a few questions from introduction to sr by rindler.

MathematicalPhysicist

we have two inertial frames, S and S' where S' is moving with speed v along the x axis.
here are a few questions about these frames:
1. if two events occur at the same point in some inertial frame S, prove that their temporal order is the same in all inertial frames, and that the least time seperation is assigned to them in S.
2. if two events occur at the same time in some inertial frame S,prove that there is no limit on the time seperations assigned to these events in other frames, but that their space seperation varies from infinity to a minimum which is measured in S.
3. in the inertial frame S' the standard lattice clocks all emit a 'flash' at noon. prove that in S this flash occurs on plane orthogonal to the x-axis and travlling in the positive x direction at speed c^2/v.

well im not sure what to do in 2 or 1.
but in three, the beam of light from S' obviously travels at speed c, and according to einstein's postulate the speed of light is constant to all observers, so shouldnt the flash travel at c?
anyway, i know that the flash should travel a distance of vt, where v is the speed of S', where t is the time in S, so we should have ct'=vt where t' is the time measured in S', but is this correct?

i would like to advise me how to solve 1 and 2.

thanks in advance.

robphy

are you familiar with 4-vectors? spacetime diagrams?

MathematicalPhysicist

im familiar with spacetime diagrams, where t is a function of x.
but i havent yet used 4-vectors.
anyway, there isn't mathematical way to prove these questions?

robphy

for 1,

Draw the two events on a spacetime diagram, with the past event at the origin.
Note that any proper Lorentz boost will slide the future event along a [future] hyperbola centered at the origin, asymptotes along the light-cone. All events on that hyperbola have the same interval with the origin t²-x²=constant > 0. Note that the time-difference [difference in t-coordinates] of any event on that hyperbola is always positive [so the causal order is preserved]... in fact, the smallest value occurs when x=0.

bernhard.rothenstein

1. The events you define are in S, E(1)[x,t(1)] and E(2)[x,t(2)]; Perform the Lorentz transformations to S' , reckon the corresponding time intervals and space separations and you recover the anticipated results.
2. The events you define are in S, E(1)[x(1),t] and E(2)[x(2),t]. Do the same thing as above.
Consider the numbers as indexes.
use soft words and hard arguments

pmb_phy

In general t is not a function of x. That happens on certain occasions such as a particle moving at constant velocity. But for particles which increase speed from 0 at x = 0 and then later decrease in speed, turns around and finally reaches x = 0 again. In this case t is not a true function of x since a function must be single valued and in the example I gave you t has two values for which x = 0. Thus t(x) is multivalued.

Pete

MathematicalPhysicist

thanks guys.
what about question three?

bernhard.rothenstein

I think you should state it with more details

MathematicalPhysicist

what isnt clear there?

bernhard.rothenstein

3. in the inertial frame S' the standard lattice clocks all emit a 'flash' at noon. prove that in S this flash occurs on plane orthogonal to the x-axis and travlling in the positive x direction at speed c^2/v.

a. in which direction are the light signals emitted (supposed simultaneously in S')?
b.on or in the plane?

George Jones

Have have written down the question exactly as it appears in the book? If not, please do so.

Suppose that noon is taken as t' = 0 in the primed frame. Then, in the primed frame, the coordinates of an arbitrary flash are (t', x', y', z') = (0, A, B, C). What are the unprimed coordinates of an arbitrary flash?

MathematicalPhysicist

well this is exactly what is written in the book, i guess my only other option is to ask rindler via email what he meant in this question.

pervect

Try looking at http://en.wikipedia.org/wiki/Relativ...ransformations

the diagram of the "line of simultaneity". Specifically http://en.wikipedia.org/wiki/Image:R...multaneity.png

The way I interpret the question, Rindler is talking about the set of events that are simultaneous in S (he says "at noon", I read "simultaneous"), and how they appear in frame S'. The Wiki article addresses the same question with two of the spatial dimensions suppressed.

George Jones

I'm trying to lead you to the answer; I just wondered whether the question was phrased a little differently in the book.

Can you answer the question I posed in my previous post?

MathematicalPhysicist

well in this case we only need to use this equation:
[tex]t'=\gamma*t(1-v*U/c^2)[/tex] where U is the velocity of flash and v is the velocity of S', at t'=0 we would have that U=c^2/v, but how would i prove that flash occurs on plane orthogonal to the x-axis?
well if it were to occur at a plane not orthogonal to the x axis of S, then it will not be orthogonal to the S' system.

George Jones

The question is not looking for the speed at which light spreads from a flash point; this is what's confusing about the question. As pervect noted, the question is about simultaneity.

Consider a bunch of cameras, one at each point in space (not spacetime) for S'. At t' = 0, all the camera flashes go off simultaneously for S'. The collection of events that represents the camera flashes going off is then

[tex]F = \{(t', x', y', z') = (0, A, B, C) \},[/tex]

where [itex]A[/itex], [itex]B[/itex], and [tex]C[/itex] are arbitrary real numbers.

What does this collection of events look like in the frame of S?

Assume that S and S' are related by a Lorentz transformation along the x-axis in the usual way. Apply a Lorentz transformation to the collection of events that represents the flashes going off gives

[tex]F = \left{ \left( t, x, y, z \right) = \left( \frac{v}{c^2} \gamma A, \gamma A, B, C \right) \right}.[/tex]

Using

[tex]t = \frac{v}{c^2} \gamma A[/tex]

gives

[tex]F = \left( t, \frac{c^2}{v} t, B, C \right) \right}.[/tex]

This indicate that all the flashes that occur simultaneously in S at time t occur in space at fixed [itex]x = (c^2/v) t[/itex] and at arbitrary [itex]y[/itex] and [itex]z[/itex]. For S, this is a spatial plane orthogonal to the x-axis.

Now, consider two times, [itex]t_1[/itex] and [itex]t_2[/itex], for S, with [itex]t_1 < t_2.[/itex] At time [itex]t_1[/itex], a bunch of flashes go off simultaneously in the spatial plane [itex]x = c^2/v t_1;[/itex] at time [itex]t_2[/itex], a bunch of flashes go off simultaneously in the spatial plane [itex]x = c^2/v t_2.[/itex] The spatial distance between the planes divided by difference in times gives that the "plane of flashes" propagates with speed [itex]c^2/v.[/itex]

bernhard.rothenstein

Thanks for bringing light in the statement of the problem. I think that we could state it as (the introduction of the light signals is confusing): (g stands for gama and b for beta)
Consider the events E'(0.r',p') in a two space dimensions approach using polar coordinates. Using the LT we obtain that one of those events is defined in S by the polar coordinates (r,p)
r=r'g[1-bb(sinp')^2]^1/2 (1)
tgp=tgp'/g[/I] (2)
Equation shows that if the events E' are located in S' on a normal on the O'X' axis the same events are located in S are located on a normal on the same axis.
Consider that the events E' take place in S' on a given curve say on the circle r'=R(0). Detected from S they take place on the curve
r=R(0)g[1-bb(sinp')^2]^1/2 . (3)
The problem can be extended.
Thanks for giiving me the opportunity to spend some pleasant time on an interesting problem. Am I correct?
Regards
Bernhard