[the_roosh (OP)]

Essentially, it is the light clock thought experiment but, instead of one observer on the platform, both are on trains moving relative to each other, and relative to the tracks.

The set-up is such that there are two points  on the tracks, A and B, which are equidistant from a mid-point M (as below):

========A==========M==========B=========

Alice is moving from left to right (A>B) and Bob in the opposite direction (B>A). Alice and Bob pass each other at point M and zero their clocks.

Is it possible for Alice to measure Bob's velocity relative to the tracks using her instruments?

Even if not, Alice measures her own velocity relative to the tracks (that Bob is running on) - getting a value of 0.4c - she measures the relative velocity between herself and Bob as 0.8c. It would seem that a measurement or calculation of 0.4c (for the relative velocity between Bob and the tracks) is a logical necessity in Alice's frame of reference.

Will Bob make similar meaurements/calculations for Alice's velocity relative to the tracks?

NOTE: there is no switching of reference frames here. Neither Alice nor Bob are trying to ascertain the measurements/co-ordinates of the others reference frame. Each simply makes measurements, using their own instruments, in their own inertial frame.

[evan_au]

she measures the relative velocity between herself and Bob as 0.8c

I'm afraid not.
0.4c + 0.4c ≠ 0.8c

At speeds that are a significant fraction of c, you cannot just do an arithmetic addition; you have to use the relativistic formula.

This becomes even more extreme when you consider bunches of protons in the LHC, which are travelling at just a smidgeon under c, in both directions around the ring. In their frame of reference, the oncoming bunch of protons is not travelling at (roughly) 2c, but scrapes in at just under 1c.

See: https://en.wikipedia.org/wiki/Velocity-addition_formula

I get 0.8c/(1+1/16) = 0.7529...c  as the speed Alice would measure for Bob (and vice-versa).

[the_roosh (OP)]

she measures the relative velocity between herself and Bob as 0.8c

I'm afraid not.
0.4c + 0.4c ≠ 0.8c

At speeds that are a significant fraction of c, you cannot just do an arithmetic addition; you have to use the relativistic formula.

This becomes even more extreme when you consider bunches of protons in the LHC, which are travelling at just a smidgeon under c, in both directions around the ring. In their frame of reference, the oncoming bunch of protons is not travelling at (roughly) 2c, but scrapes in at just under 1c.

See: newbielink:https://en.wikipedia.org/wiki/Velocity-addition_formula [nonactive]

Cheers Evan. I still need to get more familiar with the calculation for relativistic velocity addition (RVA) but it looks quite simple there.

I get 0.8c/(1+1/16) = 0.7529...c  as the speed Alice would measure for Bob (and vice-versa).

I'm not entirely sure what this figure represents though. Is it what Alice says that Bob's measurement "actually" is?

We're not really interested in that though. We're simply looking at things from Alice's reference frame, to begin with. Alice measures the relative velocity between herself and Bob to be 0.8c and she measures her speed relative to the tracks that Bob's train runs on as being 0.4c. She should therefore measure Bob's velocity relative to the tracks as being 0.4c, in her reference frame. If she cannot use her instruments to measure that value then she deduces it as a matter of logical necessity.

The issue might be when we consider things from Bob's perspective. He measures the relative velocity between him and Alice to be 0.8c also - using his equipment, in his reference frame (so no need for RVA there). What will Bob measure his own velocity relative to the tracks to be?

[Halc]

Essentially, it is the light clock thought experiment but, instead of one observer on the platform, both are on trains moving relative to each other, and relative to the tracks.

The set-up is such that there are two points  on the tracks, A and B, which are equidistant from a mid-point M (as below):

========A==========M==========B=========

Alice is moving from left to right (A>B) and Bob in the opposite direction (B>A). Alice and Bob pass each other at point M and zero their clocks.

Is it possible for Alice to measure Bob's velocity relative to the tracks using her instruments?

