Einstein's Clock Synchronization Convention

[yogi]

Aether - are you saying that summing the two velocities leads to a correct average velocity over the total round trip distance

 

[Ich]

yogi - are Aether´s tranformation equations those that you call "Selleri Transformation"?

 

[Aether]

Aether - are you saying that summing the two velocities leads to a correct average velocity over the total round trip distance

Over the total round trip distance, the sum of the reciprocal speeds of light in LET is the same as the sum of the reciprocal speeds of light in SR: $$(c_0+v)/c_0^2+(c_0-v)/c_0^2=1/c_0+1/c_0$$.

 

[DrGreg]

Aether

Here's an algebraic proof from your beloved transformation equations. Combine the LET transform

$$t_{LET} = T / \gamma$$
$$x_{LET} = \gamma (X - vT)$$

with the Lorentz transform

$$t_{SR} = \gamma (T - vX / c^2)$$
$$x_{SR} = \gamma (X - vT)$$

(where $\gamma = 1 / \sqrt{1 - v^2/c^2}$, X and T are ether co-ordinates, and c is the absolute speed of light. Quantities with an SR subscript are measured in Special Relativity theory. Quantities with an LET subscript are measured in Lorentz Ether Theory. Quantities without a subscript are measured relative to the ether and therefore are valid in both theories.)

You get

$$t_{LET} = t_{SR} + v \ x_{SR} / c^2$$
$$x_{LET} = x_{SR}$$

For a particle with velocity u (along the x-axis), substitute $x_{SR} = u_{SR} \ t_{SR}$ and $x_{LET} = u_{LET} \ t_{LET}$ to obtain

$$\displaystyle{ u_{LET} = \frac{u_{SR}} {1 + v \ u_{SR} / c^2} }$$

So, two particles of equal mass with equal and opposite values of u_SR do not have equal and opposite values of u_LET. The last four paragraphs of pervect's last post (#34) now apply.

 

[Aether]

Here's an algebraic proof from your beloved transformation equations. Combine the LET transform

$$t_{LET} = T / \gamma$$
$$x_{LET} = \gamma (X - vT)$$

with the Lorentz transform

$$t_{SR} = \gamma (T - vX / c^2)$$
$$x_{SR} = \gamma (X - vT)$$

(where $\gamma = 1 / \sqrt{1 - v^2/c^2}$, X and T are ether co-ordinates, and c is the absolute speed of light. Quantities with an SR subscript are measured in Special Relativity theory. Quantities with an LET subscript are measured in Lorentz Ether Theory. Quantities without a subscript are measured relative to the ether and therefore are valid in both theories.)

DrGreg, this equation is not the one that I gave from M&S: $$t_{SR} = \gamma (T - vX / c^2)$$, but you have mixed it in with the other three that I did give. I think that yours implies a metric signature of -2, but M&S' imply a metric signature of +2?

You get

$$t_{LET} = t_{SR} + v \ x_{SR} / c^2$$
$$x_{LET} = x_{SR}$$

For a particle with velocity u (along the x-axis), substitute $x_{SR} = u_{SR} \ t_{SR}$ and $x_{LET} = u_{LET} \ t_{LET}$ to obtain

$$\displaystyle{ u_{LET} = \frac{u_{SR}} {1 + v \ u_{SR} / c^2} }$$

So, two particles of equal mass with equal and opposite values of u_SR do not have equal and opposite values of u_LET. The last four paragraphs of pervect's last post (#34) now apply.

Starting out with all of our clocks Einstein synchronized we have this:
$$x_{SR}=u_{SR}(T/\gamma-vx_{SR}/c_0^2)$$

and then after we readjust these clocks according to M&S-I p. 502 Eq. (3.5) $$f(x,v)=-vx_{SR}/c_0^2$$ we have this:
$$x_{LET}=u_{LET}(T/\gamma)$$

$$u_{SR}$$ and $$u_{LET}$$ are completely unaffected by this, so I suspect that these issues of the conservation of momentum will go away when this clock synchronization step is properly accounted for. There really isn't any empirical difference between these two theories; the only difference is that in one the speed of light is relative and simultaneity is absolute, but in the other the speed of light is absolute and simultaneity is relative. Thank you for the algebraic proof; this is not my final word on it, only a preliminary one.

 

[pervect]

The problem isn't going away. There is a very simple proof that one, and only one clock synchronization scheme will give isotropy of momentum.

Let us have three clocks

t1......t2,t3

Assume that t1 and t2 are synchronized so that momentum is isotropic. Assume t3 has some synchronization difference delta, i.e. t3 = t2 + delta.

We have an object #1 moving left to right with some fixed momentum p that starts out on the left at t1=0. At the same time, we have an object #2 moving right to left with the same mass and the same fixed momentum p that starts ou on the right at t2=0.

Object #1 will have a velocity of d/t2 according to clock 2, and a velocity of d/t3 according to clock three. Since t3 = t2 + delta, the velocity according to clock 2 will be d/(t2+delta).

Object #2 will have a velocity of d/t1 according to clock 2 (which starts out at 0), and a velocity of d/(t1-delta) according to clock 3 (which starts out at a value of t2+delta, and we've already mentioned that t2 was zero.

So if d/t2 = d/t1, then for any delta other than zero, d/(t2+delta) cannot be equal to d/(t1-delta.

Thus if we assume that the velocities are the same for some particular clock synchronization, any other clock synchronization will make them different.

Now here is the part where experiment steps in.

Relativity makes the claim that the _same_ clock synchronization that makes the speed of light isotropic (the same in both directions) is this unique clock synchronization that makes the velocities of objects with equal momentum isotropic.

 

[DrGreg]

DrGreg, this equation is not the one that I gave from M&S: $$t_{SR} = \gamma (T - vX / c^2)$$, but you have mixed it in with the other three that I did give. I think that yours implies a metric signature of -2, but M&S' imply a metric signature of +2?

The Lorentz equations I quoted are in the form you will find them in most textbooks (maybe using different letters, etc). You can invert them to get

$$T = \gamma (t_{SR} + v \ x_{SR} / c^2)$$
$$X = \gamma (x_{SR} + v \ t_{SR})$$

The first of these equations rearranges to give the first equation you quoted in this message: https://www.physicsforums.com/showpost.php?p=755432&postcount=92. I haven't assumed any metric signature.

...so I suspect that these issues of the conservation of momentum will go away when this clock synchronization step is properly accounted for

My algebra already takes account of the clock synchronisation -- it is included within the LET transformation formula. I stand by my argument. The only way you can account for momentum is to give it an anisotropic formula in LET coordinates.

 

[Aether]

Let us have three clocks

t1......t2,t3

Assume that t1 and t2 are synchronized so that momentum is isotropic. Assume t3 has some synchronization difference delta, i.e. t3 = t2 + delta.

We have an object #1 moving left to right with some fixed momentum p that starts out on the left at t1=0. At the same time, we have an object #2 moving right to left with the same mass and equal and opposite fixed momentum p that starts ou on the right at t2=0.

Object #1 will have a velocity of d/t2 according to clock 2, and a velocity of d/t3 according to clock three. Since t3 = t2 + delta, the velocity according to clock 2 will be d/(t2+delta).

Object #2 will have a velocity of d/t1 according to clock 2 (which starts out at 0), and a velocity of d/(t1-delta) according to clock 3 (which starts out at a value of t2+delta, and we've already mentioned that t2 was zero.

So if d/t2 = d/t1, then for any delta other than zero, d/(t2+delta) cannot be equal to d/(t1-delta.

The x-coordinate of each clock determines a unique value of $delta=-vx/c_0^2$ for that clock so I have labeled each delta according to the clock that it belongs to, and restated the travel times more explicitly to show that the delta terms fall out for any time interval that is measured by a single clock. This always leads to the opposite conclusion from your example when any time interval is measured by a single clock, so I presume that you will want to specify that each of these time intervals is to be measured using two different clocks?


Let us have three clocks

t1......t2,t3

Assume that t1 and t2 are synchronized so that momentum is isotropic. Assume t3 has some synchronization difference delta, i.e. t3 = t2 + delta2.

We have an object #1 moving left to right with some fixed momentum $p1=\gamma 1 m(v+u)$ that starts out on the left at t1_0=0. At the same time, we have an object #2 moving right to left with the same mass and a fixed momentum $p2=\gamma 2 m(v-u)$ that starts out on the right at t2_0=0.

Object #1 will have a relative velocity of u12=d/(t2_1-t2_0) according to clock 2, and a relative velocity of u13=d/(t3_1-t3_0) according to clock three. Since t3 = t2 + delta2, the relative velocity according to clock 2 will be u12=d/(t2_1+delta2-t2_0-delta2).

