials might not contract differently in different frames unless the physics is the same. I find your challenge interesting, so as a way to get around the first postulate for now, let's just say that the ships and rulers are made of the same material so will contract in the same way together such that according to the moving observers, the ships are still measured the same as before, and the ticking of the clock and the biology of the moving observer and processes of the ship are all time dilating by the same amount to the observer in the reference frame also so that the moving observer notices no difference there either.

    Okay, so according to what I have demonstrated, allowing only this condition to occur such that the moving observers measure nothing anything differently about their own ships and clocks, I will see if I can find something which occurs differently using only the second postulate without relating the physics mathematically as one might normally do with Doppler or some such as I showed a few posts back in post #3 where the physics is the same in all frames in order to determine the precise values of z, Lx, and Ly, that will produce a contradiction if the values are anything but what SR predicts.
     
    Last edited: Feb 6, 2010
  16. Feb 6, 2010 #15

    JesseM

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    Well, that's already a consequence of the coordinate transformation I provided where A was different from gamma. Just as you can derive length contraction, time dilation, and the relativity of simultaneity from the Lorentz transformation, so you can derive "altered" versions of these things from the more general coordinate transformation I provided. Remember, the general transformation was as follows:

    x' = A*(x - vt)
    t' = A*(t - vx/c^2)

    x = (x' + vt')/[A*(1 - v^2/c^2)]
    t = (t' + vx'/c^2)/[A*(1 - v^2/c^2)]

    Then suppose we have a measuring-rod at rest in the primed frame, with clocks on either end and at the center which are synchronized in the primed frame. And suppose that in the unprimed frame, at t=0 the center of the measuring rod is at x=0 and the clock there reads t'=0 (which is implied by the transformation above if you plug in x=0 and t=0 to find the value of t'), while the back end of the rod is at x=-L and the front is at x=L, and since these positions are both measured at the same moment of t=0 in the unprimed frame, that means the length of the rod in the unprimed frame must be L - (-L) = 2L.

    Suppose further that we set off a flash of light at x=0 and t=0, emitting light in both directions. Since the back end starts at a distance of L from the flash and is moving towards it at speed v in the unprimed frame, in this frame the light must catch up with the back end at t=L/(c+v). And the front end starts at a distnace of L from the flash and is moving away from it at speed v in the unprimed frame, so the light must catch up with the front end at t=L/(c-v). Since the back end has position as a function of time given by x(t) = -L + vt in the unprimed frame, the position of the light hitting the back end must be -L + vL/(c+v) = [-L(c+v)/(c+v)] + [vL/(c+v)] = -Lc/(c+v). And the front end has position as a function of time given by x(t) = L + vt, so the position of the light hitting the front end must be L + vL/(c-v) = [L(c-v)/(c-v)] + [vL/(c-v)] = Lc/(c-v).

    Now, since this coordinate transformation guarantees that light moves at c in both directions in the primed frame, and since the flash was set off at the midpoint of the rod, these two events must actually be simultaneous in the primed frame. The coordinates of the light hitting the back end in the unprimed frame were x=-Lc/(c+v), t=L/(c+v), so the coordinates in the primed frame are:

    x' = A*(-Lc/(c+v) - Lv/(c+v)) = (-AL/(c+v))(c+v) = -AL
    t' = A*(L/(c+v) + Lv/(c*(c+v))) = A*(Lc/(c*(c+v)) + Lv/(c*(c+v))) = (AL/(c*(c+v)))*(c+v) = AL/c

    And the coordinates of the light hitting the front end in the unprimed frame were x=Lc/(c-v), t=L/(c-v), so the coordinates in the primed frame are:

    x' = A*(Lc/(c-v) - Lv/(c-v)) = (AL/(c-v))*(c-v) = AL
    t' = A*(L/(c-v) - Lv/(c*(c-v))) = A*(Lc/(c*(c-v)) - Lv/(c*(c-v))) = (AL/(c*(c-v)))*(c-v) = AL/c

    So you can see that the two events are indeed simultaneous in the primed frame, both happening at t'=AL/c, while the distance between them in the primed frame is AL - (-AL) = 2AL. So if we have two simultaneous events on either end of the rod in the primed frame, and the rod is at rest in the primed frame, the distance between them must be the rest length of the rod, 2AL. And remember, the length of the rod in the unprimed frame was 2L, which means that for an object at rest in the primed frame, we have the "length contraction" equation (length of object in unprimed frame) = (1/A)*(rest length of object in primed frame).