For starters, how might Alice measure her own speed relative to the tracks?  How is 0.4c arrived at?  The answer to that might give us a clue as to what is known and what isn't.  I might suggest shining a light to a mirror on the tracks and one on the other train and observe the blue shift of the reflected beam.  That's how the police do it.  I don't see the point of points A,M,B above.  We don't need to know about them to do this.

Even if not, Alice measures her own velocity relative to the tracks (that Bob is running on) - getting a value of 0.4c - she measures the relative velocity between herself and Bob as 0.8c. It would seem that a measurement or calculation of 0.4c (for the relative velocity between Bob and the tracks) is a logical necessity in Alice's frame of reference.

Surely you know better than this.  Evan points out this error.
I get .588c for Bob relative to the tracks given the above two measurements.

Will Bob make similar meaurements/calculations for Alice's velocity relative to the tracks?

Why not?  Is Alice special?  Bob can do the exact same thing and get .4c.

I'm afraid not.
0.4c + 0.4c ≠ 0.8c

I'm not entirely sure what this figure represents though. Is it what Alice says that Bob's measurement "actually" is?

Evan has added 0.4 to 0.4.

I based my calculation on this:

she measures the relative velocity between herself and Bob as 0.8c

0.8c. - 0.4c  = ~0.588c
Bob is going faster down the tracks for Alice to have measured those figures.

If she cannot use her instruments to measure that value then she deduces it as a matter of logical necessity.

She uses her instruments to get that figure.  None of it is assumed.

The issue might be when we consider things from Bob's perspective. He measures the relative velocity between him and Alice to be 0.8c also - using his equipment, in his reference frame (so no need for RVA there). What will Bob measure his own velocity relative to the tracks to be?

0.588c

[the_roosh (OP)]

For starters, how might Alice measure her own speed relative to the tracks?  How is 0.4c arrived at?  The answer to that might give us a clue as to what is known and what isn't.  I might suggest shining a light to a mirror on the tracks and one on the other train and observe the blue shift of the reflected beam.  That's how the police do it.

I suppose she measures the relative velocity between herself and the tracks in a similar manner to how she measures the relative velocity between herself and Bob. I was thinking about how police use the radar but also she might use the speedometer on her train.

Could she use the "police radar" to measure the velocity between Bob and the tracks?

  I don't see the point of points A,M,B above.  We don't need to know about them to do this.

Those are for a different purpose, but it is dependent on the first part.

Surely you know better than this.  Evan points out this error.
I get .588c for Bob relative to the tracks given the above two measurements.

I was thinking RVA would play a role but given that Alice is using her own instruments, in her own reference frame I was struggling to see why it should play a role.

The value of .588c for Bob's velocity relative to the tracks, is that the value that Alice's radar gun returns?

Will Bob make similar meaurements/calculations for Alice's velocity relative to the tracks?

Why not?  Is Alice special?  Bob can do the exact same thing and get .4c.

What will Bob measure his own velocity relative to the tracks to be?[/quote]0.588c
[/quote]
Just putting these two comments together bcos I'm confused by them. Does Bob also measures his own velocity relative to the tracks as 0.4c or 0.588c?

0.8c. - 0.4c  = ~0.588c
Bob is going faster down the tracks for Alice to have measured those figures.

This is the part I don't get. If Alice conducts all measurements within her own inertial frame, using her radar equipment, why does she apply RVA? She's not trying to determine the values that Bob gets in his intertial frame, she's doing everything in her own IF.

If she cannot use her instruments to measure that value then she deduces it as a matter of logical necessity.

She uses her instruments to get that figure.  None of it is assumed.[/quote]
How does her radar equipment measure a velocity of 0.4c between her and the tracks, 0.8c between her and Bob, but 0.588c between Bob and the tracks. Is it something to do with the path length of the light? How does her radar gun know to apply RVA for the measurement between Bob and the tracks?

[Halc]

I suppose she measures the relative velocity between herself and the tracks in a similar manner to how she measures the relative velocity between herself and Bob. I was thinking about how police use the radar but also she might use the speedometer on her train.

