# A few questions from introduction to sr by rindler.

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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.

robphy
Homework Helper
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are you familiar with 4-vectors? spacetime diagrams?

Gold Member
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
Homework Helper
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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.

lorentz transformation exercise

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.

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

im familiar with spacetime diagrams, where t is a function of x.
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

Gold Member
thanks guys.

question 3

thanks guys.
I think you should state it with more details

Gold Member
what isnt clear there?

question 3

what isnt clear there?

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
Staff Emeritus
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what isnt clear there?

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?

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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
Staff Emeritus
thanks guys.

Try looking at http://en.wikipedia.org/wiki/Relativity_of_simultaneity#Lorentz_transformations

the diagram of the "line of simultaneity". Specifically http://en.wikipedia.org/wiki/Image:Relativity_of_simultaneity.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
Staff Emeritus
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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.

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?

Gold Member
well in this case we only need to use this equation:
$$t'=\gamma*t(1-v*U/c^2)$$ 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
Staff Emeritus
Gold Member
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

$$F = \{(t', x', y', z') = (0, A, B, C) \},$$

where $A$, $B$, and $$C[/itex] are arbitrary real numbers. What does this collection of events look like in the frame of S? Assume that S and S' are related by a Lorentz transformation along the x-axis in the usual way. Apply a Lorentz transformation to the collection of events that represents the flashes going off gives [tex]F = \left{ \left( t, x, y, z \right) = \left( \frac{v}{c^2} \gamma A, \gamma A, B, C \right) \right}.$$

Using

$$t = \frac{v}{c^2} \gamma A$$

gives

$$F = \left( t, \frac{c^2}{v} t, B, C \right) \right}.$$

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

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

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events simultaneous in S'

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

$$F = \{(t', x', y', z') = (0, A, B, C) \},$$

where $A$, $B$, and $$C[/itex] are arbitrary real numbers. What does this collection of events look like in the frame of S? Assume that S and S' are related by a Lorentz transformation along the x-axis in the usual way. Apply a Lorentz transformation to the collection of events that represents the flashes going off gives [tex]F = \left{ \left( t, x, y, z \right) = \left( \frac{v}{c^2} \gamma A, \gamma A, B, C \right) \right}.$$

Using

$$t = \frac{v}{c^2} \gamma A$$

gives

$$F = \left( t, \frac{c^2}{v} t, B, C \right) \right}.$$

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

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

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

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thanks for the help.
i have just another query:
we have two photons which travel along the x axis of S with constant distance L between them.
prove that in S' the distance between them is L*(c+v)^0.5/(c-v)^0.5.
now the way i sloved is as follows:
(x1,0) and (x2,0) are the coordinates in S where x2>x1, then we have
x1=ct-L x2=ct and in S' it will be $$x1'=\gamma(ct-L-vt)$$ $$x2'=\gamma(ct-vt)$$ when $$x2'-x1'=\gamma(L)$$ but this is ofcourse not the same as what i need to prove, where did i go wrong?

solving the proposed problem

thanks for the help.
i have just another query:
we have two photons which travel along the x axis of S with constant distance L between them.
prove that in S' the distance between them is L*(c+v)^0.5/(c-v)^0.5.
now the way i sloved is as follows:
(x1,0) and (x2,0) are the coordinates in S where x2>x1, then we have
x1=ct-L x2=ct and in S' it will be $$x1'=\gamma(ct-L-vt)$$ $$x2'=\gamma(ct-vt)$$ when $$x2'-x1'=\gamma(L)$$ but this is ofcourse not the same as what i need to prove, where did i go wrong?

I have found in a paper I studied long time ago that in order to solve a problem it is advisable to start in the reference frame where it is the simplest and to find out there the significant events. Transform them via the LT to another inertial reference frame. So we start in S where the events generated by the two photons (light signals) are E(1)[a,a/c) and E(2)[a+L,(a+L)/c). Detected from S' the space coordinates of the two events are
x'(1)=ga(1-b) and x'(2)=g(a+L)(1-b)
and so the distance between the two events is
x'(2)-x'(1)=Lsqrt[(1-b)/(1+b)]
probably the desired result? Do you see some analogy with the formula whixh accounts for the Doppler Effect? Please give me the exact statement of the problem proposed by Rindler and its quotation, because my edition is quite old.

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ok, i can see that you wrote down what is t specifically, but still my question is why my approach yields a different answer than the expected one.

p.s
why do you folks think that im not giving you already the full question from the text, i.e a quoted question?

Gold Member
ok i see my mistake.

quotation

ok, i can see that you wrote down what is t specifically, but still my question is why my approach yields a different answer than the expected one.

p.s
why do you folks think that im not giving you already the full question from the text, i.e a quoted question?
what i asked for was the exact quotation of Rindler's book (name,title, page).