Object #2 will have a relative velocity of u21=d/(t1_1-t1_0) according to clock 1 (which starts out at 0), and a relative velocity of u23=d/(t1_1+delta1-t1_0-delta1) according to clock 3 (which starts out at a value of t3_0=t2_0+delta2, and we've already mentioned that t2_0=0.

So if d/(t2_1-t2_0) = d/(t1_1-t1_0), then for any delta whatsoever, d/(t2_1+delta2-t2_0-delta2) is also equal to d/(t1_1+delta1-t1_0-delta1).

 

[Aether]

You get

$$t_{LET} = t_{SR} + v \ x_{SR} / c^2$$
$$x_{LET} = x_{SR}$$

For a particle with velocity u (along the x-axis), substitute $x_{SR} = u_{SR} \ t_{SR}$ and $x_{LET} = u_{LET} \ t_{LET}$ to obtain

$$\displaystyle{ u_{LET} = \frac{u_{SR}} {1 + v \ u_{SR} / c^2} }$$

Let’s be more explicit and say that $x_{SR1}-x_{SR0}=u_{SR}(t_{SR1}-t_{SR0})$, $x_{LET1}-x_{LET0}=u_{LET}(t_{LET1}-t_{LET0})$, and $x_{LET1}-x_{LET0}=x_{SR1}-x_{SR0}$.

$u_{LET}=u_{SR}(t_{SR1}-t_{SR0})/(t_{LET1}-t_{LET0})=$
$u_{SR}(t_{SR1}-t_{SR0})/(t_{SR1}+vx/c_0^2-t_{SR0}-vx/c_0^2)=u_{SR}$.

So, two particles of equal mass with equal and opposite values of u_SR do not have equal and opposite values of u_LET. The last four paragraphs of pervect's last post (#34) now apply.

False.

 

[pervect]

Combinging two previous posts by Aether

The x value of each clock determines a unique value of $delta=-vx/c_0^2$ for that clock so I have labeled each delta according to the clock that it belongs to, and restated the travel times more explicitly to show that the delta terms fall out for any time interval that is measured by a single clock. This always leads to the opposite conclusion from your example when any time interval is measured by a single clock, so I presume that you will want to specify that each of these time intervals is to be measured using two different clocks?


Let’s be more explicit and say that $x_{SR1}-x_{SR0}=u_{SR}(t_{SR1}-t_{SR0})$, $x_{LET1}-x_{LET0}=u_{LET}(t_{LET1}-t_{LET0})$, and $x_{LET1}-x_{LET0}=x_{SR1}-x_{SR0}$.

$u_{LET}=u_{SR}(t_{SR1}-t_{SR0})/(t_{LET1}-t_{LET0})=$
$u_{SR}(t_{SR1}-t_{SR0})/(t_{SR1}+vx/c_0^2-t_{SR0}-vx/c_0^2)=u_{SR}$.

If we take your argument at face value, you have just proven that all velocities in LET are the same as all velocities in SR for any value of velocity, including 'c', therefore the speed of light is isotropic in LET.

But the speed of light is not supposed to be isotropic in LET, if I have been following your exposition of the theory correctly.

You have written vx/c_0^2, but you have also said that x is unique for each clock. If we assume that the x is different for clock 1 and clock 0, we get Dr Greg's results, and we also get an anisotropic speed of light.

Note that if we use your equations, you are adding the _same_ value of time to _every_ clock regardless of position. It is not terribly surprising that this does not give us any variation in velocity. It also gives us an isotropic speed for light.

But making the value of time added depend on position gives us

$$T_{LET1} = T_{SR1} + \frac{v X1}{c_0^2}$$
$$T_{LET0} = T_{SR0} + \frac{v X0}{c_0^2}$$

here v is a hypothetical "ether velocity", and X0 and X1 are "ether coordinates", which are different for the two different events because they don't happen at the same point in space.

This gives us directly that
$$T_{LET1} - T_{LET0} = T_{SR1} - T_{SR0} + \frac{v \left( X1-X0 \right) } {c_0^2}$$

Substituting this into your first equation (which I think is correct though I haven't double checked it) we get

$$\frac {u_{LET}}{u_{SR}} = \frac { T_{SR1} - T_{SR0} }{ T_{SR1} - T_{SR0} + \frac{v \left( X1-X0 \right) }{c^2} }$$

which gives anisotropy of velocity for v=c and for v<c.

 

[DrGreg]

pervect (post #45) is nearly correct.

The correct equations are

$$t_{LET1} = t_{SR1} + v \ x_{SR1} / c_0^2$$
$$t_{LET0} = t_{SR0} + v \ x_{SR0} / c_0^2$$

But $x_{SR1} \neq x_{SR0}$ (except if u_SR = 0) so the rest of his argument still applies.

 

[DrGreg]

We have an object #1 moving left to right with some fixed momentum $p1=\gamma 1 m(v+u)$ that starts out on the left at t1_0=0.

When you quote the momentum as $p=\gamma_{v+u} m(v+u)$ there are two things wrong with this.

1. pervect and myself are talking about momentum relative to the observer (who is moving at absolute velocity v). You seem to be talking about absolute momentum, relative to the ether. As behaviour relative to the ether is agreed between SR and LET, absolute momentum isn't an issue, only relative momentum.

2. The formula $p=\gamma_{v+u} m(v+u)$ isn't valid for absolute momentum. v is the velocity of the observer relative to the ether. u is the velocity of the particle relative to the observer. But (v+u) is not the velocity of the particle relative to the ether, neither in SR nor in LET.

(Added)

Incidentally, the reasoning that momentum is $p=\gamma_u m u$ rather than Newtonian $p=m u$ requires an application of the postulates of SR. How would you justify this formula without invoking SR?

 

[Aether]

When you quote the momentum as $p=\gamma_{v+u} m(v+u)$ there are two things wrong with this.

1. pervect and myself are talking about momentum relative to the observer (who is moving at absolute velocity v). You seem to be talking about absolute momentum, relative to the ether. As behaviour relative to the ether is agreed between SR and LET, absolute momentum isn't an issue, only relative momentum.

OK. I don't mean to ignore this thread, but the other one is keeping me occupied at the moment. I think that I proved here that velocities and momentums measured by any single clock are the same for both SR and LET, and I gather that you and pervect want to talk only about time intervals measured by two different clocks? It is not at all clear to me what either of you are saying when you specify a time interval as "t2" when there is a clock labeled t2, etc..

2. The formula $p=\gamma_{v+u} m(v+u)$ isn't valid for absolute momentum. v is the velocity of the observer relative to the ether. u is the velocity of the particle relative to the observer. But (v+u) is not the velocity of the particle relative to the ether, neither in SR nor in LET.

As long as we agree to restrict our discussion to motion along the x-axis then this is OK. Wouldn't it be a needless complication to do otherwise?

Incidentally, the reasoning that momentum is $p=\gamma_u m u$ rather than Newtonian $p=m u$ requires an application of the postulates of SR. How would you justify this formula without invoking SR?

Do you not use the Lorentz transformation to justify this? I would then use the LET transformation.

 

[pervect]

You can't measure the one-way speed of light (or anything else, for that matter) without two clocks. I was not the person who introduced "one-way" speeds into this thread.

Look at the thread title and originating post, for instance:

Einstein's Clock Synchronization Convention

Just a friendly reminder, the first postulate of the special theory of relativity, namely that the speed of light c is the same in all inertial frames, only holds true in view of Einstein's clock synchronization convention. There is no experimental basis whatsoever for preferring this convention over absolute clock synchronization

It seems a bit revisionist to me to start saying

I think that I proved here that velocities and momentums measured by any single clock are the same for both SR and LET

when the entire thread, from its inception, has been about clock synchronization, which of course implies using more than one clock. (You can't synchronize a clock to itself!).'

What I've been doing is attempting to point out the importance of Einstein's clock synchronization convention. It's not a matter of "don't use this convention and nothing will happen". It's a matter of "don't use this convention and Newton's laws will fail and the momentum of a body will depend on it's direction of travel".

 

[Aether]

You can't measure the one-way speed of light (or anything else, for that matter) without two clocks. I was not the person who introduced "one-way" speeds into this thread.

Look at the thread title and originating post, for instance:



It seems a bit revisionist to me to start saying



when the entire thread, from its inception, has been about clock synchronization, which of course implies using more than one clock. (You can't synchronize a clock to itself!).'

What I've been doing is attempting to point out the importance of Einstein's clock synchronization convention. It's not a matter of "don't use this convention and nothing will happen". It's a matter of "don't use this convention and Newton's laws will fail and the momentum of a body will depend on it's direction of travel".

