ice109 said:
yes it would be more than acceptable; i would very much appreciate it
OK, so let's assume that in terms of my own coordinate system (which is defined in terms of rulers and clocks at rest in my frame, with the clocks synchronized using the Einstein synchronization convention so that light moves at the same speed in both directions in my coordinate system) the light bulb is turned on at position x=0 light years and time t=0 years. Assume a ruler is moving along my x-axis in the positive x direction with velocity v, with clocks at either end as well as at its center; at t=0 the center of the moving ruler is at position x=0, where the light is switched on, and the clocks on the ruler have been synchronized using the Einstein synchronization convention, so that they the clocks at either end will read the same time at the moment the light from the bulb being turned on reaches them.
Now we don't know how length contraction or time dilation works from the start, so let's say that if marks on the moving ruler are 1 meter apart in its own frame, in my frame they'll be L meters apart, and if the clocks on the moving ruler tick at a rate of 1 tick per second, in my frame they'll tick at T ticks per second. We need not assume that the L and T factors are equal, nor that their value is smaller than 1 as opposed to greater than 1 or equal to 1.
Assume without loss of generality that the ruler is 2 light years long in its own frame, so that in my frame its length is 2L light years. In this case, at t=0 the back end is at position x=-L ly, while the front end is at position x=L ly. Since the rod is moving at velocity v in the positive x direction, the back end's position as a function of time is given by x=vt - L, while the front end's position is given by x=vt + L. So, to figure out when the light from the bulb first reaches the back end in my coordinate system, take vt - L = -ct and solve for t, giving t = L/(c + v). Likewise, to figure out when the light from the bulb first reaches the front end in my coordinate system, take vt + L = ct and solve for t, giving t = L/(c - v). We can also plug these times back into the equations for position as a function of time to find the position of the front and back ends when the light first reaches them; the back end will be at position v*[L/(c + v)] - L = L*v/(c + v) - L*(c + v)/(c + v) = -Lc/(c + v), while the front end will be at position v*[L/(c - v)] + L = L*v/(c - v) + L*(c - v)/(c - v) = Lc/(c - v).
Now, suppose I have my own ruler which is at rest in my frame, and whose back end is permanently at position x = -Lc/(c + v) while its front end is permanently at position x = Lc/(c - v). In my frame the length of this ruler must just be [Lc/(c - v)] - [-Lc/(c + v)] = [Lc*(c + v) + Lc*(c - v)]/[(c - v)*(c + v)] = (2Lc^2)/(c^2 - v^2). The back end of my ruler will be at the same position as the back end of the moving ruler at the moment the light from the bulb reaches the back end of the moving ruler, while the front end of my ruler will be at the same position as the front end of the moving ruler at the moment the light from the bulb reaches the front end of the ruler. Remember that the clocks on the moving ruler were synchronized using the Einstein synchronization convention, so these events are simultaneous in the moving ruler's frame. Since each frame defines the "length" of a moving object as "the position of the front end at a given time minus the position of the back end at the same time", this means that in the moving ruler's frame, my ruler must be exactly the same length as the moving ruler, which we assumed earlier was 2 light years in its own frame.
Now we can use the fact that one of the postulates of SR says the laws of physics must be identical in different inertial frames, which means that if I see the moving ruler's length as different by a factor of L from its rest length, then in the moving frame *my* ruler must be different by a factor of L from its rest length in my frame (I guess to be really rigorous here, to justify this we'd also have to show that speeds are reciprocal, so that if I measure a moving ruler to be moving at speed v, then in the moving ruler's frame I must be moving at speed v as well; I'll give a little justification of this at the end of the post). So since my ruler has a length (2Lc^2)/(c^2 - v^2) in my frame, its length in the moving frame must be L times that, or (2L^2 * c^2)/(c^2 - v^2). But I also showed that its length in the moving frame must be equal to that of the moving ruler in its own frame, which was assumed to be 2 light years; this means that (2L^2 * c^2)/(c^2 - v^2) = 2, which if we solve for L gives L^2 = (c^2 - v^2)/c^2 = (1 - v^2/c^2), which shows that L is equal to the familiar lorentz contraction factor sqrt(1 - v^2/c^2).
Once you have the lorentz contraction factor I can think of two different ways you might go about finding the time dilation factor T. One way is to place a mirror at one end of the moving ruler, send a light signal from an emitter at the other end, and then impose the condition that the amount of time t for the light to go to the mirror and bounce back to where it was emitted, as measured on a clock next to the emitter, must satisfy d/t = c, where d is twice the ruler's length in its rest frame (the distance the light travels between leaving the emitter and returning to it). If the ruler's length in its rest frame is 2 light years and its length in my frame is 2L, then if the signal first travels in the same direction the ruler is moving in my frame (from back to front), the time for the signal to go from the emitter to the mirror in my frame will be given by the equation vt + 2L = ct, so t = 2L/(c - v). Then the time for the signal to return from the mirror to the emitter, this time traveling in the opposite direction as the ruler in my frame (front to back) will be given by vt = 2L - ct, so t = 2L/(c + v). So, the total time in my frame is 2L/(c - v) + 2L/(c + v) = [2L*(c + v) + 2L*(c - v)]/[(c + v)*(c - v)] = 4Lc/(c^2 - v^2) = 4L/[c*(1 - c^2/v^2)]. If we make use of the fact that we know L = sqrt(1 - v^2/c^2), then that means the time in my frame must be 4/[c*sqrt(1 - v^2/c^2)]. But we also know that whatever the time in my frame, the time as measured by a clock that's moving with the ruler must be T times that, so the time as measured by a clock next to the emitter must be 4T/[c*sqrt(1 - v^2/c^2)]. Meanwhile, the distance the light travels in the ruler's frame is twice the ruler's rest length, or 4 light years. So, d/t = c gives 4/(4T/[c*sqrt(1 - v^2/c^2)]) = c, or [c*sqrt(1 - v^2/c^2)]/T = c, which gives T = sqrt(1 - v^2/c^2), the familiar time dilation factor.
