#### NoahsArk

Gold Member

- 136

- 7

t = γt

^{1}+ (γv/c

^{2}/)x

^{1}

It's the second part of this equation, the (γv/c

^{2}/)x

^{1}, which is throwing me way off. The first part of the equation represents time dilation. The second part of the equation, if I'm not mistaken, represents the fact that the distance in the direction of motion in which an event occurred in one frame of reference will also play a role in what time is recorded for that same event in another frame of reference. In other words, this demonstrates the relativity of simultaneity. It seems like the wrong formula though based on what's actually happening.

Say t is the earth frame and t

^{1}is the rocket frame. Say t

^{1}is moving at a speed of 3/4 c relative to t. If event 1 is a beam of light shooting from the tale of the rocket (which is the origin of the rocket frame), and event two is the beam of light striking the nose end of the rocket, and the rocket has a length of one light second, then the rocket frame will measure one light second of time between the events. In the earth frame, because the nose of the rocket has been moving according to an earth observer, the beam of light is catching up with the nose at a speed of only 1/4 c every second, and it will take 4 seconds for the light to strike the nose according to the earth observer. This is 3 seconds more than it took according to the rocket's measurement. I should also take into account who is measuring the distance from the tail to the nose to be 1 light second. If it is someone in the rocket making this measurement, then to get an earth observer's measurement I'd need to multiply 1 by γ. So, here is what the correct formula seems to be for the right part of the Lorentz Transformation for time:

(x

^{1}γ / c - v) - t

^{1}. Basically I divided the distance, from earth's measurement, by the difference in speed between the beam of light and the rocket to get earth's time for event two and then subtracted that by rocket's measured time to see how much more time elapsed according to an earth observer. The actual formula though is very different, and I don't see the logic of it based on what's actually happening. The actually formula says you take the speed of the rocket then multiply that by γ, then divide by C squared then multiply by x

^{1}?? I understand that when you do the derivation of the Lorentz Transformation from it's general form, that's what comes out, but still it doesn't make sense to me physically. My questions are:

1) How to make sense of the actual form of the second part of the LT for time, and where did I go wrong in thinking it should be in the other form?

2) Do we also need to take into account whether the event in question is happening inside the rocket (which would make it part of the rocket frame and therefore moving with respect to an earth observer), or outside the rocket which would might make it stationary with respect to the earth frame and moving with respect to the rocket? If so, shouldn't the formula for the LT change based on where the event is happening?

Thank you.