I'm only asking for you to label the coordinates in your example so that I can see which clock they came from at which snapshot in time, and where you specify a time interval to label it explicitly so that I can see exactly how it is supposed to be measured. My reality check has been to measure velocities using a radar pulse and a single clock, so I know that LET must give the same velocity as that method. If we use two clocks, then the synchronizations of those clocks will have to be accounted for, and therefore the two clocks will have to be explicitly represented within any time interval.

 

[JesseM]

My reality check has been to measure velocities using a radar pulse and a single clock, so I know that LET must give the same velocity as that method.

How does this method work, exactly--are you bouncing multiple pulses off an object and seeing the difference between the time interval that the pulses are emitted and the time interval that the pulses are received? If so, what equation do you solve to find the speed of the object? If you assume the radar signals travel at c in both directions as in SR, that the object's velocity is v and its distance at t=0 is d, and that there is a time interval of $$t_0$$ between when two pulses are emitted, then the time the first pulse catches up to the object and bounces back can be found by solving this equation for t:

$$ct = vt + d$$

and the time the second pulse bounces back can be found by solving this equation:

$$c (t - t_0 ) = vt + d$$

Solving the first equation gives $$t = d/(c-v)$$, solving the second gives $$t = (d + ct_0 )/(c-v)$$. So, the difference between these times is $$ct_0 / (c - v)$$, during which time the object will have moved further away by a distance of $$vct_0 / (c - v)$$, so the second pulse has that much further to get back, and since it also travels back at c according to SR, this will add another $$vt_0 / (c - v)$$ to the time it takes to return. So, the total time interval between the return of the two pulses will be $$(ct_0 / (c - v)) + (vt_0 / (c - v)) = t_0 (c + v)/(c - v)$$. So if you measure the time interval between the pulses being emitted as $$t_0$$, and the time interval between them returning as $$t_1$$, then you can solve $$t_1 = t_0 (c + v)/(c - v)$$ for v to get an equation that tells you the object's velocity in terms of these two time intervals, giving $$v = c (t_1 - t_0) / (t_1 + t_0 )$$. So that's how I think you could use radar signals to measure velocities in SR, tell me if you see any mistakes in my reasoning or my math. (edit: I did make a conceptual mistake, but it's fixed now)

But if you can't assume the radar signals moved at c in both directions, it's not so obvious to me how you'd use radar signals to measure an object's velocity in the LET, or why the fact that the round-trip velocity of light is still c would imply that the answer you'd get for a given object's one-way speed would be the same as the SR answer. In fact, based on the numerical example I did on the "relativity without aether" thread, I'm pretty sure the coordinate velocity of an object would not be the same in a given observer's frame if he was using the LET transform equations as it would if he was using the Lorentz transform equations. If you think it would be, what's your reasoning?

 

[Aether]

How does this method work, exactly--are you bouncing multiple pulses off an object and seeing the difference between the time interval that the pulses are emitted and the time interval that the pulses are received? If so, what equation do you solve to find the speed of the object? If you assume the radar signals travel at c in both directions as in SR, and there is a time interval of $$t_0$$ between when two pulses are emitted, then the object's distance will have increased by $$v t_0$$ between the time the first and second pulse are emitted (assume the initial distance was d), so the time the first pulse catches up to the object and bounces back can be found by solving this equation for t:

$$ct = vt + d$$

and the time the second pulse bounces back can be found by solving this equation:

$$c (t - t_0 ) = vt + (d + v t_0 )$$

Solving the first equation gives $$t = d/(c-v)$$, solving the second gives $$t = (d + t_0 + v t_0 )/(c-v)$$. So, the difference between these times is $$t_0 (1 + v) / (c - v)$$, and since the pulses take the same amount of time to return in SR, the time interval between the return of the two pulses to where they were emitted should be $$2(1+v) t_0 / (c-v)$$. So if you measure the time interval between the pulses being emitted as $$t_0$$, and the time interval between them returning as $$t_1$$, then you can solve $$t_1 = 2(1+v) t_0 / (c-v)$$ for v to get an equation that tells you the object's velocity in terms of these two time intervals, giving $$v = c (t_1 - 2t_0 ) / (t_1 + 2t_0 )$$. So that's how I think you could use radar signals to measure velocities in SR, tell me if you see any mistakes in my reasoning or my math.

But if you can't assume the radar signals moved at c in both directions, it's not so obvious to me how you'd use radar signals to measure an object's velocity in the LET, or why the fact that the round-trip velocity of light is still c would imply that the answer you'd get for a given object's one-way speed would be the same as the SR answer. In fact, based on the numerical example I did on the "relativity without aether" thread, I'm pretty sure the coordinate velocity of an object would not be the same in a given observer's frame if he was using the LET transform equations as it would if he was using the Lorentz transform equations. If you think it would be, what's your reasoning?

Please see posts #33 & 35. I'm not saying that I'm sure that the coordinate velocity would be the same in LET as in SR, I'm saying that here's an experiment (like Michelson-Morley) that gives the same result for the velocity of an object using either LET or SR because there is no question of clock synchronization (e.g., only one clock is involved). So, if we can prove that SR gives this same velocity using two clocks but LET doesn't, then that's an example where they are not empirically equivalent. That's not supposed to happen. Unless of course the difference turns out to be something entirely trivial like (v+u) in one direction, and (v-u) in the other.

 

[Hans de Vries]

What I've been doing is attempting to point out the importance of Einstein's clock synchronization convention. It's not a matter of "don't use this convention and nothing will happen". It's a matter of "don't use this convention and Newton's laws will fail and the momentum of a body will depend on it's direction of travel".

I once confronted him with the problem of a wheel and anisotropic speed.
Imagine a rigid wheel with different angular velocities at different places..

The wheel would also have a non-zero momentum into some direction while
rotating in place...

Actually. One could easily create a situation where the rotating wheel moves
in one direction while the momentum points in the other direction. A situation
which corresponds with negative mass..


Regards, Hans

 

[pervect]

pervect (post #45) is nearly correct.

The correct equations are

$$t_{LET1} = t_{SR1} + v \ x_{SR1} / c_0^2$$
$$t_{LET0} = t_{SR0} + v \ x_{SR0} / c_0^2$$

But $x_{SR1} \neq x_{SR0}$ (except if u_SR = 0) so the rest of his argument still applies.

Sorry to muddle the waters, you're right. If we invert the standard textbook Lorentz transforms (given in your post), we can solve for T and X, getting the standard result.

$T = \gamma(t_{sr} + v x_{sr} / c^2)$

Substituting in the equation $t_{let} = T / \gamma$, which defines t_let, (and I think everyone has agreed to this as being the proper equation to define t_let), we get

$t_{let} = t_{sr} + v x_{sr}/c^2$

from which follows your results directly, and the rest of my original argument.

 

[JesseM]

Please see posts #33 & 35.

OK, in post #33 you provide an example:

Suppose that we're on a ship using radar to monitor the approach of two unidentified aircraft coming in from opposite directions. At time $t_1$ we transmit a microwave pulse in one direction, and at time $t_2$ we receive an echo from the first plane; then at time $t_3$ we transmit a microwave pulse in the opposite direction, and at time $t_4$ we receive an echo from the second plane: $t_2-t_1=t_4-t_3=0.001 000000\ seconds$. Then one second later we repeat the process and get $t_2-t_1=t_4-t_3=0.000 996 998$. Evidently, both aircraft are approaching our ship at v=450m/s from a range of 149.446km

But what calculation did you do to get this approach speed? This works if I plug in $$t_0 = 1$$ and $$t_1 = 1.000996998 - 0.001000000 = 0.999996998$$ into my equation $$v = c (t_1 - t_0) / (t_1 + t_0 )$$ (the original equation I had in my last post was wrong, but I edited it), but this equation was derived using the assumption that the signal moved at c in both directions, I don't see how you could get an approach speed of 450 m/s without making this assumption.

I'm not saying that I'm sure that the coordinate velocity would be the same in LET as in SR, I'm saying that here's an experiment (like Michelson-Morley) that gives the same result for the velocity of an object using either LET or SR

Every experiment gives the same result using either LET or SR, that's what is meant when it's said they are "empirically equivalent". But the interpretation of the experiment is different--the timing of the radar pulses can no longer tell you the "velocity" of the object in your reference frame in LET, because "velocity" means something different.

So, if we can prove that SR gives this same velocity using two clocks but LET doesn't, then that's an example where they are not empirically equivalent.

You're still confusing physical facts with coordinate-dependent statements. All the physical facts involving the reading on your clock when the radar pulses will be the same, but you can no longer use the timing of these pulses to determine the "velocity" of the object in the same way (in particular, you can't plug the time intervals $$t_0$$ and $$t_1$$ into the equation $$v = c (t_1 - t_0) / (t_1 + t_0 )$$) because the notion of "velocity" itself is coordinate-dependent.

