C                     /\       A             |  (velocity)        B
1. inertial
2. gravitational
• t' is the time as measured by the moving frame
• t is the time as measured by the absolute frame
• L' is the length as measured by the moving frame
• L is the length as measured by the absolute frame
• v is the (constant) speed at which the moving frame is travelling with respect to the absolute frame; of course |v|<c.
I'm going to re-derive the above equations from an absolute reference frame, which I've previously done (though not all the way) in the "Unrelative Relativity 2" thread. This time I'm going to be quite a bit more explicit and rigorous. Here goes: There are three bodies A, B, and C, all travelling with constant speed v<c along parallel vectors (the distance between them doesn't change.) The bodies are arranged as follows: A and B are arranged along a line orthogonal to their direction of motion, while A and C are arranged precisely along the direction of motion. |AB| == |AC| == L (L is the distance as measured by the absolute observer), so that ABC is an equilateral right triangle in the absolute frame. Here's an illustration for visual reference:
C           /\                                |  (direction of motion)                    A  B        |

Case (1)

A wants to measure the distance to B. To perform the measurement, A bounces a photon off B and uses the speed of light to calculate the distance (remember, A assumes the speed of light to be constant in all directions.) But as A bounces a photon off B, in the absolute reference frame the photon is actually describing an equilateral triangle with height L since A and B are co-moving. The trip from A to B will take exactly the same amount of time as the trip from B to A, and the total roundtrip time will be twice that. To solve for the one-way-trip time, we take advantage of the right triangle formed by A at the moment of light emission, A at the moment B receives the light and B at the moment B receives the light: v^2t^2 + L^2 = c^2t^2. Solving for t, we have: t^2(v^2 - c^2) = -L^2 t = L/sqrt(c^2 - v^2) The roundtrip time t_r(AB) is just twice that: (1) t_r(AB) = 2L/sqrt(c^2-v^2). As a check, if v was 0, then we would have 2t = 2L/c which is what we would expect. Also, (2) |v| > 0 => t_r(AB) > 2L/c The reason I constructed ABC the way I did, is that AB will correspond to a unit on an X axis and AC to a unit on the Y axis of the moving reference frame. The Z axis sticks out of the page, but with respect to the movement behaves identically to the X axis in this case (hopefully, you can see that.) Any interaction in the moving frame will have to occur along a vector in the XYZ coordinate system I just defined. The vector will either be parallel to the Y axis or have a component orthogonal to the Y axis.

Case (2)

A wants to determine the distance to C. To perform the measurement, A is going to bounce a photon off C and use the speed of light to calculate the distance (remember, A assumes the speed of light to be constant in all directions.) In the absolute frame, going from A to C the photon is going to take L/(c - v) seconds to reach C, and then from C to A it's going to take L/(c+v) seconds. Add these fractions and simplify to get the total round-trip time t_r(AC): t_r(AC) = L/(c-v) + L/(c+v) = (Lc + Lv + Lc - Lv)/(c^2 - v^2) (3) t_r(AC) = 2Lc/(c^2 - v^2) Rewriting t_r(AC) as 2L/(c - v^2/c), we note that (4) |v| > 0 => t_r(AC) > 2L/c Now let's compare t_r(AB) and t_r(AC) by subtracting the former from the latter: 2Lc/(c^2 - v^2) - 2L/sqrt(c^2 - v^2) = (2Lc - 2Lsqrt(c^2 - v^2))/(c^2 - v^2) The denominator is always positive. The numerator can be re-written as 2L(c - sqrt(c^2 - v^2)). From which we see that: (5) |v| > 0 => t_r(AC) > t_r(AB)

