Einstein's Clock Synchronization Convention

[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?

 

[JesseM]

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.

Sure. But do you disagree with my speculation that this would require us to have separate metrics for the two types of clocks, rather than modifying the original metric?

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.

Are you sure about that? I thought the concept of gravitational potential didn't make sense in GR. But either way, what does this have to do with aether? If you're saying the "something physical" the metric represents is gravitational potential, then doesn't it already represent that even if there is no aether?

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.

That still doesn't help me make sense of your original comment that you were acknowledging something was true "not only for speeds, but for every possible empirical measurement". If you're now saying the thing you were acknowledging is that speed is a dimensionful quantity, does this sentence say that all empirical measurements are dimensionful quantities? That certainly isn't true--for instance, the ratio between the number of ticks on two clocks which take different paths through spacetime is an empirical quantity that's not dimensionful, and likewise the empirical determination of whether two events coincide at the same point in spacetime is not a dimensionful quantity (it's not a 'quantity' at all). edit: also, I just wanted to add that even dimensionful quantities need not be coordinate-dependent--if we ask how many seconds are ticked by a clock that takes a certain path through spacetime, different observers in different frames systems won't disagree on this, assuming they all define "1 second" in the same way (say, as 9,192,631,770 oscillations of a cesium atom). Do you understand the difference between saying a quantity is dependent on your choice of coordinate system and saying it is dimensionful? They are not the same thing at all.

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.

Fair enough. But again, this isn't an argument original to me, it's the standard way that I've always seen the Lorentz transform derived.

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'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.

When you say "rings a bell", does that mean it's just sort of a vague intuition that there's some analogy there but don't have a clear idea of what the analogy is? Because this explanation still doesn't give me any clear idea of what you meant when you said "I am now viewing the Lorentz transform as an analogy to the homogeneity and isotropy of the Universe".

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?

But the aether is not the same thing as absolute simultaneity, that's what I just said in the paragraph you were responding to! Simultaneity is purely a question of your choice of coordinate systems, and you are free to use any set of coordinate systems you want regardless of what the laws of physics are. Again, the laws of physics may make one choice of coordinate system more "natural" than another, but that doesn't mean you can't use unnatural coordinate systems. Even if there is no aether, you can still pick an arbitrary observer to be the "preferred" one and have all other observers synchronize their clocks so that their definition of simultaneity agrees with his; likewise, even if there is an aether you are still free to have each observer synchronize their clocks under the assumption that light moves at c in their rest frames, which will lead different observers to have different definitions of simultaneity.

 

[Aether]

Sure. But do you disagree with my speculation that this would require us to have separate metrics for the two types of clocks, rather than modifying the original metric?

No, I don't disagree, but I'm not settled on how to model every possible Lorentz violation yet. The Minkowski metric is never going to change no matter what new experiments might prove, so an asymmetrical metric would have to be a new metric. However, I like the Minkowski metric just fine as it is. So I may end up keeping the Minkowski metric but treating ds as a vector rather than as a scalar for example. I may also choose to leave both of those alone and just use different coordinates.

Are you sure about that? I thought the concept of gravitational potential didn't make sense in GR. But either way, what does this have to do with aether? If you're saying the "something physical" the metric represents is gravitational potential, then doesn't it already represent that even if there is no aether?

One of my GR textbooks says that "In a more general coordinate system, this Newtonian potential would be dispersed throughout the g_{\mu \nu}, so there is a sense in which all the components g_{\mu \nu} can be regarded as gravitational potentials."

That still doesn't help me make sense of your original comment that you were acknowledging something was true "not only for speeds, but for every possible empirical measurement". If you're now saying the thing you were acknowledging is that speed is a dimensionful quantity, does this sentence say that all empirical measurements are dimensionful quantities? That certainly isn't true--for instance, the ratio between the number of ticks on two clocks which take different paths through spacetime is an empirical quantity that's not dimensionful, and likewise the empirical determination of whether two events coincide at the same point in spacetime is not a dimensionful quantity (it's not a 'quantity' at all).

Dimensionless ratios are measurable, and they are coordinate independent. Nothing else is directly measurable.

edit: also, I just wanted to add that even dimensionful quantities need not be coordinate-dependent--if we ask how many seconds are ticked by a clock that takes a certain path through spacetime, different observers in different frames systems won't disagree on this, assuming they all define "1 second" in the same way (say, as 9,192,631,770 oscillations of a cesium atom). Do you understand the difference between saying a quantity is dependent on your choice of coordinate system and saying it is dimensionful? They are not the same thing at all.