    Now say we have rod #2 at rest in the unprimed frame, whose back end is at position x=-Lc/(c+v), and whose front end is at position x=Lc/(c-v), so the back end of this rod #2 is at the same position as the back end of the rod #1 discussed above (the one at rest in the primed frame) when the light hits it, and the front end of rod #2 is at the same position as the front end of the rod #1 when the light hits it. Since these events are simultaneous in the primed frame, if rod #1 has length 2AL in the primed frame and both ends of rod #2 line up with both ends of rod #1 at a single moment, that means rod #2 must also have length 2AL in the primed frame. Meanwhile, in the unprimed frame the length of rod #2 must be Lc/(c-v) - (-Lc/(c+v)) = [Lc(c+v) + Lc(c-v)]/[(c-v)*(c+v)] = 2Lc^2/(c^2 - v^2) = 2L/(1 - v^2/c^2). So, this tells us that for an object at rest in the unprimed frame, we have the "length contraction" equation (length of object in the primed frame) = (A*(1 - v^2/c^2))*(rest length of object in the unprimed frame).

    You can see from this that if A is equal to the gamma factor of 1/sqrt(1 - v^2/c^2), then both "length contraction" equations reduce to the standard relativistic length contraction equation (length of object in frame where it's moving) = sqrt(1 - v^2/c^2)*(length of object in frame where it's at rest). But if A is not equal to gamma, the two frames will get different answers for the amount that an object at rest in the other frame is shrunk (or possibly stretched, depending on the value of A and v) in their own frame.

    You can derive analogues of time dilation and relativity of simultaneity in similar ways. Relativity of simultaneity in SR says that if two clocks which are moving inertially at the same speed v in your frame are a distance of L apart in your frame, and they are synchronized in their own rest frame, then in your frame they'll be out-of-sync by Lv/(c^2*sqrt(1 - v^2/c^2)). The analogue of this for my equation above is that the two clocks will be out-of-sync by ALv/c^2, which you can see reduces to the relativistic version if A = 1/sqrt(1 - v^2/c^2). So, in my example above where rod #1 is at rest in the primed frame and is centered at x=0 at t=0 in the unprimed frame, this means that if all the clocks on rod #1 are synchronized in the primed frame and the clock at the center reads t'=0 when at t=0 in the unprimed frame, then at t=0 in the unprimed frame the clock at the back end reads ALv/c^2, and the clock at the front end reads -ALv/c^2. And in the unprimed frame the two moving clocks on rod #1 are slowed down by a factor of (A*(c^2 - v^2))/c^2, so after a time of t=L/(c+v) has passed in the unprimed frame, the moment when the light hits the back end, the clock at the back end has only ticked forward by (AL*(c^2-v^2))/(c^2*(c+v)) = (AL*(c+v)*(c-v))/(c^2*(c+v)) = (AL*(c-v))/c^2. If you add that to the back clock's initial reading of ALv/c^2, you find that at the moment the light reaches the back clock it reads a time of ALc/c^2 = AL/c, which I showed earlier is in fact the t'-coordinate of this event in the primed frame. Likewise, at time t=L/(c-v) in the unprimed frame when the light hits the front end, the clock at the front end has only ticked forward by (AL*(c^2-v^2))/(c^2*(c-v)) = (AL*(c+v)*(c-v))/(c^2*(c-v)) = (AL*(c+v))/c^2. Add this to the front clock's initial reading of -ALv/c^2 and you find that at the moment the light reaches the front clock it reads a time of ALc/c^2 = AL/c as well.
    Length contraction has nothing to do with physical properties of the material, it is derived directly from the coordinate transformation you are using. Even if we lived in a universe that obeyed Newtonian laws rather than relativistic ones, we'd be free to pick a particular Newtonian inertial frame to be the "unprimed" one, and then define the coordinate systems for other observers moving at v in the unprimed frame using the standard Lorentz transformation equations:

    x' = gamma*(x - vt)
    t' = gamma*(t - vx/c^2)