 

[Aether]

OK, in post #33 you provide an example: But what calculation did you do to get this approach speed? This works if I plug in $$t_0 = 1$$ and $$t_1 = 1.000996998 - 0.001000000 = 0.999996998$$ into my equation $$v = c (t_1 - t_0) / (t_1 + t_0 )$$ (the original equation I had in my last post was wrong, but I edited it), but this equation was derived using the assumption that the signal moved at c in both directions, I don't see how you could get an approach speed of 450 m/s without making this assumption.

Since we know (see post #35) that the average round trip speed of light is equal to $$c_0$$ then I did use that as a simplifying assumption when I calculated this velocity.

Every experiment gives the same result using either LET or SR, that's what is meant when it's said they are "empirically equivalent". But the interpretation of the experiment is different--the timing of the radar pulses can no longer tell you the "velocity" of the object in your reference frame in LET, because "velocity" means something different. You're still confusing physical facts with coordinate-dependent statements. All the physical facts involving the reading on your clock when the radar pulses will be the same, but you can no longer use the timing of these pulses to determine the "velocity" of the object in the same way (in particular, you can't plug the time intervals $$t_0$$ and $$t_1$$ into the equation $$v = c (t_1 - t_0) / (t_1 + t_0 )$$) because the notion of "velocity" itself is coordinate-dependent.

Is what you're saying equivalent to what DrGreg and pervect are saying?

 

[JesseM]

Since we know (see post #35) that the average round trip speed of light is equal to $$c_0$$ then I did use that as a simplifying assumption when I calculated this velocity.

Yeah, but in my derivation of the equation $$v = c (t_1 - t_0) / (t_1 + t_0 )$$ I assumed that the one-way velocity of light in each direction was c, not just that the average round-trip velocity was c. I'm pretty sure that in the non-preferred coordinate systems of the LET transform, the round-trip velocity of light would still be c but the velocity of slower-than-light objects would not necessarily be the same as what my formula says, therefore in your example it would no longer be true that the velocity of both aircraft must be 450 m/s in such a coordinate system. I think that in calculating that number, you must have implicitly or explicitly assumed that the one-way velocity of light was c...but if you think I'm wrong, could you show me the steps of how you got that number for the velocity of each airplane?

Is what you're saying equivalent to what DrGreg and pervect are saying?

I dunno, I haven't been following this thread as carefully as I have the "relativity without aether" thread...I'll look over their comments later on and see if I agree with their arguments.

 

[Aether]

Yeah, but in my derivation of the equation $$v = c (t_1 - t_0) / (t_1 + t_0 )$$ I assumed that the one-way velocity of light in each direction was c, not just that the average round-trip velocity was c. I'm pretty sure that in the non-preferred coordinate systems of the LET transform, the round-trip velocity of light would still be c but the velocity of slower-than-light objects would not necessarily be the same as what my formula says, therefore in your example it would no longer be true that the velocity of both aircraft must be 450 m/s in such a coordinate system. I think that in calculating that number, you must have implicitly or explicitly assumed that the one-way velocity of light was c...but if you think I'm wrong, could you show me the steps of how you got that number for the velocity of each airplane?

OK. It may be tomorrow before I am able to put all the steps in here, but the jist is that the round-trip speed of light is isotropic (rotation invariance) in LET as well as SR. So, I ping an airplane twice to measure $$\Delta x/\Delta t$$.

I didn't calculate this example using LET; I calculated it to show that I could measure a velocity with one clock.

I dunno, I haven't been following this thread as carefully as I have the "relativity without aether" thread...I'll look over their comments later on and see if I agree with their arguments.

OK. Thanks.

 

[pervect]

I'm only asking for you to label the coordinates in your example so that I can see which clock they came from at which snapshot in time, and where you specify a time interval to label it explicitly so that I can see exactly how it is supposed to be measured. My reality check has been to measure velocities using a radar pulse and a single clock, so I know that LET must give the same velocity as that method. If we use two clocks, then the synchronizations of those clocks will have to be accounted for, and therefore the two clocks will have to be explicitly represented within any time interval.

I hadn't realized that you were using different methods to measure velocities for light and for material objects.

I'm not totally sure I understand all the details of your radar method, because I didn't look at it very closely - it seemed like just another complication to me. Why introduce yet another means of measuring velocities, one that is different for light and material objects? Why not just use the same definition of velocity for both? If you have a method that can measure the speed of light (one-way), it should also (if it is any good!) be able to measure the speed of non-light (one-way). If it is not any good, maybe it is better not to use it at all.

As far as subscripts go, there are two events

Event 0 is when the moving object crosses the position of clock 0. The time at which this occurs will be measured by clock 0 and is called t_sr0 or t_let0, depending on the clock synchronization method used.

Event 1 is when the moving object crosses the position of clock 1. The time will be measured by clock 1. The time at which this occurs will be measured by clock 1 and is called t_sr1 or t_let1, depending again on the clock synchronization method.

The velocity of the moving object is given by v_sr = d_sr/(t_sr1-t_sr0) or v_let = d_let/(t_let1 - t_let0). These numbers will in general be different.

 

[Aether]

I hadn't realized that you were using different methods to measure velocities for light and for material objects.

I'm not totally sure I understand all the details of your radar method, because I didn't look at it very closely - it seemed like just another complication to me. Why introduce yet another means of measuring velocities, one that is different for light and material objects? Why not just use the same definition of velocity for both? If you have a method that can measure the speed of light (one-way), it should also (if it is any good!) be able to measure the speed of non-light (one-way). If it is not any good, maybe it is better not to use it at all.

My apologies for not making that clear. One property of LET is that the round-trip speed of light is isotropic, so I presume that I can use a radar to measure a velocity with one clock just as in SR. I don't expect for two clocks to give me a different answer than the radar, so if they do then I would be worried. That's the only point of that.

As far as subscripts go, there are two events

Event 0 is when the moving object crosses the position of clock 0. The time at which this occurs will be measured by clock 0 and is called t_sr0 or t_let0, depending on the clock synchronization method used.

Event 1 is when the moving object crosses the position of clock 1. The time will be measured by clock 1. The time at which this occurs will be measured by clock 1 and is called t_sr1 or t_let1, depending again on the clock synchronization method.

The velocity of the moving object is given by v_sr = d_sr/(t_sr1-t_sr0) or v_let = d_let/(t_let1 - t_let0). These numbers will in general be different.

OK. At first I was looking for something like (t_let11-t_let10)=light signal travel-time interval on clock 1.

 

[JesseM]

OK. It may be tomorrow before I am able to put all the steps in here, but the jist is that the round-trip speed of light is isotropic (rotation invariance) in LET as well as SR. So, I ping an airplane twice to measure $$\Delta x/\Delta t$$.

I didn't calculate this example using LET; I calculated it to show that I could measure a velocity with one clock.

I didn't say that you calculated it using LET, my point was that your calculation must have implicitly snuck in an assumption about the one-way speed of light being c, because LET shows that you can have coordinate systems where the round-trip speed of light is always c but the speed you get for the airplane will be different. However, the speed of that airplane was very small compared to the speed of light so the difference would be small--let me introduce a new numerical example, where I send out a pulse at an approaching spaceship and it bounces back to me after 0.8 seconds, then I send out another pulse 1 second after the first one, and it returns after 0.4 seconds. In this case, if I am in a reference frame constructed in the standard way in relativity, I will conclude that the ship is moving towards me at 0.25c. In my coordinate system, the coordinates of each pulse's emission, reflection and return will be (working in units of light-seconds and seconds, so c=1):

first pulse emitted: x'=0, t'=0
first pulse hits ship: x'=0.4, t'=0.4
first pulse returns: x'=0, t'=0.8
second pulse emitted: x'=0, t'=1
second pulse hits ship: x'=0.2, t'=1.2
second pulse returns: x'=0, t'=1.4

So you can see that between the first pulse hitting the ship and the second one hitting it, 1.2-0.4=0.8 seconds have passed, and the ship has moved closer to me by 0.4-0.2=0.2 light-seconds, so its speed is (0.2/0.8) c = 0.25c.