Defining "1 second" that way makes your measurement a dimensionless ratio of the number of oscillations of a cesium atom in your traveling clock per 9,192,631,770 oscillations of a reference cesium atom in one second. Isn't the frame of this reference cesium atom coordinate dependent?

When you say "rings a bell", does that mean it's just sort of a vague intuition that there's some analogy there but don't have a clear idea of what the analogy is? Because this explanation still doesn't give me any clear idea of what you meant when you said "I am now viewing the Lorentz transform as an analogy to the homogeneity and isotropy of the Universe".

At this point it is more than vague intuition, but not quite a clear and convincing understanding. I understand that the homogeneity and isotropy of the universe is not something that is generally true for an observer; to make this true, he must take great pains to seek out a unique local frame. Once he is in that particular frame, then the universe looks homogeneous and isotropic. The Lorentz transform is analogous in that there is one and only one local frame in which the observer sees "something" in the same way as every other observer, and this happens to be his own rest frame (which happens to be much easier to locate than the comoving frame).

But the aether is not the same thing as absolute simultaneity, that's what I just said in the paragraph you were responding to! Simultaneity is purely a question of your choice of coordinate systems, and you are free to use any set of coordinate systems you want regardless of what the laws of physics are.

OK, well I'm using "aether" to mean "absolute simultaneity" as opposed to a luminiferous fluid. I suppose that we may discuss "objective absolute simultaneity" which is a physical consequence of any Lorentz violation, and "subjective absolute simultaneity" which is merely a coordinate system trick. So may we agree that "relativity without at least the 'subjective aether' is pseudoscience"?

Again, the laws of physics may make one choice of coordinate system more "natural" than another, but that doesn't mean you can't use unnatural coordinate systems. Even if there is no aether, you can still pick an arbitrary observer to be the "preferred" one and have all other observers synchronize their clocks so that their definition of simultaneity agrees with his; likewise, even if there is an aether you are still free to have each observer synchronize their clocks under the assumption that light moves at c in their rest frames, which will lead different observers to have different definitions of simultaneity.

This does not describe SR per se. It shows that SR is one carefully chosen slice of a more general science which also allows for LET.

 

[Aether]

From post #45: <br /> \frac {u_{LET}}{u_{SR}} = \frac { T_{SR1} - T_{SR0} }{ T_{SR1} - T_{SR0} + \frac{v \left( X1-X0 \right) }{c^2} }<br />

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

OK, but SR and LET are supposed to be empirically equivalent. So, I wonder if we aren't also supposed to synchronize the two SR clocks before using them with the LET transform and thereby annihilate any differences between the two.

DrGreg, after studying the Mansouri-Sexl papers, do you still think that this is exactly how they intended for their ether transform to be applied? If so, then I'll agree that velocities are anisotropic in LET.

 

[Hans de Vries]

Speeds are anisotropic in LET.

OK, but SR and LET are supposed to be empirically equivalent. So, I wonder if we aren't also supposed to synchronize the two SR clocks before using them with the LET transform and thereby annihilate any differences between the two.

DrGreg, after studying the Mansouri-Sexl papers, do you still think that this is exactly how they intended for their ether transform to be applied? If so, then I'll agree that velocities are anisotropic in LET.

Speeds are anisotropic in LET


Take an SR frame moving at v relative to the preferred frame. Define two
velocities w and -w in this frame moving in the same and in the opposite
direction of v. Via Einsteins velocity addition rule we can calculate what
these two speeds are in the preferred frame:

<br /> v_1\ = \ \frac{v+w}{1+\frac{vw}{c^2}}, \ \ \ \ \ \ \ v_2\ = \ <br /> \frac{v-w}{1-\frac{vw}{c^2}} <br />

(see http://www.theory.caltech.edu/people/patricia/minkc.html or
http://www.phys.unsw.edu.au/einsteinlight/jw/module4_Lorentz_transforms.htm)

Now take the M&S formula's for the transformation to the moving LET frame:

<br /> t\ =\ \sqrt{1-v^2/c^2}\ \ T,\ \ \ \ \ \ \ \ x\ =\ <br /> \frac{1}{\sqrt{1-v^2/c^2}}\ (X-vT) <br />

Which we may write in differential form:

<br /> dt\ =\ \sqrt{1-v^2/c^2}\ \ dT,\ \ \ \ \ \ \ \ dx\ =\ <br /> \frac{1}{\sqrt{1-v^2/c^2}}\ (dX-vdT) <br />

From this we can derive the transformation of velocities from the preferred
frame to the moving LET frame:

<br /> \frac{dx}{dt}\ =\ \frac{1}{1-v^2/c^2}\ \left(\frac{dX}{dT}-v <br /> \right) <br />

Now, substitute with v1 and v2:

<br /> w_1\ = \ \frac{1}{1-v^2/c^2}\ \left(\frac{v+w}{1+\frac{vw}{c^2}} <br /> -v \right), \ \ \ \ \ \ \ w_2\ = \ \frac{1}{1-v^2/c^2}\ <br /> \left(\frac{v-w}{1-\frac{vw}{c^2}}-v \right) <br />

Which can be simplified further. Now, w and -w in the moving SR frame become:

<br /> w_1\ = \ \frac{w}{1+\frac{vw}{c^2}}, \ \ \ \ \ \ \ w_2\ = \ <br /> \frac{-w}{1-\frac{vw}{c^2}} <br />

In the moving LET frame. (You better believe it )

------------------------------------


Regards, Hans

 

[JesseM]

No, I don't disagree, but I'm not settled on how to model every possible Lorentz violation yet. The Minkowski metric is never going to change no matter what new experiments might prove, so an asymmetrical metric would have to be a new metric. However, I like the Minkowski metric just fine as it is. So I may end up keeping the Minkowski metric but treating ds as a vector rather than as a scalar for example.

How can ds be a vector? And since the whole concept of a metric is to define a notion of distance between points on the manifold, you still need a scalar rather than a vector to use in the definition of the metric, don't you?

I may also choose to leave both of those alone and just use different coordinates.

But again, using different coordinates doesn't involve any different physical assumptions in itself.

Are you sure about that? I thought the concept of gravitational potential didn't make sense in GR. But either way, what does this have to do with aether? If you're saying the "something physical" the metric represents is gravitational potential, then doesn't it already represent that even if there is no aether?

One of my GR textbooks says that "In a more general coordinate system, this Newtonian potential would be dispersed throughout the g_{\mu \nu}, so there is a sense in which all the components g_{\mu \nu} can be regarded as gravitational potentials."

Well, I'd like to know the details of what that "sense" is, whether it only works in some limiting case or whether it these components could rigourously be viewed as potentials in an arbitrary curved spacetime. In any case, you didn't answer the second part of my question, the one beginning with "But either way, what does this have to do with aether?"

That still doesn't help me make sense of your original comment that you were acknowledging something was true "not only for speeds, but for every possible empirical measurement". If you're now saying the thing you were acknowledging is that speed is a dimensionful quantity, does this sentence say that all empirical measurements are dimensionful quantities? That certainly isn't true--for instance, the ratio between the number of ticks on two clocks which take different paths through spacetime is an empirical quantity that's not dimensionful, and likewise the empirical determination of whether two events coincide at the same point in spacetime is not a dimensionful quantity (it's not a 'quantity' at all).

Dimensionless ratios are measurable, and they are coordinate independent. Nothing else is directly measurable.

I agree, but you pretty much ignored the point I was making in that paragraph, which is that plenty of "empirical measurements" are dimensionless ratios, like the two I mentioned above. So it still seems wrong for you to have said "not only for speeds, but for every possible empirical measurement" there.

edit: also, I just wanted to add that even dimensionful quantities need not be coordinate-dependent--if we ask how many seconds are ticked by a clock that takes a certain path through spacetime, different observers in different frames systems won't disagree on this, assuming they all define "1 second" in the same way (say, as 9,192,631,770 oscillations of a cesium atom). Do you understand the difference between saying a quantity is dependent on your choice of coordinate system and saying it is dimensionful? They are not the same thing at all.

Defining "1 second" that way makes your measurement a dimensionless ratio of the number of oscillations of a cesium atom in your traveling clock per 9,192,631,770 oscillations of a reference cesium atom in one second. Isn't the frame of this reference cesium atom coordinate dependent?

What measurement are we talking about here? I wasn't talking about measuring the ratio of ticks of two clocks moving at different velocities, I was just talking about measuring the amount of proper time ticked on a single clock which takes a certain path through spacetime. If everyone agrees to define "1 second" the same way--say, by 9,192,631,770 oscillations of a cesium atom moving alongside the clock--then don't you agree that although proper time is a dimensionful quantity, it is coordinate-independent in the sense that all frames will agree on the amount of proper time ticked by a clock that takes a certain path through spacetime?