However, you may prefer to find the value of T using your original thought-experiment where, instead of light starting from one end and bouncing off a mirror on the other end, we instead have a light bulb which turns on when it is next to the center of the moving ruler, and clocks at either end of the ruler are synchronized so that they read the same time when the light from the bulb reaches them. As mentioned above in the third paragraph, if in my frame the bulb is turned on at time coordinate t=0, then the light reaches the back end at t = L/(c + v) in my frame, while the light reaches the front end at t=L/(c - v). So, the time interval between these two events in my frame is just [L/(c - v)] - [L/(c + v)] = [L*(c + v) - L*(c - v)]/[(c + v)*(c - v)] = 2Lv/(c^2 - v^2). Since both clocks are synchronized in their rest frame so that they read the same time at the moment the light hits them, in my frame the time on the clock at the back end will have advanced by T times this amount by the moment the light reaches the clock at the front end, so in my frame the two clocks will be out of sync by 2TLv/(c^2 - v^2). This means that if there is another clock at the center of the moving ruler, and it reads a time of 0 years at the moment it's next to the light bulb and the light bulb turns on, then in my frame the clock at the back end will read TLv/(c^2 - v^2) at the moment the bulb is turned on, while the clock at the front end will read -TLv/(c^2 - v^2) at the moment the bulb is turned on.
As mentioned earlier, in my frame the light reaches the back end at a time of L/(c + v) after the bulb was turned on. The clock at the back end will have advanced forward by T times this amount between the time the bulb turned on and the time the light reached it, and in my frame it started at TLv/(c^2 - v^2) when the bulb was turned on, so when the light reaches it the time on the clock at the back end will be TLv/(c^2 - v^2) + T*L/(c + v) = [TLv + TL*(c - v)]/(c^2 - v^2) = TLc/(c^2 - v^2). Since the clock next to the bulb read 0 at the moment it was turned on, this must be the actual amount of time for the light to travel from the bulb to the back end of the ruler as measured in the ruler's own rest frame. And we know the distance from the center of the ruler to either end is 1 light year in its own rest frame, so if distance/time must equal c in this frame, then we have [1]/[TLc/(c^2 - v^2)] = c, which gives (c^2 - v^2)/TLc = c which in turn gives TL = (c^2 - v^2)/c^2 = (1 - v^2/c^2).
If we know that L = sqrt(1 - v^2/c^2), then TL = (1 - v^2/c^2) is enough to prove that T = sqrt(1 - v^2/c^2). However, there's one missing step here, because as I said in my original derivation of L, we really need to prove that speeds are reciprocal, so that if the ruler's speed is v in my frame, then my speed is also v as measured in the ruler's frame. To show this, imagine I place a marker at the 0 mark of my own ruler, and the moving ruler notes the time according to its own clocks that the marker is next to its center and the time the marker is passing the back end of the ruler. We already assumed that the clock at the middle of the moving ruler reads 0 at the moment it coincides with position x=0 on my ruler. Then since the distance between the middle of the moving ruler and the back end is L in my frame, and the ruler's speed is v, the time in my frame for the back end of the moving ruler to reach the marker must just be L/v, so the clocks on the moving ruler must advance forward by T times this amount, or TL/v. But remember that in my frame the clocks at different points on the ruler are out of sync, so the clock at the back end didn't start out with a time of zero at the moment the clock in the middle read zero; instead the initial time on the clock at the back end was TLv/(c^2 - v^2) in my frame. So, by the time the back end passes my marker, its time must read TLv/(c^2 - v^2) + TL/v, which can be rewritten as [TLv^2 + TL*(c^2 - v^2)]/v*(c^2 - v^2) = (TLc^2)/[v*(c^2 - v^2)]. So, this is the time for the marker to move from the middle of the moving ruler to its back end as measured in the moving ruler's frame. Since the distance between the middle and the back end is 1 light year in the moving ruler's frame, the speed of the marker as measured in the moving ruler's frame must be distance/time = 1/[(TLc^2)/[v*(c^2 - v^2)]] = v*(c^2 - v^2)/TLc^2 = vc^2*(1 - v^2/c^2)/TLc^2 = v*(1 - v^2/c^2)/TL. And recall that in the previous paragraph I found TL = (1 - v^2/c^2), so plugging this in we find that the speed of the marker as measured by the moving ruler is v*(1 - v^2/c^2)/(1 - v^2/c^2) = v. So, this shows that my speed as measured in the moving ruler's frame must be equal to the moving ruler's speed in my frame. Note that we didn't need to know the value of L to get either this result or the earlier result that TL = (1 - v^2/c^2), so there isn't any circular reasoning in using this result to find L as I did in the fifth paragraph above.
Let me know if you have any questions about these derivations...if you have trouble following some of it, it may help to draw a diagram, or to plug in some specific numbers like v=0.6c.
