'Speed of Light' Thought Experiment

[Thomas2]

Consider the following situation: a car is moving away from a stationary observer with velocity V. At distance S it triggers a contact in the road which sends a light signal back to the observer from the moving car. The question is how long will it take for the signal to reach the observer ? (The time could be measured by having synchronized clocks at distance S (the trigger point) and at the observer, the former being stopped when the car passes the contact (and emits the light signal), the latter when the light signal arives at the observer; the difference of the clock readings is then the light travel time).

 

[James R]

The light signal has to travel distance S at speed c, so the time of arrival of the signal is S/c, right?

Of course, the question of how the trigger manages to start the observer's clock instantaneously is a potential problem, but I assume that's not important.

 

[Thomas2]

Just to avoid confusion: the light signal is sent out from the moving car (I have clarified the opening post in this respect).

 

[James R]

I already assumed the light signal was sent from the moving car, so my previous answer stands.

Another problem is that you haven't specified the reference frame in which the observer's clock is started simultaneously with the emission of the light signal. My answer assumes the emission and the clock start are simultaneous in the observer's frame.

What are you having trouble with?

 

[Thomas2]

Another problem is that you haven't specified the reference frame in which the observer's clock is started simultaneously with the emission of the light signal. My answer assumes the emission and the clock start are simultaneous in the observer's frame.

I have modified the example in my opening post somewhat so that there are no 'simultaneity' issues with the timing procedure.

What are you having trouble with?

OK, if you don't have any problems with it and your suggestion is the time is S/c then we would agree (the light travel time depends only on S but not on the velocity V).
But one can consider the whole thing from the viewpoint of the car as well : when passing the contact at distance S from the observer, the car emits the light signal. The observer is receding now with velocity V, but still the signal should reach him within a time S/c despite the fact that the distance has increased in the meanwhile from S to S*(1+V/c).

 

[Janus]

I have modified the example in my opening post somewhat so that there are no 'simultaneity' issues with the timing procedure.

OK, if you don't have any problems with it and your suggestion is the time is S/c then we would agree (the light travel time depends only on S but not on the velocity V).
But one can consider the whole thing from the viewpoint of the car as well : when passing the contact at distance S from the observer, the car emits the light signal. The observer is receding now with velocity V, but still the signal should reach him within a time S/c despite the fact that the distance has increased in the meanwhile from S to S*(1+V/c).

No, from the car's viewpoint, the light will take

\frac{S\sqrt{1-\frac{V^2}{c^2}}}{c-V}

to reach the observer, assuming that S is the proper distance between the contact and observer as measured by the observer.

 

[RandallB]

Wow when & how did S to Observer get longer from the car view??
I thought it got shorter from the moving view.
Plus time runs slower in the moving car.
The little shorter length divided by the shorter time still = c

What is the speed (c-V)
Is there something going near the speed of light here?

RB

 

[Janus]

Wow when & how did S to Observer get longer from the car view??
I thought it got shorter from the moving view.

Where did anyone say differently?

Plus time runs slower in the moving car.
The little shorter length divided by the shorter time still = c

Time runs slower for the car as measured from the frame of the contact and observer, but at this point we are talking about the frame of the Car, in which the time oif the car runs normally.

What is the speed (c-V)
Is there something going near the speed of light here?

RB

(c-V) is the difference between the speed of light and the relative speed between the car and observer. The car sees the light travel at c wrt itself at c and the observer recede at V. Thus the "apparent" relative speed between the light and observer according to the car is (c-V), Divide this into the distance that the car measures between contact and observer and you get how long it takes from the car's frame for the light to "catch up" to the "receding" observer.

 

[James R]

The travel time for the light will be different according to the ground-based observer and a car-based observer.

The time for the ground observer is just S/c, as I said before.

In the car's reference frame, the distance S is contracted to a distance of

S\sqrt{1-(v/c)^2}

where v is the car's speed, and c is the speed of light. Therefore, the time taken, according to the car, is the above distance divided by c.

It is NOT divided by (c-v), as Janus said, since the car observer still sees the light traveling at c, despite the car's motion.

 

[Thomas2]

I should have made the reversal of the situation differently:
assume that the light signal is sent from the observer to the car rather than the other way around (the signal is sent out (and the observer clock stopped) when the car is at distance S and the car stops one of a number of synchronized clocks alongside the road when the signal is received).
Since it cleary can not make any difference for the light travel time whether you send a signal from A to B or B to A (even if both are moving relatively to each other) the travel time should then still be S/c here.