Gold Member
it's called "Introduction to Special Relativity", i think have the second edition of this book, and thus far the questions were from chapter one.

superluminal velocity

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

As I see the speed at which the "plane of flashes" propgates is superluminal. Have you an explanation for that fact. cc/V reminds me the concept of phase velocity w (wV=cc). Is w the phase velocity of the light signals that perform the synchronization of the involved clolcks of I and I' respetively?
soft words and hard arguments

The answer to the two photon distance question can be obtained more simply and fully by emitting the photons from the same point in S, a time L/c apart. From S' the clocks in S run slow so the time in S' will be gamma.L/c corresponding to a distance gamma.L. During this time the point of emission in S will have moved a distance gamma.L.v/c so the total distance from S' between the photons will be simply gamma.L(1 +/- v/c), depending on which direction S' is moving. Thus there are two answers - the plus sign for S' moving towards, and minus for S' moving opposite to, photon direction. The answer apparently given by Rindler corresponds to the plus sign but if S' is moving opposite to photon propagation the minus sign gives a smaller distance gamma.L(1 - v/c) which is equal to L.sqrt[(1 - v/c)/(1 + v/c)].

superluminal non physical motion

The question is not looking for the speed at which light spreads from a flash point; this is what's confusing about the question. $$F = \left( t, \frac{c^2}{v} t, B, C \right) \right}.$$

Now, consider two times, $t_1$ and $t_2$, for S, with $t_1 < t_2.$ At time $t_1$, a bunch of flashes go off simultaneously in the spatial plane $x = c^2/v t_1;$ at time $t_2$, a bunch of flashes go off simultaneously in the spatial plane $x = c^2/v t_2.$ The spatial distance between the planes divided by difference in times gives that the "plane of flashes" propagates with speed $c^2/v.$
The speed cc/V is superluminal but the supposed motion is not physical. A Lorentz transformation relates the spece-time coordinates of two events which take place at the same point in space and performing it the point in space does not move!
Regards

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:
..............
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.
..............
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?
Of course you're right that the flashes cannot possibly travel at speed c.c/v. If you have quoted the problem correctly it is clearly misleading since as v approaches zero the velocity would become infinite. Rindler is either confused or being confusing. Two different simultaneous flashes along x' in S' would appear separated in S by a time x.v/(c.c) over a distance x, so if one 'imagined' they were the same flash propagating along x the velocity might 'appear' to be c.c/v. The illusion would be spoiled by the arrival of the earlier flash at normal light speed a moment later.
It's trivially like a "mexican wave" at a football match, which can "travel" at any speed you like, even superluminal.

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quotation

it's called "Introduction to Special Relativity", i think have the second edition of this book, and thus far the questions were from chapter one.

I remind you that the standard way to quote a book is
Author Title (Editor year) page
Helping me please quote the source in that way. The problem has a high pedagogical potential illustrating a superluminat but not physical motion.
Thanks

Gold Member
Wolfgang Rindler, Introduction to Special Relativity, second edition,page 21.( i think the editor year is 1995).

I have found in a paper I studied long time ago that in order to solve a problem it is advisable to start in the reference frame where it is the simplest and to find out there the significant events. Transform them via the LT to another inertial reference frame. So we start in S where the events generated by the two photons (light signals) are E(1)[a,a/c) and E(2)[a+L,(a+L)/c). Detected from S' the space coordinates of the two events are
x'(1)=ga(1-b) and x'(2)=g(a+L)(1-b)
and so the distance between the two events is
x'(2)-x'(1)=Lsqrt[(1-b)/(1+b)]
probably the desired result? Do you see some analogy with the formula whixh accounts for the Doppler Effect? Please give me the exact statement of the problem proposed by Rindler and its quotation, because my edition is quite old.

Hi,

I know this is a very old post, but the answer x'(2)-x'(1)=Lsqrt[(1-b)/(1+b)] is obviously wrong, as indicated by the correct solution: [..]prove that in S' the distance between them is L*(c+v)^0.5/(c-v)^0.5[..]

To obtain that, you have to pick a rest frame at first. If we take S' as a rest frame the correct Lorentztransformation is

$$x' = \gamma (x + vt)$$

and not

$$x' = \gamma (x - vt)$$ since S is moving.

Therefore we get:

$$\Delta x' = \gamma \Delta x (1 + \frac{v}{c})$$

with t=c*x.

Now we have
$$\Delta x = L = const.$$ in S and this gives the desired result in S':

$$\Delta x' = L\sqrt{\frac{c+v}{c-v}}$$.

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