At this point it is more than vague intuition, but not quite a clear and convincing understanding. I understand that the homogeneity and isotropy of the universe is not something that is generally true for an observer; to make this true, he must take great pains to seek out a unique local frame. Once he is in that particular frame, then the universe looks homogeneous and isotropic. The Lorentz transform is analogous in that there is one and only one local frame in which the observer sees "something" in the same way as every other observer, and this happens to be his own rest frame (which happens to be much easier to locate than the comoving frame).

Wait, what do you mean when you say there's only one local frame in which the observer sees "something"? What is this "something"? The whole point of Lorentz-symmetry is that the laws of physics will look the same in every frame, there's no frame that's "special" in any way. Also, even leaving aside the question of the laws of physics, the Lorentz transform is also symmetric in the sense that any observer who sees another observer moving at velocity v relative to him will use the same equations to transform into that observer's frame, unlike in the LET transform where you don't use the same equation to transform from the preferred frame to another frame as you would to transform from a non-preferred frame to the preferred frame, or to another non-preferred frame.

OK, well I'm using "aether" to mean "absolute simultaneity" as opposed to a luminiferous fluid. I suppose that we may discuss "objective absolute simultaneity" which is a physical consequence of any Lorentz violation, and "subjective absolute simultaneity" which is merely a coordinate system trick.

But all statements about simultaneity are just properties of your coordinate system, I have no idea what you mean by "objective absolute simultaneity". Again, I agree that Lorentz violations might make a coordinate transform which preserves absolute simultaneity to be more "natural" than the Lorentz transform, but it seems like you're talking about something more than just which type of coordinate transform is most "natural".

So may we agree that "relativity without at least the 'subjective aether' is pseudoscience"?

I don't understand what you mean by "relativity without at least the 'subjective aether'". Do you mean relativity that denies the possibility of using coordinate transforms other than the Lorentz transform? I would agree that it is wrong to deny that you're allowed to use any set of coordinate systems you like, if that's what you're saying.

Again, the laws of physics may make one choice of coordinate system more "natural" than another, but that doesn't mean you can't use unnatural coordinate systems. Even if there is no aether, you can still pick an arbitrary observer to be the "preferred" one and have all other observers synchronize their clocks so that their definition of simultaneity agrees with his; likewise, even if there is an aether you are still free to have each observer synchronize their clocks under the assumption that light moves at c in their rest frames, which will lead different observers to have different definitions of simultaneity.

This does not describe SR per se. It shows that SR is one carefully chosen slice of a more general science which also allows for LET.

I think we might disagree on the meaning of "SR". I would say that SR just refers to the hypothesis that the laws of physics are all Lorentz-symmetric (including any unobserved entities which play a fundamental role in the laws of physics, ruling out 'aether'), but does not specify that you must use the coordinate systems produced by the Lorentz transform. For example, I have seen physicists say that SR allows for accelerated coordinate systems, which are clearly not allowed under the Lorentz transform.

 

[Aether]

How can ds be a vector?And since the whole concept of a metric is to define a notion of distance between points on the manifold, you still need a scalar rather than a vector to use in the definition of the metric, don't you? But again, using different coordinates doesn't involve any different physical assumptions in itself.

ds^2=dA^2+dB^2+dC^2

Well, I'd like to know the details of what that "sense" is, whether it only works in some limiting case or whether it these components could rigourously be viewed as potentials in an arbitrary curved spacetime. In any case, you didn't answer the second part of my question, the one beginning with "But either way, what does this have to do with aether?"

That's only intended as an example of what it might mean for the metric to "represent" something physical.

I agree, but you pretty much ignored the point I was making in that paragraph, which is that plenty of "empirical measurements" are dimensionless ratios, like the two I mentioned above. So it still seems wrong for you to have said "not only for speeds, but for every possible empirical measurement" there.

My original statement may not have been self explanatory, but I don't see how there can still be any room left for doubt that I'm saying that all empirical measurements are dimensionless ratios.

What measurement are we talking about here? I wasn't talking about measuring the ratio of ticks of two clocks moving at different velocities, I was just talking about measuring the amount of proper time ticked on a single clock which takes a certain path through spacetime. If everyone agrees to define "1 second" the same way--say, by 9,192,631,770 oscillations of a cesium atom moving alongside the clock--then don't you agree that although proper time is a dimensionful quantity, it is coordinate-independent in the sense that all frames will agree on the amount of proper time ticked by a clock that takes a certain path through spacetime?