Now let's imagine that I am moving at 0.6c in the +x direction in your frame, and we want to know what the coordinates of these events will be in your own rest frame. We can use the Lorentz transform here:

x = 1.25(x' + 0.6t')
t = 1.25(t' + 0.6x'/c^2)

This gives the coordinates:

first pulse emitted: x=0, t=0
first pulse hits ship: x=0.8, t=0.8
first pulse returns: x=0.6, t=1
second pulse emitted: x=0.75, t=1.25
second pulse hits ship: x=1.15, t=1.65
second pulse returns: x=1.05, t=1.75

But now imagine that your frame is actually the rest frame of the ether, and we want to know what the coordinates of these events would be in my rest frame if we used the LET transform instead of the Lorentz transform:

x'=1.25(x - 0.6t)
t'=0.8t

This gives the following coordinates in my LET-transform-rest-frame:

first pulse emitted: x'=0, t'=0
first pulse hits ship: x'=0.4, t'=0.64
first pulse returns: x'=0, t'=0.8
second pulse emitted: x'=0, t'=1
second pulse hits ship: x'=0.2, t'=1.32
second pulse returns: x'=0, t'=1.4

You can see it's still true that the first pulse returned to me 0.8 seconds after it was emitted, that the second pulse was emitted 1 second later, and that the second pulse returned to me after 0.4 seconds. The first pulse had a round-trip distance of 0.8 and a round-trip time of 0.8, while the second had a round-trip distance of 0.4 and a round-trip time of 0.4, so it's true that both pulses have a round-trip speed of c in my coordinate system. But it's no longer true that the ship is traveling at 0.25c towards me; between the time of the first pulse hitting it and the second hitting it, 1.32-0.64=0.68 seconds had passed and the ship had gotten closer by 0.4-0.2=0.2 light-seconds, so its speed is 0.2/0.68 = 0.294c in my coordinate system. So you see, just knowing the timing of the pulses and the round-trip speed of light is not enough to uniquely determine the speed of the ship in any coordinate system; both these things are the same in both my Lorentz-transform-rest-frame and my LET-transform-rest-frame, but the coordinate velocity of the ship is different in these two cases.

 

[Aether]

I think that in calculating that number, you must have implicitly or explicitly assumed that the one-way velocity of light was c...but if you think I'm wrong, could you show me the steps of how you got that number for the velocity of each airplane?

The first ping has a round-trip travel time of 0.001 000 000 000 000 seconds, so I conclude that the range to the target is 149,896.229 meters using the assumtion that the speed of light has the SI value of exactly $$c_0=299,792,458 m/s$$. The second ping is initiated one second after the first ping and is found to have a round-trip travel time of 0.000 996 998 003 198 seconds, so I conclude that the new range to the target is 149,446.241 meters: $$\Delta x/\Delta t=449.988 meters/second$$. The second plane is tracked in exactly the same way.

In this example v=368km/s, so the speed of light in one direction using LET is $$c(v,0)=c_0^2/(c_0+v)=299,424,909.172 m/s$$ and the speed of light in the other direction is $$c(v,\pi)=c_0^2/(c_0-v)=300,160,910.281 m/s$$. This means that the travel time of the first ping to the target at $$x=149,896.229 meters$$ is $$(t_1-t_0)=0.000 500 613 757 935 seconds$$, and the return travel-time is $$(t_2-t_1)=0.000 499 386 242 065 seconds$$, and $$(t_2-t_0)=0.001 000 000 000 000 seconds$$.

The travel time of the second ping to the target at $$x=149,446.241meters$$ is $$(t_1-t_0)=0.000 499 110 917 035 seconds$$, and the return travel-time is $$(t_2-t_1)=0.000 497 887 086 164 seconds$$, and $$(t_2-t_0)=0.000 996 998 003 198 seconds$$: $$\Delta x/\Delta t=449.988 meters/second$$. The second plane is tracked in exactly the same way.

 

[JesseM]

In this example v=368km/s, so the speed of light in one direction using LET is $$c(v,0)=c_0^2/(c_0+v)=299,424,909.172 m/s$$ and the speed of light in the other direction is $$c(v,\pi)=c_0^2/(c_0-v)=300,160,910.281 m/s$$. This means that the travel time of the first pulse to the target at $$x=149,896.229 meters$$ is $$(t_1-t_0)=0.000 500 613 757 935 seconds$$, and the return travel-time is $$(t_2-t_1)=0.000 499 386 242 065 seconds$$, and $$(t_2-t_0)=0.001 000 000 000 000 seconds$$.

OK, I take back what I said about you implicitly using the assumption that the one-way trip speed was c, I hadn't noticed that you explicitly assumed your velocity relative to the ether frame was 368 km/s. It turns out that since the speeds here are so small compared to the speed of light, the answer you get for the speed of the plane in your LET-rest-frame is almost identical to the answer you'd get for the speed of the plane in your Lorentz-rest-frame, which is why I seemed to get the same answer using my formula which was derived using the assumption that the speed of light is c in both directions.

But if you don't even believe there's an actual physical substance called "ether" and an objective truth about your velocity relative to it, then the choice of which frame to use as the preferred "ether frame" is totally arbitrary, no? And if you had picked some other frame you'd get a different answer for the velocity of the plane in your rest frame, right? And my example above showed that you get a different answer for the velocity of moving objects depending on whether you use the Lorentz transform or the LET transform.

Assuming you agree with all this, it shows that all velocities of objects depend on which coordinate system you use, velocities of objects can't be uniquely determined in a coordinate-free way using a single clock and radar pulses. When you say "it's false that the one-way speed of light has been measured to be constant" because you can pick alternate coordinate systems where all the physical facts are the same but the value of light's speed isn't constant, what you really should say is "it's false that the speed of anything in the universe relative to anything else has been measured", because for any two objects you can pick alternate coordinate systems where the speed of the first object in the second's rest frame is different. So I think you should stop harping on the speed of light and acknowledge that you are making a general point that all speeds are coordinate-dependent. And while this is true, I think it's understood that any statement about speed are implicitly assuming the most physically natural coordinate system, which in our universe is the set of coordinate systems provided by the Lorentz transform. Because of this, I don't think it's really fair to say all statements about speed (including the statement that light has a constant speed) are "false", just that they don't usually state explicitly this implicit assumption.

 

[Aether]

It turns out that since the speeds here are so small compared to the speed of light, the answer you get for the speed of the plane in your LET-rest-frame is almost identical to the answer you'd get for the speed of the plane in your Lorentz-rest-frame, which is why I seemed to get the same answer using my formula which was derived using the assumption that the speed of light is c in both directions.

I thought that they should be exactly the same. If you still think that they are not, I can recalculate to 100 digits of precision and see if there is a residual velocity.

But if you don't even believe there's an actual physical substance called "ether" and an objective truth about your velocity relative to it, then the choice of which frame to use as the preferred "ether frame" is totally arbitrary, no?

I think that the metric represents a physical thing/environment aka "the aether", but I agree for now that any choice of a preferred "ether frame" would be totally arbitrary.

And if you had picked some other frame you'd get a different answer for the velocity of the plane in your rest frame, right?

No, at least not at the precision of these calculations. The speed of light would be different, but the plane's velocity wouldn't.

And my example above showed that you get a different answer for the velocity of moving objects depending on whether you use the Lorentz transform or the LET transform.

I haven't looked at your example yet because I thought that it assumed something about the one-way speed of light being constant...if it still applies then I'll go back and look at it.

Assuming you agree with all this, it shows that all velocities of objects depend on which coordinate system you use, velocities of objects can't be uniquely determined in a coordinate-free way using a single clock and radar pulses. When you say "it's false that the one-way speed of light has been measured to be constant" because you can pick alternate coordinate systems where all the physical facts are the same but the value of light's speed isn't constant, what you really should say is "it's false that the speed of anything in the universe relative to anything else has been measured", because for any two objects you can pick alternate coordinate systems where the speed of the first object in the second's rest frame is different. So I think you should stop harping on the speed of light and acknowledge that you are making a general point that all speeds are coordinate-dependent.

Acknowledged. Not only for speeds, but every possible empirical measurement.

And while this is true, I think it's understood that any statement about speed are implicitly assuming the most physically natural coordinate system, which in our universe is the set of coordinate systems provided by the Lorentz transform. Because of this, I don't think it's really fair to say all statements about speed (including the statement that light has a constant speed) are "false", just that they don't usually state explicitly this implicit assumption.

I think that they are "false" when they step outside of this box where they are understood to be coordinate dependent stantements, and are claimed as some kind of proof that an alernate coordinate system is invalid.

 

[JesseM]

I thought that they should be exactly the same. If you still think that they are not, I can recalculate to 100 digits of precision and see if there is a residual velocity.

No, they will be slightly different. This is easier to see in my spaceship example where the velocities are higher.

I think that the metric represents a physical thing aka "the aether", but I agree for now that any choice of a preferred "ether frame" would be totally arbitrary.

OK, I thought you had said earlier that your notion of LET was only about a different coordinate transform and included no new physical assumptions, but it doesn't matter too much either way. By the way, I think you're misusing the term "metric" here, my understanding is that the metric is just the thing that tells you the coordinate-independent proper time between two points in spacetime, I don't think a different choice of coordinate systems qualifies as a different metric.

No, at least not at the precision of these calculations. The speed of light would be different, but the plane's velocity wouldn't.

Maybe not at the precision of your calculations, but at a high enough precision it would be. Again, look at my spaceship example.