 

[Doc Al]

I should have made the reversal of the situation differently:
assume that the light signal is sent from the observer to the car rather than the other way around (the signal is sent out (and the observer clock stopped) when the car is at distance S and the car stops one of a number of synchronized clocks alongside the road when the signal is received).

Is this your setup: A ground observer at location x = 0. When a car traveling at speed v passes position x = S, the ground observer initiates a light signal from x = 0 to the car. The signal reaches the car when the car is at a position x = S' (where S' > S). (All distances measured in the ground frame.)

Since it cleary can not make any difference for the light travel time whether you send a signal from A to B or B to A (even if both are moving relatively to each other) the travel time should then still be S/c here.

Huh? How could it not make a difference? If the signal is sent from ground observer to car, it must travel a distance S'. But if sent from car to ground observer, it need only travel S. (All distances measured in the ground frame.)

 

[Thomas2]

Huh? How could it not make a difference? If the signal is sent from ground observer to car, it must travel a distance S'. But if sent from car to ground observer, it need only travel S. (All distances measured in the ground frame.)

The question was not at what distance the car is when the light signal reaches it, but what the clock shows at this moment.
Your conclusion that the larger distance S'>S is also associated with a larger time T'>T is based on the usual velocity addition rule, which however (as I am sure you are aware) can not be applied to light signals (any such expression as "c-v" contradicts the invariance property for the propagation of light). Only a velocity independent travel time T'=T= S(t=0)/c would be truly independent of any motion of emitter or observer (as the invariance principle for light requires).

 

[Doc Al]

The question was not at what distance the car is when the light signal reaches it, but what the clock shows at this moment.
Your conclusion that the larger distance S'>S is also associated with a larger time T'>T is based on the usual velocity addition rule, which however (as I am sure you are aware) can not be applied to light signals (any such expression as "c-v" contradicts the invariance property for the propagation of light). Only a velocity independent travel time T'=T= S(t=0)/c would be truly independent of any motion of emitter or observer (as the invariance principle for light requires).

Nonsense. The car moves, so of course it takes longer (as measured by the ground observer) for a signal from the ground observer to the reach the car than it would for a signal from the car to reach the ground observer. (I assume that in both scenarios the car is the same distance from the ground observer when the signal is initiated.)

And what makes you think that figuring out the distance involves some kind of "velocity addition rule" or that having an expression of "c-v" somehow contradicts the invariance of the speed of light?? Get a grip! In figuring out the time of travel of the signal, both frames would make use of the invariance of light speed. Show me where someone is claiming otherwise.

It's trivial. If the ground observer measures the time for a light signal from him to the car he must start with the distance the light must travel in that time, thus: ct = S + vt, so t = S/(c-v).

Of course, if the signal went from car to ground observer, it would only take t = S/c (according to the ground observer), since the car's motion would not affect the distance the light has to travel.

 

[Thomas2]

It's trivial. If the ground observer measures the time for a light signal from him to the car he must start with the distance the light must travel in that time, thus: ct = S + vt, so t = S/(c-v).

This is the formula you would use to take into account the relative velocity of normal objects: if you have one object moving with velocity v1 and a second moving with v2 (starting when the other object is at S), you would indeed have v2*t = S + v1*t if you want to find the time when both have the same location, but this is the usual vectorial velocity addition (Galilei transformation) which can not be applied to light signals, i.e. you can not set v2=c here.

Of course, if the signal went from car to ground observer, it would only take t = S/c (according to the ground observer), since the car's motion would not affect the distance the light has to travel.

As I said already, the time for the signal to get from A to B should always be equal to the time to get from B to A, whether it is for normal objects or light (this should be a straightforward consequence of the principle of relativity and the isotropy of space and time).
For light signals, a velocity dependence appears only in form of the Doppler shift of the light frequency but it should not appear in form of different light travel times.

 

[pervect]

As I said already, the time for the signal to get from A to B should always be equal to the time to get from B to A, whether it is for normal objects or light

But if I'm understanding this problem correctly, there are three points involved, A,B, and C - not two points. So I don't see how this comment is relevant.