OK, so there is no reference cesium atom needed in this clock, you're counting oscillations and dividing by 9,192,631,770. Perhaps each oscillation of the cesium atom is what is actually measured as a dimensionless ratio against a voltage standard or something like that, and each count is its own dimensionless ratio.

Wait, what do you mean when you say there's only one local frame in which the observer sees "something"? What is this "something"? The whole point of Lorentz-symmetry is that the laws of physics will look the same in every frame, there's no frame that's "special" in any way. Also, even leaving aside the question of the laws of physics, the Lorentz transform is also symmetric in the sense that any observer who sees another observer moving at velocity v relative to him will use the same equations to transform into that observer's frame, unlike in the LET transform where you don't use the same equation to transform from the preferred frame to another frame as you would to transform from a non-preferred frame to the preferred frame, or to another non-preferred frame.

This is not a developed notion just a beginning of appreciation.

But all statements about simultaneity are just properties of your coordinate system, I have no idea what you mean by "objective absolute simultaneity".

"Objective" would apply if I could actually detect a locally preferred frame, but "subjective" applies while I still can't.

Again, I agree that Lorentz violations might make a coordinate transform which preserves absolute simultaneity to be more "natural" than the Lorentz transform, but it seems like you're talking about something more than just which type of coordinate transform is most "natural". I don't understand what you mean by "relativity without at least the 'subjective aether'". Do you mean relativity that denies the possibility of using coordinate transforms other than the Lorentz transform? I would agree that it is wrong to deny that you're allowed to use any set of coordinate systems you like, if that's what you're saying. I think we might disagree on the meaning of "SR". I would say that SR just refers to the hypothesis that the laws of physics are all Lorentz-symmetric (including any unobserved entities which play a fundamental role in the laws of physics, ruling out 'aether'), but does not specify that you must use the coordinate systems produced by the Lorentz transform. For example, I have seen physicists say that SR allows for accelerated coordinate systems, which are clearly not allowed under the Lorentz transform.

One postulate specifies that the laws of physics are all Lorentz symmetric, but the other specifies that the speed of light is a constant; that second postulate seems to specify that you must use the coordinate systems produced by the Lorentz transform. So, by "relativity without the aether", I'm referring to the apparent specification in SR that "you must use the coordinate systems produced by the Lorentz transform". That's OK as long as you only apply it to SR per se, but when that is understood as "experimental proof that LET is false" then that is at best unscientific, and possibly even pseudoscientific depending on the circumstances.

 

[DrGreg]

DrGreg, after studying the Mansouri-Sexl papers, do you still think that this is exactly how they intended for their ether transform to be applied? If so, then I'll agree that velocities are anisotropic in LET.

t_LET means time measured by a clock synchronized to the observer's proper-time clock via "absolute simultaneity".

t_SR means time measured by a clock synchronized to the observer's proper-time clock via the Einsteinian method.

The transformation formulas all take into account the method of synchronization and so no further adjustment is necessary.

I'm glad we now agree on something.

 

[DrGreg]

ds^2=dA^2+dB^2+dC^2

You seem to think this implies that ds is a vector.

Everything in this equation is a real number (i.e. one-dimensional) even if dA, dB, and dC are components of a vector.

This particular equation can be interpreted as the metric for 3D Euclidean geometry

ds^2=dx^2+dy^2+dz^2

which is essentially Pythagoras's Theorem in 3 dimensions.

 

[Aether]

You seem to think this implies that ds is a vector.

Everything in this equation is a real number (i.e. one-dimensional) even if dA, dB, and dC are components of a vector.

This particular equation can be interpreted as the metric for 3D Euclidean geometry

ds^2=dx^2+dy^2+dz^2

which is essentially Pythagoras's Theorem in 3 dimensions.

In post #14 I gave a quote from one of Paul Dirac's books which indicates that the instantaneous velocity of any particle is c, but that the direction of this vector oscillates very rapidly (&gt;10^{20} cycles per second) about a mean value of dv=dx+dy+dz. When dv=0, then ds=cdt, and at any given instant this is evidently a vector ds=dA+dB+dC=cdt (where dA is parallel to dx, dB is parallel to dy, and dC is parallel to dz). It only appears to be a scalar when you smear it over a relatively long time period. That looks like a violation of Lorentz symmetry to me, but it is changing so fast that the effect can easily be overlooked in experiments where speeds are averaged over any suitably long time period. Wouldn't it be interesting if two particles 1km apart had ds vectors that pointed instantaneously in the same direction at all times? I don't know that they actually do that, but would like to have metrics and transforms which are general enough to allow me to examine all such possibilities. Lorentz transforms and LET transforms each preserve the magnitude of the line interval ds, but I prefer to have a transform handy that preserves both the magnitude and direction of ds.