I haven't looked at your example yet because I thought that it assumed something about the one-way speed of light being constant...if it still applies then I'll go back and look at it.

No, my example doesn't assume the one-way speed of light is constant, at least not in all cases. I calculated the ship's velocity and the coordinates of different events using two separate assumptions, the first being the usual SR assumption that the one-way speed is constant and the second being the assumption that I am moving at 0.6c relative to the ether frame and using the LET coordinate system, and I showed that the coordinate velocity of the ship is different in these two cases.

Assuming you agree with all this, it shows that all velocities of objects depend on which coordinate system you use, velocities of objects can't be uniquely determined in a coordinate-free way using a single clock and radar pulses. When you say "it's false that the one-way speed of light has been measured to be constant" because you can pick alternate coordinate systems where all the physical facts are the same but the value of light's speed isn't constant, what you really should say is "it's false that the speed of anything in the universe relative to anything else has been measured", because for any two objects you can pick alternate coordinate systems where the speed of the first object in the second's rest frame is different. So I think you should stop harping on the speed of light and acknowledge that you are making a general point that all speeds are coordinate-dependent.

Acknowledged. Not only for speeds, but every possible empirical measurement.

Not true, there are plenty of empirical measurements that are not affected by your choice of coordinate system--this was the point I tried to make in my last long post to you on the "relativity without aether" thread (#85, which I hope you'll respond to when you get the chance) where I accused you of being confused about the distinction between physical facts and coordinate-dependent statements. For example, the question of whether two events coincide at the same point in spacetime is such a physical fact--if there is one coordinate system that says that at the moment two clocks pass arbitrarily close to each other, one reads 12:30 while the second reads 1:00, then all coordinate systems must agree that this is what the clocks read at the moment they pass next to each other. Likewise, all coordinate systems must agree on the amount of proper time between two points on an object's worldline (they agree on how much time is ticked by a clock that moves along that path).

Einstein said at one point that he should have called his theory the "theory of invariants" because the name "relativity" gets people confused, the essence of the theory is noting the quantities that do not vary by coordinate systems, those are the ones that can be considered genuinely "physical" quantities.

I think that they are "false" when they step outside of this box where they are understood to be coordinate dependent stantements, and are claimed as some kind of proof that an alernate coordinate system is invalid.

Except this is a strawman, physicists would never claim any coordinate system is "invalid" (what would that even mean?) although they might call some coordinate systems more "unphysical" than others, depending on how easily they could be constructed using actual physical clocks and rulers.

 

[Aether]

No, they will be slightly different. This is easier to see in my spaceship example where the velocities are higher...Maybe not at the precision of your calculations, but at a high enough precision it would be. Again, look at my spaceship example. No, my example doesn't assume the one-way speed of light is constant, at least not in all cases. I calculated the ship's velocity and the coordinates of different events using two separate assumptions, the first being the usual SR assumption that the one-way speed is constant and the second being the assumption that I am moving at 0.6c relative to the ether frame and using the LET coordinate system, and I showed that the coordinate velocity of the ship is different in these two cases.

OK, that's not what I expected. I'm calculating the original example to higher precision, and then I'll move on to your new example and DrGreg & pervect's example.

OK, I thought you had said earlier that your notion of LET was only about a different coordinate transform and included no new physical assumptions, but it doesn't matter too much either way.

I'm using LET as a label for what Mansouri-Sexl are talking about, and I don't want spoil their analysis by adding assumptions to it. So, I would answer "does LET assume an ether" and "do you think there is an aether" slightly differently.

By the way, I think you're misusing the term "metric" here, my understanding is that the metric is just the thing that tells you the coordinate-independent proper time between two points in spacetime, I don't think a different choice of coordinate systems qualifies as a different metric.

I think that the Minkowski metric $$\eta_{\mu \nu}$$ is what embodies Lorentz symmetry in both SR and LET, and that both transforms use it as a catalyst. I think that all empirical results apply to the metric, and that the Lorentz transform only stands out when the metric is completely symmetric, and the LET transform only stands out when it is not. Is that misusing the term?

Not true, there are plenty of empirical measurements that are not affected by your choice of coordinate system--this was the point I tried to make in my last long post to you on the "relativity without aether" thread where I accused you of being confused about the distinction between physical facts and coordinate-dependent statements. For example, the question of whether two events coincide at the same point in spacetime is such a physical fact--if there is one coordinate system that says the event of one clock ticking 12:30 happens arbitrarily close to the event of another passing clock ticking 1:00, then all coordinate systems must agree on this. Likewise, all coordinate systems must agree on the amount of proper time between two points on an object's worldline (they agree on how much time is ticked by a clock that moves along that path).

I'm saying that all measurements are of dimensionless ratios, and that they are independent of coordinate systems. Did you think that I meant the opposite?

Einstein said at one point that he should have called his theory the "theory of invariants" because the name "relativity" gets people confused, the essence of the theory is noting the quantities that do not vary by coordinate systems, those are the ones that can be considered genuinely "physical" quantities.

Ha, ha, ha...but Lorentz had already taken the name?

Except this is a strawman, physicists would never claim any coordinate system is "invalid" (what would that even mean?) although they might call some coordinate systems more "unphysical" than others, depending on how easily they could be constructed using actual physical clocks and rulers.

Is LET a coordinate system? If I tell a physicist that in LET the speed of light is a variable, and then he looks at his watch and says that he's late for a meeting with the Hopi elders, does that not imply that he thinks my coordinate system is invalid?

 

[JesseM]

OK, that's not what I expected. I'm calculating the original example to higher precision, and then I'll move on to your new example and DrGreg & pervect's example.

OK, let me know if you see any problems when you look it over. Why do you find this unexpected, by the way? We know it's true that if something is moving at c relative to the LET transform preferred frame, then in another frame its one-way velocity will be different depending on whether we use the LET transform or the Lorentz transform, So it would be pretty weird if this wasn't also true for things moving at sublight velocities...would you naturally expect that if something is moving at 0.999999c relative to the preferred frame, in another frame its velocity will be the same regardless of whether we use the LET transform or the Lorentz transform, but this only ceases to be true if the thing is moving at exactly c?

OK, I thought you had said earlier that your notion of LET was only about a different coordinate transform and included no new physical assumptions, but it doesn't matter too much either way.

I'm using LET as a label for what Mansouri-Sexl are talking about, and I don't want spoil their analysis by adding assumptions to it. So, I would answer "does LET assume an ether" and "do you think there is an aether" slightly differently.

OK, but I already provided a quote by Mansouri and Sexl in that post #85 from the other thread that I think shows they do mean "LET" to include a physical assumption about there being a substance called ether which causes clocks to slow down and rulers to shrink when they move relative to its rest frame. Here's that quote again:

System-internal methods of synchronization are not the only conceivable ones. In section 3 we shall discuss in detail an alternative procedure belonging to the class of system-external synchronization methods. Here one system of reference is singled out ("the ether system") and clocks in all systems are synchronized by comparing them with standard clocks in the preferred system of reference. Infinitely many inequivalent system-external procedures are possible. Among these, one is of special interest: A convention about clock synchronization can be chosen that does maintain absolute simultaneity. Based on this convention an ether theory can be constructed that is, as far as kinematics is concerned (dynamics will be studied in a later paper in this series) equivalent to special relativity. In this theory measuring rods show the standard Fitzgerald-Lorentz contraction and clocks the standard time dilation when moving relative to the ether. Such a theory would have been the logical consequence of the development along the lines of Lorentz-Larmor-Poincaré. That the actual development went along different lines was due to the fact that "local time" was introduced at the early stage in considering the covariance of the Maxwell equations.

By the way, I think you're misusing the term "metric" here, my understanding is that the metric is just the thing that tells you the coordinate-independent proper time between two points in spacetime, I don't think a different choice of coordinate systems qualifies as a different metric.

I think that the Minkowski metric $$\eta_{\mu \nu}$$ is what embodies Lorentz symmetry in both SR and LET, and that both transforms use it as a catalyst. I think that all empirical results apply to the metric, and that the Lorentz transform only stands out when the metric is completely symmetric, and the LET transform only stands out when it is not. Is that misusing the term?

No, but what you said earlier was "I think that the metric represents a physical thing aka 'the aether'"--how does that relate to your comments above? If the metric is just a function that gives you some coordinate-invariant notion of "distance" between points (whether proper time or the spacetime interval), what does it mean to say this function "represents" the aether, when the function is unchanged regardless of whether we assume there's an aether or not, and when even if we do assume on aether, the function is unchanged regardless of what frame happens to be the aether's rest frame? Maybe you just mean that the aether is the cause of rulers shrinking and clocks slowing down, and that once you assume they shrink/slow down in that way you automatically get this metric?