Let A be the fixed observer, and B be the car, and S be a bump in the road.


In one case we have


A--------------B-------C
---------------S

When the car B reaches point S, a signal is sent from point A towards B. This has to be pre-arranged based on knowledge of how fast the car is moving, the car reaching point B cannot cause a signal to be sent from point A instantaneously.

The signal reaches the car at some later point C after the car has moved.


In the second case we have


A----------------B
-----------------S

When car B reaches point S, it causes the car to emit a signal. This signal travels from point B back to point A. Here, there is an actual cause and effect relatinoship between the car being at point S and the signal being emitted. And there is no point C, there is only point A and point B.

The time it takes for light to travel from point B to point A is not the same as it takes light to travel from point A to point C.

 

[Doc Al]

It's trivial. If the ground observer measures the time for a light signal from him to the car he must start with the distance the light must travel in that time, thus: ct = S + vt, so t = S/(c-v).

This is the formula you would use to take into account the relative velocity of normal objects: if you have one object moving with velocity v1 and a second moving with v2 (starting when the other object is at S), you would indeed have v2*t = S + v1*t if you want to find the time when both have the same location, but this is the usual vectorial velocity addition (Galilei transformation) which can not be applied to light signals, i.e. you can not set v2=c here.

Sorry, but this has nothing whatsoever to do with addition of velocities. Both the speed of light (v2 = c) and the speed of the car (v1 = v) are measured in the same frame. (It is just an application of "distance = speed x time"; as long as all measurements are made from the same frame, no problem.)

Of course, if the signal went from car to ground observer, it would only take t = S/c (according to the ground observer), since the car's motion would not affect the distance the light has to travel.

As I said already, the time for the signal to get from A to B should always be equal to the time to get from B to A, whether it is for normal objects or light (this should be a straightforward consequence of the principle of relativity and the isotropy of space and time).

Just because you keep saying it, doesn't make it true. See pervect's post illustrating that the distance traveled is different in each case.

For light signals, a velocity dependence appears only in form of the Doppler shift of the light frequency but it should not appear in form of different light travel times.

The speed of light is invariant, but the time it takes for light to travel depends on the distance involved.

 

[Thomas2]

But if I'm understanding this problem correctly, there are three points involved, A,B, and C - not two points. So I don't see how this comment is relevant.

Let A be the fixed observer, and B be the car, and S be a bump in the road.


In one case we have


A--------------B-------C
---------------S

When the car B reaches point S, a signal is sent from point A towards B. This has to be pre-arranged based on knowledge of how fast the car is moving, the car reaching point B cannot cause a signal to be sent from point A instantaneously.

The signal reaches the car at some later point C after the car has moved.

No, there are in fact only two points involved here: the observer A and the car B (betweeen which the light signal is exchanged). Point C only enters into the picture if you assume that the light signal is an independent entity that can be classically described as having a velocity c with regard to A and c-v with regard to B (see also my reply to Doc Al below).

 

[Thomas2]

Sorry, but this has nothing whatsoever to do with addition of velocities. Both the speed of light (v2 = c) and the speed of the car (v1 = v) are measured in the same frame. (It is just an application of "distance = speed x time"; as long as all measurements are made from the same frame, no problem.)

This is exactly the point: by plotting the velocity of the car v and the velocity of the light signal c in the same frame, you are implying that the light signal moves relative to the car with velocity c-v, which contradicts the invariance of c.
As I said already, it is not the question here when a light signal in the observers frame reaches a point at which the car happens to be as well at that moment, but when the light signal reaches the car in its frame. These two things would only be identical if you have the usual vectorial velocity addition (which however does not hold for light signals). The usual definition of 'speed'=distance/time does simply not apply here.

 

[Doc Al]

No, there are in fact only two points involved here: the observer A and the car B (betweeen which the light signal is exchanged).

The car moves!

Point C only enters into the picture if you assume that the light signal is an independent entity that can be classically described as having a velocity c with regard to A and c-v with regard to B (see also my reply to Doc Al below).

Nonsense! Both ground observers and car observers measure the speed of light to be c with respect to themselves. (That's what invariance of light speed means!) That "c-v" is not the speed of light as measured in the car frame. That "c-v" is not the speed of any "thing", it's just the rate at which the ground observer sees the light gain on the car. You do realize that no matter how fast the car moves, the light will catch up... don't you?