    Despite the fact that this is a purely Newtonian universe, the mere fact that we are using this family of coordinate systems automatically implies that a ruler at rest in one frame and with coordinate length Lrest in that frame will be found to have a contracted length of Lmoving given by Lmoving = Lrest/gamma. But there will be an important difference: because the laws of this Newtonian universe are not Lorentz-invariant, if different observers use identical coordinate-independent physical procedures to construct measuring-rods at rest relative to themselves (for example, defining the measuring-rod's length in terms of some fixed multiple of the spacing between atoms in a diamond that's at rest relative to themselves), they will not find that their physically identical measuring rods actually have identical coordinate lengths in their own rest frames--an observer at rest in the primed frame will find that the coordinate length of his measuring rod in his own frame is greater than it would be if he was using a standard Newtonian inertial frame, while an observer at rest in the unprimed frame will find that the coordinate length of his measuring rod in his own frame is exactly the same as it would be in a Newtonian inertial frame (because we defined the unprimed frame to just be a standard Newtonian inertial frame). Thus in this case the unprimed coordinate systems are giving "lengths" which are divorced from normal physical measurement procedures, which explains how length contraction can still apply in coordinate terms even though we are assuming a universe which follows Newtonian laws of physics.

    By the same token, if you choose to use the family of coordinate systems defined by the generalized transformation I gave above with an arbitrary dimensionless constant A in place of gamma, then different frames will have the different equations for "length contraction" I gave above (although they become identical in the special case where A = 1/sqrt(1 - v^2/c^2)), regardless of the physical properties of their ships and measuring-rods.
     
    Last edited: Feb 6, 2010
  17. Feb 6, 2010 #16
    I am still working through the rest of your post, but this part caught my eye. That's true, isn't it? All we have to do is to say that each observer measures the length of their own ship as the same as a given and then apply the coordinate transformations in the same way as measured from the reference frame. Cool, so my threads still stands. Thank you, JesseM. :)
     
  18. Feb 6, 2010 #17

    JesseM

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    But do you agree there's nothing in the second postulate that requires that length contraction work the same way in different frames? In other words, if both ships have a length L in their own frame, but the unprimed frame sees the ship at rest in the primed frame as having a length of L/A, while the primed frame sees the ship at rest in the unprimed frame as having a length of AL*(1 - v^2/c^2), then this does not conflict with the second postulate, even if we assume both ships were constructed using identical physical procedures in their own frame (same material, same multiple of spacing between individual atoms, etc.)--do you agree? On the other hand, the first postulate does imply that length contraction for identically-constructed objects in each frame should obey the same equation in each frame (likewise for time dilation and identically-constructed clocks).
     
  19. Feb 7, 2010 #18
    I thought the second postulate actually were: "the speed of light is independent on the speed of the source of light", which is not exactly the same.
     
  20. Feb 7, 2010 #19

    JesseM

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    I've always seen the second postulate stated in terms of light moving at c in every inertial frame--saying only that it's independent of the source's speed doesn't rule out the possibility that the speed of light rays could vary with time or spatial region, for example. Einstein did state the postulate in terms of light moving at c in section 2 of his original 1905 paper: 2. Any ray of light moves in the "stationary'' system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body. (technically it seems he is saying here only that all light moves at c in one inertial frame, the one he has labeled the "stationary" system, but if you combine that with the first postulate it of course implies that light must move at c in every inertial frame)
     
  21. Feb 7, 2010 #20

    bcrowell

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    I don't think you can take the two postulates in the 1905 paper as a logically rigorous axiomatization of SR. E.g., in section 3 he has: "In the first place it is clear that the equations must be linear on account of the properties of homogeneity which we attribute to space and time." This is a new assumption about spacetime that is logically independent of the ones he's stated earlier. IMO the purpose of the postulates is just to lay out a philosophical set of criteria to apply to candidate theories.

    You get a similar situation with Newton's laws. Newton's first law, as originally written, is a special case of the second law. Modern textbook authors tie themselves up in knots trying to reword them or reinterpret them so that the first law is logically independent of the second. In order to do that, they often interpret the first law in terms of the existence of inertial frames, which is simply not a possible reading of the Principia.

    Newton and Einstein were both writing for audiences who had strong preconceptions. Newton's audience had Aristotelian preconceptions, and Einstein's audience had a preconception of Maxwell's equations as a partial mathematical picture of an underlying aether theory. Neither was trying to create a rigouous axiomatic system like Euclid's postulates. Newton stated the first law separately, rather than taking it as a consequence of the second law, because he wanted to beat it into his readers' heads that his theory was inertial. Einstein stated the second postulate separately, rather than taking it as a consequence of the first, because he wanted to beat it into his readers' heads that he was willing to accept that this conclusion was "only apparently irreconcilable" with the first postulate.
     
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