 

[DrGreg]

In post #14 I gave a quote from one of Paul Dirac's books...

I can't comment on this quote because

• I don't have a copy of this book and I can't see the context of the quote
• I don't know a great deal about quantum theory anyway

But you are misunderstanding the notation.

ds is always a scalar, never a vector. In the context of a (t, x, y, z) co-ordinate system, it is the length of the vector (dt, dx, dy, dz).

The equations in your last post make no sense to me and I can't understand what you are trying to say.


Note that in a printed book, scalars and other 1-dimensional quantities are written in italics, while vectors are often written in bold font. (Though not all authors follow this convention, especially in pure maths. In GR, vectors may be indicated instead by Greek superscripts such as X^\mu. (But, just to confuse you further, a numerical superscript like X^0 indicates a single component of a vector.))

 

[JesseM]

ds^2=dA^2+dB^2+dC^2

What do dA, dB and dC represent? In any case, this equation would still indicate that ds is a scalar rather than a vector, since it's the norm of the vector (dA, dB, dC).

That's only intended as an example of what it might mean for the metric to "represent" something physical.

But your original comment was not just that it represents "something" physical, but that it represents the aether--you said "I think that the metric represents a physical thing aka 'the aether'". So I still don't understand how a function defining distance in spacetime would "represent" the aether.

My original statement may not have been self explanatory, but I don't see how there can still be any room left for doubt that I'm saying that all empirical measurements are dimensionless ratios.

I agree that all truly empirical measurements are dimensionless, or can be stated in units which are defined in a dimensionless way...I wouldn't say all empirical measurements are dimensionless ratios though (I don't see how counting the number of oscillations of a cesium atom moving along a particular path is a ratio, for example). In any case, this still doesn't help me understand what it was you thought you were acknowledging in your original comment, the thing you were saying was true "not only for speeds, but for every possible empirical measurement"--surely you weren't acknowledging that both empirical measurements and speeds are all dimensionless ratios, because speeds clearly aren't, and the comment of mine you were responding to was one about how speeds are coordinate-dependent. Perhaps we can just agree that your original comment here was confused and move on?

But all statements about simultaneity are just properties of your coordinate system, I have no idea what you mean by "objective absolute simultaneity".

"Objective" would apply if I could actually detect a locally preferred frame, but "subjective" applies while I still can't.

But as I keep saying over and over, even if we did detect a locally preferred frame there would be nothing stopping us from using a coordinate transform that led different reference frames to define simultaneity differently. A coordinate transform where all frames agreed on simultaneity might be more "natural" in this case, but other coordinate transforms would still be permissable. Do you agree or disagree?

I think we might disagree on the meaning of "SR". I would say that SR just refers to the hypothesis that the laws of physics are all Lorentz-symmetric (including any unobserved entities which play a fundamental role in the laws of physics, ruling out 'aether'), but does not specify that you must use the coordinate systems produced by the Lorentz transform. For example, I have seen physicists say that SR allows for accelerated coordinate systems, which are clearly not allowed under the Lorentz transform.

One postulate specifies that the laws of physics are all Lorentz symmetric, but the other specifies that the speed of light is a constant; that second postulate seems to specify that you must use the coordinate systems produced by the Lorentz transform.

I think it's implicit in both of the postulates that they only refer to what would be true in inertial coordinate systems where the clocks are synchronized according to the Einstein synchronization procedure. After all, in Einstein's original paper he spends all of section 1 defining how to construct such coordinate systems out of physical measuring rods and clocks, and then only in section 2 does he lay out the two fundamental postulates. So the postulates don't deny the possibility of creating other types of coordinate systems where the speed of light is no longer constant, as I understand them. And like I said, I have seen physicists say that SR can handle accelerating coordinate systems, and the speed of light would not in general be a constant in such coordinate systems.

So, by "relativity without the aether", I'm referring to the apparent specification in SR that "you must use the coordinate systems produced by the Lorentz transform".

I don't think there is any such specification in "SR" as I understand it.