 

[Doc Al]

Sorry, but this has nothing whatsoever to do with addition of velocities. Both the speed of light (v2 = c) and the speed of the car (v1 = v) are measured in the same frame. (It is just an application of "distance = speed x time"; as long as all measurements are made from the same frame, no problem.)

This is exactly the point: by plotting the velocity of the car v and the velocity of the light signal c in the same frame, you are implying that the light signal moves relative to the car with velocity c-v, which contradicts the invariance of c.

Nonsense. See my post above to learn what "c-v" actually means.

As I said already, it is not the question here when a light signal in the observers frame reaches a point at which the car happens to be as well at that moment, but when the light signal reaches the car in its frame.

In the car's frame, the light travels a distance of S/\gamma, so the travel time is just S/(\gamma c) according to the car frame.

These two things would only be identical if you have the usual vectorial velocity addition (which however does not hold for light signals).

Nonsense. If the signal is now sent from car to ground observer, then according to the car frame the distance the light must travel is: ct = S/\gamma + vt. So, in that case, the travel time according to the car frame is S/[\gamma (c-v)].

The usual definition of 'speed'=distance/time does simply not apply here.

Nonsense. You just don't know what you're doing, I'm afraid.

 

[pervect]

No, there are in fact only two points involved here: the observer A and the car B (betweeen which the light signal is exchanged). Point C only enters into the picture if you assume that the light signal is an independent entity that can be classically described as having a velocity c with regard to A and c-v with regard to B (see also my reply to Doc Al below).

Sorry, that's not how it works. Refering to the diagram below, let us assign coordinates to analyze the problem. We will assign the coordinates t=0 and x=0 to point A. We will assign the coordinates t=0 and x=S to point B.

A--------------B-------C
---------------S

Given that the speed of light is 'c', we can describe the position of the light signal as a function of time via an equation.

The general equation for ANY moving body moving at a constant velocity will be a linear equation

position = velocity * time + offset

The value of offset is the starting position of the body, because at t=0, position = constant.

The velocity of the object is defined as the slope (derivative) of the straight line which plots the position as a function of time.

Using this general result, we can then write the equation which describes the motion of the light as a function of time:

light(t) = c*t (eq 1)

We can do this because the velocity of light is 'c', and because the position of the light at t=0 is 0, setting the value of the offset in the general equation.

We can in a similar manner describe the position of the car as a function of time

car(t) = S + v*t. (eq 2)

We can do this because we know that at time t=0, the car is at x=S, and that it moves at a velocity v.

We can find the time and position at which the car meets the light beam by solving the equation for the value of time at which

car(t) = light(t)

We substitute in the results from eq(1) and eq(2)

S+v*t = c*t

This is the equation that Doc Al derived. I am just deriving it more slowly and explaining it in full detail, one small step at a time.

We can re-write the equation

S = c*t - v*t = (c-v)*t

thus t = S / (c-v)

We can find the position of the car and light at time t, and insure that they are the same, this will also solve for the location of point 'C".

car = S+v*(S/(c-v)) = S*(c-v)+v*S / (c-v) = S*c - S*v + v*s / c-v =
S*c/(c-v)

light = c*S/(c-v)

Thus we confirm that the position of both the car and the light signal at time

t = S/(c-v) is the positoin c*S/(c-v).

 

[Thomas2]

​

In the car's frame, the light travels a distance of S/\gamma, so the travel time is just S/(\gamma c) according to the car frame.

I am afraid I don't follow you here and I don't quite see how you want to make this consistent with your previous values obtained in the ground based frame.

Let's try to make the thought experiment even clearer by just sticking to the physical facts and avoiding any unfounded and inconsistent assertions:

An observer in a car is moving away from a ground based oberver (GBO) with velocity V. Assume that both carry synchronized clocks (these could have been started for instance through a contact when the car passed the GBO). Assume also that the car and the GBO are connected by two ropes, one fixed to the car and reeling off at the GBO, the other fixed to the GBO and reeling off at the car. Both ropes carry markings which indicate the length of rope that has reeled off (i.e. the distance between the both). When the reel at the GBO has reached the marking S1, a contact triggers a light signal to be sent after the car. The clock at the GBO at this moment reads T1=S1/V . Since obviously the same amount of rope has reeled off at the car as well, the marking at the latter does at this moment also read S1 and since the rope has reeled off with the same speed, the car-clock does also shows T1'=T1=S1/V. Now, the observer in the car knows (by agreement) that at this moment the GBO (who is receding with velocity V relatively to him) sends a light signal to him which should reach him at his time T2'=T1'+S1/c = T1 +S1/c as the speed of light does not depend on the relative velocity of the GBO. For the observer in the car the signal has therefore taken an interval T2'-T1'=S1/c to reach him. During this time the rope has reeled off by a further S2'-S1'=V*(T2'-T1')=S1*V/c and reads therefore the distance S2'=S1*(1+V/c). At this moment (when the light signal reaches the car) the car-reel blocks and stops the relative motion of the two. Now, for the GBO the same amount of rope must have reeled off, so he knows that the light signal reached the car when the latter was at a distance S2=S2'=S1*(1+V/c) and since for him the rope also reeled off with velocity V, his clock also shows the time T2=S2/V = S1*(1/V+1/c)=T1+S1/c i.e. T2-T1=S1/c.

 

[Thomas2]

Given that the speed of light is 'c', we can describe the position of the light signal as a function of time via an equation.

S+v*t = c*t

As I said already, this implies that the propagation of light is in no way different from the propagation of material bodies, which is not the case.

 

[Doc Al]

I am afraid I don't follow you here and I don't quite see how you want to make this consistent with your previous values obtained in the ground based frame.

Where's the problem? All the values and calculations that I presented are perfectly consistent.

Let's try to make the thought experiment even clearer by just sticking to the physical facts and avoiding any unfounded and inconsistent assertions:

The thought experiment is already quite clear. Introducing mechanical contrivances such as "reeling ropes" will only serve to complicate things. But it won't change the answer! I suspect that what you are calling "unfounded and inconsistent assertions" are just the basic facts of life according to relativity, which you seem to neither understand nor accept.

An observer in a car is moving away from a ground based oberver (GBO) with velocity V. Assume that both carry synchronized clocks (these could have been started for instance through a contact when the car passed the GBO).

The two clocks can read the same at the moment they pass each other. But they will not remain synchronized.

Assume also that the car and the GBO are connected by two ropes, one fixed to the car and reeling off at the GBO, the other fixed to the GBO and reeling off at the car. Both ropes carry markings which indicate the length of rope that has reeled off (i.e. the distance between the both).

The markings just serve to indicate the proper length of the rope when it was measured at rest. If the car is going to be "reeling off" the rope, then the rope will be moving with respect to the car and at rest with respect to the GBO.

When the reel at the GBO has reached the marking S1, a contact triggers a light signal to be sent after the car. The clock at the GBO at this moment reads T1=S1/V .

Right, because the rope is at rest with respect to the GBO, so the GBO agrees that it has a length of S1.

Since obviously the same amount of rope has reeled off at the car as well, the marking at the latter does at this moment also read S1 and since the rope has reeled off with the same speed, the car-clock does also show T1'=T1=S1/V.

Nope. Yes, "the same amount of rope" has reeled off, but the two frames disagree on its length. Since the rope is now moving with respect to the car, the car frame will measure its length as S1' = S1/\gamma, so the car clock reads T1' = S1/(\gamma v). So T1' does not equal T1.

Now, the observer in the car knows (by agreement) that at this moment the GBO (who is receding with velocity V relatively to him) sends a light signal to him which should reach him at his time T2'=T1'+S1/c = T1 +S1/c as the speed of light does not depend on the relative velocity of the GBO.

Correcting your earlier mistake, we get: T2' = T1' + S1'/c.

For the observer in the car the signal has therefore taken an interval T2'-T1'=S1/c to reach him.

Nope, T2' - T1' = S1'/c = S1/(\gamma c).

During this time the rope has reeled off by a further S2'-S1'=V*(T2'-T1')=S1*V/c and reads therefore the distance S2'=S1*(1+V/c).

The proper length of the additional rope reeled off is S2 - S1 = V(T2 - T1). Of course, the car will measure this as being contracted by a factor of 1/\gamma.

At this moment (when the light signal reaches the car) the car-reel blocks and stops the relative motion of the two. Now, for the GBO the same amount of rope must have reeled off, so he knows that the light signal reached the car when the latter was at a distance S2=S2'=S1*(1+V/c) and since for him the rope also reeled off with velocity V, his clock also shows the time T2=S2/V = S1*(1/V+1/c)=T1+S1/c i.e. T2-T1=S1/c.