How might the train speedometer work?  That is a poor solution since she can't measure her speed relative to the Bob train that way.  I'm presuming a radar gun that works with relativistic speeds.

Could she use the "police radar" to measure the velocity between Bob and the tracks?

Only indirectly, as we have shown. One must do some calculations, but the radar gun wouldn't work in the first place (nor the speedometer) if calculations were not allowed.  The speedometer in your car is very much running calculations in order to display the car speed.

The value of .588c for Bob's velocity relative to the tracks, is that the value that Alice's radar gun returns?

No.  Alices's gun returns .4c if she points it at the trees, and .8c if she points it at the Bob train.  That was in your OP.  We're assuming she has a device that displays a speed here relative to the device (like they use to measure the speed of a baseball pitch), unlike police radar guns which display the speed of the target relative to the background.

Just putting these two comments together bcos I'm confused by them. Does Bob also measures his own velocity relative to the tracks as 0.4c or 0.588c?

The latter.  I think I said it multiple times in my posts.

Bob is going faster down the tracks for Alice to have measured those figures.

This is the part I don't get. If Alice conducts all measurements within her own inertial frame, using her radar equipment, why does she apply RVA? She's not trying to determine the values that Bob gets in his intertial frame, she's doing everything in her own IF.

You asked for Bob's speed relative to the tracks.  RVA is needed to determine that.  All speed is relative to something, but not relative to a third thing.  Bob's speed relative to the tracks is exactly that.  There is no additional 'Bob's speed relative to the tracks relative to some other frame'.  That's a 3-way relation that is something else.  I suppose it can be computed (0.4, simple subtraction), but I'm not sure what meaning you would find in it.  It certainly isn't what Bob would measure or what an observer on the tracks would measure.

How does her radar equipment measure a velocity of 0.4c between her and the tracks, 0.8c between her and Bob, but 0.588c between Bob and the tracks.

It doesn't measure any of those things.  It measures the frequency of reflected light, and all the rest is done by computations.  All 3 figures are the result of mathematics being performed.  I cannot think of a way to directly measure speed without performing a calculation along the way.  You seem to be pushing for a direct measurement of speed, but I don't know of one.  Perhaps you can suggest one.

Is it something to do with the path length of the light? How does her radar gun know to apply RVA for the measurement between Bob and the tracks?

I imagine the simple ones used by the police do not do the full computation and thus can only be used for non-relativistic speeds.  We're assuming the Alice-Cop is armed with the high end model here, and it would need to employ RVA to get an accurate speed for Bob.
Actual police radar measures but doesn't display the police speed relative to the background (she's not interested in her own speed), nor does it display Bob's measured speed relative to the moving police car.  Alice is interested in Bob's speed relative to the road/tracks, and it might need to determine this even if Bob is not moving straight at or away from the gun.  That's pretty complicated, but the gun is designed to display Bob's speed relative to the road.  That's all done with mathematics inside the device.  For the simple 1-dimensional case outlined in this post, the calculation is pretty simple, and yes, RVA needs to be employed to get an accurate figure.

[Janus]

I imagine the simple ones used by the police do not do the full computation and thus can only be used for non-relativistic speeds.  We're assuming the Alice-Cop is armed with the high end model here, and it would need to employ RVA to get an accurate speed for Bob.
Actual police radar measures but doesn't display the police speed relative to the background (she's not interested in her own speed), nor does it display Bob's measured speed relative to the moving police car.  Alice is interested in Bob's speed relative to the road/tracks, and it might need to determine this even if Bob is not moving straight at or away from the gun.  That's pretty complicated, but the gun is designed to display Bob's speed relative to the road.  That's all done with mathematics inside the device.  For the simple 1-dimensional case outlined in this post, the calculation is pretty simple, and yes, RVA needs to be employed to get an accurate figure.

