Special Relativity Simple Problem

[IDumb]

Homework Statement

Consider the usual two inertial frames S and S' in standard configuration. In S' the standard lattice clocks all emit a 'flash' at noon. Prove that in S this flash occurs on a plane orthogonal to the x-axis and traveling in the positive x-direction at (de-Broglie-) speed c^2/v.

Homework Equations

Lorentz transforms. The invariant interval (ct^2 - x^2 - y^2 - z^2 = ct'^2 - x'^2 - y'^2 - z'^2). To clarify standard configuration is just an orthogonal x y z for s and then an orthogonal x' y' for s' where x and y are parallel to x' and y' respectively and s' is moving away from s at speed v along the x and x' axis.

The Attempt at a Solution

I'm struggling to begin. I think I'm misunderstanding something... because in thinking about limiting cases. Say we have an S' frame that happens to be the same as S (so v = 0), then the problem makes no sense. I feel like in that case the flash should be moving at a speed of 0. Perhaps this is misunderstanding what it means by a "flash occurring on a plane".

I just really need a hint on how to get started on this : (.

 

[Orodruin]

If v = 0, the flash is happening everywhere at once and thus, in some sense, travels with infinite speed (the time between reaching x = 0 and x = 2000 lyr is zero). In Minkowski space, the flashes happen at the three dimensional hyper surface t' = 0. In order to see how this hyper surface looks in S, all you need to do is to use the Lorentz transforms. You should get a new three dimensional hyper surface, which you can parameterise with three of the coordinates in S.

 

[IDumb]

Thanks for the response!

So I have gotten the velocity but I have a question about it.

So at t' = 0, we have *dt = gamma*(v*dx'/c^2). Then we say dx = gamma*dx' at t' = 0. So at t' = 0 we have dx/dt = c^2/v. I'm having trouble... figuring out what "velocity of the flash" is. I don't really know what a "hyper surface" is (no manifold stuff for me yet), so I don't understand. Is my.. process described above the correct way of looking at it? I feel weird only looking at t' = 0.

*Also as a side note - this isn't actually homework - I'm just trying to teach myself introductory relativity using Rinder's book. Is the "Homework help" section the correct place to post for me?*

 

[Orodruin]

Start thinking in two dimensions only, i.e., forget about the y and z directions, which are not affected by the Lorentz transformations anyway. The set of flash events are now a line and the line is given by t' = 0. If you look at a given time t in S, only one point will be flashing at that time, the "velocity" you are asked to find is the velocity with which this point "travels". I say it with quotes because nothing is really travelling.

 

[TSny]

To help with the interpretation of the problem, look at the attached figure which shows frame S. Consider a plane that is fixed in frame S that is oriented perpendicular to the x axis, such as plane A. You want to show that all points on this plane get illuminated simultaneously. If plane B is located some distance from plane A, then you need to show that plane B gets illuminated at a time later than A such that it would appear that light has traveled from A to B at a speed of c²/v.

 

[Chestermiller]

The people in the S frame of reference don't see all the clocks flash at once. They see the lights flashing in sequence along the x-axis like an array of dominos.

Chet