I'm saying that all measurements are of dimensionless ratios, and that they are independent of coordinate systems. Did you think that I meant the opposite?

Yes, I did. The exchange that led up to this was:

So I think you should stop harping on the speed of light and acknowledge that you are making a general point that all speeds are coordinate-dependent.

Acknowledged. Not only for speeds, but every possible empirical measurement.

If you were "acknowledging" there that all speeds are coordinate-dependent, and then adding that this applies not only to speeds but to "every possible empirical measurement", then I interpreted that to mean you were saying every empirical measurement is coordinate-dependent too. If that's not what you meant, what property about speeds were you acknowledging above, and did you mean to say that this property also applies to all empirical measurements?

Except this is a strawman, physicists would never claim any coordinate system is "invalid" (what would that even mean?) although they might call some coordinate systems more "unphysical" than others, depending on how easily they could be constructed using actual physical clocks and rulers.

Is LET a coordinate system?

I didn't mention LET there. But any statements about the one-way velocity of light depend on your choice of coordinate systems--even if the physical aspect of LET were true you'd still be free to use the coordinate systems defined by the Lorentz transform, just as in the real world you're free to use the coordinate systems defined by what we have been calling the "LET transform". Of course, the coordinate systems defined by the LET transform might be more "natural" to use in a universe where the physical assumptions of LET are true, just as the coordinate systems defined by the Lorentz transform are more "natural" for us to use given what we currently know about the laws of physics (since you haven't responded to my comments about constructing coordinate systems in windowless boxes from post #85 I'm not sure whether you agree with this last point about the Lorentz coordinate systems being more natural to use given what we know today).

If I tell a physicist that in LET the speed of light is a variable

Again, by "LET" do you mean to refer to some physical assumptions about the existence of ether (as I think Mansouri and Sexl do), or do you just mean a new coordinate transformation without any new physical assumptions whatsoever?

and then he looks at his watch and says that he's late for a meeting with the Hopi elders, does that not imply that he thinks my coordinate system is invalid?

Is the "Hopi elders" thing supposed to mean he's calling you a crackpot or something? If so, it probably just means he hasn't looked into what you mean by "LET" in detail--if you do just mean it to refer to a different set of coordinate systems without any novel physical assumptions, then if you made this clear I'm sure he'd agree it's permissible to use any crazy coordinate system you want as long as you adjust the equations you use to express the laws of physics to fit this coordinate system.

 

[Aether]

OK, let me know if you see any problems when you look it over. Why do you find this unexpected, by the way? If we know it's true that if something is moving at c relative to the LET transform preferred frame, then in another frame its one-way velocity will be different depending on whether we use the LET transform or the Lorentz transform, then it would be pretty weird if this wasn't also true for things moving at sublight velocities...would you naturally expect that if something is moving at 0.999999c relative to the preferred frame, in another frame its velocity will be the same regardless of whether we use the LET transform or the Lorentz transform, but this only ceases to be true if the thing is moving at exactly c?

Why didn't you use the Mansouri-Sexl Lorentz transform equations? I suppose that LET should be identical to two Lorentz transforms: first transform from frame A to the ether frame, and then transform from the ether frame to frame B. I'm going try it that way too.

OK, but I already provided a quote by Mansouri and Sexl in that post #85 from the other thread that I think shows they do mean "LET" to include a physical assumption about there being a substance called ether which causes clocks to slow down and rulers to shrink when they move relative to its rest frame. Here's that quote again:

OK, LET is my label for Mansouri&Sexl's ether theory. I presume that it is the same thing as H.A. Lorentz' ether theory with a few extra parameters added to make it a "test theory of SR", but I haven't suggested that those extra parameters should be set to anything other than their SR expectation values.

No, but what you said earlier was "I think that the metric represents a physical thing aka 'the aether'"--how does that relate to your comments above? If the metric is just a function that gives you some coordinate-invariant notion of "distance" between points (whether proper time or the spacetime interval), what does it mean to say this function "represents" the aether, when the function is unchanged regardless of whether we assume there's an aether or not, and when even if we do assume on aether, the function is unchanged regardless of what frame happens to be the aether's rest frame? Maybe you just mean that the aether is the cause of rulers shrinking and clocks slowing down, and that once you assume they shrink/slow down in that way you automatically get this metric?

If any Lorentz violation is ever found, besides gravity, then I suppose that the metric is what is going to have to bend to account for that. In that case it's not just a function, it represents something physical. To look for a Lorentz violation you first have to imagine what one might look like, and that's what the Mansouri-Sexl and Kostelekey-Mewes test theories are for.

Yes, I did. The exchange that led up to this was: If you were "acknowledging" there that all speeds are coordinate-dependent, and then adding that this applies not only to speeds but to "every possible empirical measurement", then I interpreted that to mean you were saying every empirical measurement is coordinate-dependent too. If that's not what you meant, what property about speeds were you acknowledging above, and did you mean to say that this property also applies to all empirical measurements?

Speed has dimension and is coordinate dependent, and all other dimensionful quantities are also coordinate dependent. Empirical measurements always start out as coordinate independent dimensionless ratios.

I didn't mention LET there. But any statements about the one-way velocity of light depend on your choice of coordinate systems--even if the physical aspect of LET were true you'd still be free to use the coordinate systems defined by the Lorentz transform, just as in the real world you're free to use the coordinate systems defined by what we have been calling the "LET transform". Of course, the coordinate systems defined by the LET transform might be more "natural" to use in a universe where the physical assumptions of LET are true, just as the coordinate systems defined by the Lorentz transform are more "natural" for us to use given what we currently know about the laws of physics (since you haven't responded to my comments about constructing coordinate systems in windowless boxes from post #85 I'm not sure whether you agree with this last point about the Lorentz coordinate systems being more natural to use given what we know today).

I haven't been convinced by the math yet, but I am now viewing the Lorentz transform as an analogy to the homogeneity and isotropy of the Universe; e.g., that there is one and only one local frame in which the universe looks homogeneous and isotropic. I can appreciate both of these concepts as brilliant coping strategies for life in a world where there is no locally preferred frame, but not as a deterrent to doing experiments to detect a preferred frame.

Again, by "LET" do you mean to refer to some physical assumptions about the existence of ether (as I think Mansouri and Sexl do), or do you just mean a new coordinate transformation without any new physical assumptions whatsoever?

I want to have the tools to model Lorentz symmetry violations so that I will know where to look for them, and so that I would recognize it in case I ever saw one. If a Lorentz symmetry violation automatically equates to the existence of ether, then I'm looking for ether; if not, then I'm just looking for Lorentz symmetry violation.

Is the "Hopi elders" thing supposed to mean he's calling you a crackpot or something? If so, it probably just means he hasn't looked into what you mean by "LET" in detail--if you do just mean it to refer to a different set of coordinate systems without any novel physical assumptions, then if you made this clear I'm sure he'd agree it's permissible to use any crazy coordinate system you want as long as you adjust the equations you use to express the laws of physics to fit this coordinate system.

That's how I interpret that, yes. Why are there crackpots identified with this issue in the first place? They might not be out in the cold if they got the right answers to these questions about coordinate systems way back when when they first asked their teachers: "teacher, why can't simultaneity be absolute?"..."well, it could be if we could find some signal that we could use as a standard reference for time, but no experiment has ever been able to detect such a thing so far." Instead they get "because it just isn't that way, and experiments are proving that every day in particle accelerators all over the world...".

 

[JesseM]

Why didn't you use the Mansouri-Sexl Lorentz transform equations? I suppose that LET should be identical to two Lorentz transforms? First transform from frame A to the ether frame, and then transform from the ether frame to frame B?

I did use the Mansouri-Sexl equations! I started out by figuring out what things would look like in my frame if I used the Einstein synchronization procedure, then I used the ordinary Lorentz transform to see what the coordinates of the same events would be in the frame of an observer moving at 0.6c relative to me who also synchronized his clocks using the Einstein procedure, then I assumed that observer was at rest relative to the ether (remember, in LET the observer at rest relative to the ether is the one who still uses the Einstein synchronization procedure) and I used the Mansouri-Sexl equations with v=0.6c to see what coordinates the events would have in my rest frame if I synchronized my clocks with the ether frame.

If any Lorentz violation is ever found, besides gravity, then I suppose that the metric is what is going to have to bend to account for that.

I don't see why it would--if the metric just tells you the proper time ticked by a clock moving along any given path through spacetime, this shouldn't change even if we discover a preferred frame (unless we discover a new type of 'clock' that does not exhibit the same type of time dilation as normal clocks, in which case I'm not sure what you'd have to do then--have two separate metrics for the different classes of clocks, maybe?)

In that case it's not just a function, it represents something physical.