discussion

For conceptual simplicity, assume that any mechanism in the moving frame is going to operate via exchange of information between its components through the electromagnetic field (the argument can be identically repeated for all other fields.) Note that "exchange" means information flows in both directions. I cannot overemphasize enough the significance of the word "exchange"; it is inherent in the notion of interaction. As can be concluded from (2) and (4) above, such information exchanges will take longer in the moving frame than they would if the mechanism was stationary. This is uniformly true of all mechanisms, including clocks. This means that all moving clocks are slowed down as a function of their speed relative to the absolute frame. Therefore, let us define t' to be the time as measured by the moving frame. We now know that t' lags behind t, so it makes sense for us to define it separately. We're going to define t' using the results of (1) and (3) above. But because of (5), we see that the moving frame's time is not distorted equally along all vectors of interaction. We have a choice here. We can define t' via two separate equations and apply each of those equations to corresponding components of interaction vectors. Or we can choose to define t' via one equation for all interaction vectors, and define a separate correction factor to allow for (5) above based on interaction vector components. Einstein, it turns out, takes this latter approach. Define t = t_r(AB)/2 = L/sqrt(c^2-v^2). To express t' in terms of t, we know L=ct' as measured by the moving frame, so: t = t'c/sqrt(c^2-v^2) = t'/sqrt(c^2/c^2-v^2/c^2) = t'/sqrt(1-v^2/c^2) (6) t' = t*sqrt(1-v^2/c^2) Which is exactly the time dilation equation given waay above. Keep in mind that L=ct' used here is the measurement of distance by the moving frame in a direction orthogonal to the moving frame's velocity (since we are working with t_r(AB), or in the XZ plane of the moving frame using the coordinate axes I defined. We know that t_r(AB) is less than t_r(AC) so we'll need to correct t' as just defined for interactions along the moving frame's Y axis. We know t_r(AC) should be equal to L/c from the perspective of the moving frame. But t_r(AC) is larger than it should be, if L = ct'. Since we allow the moving frame to assume that speed of propagating lightfronts with respect to it is constant no matter the direction, and we've already fixed t', we are forced to conclude that the moving frame will measure a distance L' such that L' > L (where L is the distance the moving frame would measure in directions orthogonal to its velocity.) Note that this effective lengthening of the Y axis in the moving frame is not real in absolute terms; it is merely an artifact of the way we choose to define t'. Rather than saying that for the moving frame time is stretched out globally and distance stretched out in the direction of motion, we could have said that for the moving frame time is stretched out, but more so along the direction of motion. However, since the moving frame cannot perceive such anisotropy, and to make it more mathematically homologous to the absolute reference frame, we define a global notion of time as per (6) above but then are forced to introduce an anisotropy of distance to compensate. Additionally, actual solid bodies will indeed experience contraction along line of motion as observed from the absolute reference frame, due to (5), as they must maintain geometry within the moving reference frame (as per principle of relativity.) And rather than going through a circumlocution of anisotropic directional interaction rates, it is more convenient for the absolute observer to simply describe the effect as a virtual compression of the moving frame along its velocity. An interesting question arises: why did we choose t_r(AB) to define t' as opposed to t_r(AC) or some combination of the two? The answer is that if from the moving frame's perspective it took a certain amount of time for light to bounce back from B, it is going to take longer for light to bounce back from C (as per (5) above), forcing the moving observer to conclude that |AC|, as set up from the absolute frame, is greater than |AB| due to the fact that the moving observer assumes constant velocity of light, regardless of direction, and has no reason for suspecting otherwise. By this argument t_r(AB) is the "norm", while the additional elapsed time in t_r(AC) is the "aberration". Now all that's left, is to express the measurement of |AC| from the moving frame (L') in terms of L. To do this, we first determine what t_r(AC) is in terms of t' (using (6)), multiply the result by c (to get roundtrip distance) and divide by 2: L' = (t_r(AC)*sqrt(1-v^2/c^2))*c/2 = Lc^2/(c^2-v^2)*sqrt(1-v^2/c^2) = L*sqrt(1-v^2/c^2)/(1-v^2/c^2) (7) L' = L/sqrt(1-v^2/c^2) Which, of course, is the length contraction formula.