Just to add to what Halc has said here.
a radar gun, such as used by Police calculates the velocity of an object relative to the gun by using the formula
v= cΔf/2f_s
where c is the speed of light, Δf is the difference in frequency between original transmission and reception, and f_s is the original transmission frequency.   This works well enough when v is small compared to c and is simple, as v is proportional to Δf.

The more correct and accurate equation is

v = c(1-(f_o/f_s)²)/2(1+(f_o/f_s)²)
where f_o  is the observed return frequency and f_s the original source frequency.

For comparison, a typical radar gun using the first formula and giving a velocity of 67.5 mph should return 67.5000135 mph using the accurate equation.  Hardly worth the extra work.

However, a standard radar gun using the first equation and giving an answer of v=0.25c  would be significantly inaccurate, as the correct value of v for the same difference between transmission and return frequencies (using the accurate equation) is 0.3c.

For police radar used in a moving car, the radar calculates the speed of the car by comparing the frequencies returned from objects stationary with respect to the road and those returned from the car.   It then just takes a simple difference. At low values of v, this simple difference is not going to differ significantly from the answer you would get using RVA.   Again, once the velocities begin to become significant relative to c, RVA becomes more and more important for getting the correct answer.

[jeffreyH]

Essentially, it is the light clock thought experiment but, instead of one observer on the platform, both are on trains moving relative to each other, and relative to the tracks.

The set-up is such that there are two points  on the tracks, A and B, which are equidistant from a mid-point M (as below):

========A==========M==========B=========

Alice is moving from left to right (A>B) and Bob in the opposite direction (B>A). Alice and Bob pass each other at point M and zero their clocks.

Is it possible for Alice to measure Bob's velocity relative to the tracks using her instruments?

Even if not, Alice measures her own velocity relative to the tracks (that Bob is running on) - getting a value of 0.4c - she measures the relative velocity between herself and Bob as 0.8c. It would seem that a measurement or calculation of 0.4c (for the relative velocity between Bob and the tracks) is a logical necessity in Alice's frame of reference.

Will Bob make similar meaurements/calculations for Alice's velocity relative to the tracks?

NOTE: there is no switching of reference frames here. Neither Alice nor Bob are trying to ascertain the measurements/co-ordinates of the others reference frame. Each simply makes measurements, using their own instruments, in their own inertial frame.

The points A and B are actually quite important in this scenario. Say Alice passes A at time t0 and Bob passes B at time t1. At some point X they will pass each other and be able to synchronise clocks. The point X does not have to be coincident with M. This can tell us nothing about time t0 or t1. However they can measure the time between point of synchronisation and passing M. They also have the tracks as a yardstick. Since points A and B are fixed points they can each record when they pass A (Bob) and B (Alice). At some point in the future Alice and Bob could accelerate and meet up to compare measurements.

BTW That is called observation and it is what every theory lives or dies by.

[the_roosh (OP)]

Thanks for the replies here. I understand the issue much clearer now.

[alancalverd]

If that's all it does, it seems to only measure my speed component in the direction the gun is pointed, a fraction of my actual speed if I'm not driving directly towards or away from the gun.

Which is why it is very difficult to defend a charge of speeding against a Doppler radar gun - it can't indicate a higher speed than the target.

Alice could count the rate of rotation of her wheel (I've reduced them to unicycles to save engineering) which is of course how surface vehicles measure speed. Now suppose Bob's vehicle emits a light pulse for each rotation of the wheel. As they approach, Bob appears to be going faster than his measured speed, and as they recede, slower.

Suppose we fit an elastic cylinder to Bob's wheel. As it spins, it stretches, by centrifugal force. At constant speed, its diameter is constant. So Alice can estimate Bob's speed by the diameter of the cylinder, either by looking or
 by asking Bob to look and radio her, with no need for a time base. She could have an identical cylinder on her wheel, and using her own clock, can calibrate the speedometer. I can't think of a relativistic reason why Bob's cylinder should appear to contract perpendicularly to the velocity vector, so she now has two different values for their relative velocity.