Even if you say it's not "just" a function, mathematically it would still be a function defining a notion of distance in spacetime, and I don't understand what it means to say this function would "represent something physical". Can you pin down what you mean by "represent"?

To look for a Lorentz violation you first have to imagine what one might look like, and that's what the Mansouri-Sexl and Kostelekey-Mewes test theories are for.

Do Mansouri and Sexl say that? I thought they were just showing that it was possible to have a non-Lorentz-violating ether theory which was empirically equivalent to SR.

If you were "acknowledging" there that all speeds are coordinate-dependent, and then adding that this applies not only to speeds but to "every possible empirical measurement", then I interpreted that to mean you were saying every empirical measurement is coordinate-dependent too. If that's not what you meant, what property about speeds were you acknowledging above, and did you mean to say that this property also applies to all empirical measurements?

Speed has dimension and is coordinate dependent, and all other dimensionful quantities are also coordinate dependent. Empirical measurements always start out as coordinate independent dimensionless ratios.

That doesn't really answer my question of why you said "Acknowledged. Not only for speeds, but every possible empirical measurement", which seems to suggest the exact opposite (namely, that whatever is true about speeds is also true for every possible empirical measurement). But I suppose it's not really important, now that we're clear on the fact that you agree all measurements of speed are coordinate-dependent while other, more truly "physical" quantities are not.

I didn't mention LET there. But any statements about the one-way velocity of light depend on your choice of coordinate systems--even if the physical aspect of LET were true you'd still be free to use the coordinate systems defined by the Lorentz transform, just as in the real world you're free to use the coordinate systems defined by what we have been calling the "LET transform". Of course, the coordinate systems defined by the LET transform might be more "natural" to use in a universe where the physical assumptions of LET are true, just as the coordinate systems defined by the Lorentz transform are more "natural" for us to use given what we currently know about the laws of physics (since you haven't responded to my comments about constructing coordinate systems in windowless boxes from post #85 I'm not sure whether you agree with this last point about the Lorentz coordinate systems being more natural to use given what we know today).

I haven't been convinced by the math yet

Convinced of what? Of my points about observers in windowless boxes being able to construct devices whose measurements correspond to the coordinates of the Lorentz transform? The idea of constructing coordinate systems out of rulers and clocks synchronized using the Einstein synchronization procedure (which does not require any knowledge of the outside world) is the whole basis for the derivation of the Lorentz transformation, so it's pretty important that you understand it--if you're in doubt about this, please respond to my last comments in that post.

but I am now viewing the Lorentz transform as an analogy to the homogeneity and isotropy of the Universe; e.g., that there is one and only one local frame in which the universe looks homogeneous and isotropic.

Are you suggesting that the "homogeneity and isotropy of the universe" defines a preferred frame? Doesn't a "local frame" in GR by definition only deal with an arbitrarily small patch of spacetime, so a statement about what the universe as a whole looks like could only be made in a global coordinate system? Also, why would the Lorentz transform be an "analogy" for the idea of a global preferred frame? An analogy in what sense? And what does this have to do with the question of whether the Lorentz transform is the most "natural" one for us to use given what we know about the laws of physics?

I want to have the tools to model Lorentz symmetry violations so that I will know where to look for them, and so that I would recognize it in case I ever saw one. If a Lorentz symmetry violation automatically equates to the existence of ether, then I'm looking for ether; if not, then I'm just looking for Lorentz symmetry violation.

A lorentz symmetry violation need not lead to the idea of an "ether" in the sense of a substance that electromagnetic waves are actually sound waves in, but I think (though I'm not sure) that it would always lead you to the idea of a preferred frame of some sort.

But anyway, a different coordinate system won't exactly help you "model" new physical phenomena--any physical phenomena can be analyzed in any coordinate system you wish--but it may help you in the sense that the equations of any new laws that are discovered will have a simpler form when expressed in one set of coordinate systems than another. But it seems no less conceivable that a new set of laws would be most simply expressed in terms of the coordinate systems given by the Galilei transform than the ones given by the Mansouri-Sexl transformation equations.

That's how I interpret that, yes. Why are there crackpots identified with this issue in the first place? They might not be out in the cold if they got the right answers to these questions about coordinate systems way back when when they first asked their teachers: "teacher, why can't simultaneity be absolute?"..."well, it could be if we could find some signal that we could use as a standard reference for time, but no experiment has ever been able to detect such a thing so far." Instead they get "because it just isn't that way, and experiments are proving that every day in particle accelerators all over the world...".

Again, the issue of absolute vs. relative simultaneity is fundamentally just a matter of what coordinate systems you use, even with the known laws nothing is stopping you from using the Mansouri-Sexl coordinate systems to describe the world...experimental results only come into play in deciding which set of coordinate systems is more of a "natural" choice, and in that sense the results seen in particle accelerators are relevant because they keep showing that all the fundamental laws have the property of Lorentz-symmetry rather than "LET transform symmetry" (or whatever you want to call it).

 

[Aether]

I don't see why it would--if the metric just tells you the proper time ticked by a clock moving along any given path through spacetime, this shouldn't change even if we discover a preferred frame (unless we discover a new type of 'clock' that does not exhibit the same type of time dilation as normal clocks, in which case I'm not sure what you'd have to do then--have two separate metrics for the different classes of clocks, maybe?)

If a Lorentz violation leads to a preferred frame and that leads to absolute simultaneity, then I suppose that a Lorentz violation leads to an absolute clock.

Even if you say it's not "just" a function, mathematically it would still be a function defining a notion of distance in spacetime, and I don't understand what it means to say this function would "represent something physical". Can you pin down what you mean by "represent"?

The components of the metric can be regarded as representing gravitational potentials.

Do Mansouri and Sexl say that? I thought they were just showing that it was possible to have a non-Lorentz-violating ether theory which was empirically equivalent to SR.

The second paragraph of their first paper starts with "Here we shall investigate the effects of deviations from special relativity on the outcome of various experiments. For this purpose a class of rival theories has to be defined to be compared with relativity."

That doesn't really answer my question of why you said "Acknowledged. Not only for speeds, but every possible empirical measurement", which seems to suggest the exact opposite (namely, that whatever is true about speeds is also true for every possible empirical measurement). But I suppose it's not really important, now that we're clear on the fact that you agree all measurements of speed are coordinate-dependent while other, more truly "physical" quantities are not.

I meant that speed is a dimensionful quantity, and no experiment can directly measure any dimensionful quantity.

Convinced of what? Of my points about observers in windowless boxes being able to construct devices whose measurements correspond to the coordinates of the Lorentz transform? The idea of constructing coordinate systems out of rulers and clocks synchronized using the Einstein synchronization procedure (which does not require any knowledge of the outside world) is the whole basis for the derivation of the Lorentz transformation, so it's pretty important that you understand it--if you're in doubt about this, please respond to my last comments in that post.

I haven't gone through everything carefully yet, so I'm not saying that it's unconvincing; I'm saying that I haven't completed a thorough review of it all yet.

Are you suggesting that the "homogeneity and isotropy of the universe" defines a preferred frame?Doesn't a "local frame" in GR by definition only deal with an arbitrarily small patch of spacetime, so a statement about what the universe as a whole looks like could only be made in a global coordinate system? Also, why would the Lorentz transform be an "analogy" for the idea of a global preferred frame? An analogy in what sense? And what does this have to do with the question of whether the Lorentz transform is the most "natural" one for us to use given what we know about the laws of physics?

I'm saying that what I read in MTW about what is special about comoving coordinates are that the universe looks the same from every perspective because there is no "handle" rings a bell here.

A lorentz symmetry violation need not lead to the idea of an "ether" in the sense of a substance that electromagnetic waves are actually sound waves in, but I think (though I'm not sure) that it would always lead you to the idea of a preferred frame of some sort.But anyway, a different coordinate system won't exactly help you "model" new physical phenomena--any physical phenomena can be analyzed in any coordinate system you wish--but it may help you in the sense that the equations of any new laws that are discovered will have a simpler form when expressed in one set of coordinate systems than another. But it seems no less conceivable that a new set of laws would be most simply expressed in terms of the coordinate systems given by the Galilei transform than the ones given by the Mansouri-Sexl transformation equations.

I like Hurkyl's input on Minkowski geometry, and hope to find something along those lines.

Again, the issue of absolute vs. relative simultaneity is fundamentally just a matter of what coordinate systems you use, even with the known laws nothing is stopping you from using the Mansouri-Sexl coordinate systems to describe the world...experimental results only come into play in deciding which set of coordinate systems is more of a "natural" choice, and in that sense the results seen in particle accelerators are relevant because they keep showing that all the fundamental laws have the property of Lorentz-symmetry rather than "LET transform symmetry" (or whatever you want to call it).

If "the aether" is the same thing as "absolute simultaneity", then what you are describing is not "relativity without the aether" is it?