 

[Aether]

I can't comment on this quote because

• I don't have a copy of this book and I can't see the context of the quote
• I don't know a great deal about quantum theory anyway

The relevant section from the book is section "69. The motion of a free electron" which is only a little over two pages long. If you would like to download a scanned-in copy, let me know and I'll tell you where you can go to download it.

But you are misunderstanding the notation.

ds is always a scalar, never a vector. In the context of a (t, x, y, z) co-ordinate system, it is the length of the vector (dt, dx, dy, dz).

The equations in your last post make no sense to me and I can't understand what you are trying to say.

If I understand correctly what Paul Dirac is saying (and I think I do because I have several of his subsequently published papers and articles in Nature where he says explicitly that "an ether is rather forced upon us" or words to that effect) then I don't see why we can't represent \bold{c_0} dt as a vector, and d \bold{s}=\bold{c_0} dt-dx-dy-dz as a vector. I understand that for many practical purposes this would be a meaningless complication, but for my purposes I would like to figure out how to do it correctly. On second thought, it may be the vector \bold{c_0} dt that should be held invariant in the transformation that I'm trying to develop.

Mansouri-Sexl add three arbitrary synchronization parameters to the LET transformation equation for the t coordinate; one for each direction in space. Making \bold{c_0} dt a vector may imply three time coordinates; one for each direction in space.

This relates to my own personal theory, and you are free to ignore it. I'm only mentioning it in response to the questions that you and JesseM have asked.

 

[Aether]

What do dA, dB and dC represent? In any case, this equation would still indicate that ds is a scalar rather than a vector, since it's the norm of the vector (dA, dB, dC).

They are the components of this vector: d \bold{s}=\bold{c_0} dt-dx-dy-dz, where \bold{c_0} is the instananeous velocity of an electron as described by Paul Dirac.

But your original comment was not just that it represents "something" physical, but that it represents the aether--you said "I think that the metric represents a physical thing aka 'the aether'". So I still don't understand how a function defining distance in spacetime would "represent" the aether.

The discussion of d \bold{s} as a vector speaks directly and with some precision to the point of what I think that the metric may physically represent.

I agree that all truly empirical measurements are dimensionless, or can be stated in units which are defined in a dimensionless way...I wouldn't say all empirical measurements are dimensionless ratios though (I don't see how counting the number of oscillations of a cesium atom moving along a particular path is a ratio, for example). In any case, this still doesn't help me understand what it was you thought you were acknowledging in your original comment, the thing you were saying was true "not only for speeds, but for every possible empirical measurement"--surely you weren't acknowledging that both empirical measurements and speeds are all dimensionless ratios, because speeds clearly aren't, and the comment of mine you were responding to was one about how speeds are coordinate-dependent. Perhaps we can just agree that your original comment here was confused and move on?

You said "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 I said "Acknowledged. Not only for speeds, but every possible empirical measurement." This is an acknowledgment that I am making a general point which applies "not only for speeds, but for every possible empirical measurement" namely that no experiment can directly measure a dimensionful quantity.

But as I keep saying over and over, even if we did detect a locally preferred frame there would be nothing stopping us from using a coordinate transform that led different reference frames to define simultaneity differently. A coordinate transform where all frames agreed on simultaneity might be more "natural" in this case, but other coordinate transforms would still be permissable. Do you agree or disagree?

Agree.

I think it's implicit in both of the postulates that they only refer to what would be true in inertial coordinate systems where the clocks are synchronized according to the Einstein synchronization procedure. After all, in Einstein's original paper he spends all of section 1 defining how to construct such coordinate systems out of physical measuring rods and clocks, and then only in section 2 does he lay out the two fundamental postulates. So the postulates don't deny the possibility of creating other types of coordinate systems where the speed of light is no longer constant, as I understand them. And like I said, I have seen physicists say that SR can handle accelerating coordinate systems, and the speed of light would not in general be a constant in such coordinate systems.

My textbook prefaces the postulates by "The fundamental postulates of the theory concern inertial reference systems or intertial frames. Such a reference system is a coordinate system...such that when particle motion is formulated in terms of this reference system Newton's first law holds." When the scope of the postulates is limited in this way, then I welcome them with open arms. I suppose that every formal description of SR would be limited in this way, so my concerns would only ever arise when someone attempts to apply SR outside of this particular coordinate system.

I don't think there is any such specification in "SR" as I understand it.