Nope. Do it over, correctly, and you'll get the same results as already discussed.

I highly recommend dropping this "reeling rope" version of this exercise, since it introduces irrelevant complications. (For example, the rope needs to be accelerated in order to be "reeled out".)

 

[Doc Al]

Given that the speed of light is 'c', we can describe the position of the light signal as a function of time via an equation.

S+v*t = c*t

As I said already, this implies that the propagation of light is in no way different from the propagation of material bodies, which is not the case.

What are you talking about? That equation merely uses the fact that the speed of light is c. Where's the problem?

 

[pervect]

This is exactly the point: by plotting the velocity of the car v and the velocity of the light signal c in the same frame, you are implying that the light signal moves relative to the car with velocity c-v, which contradicts the invariance of c.
As I said already, it is not the question here when a light signal in the observers frame reaches a point at which the car happens to be as well at that moment, but when the light signal reaches the car in its frame. These two things would only be identical if you have the usual vectorial velocity addition (which however does not hold for light signals). The usual definition of 'speed'=distance/time does simply not apply here.

You seem to be confused about relativity, and unwilling to listen to an explanation of why you are confused.

The definition "distance = speed * time + offset always applies in any given coordinate system.

The addition-of-velocities problem involves multiple coordinate systems, and is in noways in conflict with the idea that distance = speed * time + offset in any single coordinate system.

 

[Thomas2]

You seem to be confused about relativity, and unwilling to listen to an explanation of why you are confused.
The definition "distance = speed * time + offset always applies in any given coordinate system.
The addition-of-velocities problem involves multiple coordinate systems, and is in noways in conflict with the idea that distance = speed * time + offset in any single coordinate system.

It seems it is you who is unwilling to listen: I told you already several times that you cannot calculate the offset between the positions of a light signal and a moving material object in the usual way. Otherwise you would have in your ground based reference frame x(t)=c*t-v*t and hence the relative velocity dx(t)/dt=c-v which contradicts the invariance of c.

 

[Thomas2]

The thought experiment is already quite clear. Introducing mechanical contrivances such as "reeling ropes" will only serve to complicate things.

It does not complicate things but merely enforces the required basic definitions without which any such consideration is meaningless (in fact, having left out these definitions is the reason for your frame dependent results in the first place).

The definition here is that distance of the car is indicated by the marking on the rope reeling off at the ground based observer. Because the distance S between to points A and B is commutative (S(A,B)=S(B,A)), the marking on the rope reeling off from the car must be identical to this at any moment. Also, you know that the relative velocity between two objects is commutative (i.e V(A,B)=V(B,A) )and, as a logical consequence, the related times T=S/V must also be identical (i.e. T(A,B)=T(B,A)), with the rest of the argument as given in my post above.

I can not see how somebody could possibly disagree with this, so I would suggest that we end our discussion at this point and leave it up to the reader to make up his own mind in this matter.

 

[James R]

You just need to be careful to get the speed right when you switch reference frames. You can't just add speeds in a naive way. You need to use the relativistic velocity addition formula if you want to compare speeds in different reference frames.

 

[Doc Al]

It seems it is you who is unwilling to listen: I told you already several times that you cannot calculate the offset between the positions of a light signal and a moving material object in the usual way. Otherwise you would have in your ground based reference frame x(t)=c*t-v*t and hence the relative velocity dx(t)/dt=c-v which contradicts the invariance of c.

You apparently have no clue what the invariance of c means! In fact, the invariance of light speed requires that one calculate the offset in the manner pervect (and I) explained. You just mistake the "separation rate" of light and object as seen from the frame in which the object moves as being a "relative velocity" of light with respect to the moving object. Not so. The speed of light measured by the moving object is still c.

Here's another example for you to ponder. Shine a lightbeam to the left. It moves at speed c according to us on the ground. In one second, it will travel 1 light-sec = 3E8 meters. Now have a rocket moving to the right at speed 0.99c. The light and the rocket separate according to ground observers at the rate of 1.99c. But the speed of the light as seen by an observer on the rocket is still just c. Everyone will see the speed of light as just c.

This is basic stuff. Pick up any book on relativity.