OK. It's not an issue with SR per se, and only becomes an issue when the postulates aren't kept within the context of their unique coordinate system.

 

[JesseM]

They are the components of this vector: d \bold{s}=\bold{c_0} dt-dx-dy-dz, where \bold{c_0} is the instananeous velocity of an electron as described by Paul Dirac.

But that vector has 4 components, while (dA, dB, dC) has only three. Why not just spell out what the vector is in terms of the quantities dt,dx,dy,dz instead of inventing a new set of symbols with no established meaning? And if you're just talking about the vector (dt,dx,dy,dz), that vector is already used in relativity, I don't understand what new idea you're trying to introduce. Like I said, the norm ds of this vector is still a scalar.

As for Dirac's comments, remember that in quantum physics particles don't have a well-defined velocity at all moments, and our measurements of particles affects their behavior. As I understood him, Dirac wasn't saying that all particles "naturally" move at c at all times, just that any attempt to measure "instantaneous velocity" by taking two position measurements arbitrarily close together will yield a velocity arbitrarily close to c, by some application of the uncertainty principle that I don't really understand (probably because I haven't studied relativistic quantum mechanics).

Anyway, I don't understand what connection you're trying to draw between Dirac's point about the instantaneous velocity of particles and relativity. If you're suggesting this might somehow imply a preferred frame, remember that his comments are based on theoretical calculations from relativistic QM, a Lorentz-symmetric theory.

The discussion of d \bold{s} as a vector

But it isn't a vector, it's a scalar which is the norm of a vector.

You said "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 I said "Acknowledged. Not only for speeds, but every possible empirical measurement." This is an acknowledgment that I am making a general point which applies "not only for speeds, but for every possible empirical measurement" namely that no experiment can directly measure a dimensionful quantity.

This is still totally convoluted...when you said "acknowledged", you mean you weren't acknowledging the thing I said I thought you should acknowledge ('that you are making a general point that all speeds are coordinate-dependent'), but rather that you were acknowledging a different point of your own ('that no experiment can directly measure a dimensionful quantity') which you didn't spell out? Usually if one person says "you should acknowledge X" and the other person says "acknowleged", it's assumed that the second person is still referring to X.

And like I said, the issue of dimensionful vs. dimensionless quantities is entirely separate from the issue of coordinate-dependent vs. coordinate-independent quantities, as there are plenty of dimensionful quantities that are not coordinate-dependent, like the proper time of a particular path through spacetime. Likewise, it's possible to come up with dimensionless quantities that are coordinate-dependent, like the ratio of times between one tick of my clock and one tick of your clock (if we use the coordinate systems specified by the Lorentz transform, each of us will say it's the other guy's clock ticks which are longer).

But as I keep saying over and over, even if we did detect a locally preferred frame there would be nothing stopping us from using a coordinate transform that led different reference frames to define simultaneity differently. A coordinate transform where all frames agreed on simultaneity might be more "natural" in this case, but other coordinate transforms would still be permissable. Do you agree or disagree?

Agree.

OK, glad we agree on this.

My textbook prefaces the postulates by "The fundamental postulates of the theory concern inertial reference systems or intertial frames. Such a reference system is a coordinate system...such that when particle motion is formulated in terms of this reference system Newton's first law holds." When the scope of the postulates is limited in this way, then I welcome them with open arms. I suppose that every formal description of SR would be limited in this way, so my concerns would only ever arise when someone attempts to apply SR outside of this particular coordinate system.

OK, so as long as SR is described in a way that makes it clear the two postulates only apply in coordinate systems built to Einstein's specifications, then maybe you don't have a problem with SR at all. I think physicists all have this implicit understanding of what is meant by SR even if they don't always state it with complete clarity.

 

[Aether]

But that vector has 4 components, while (dA, dB, dC) has only three. Why not just spell out what the vector is in terms of the quantities dt,dx,dy,dz instead of inventing a new set of symbols with no established meaning? And if you're just talking about the vector (dt,dx,dy,dz), that vector is already used in relativity, I don't understand what new idea you're trying to introduce. Like I said, the norm ds of this vector is still a scalar.

That vector doesn't really have four components. I'll rewrite it this way: d \textbf{s} = \textbf{c} _0 dt- \textbf{v} dt.

As for Dirac's comments, remember that in quantum physics particles don't have a well-defined velocity at all moments, and our measurements of particles affects their behavior. As I understood him, Dirac wasn't saying that all particles "naturally" move at c at all times, just that any attempt to measure "instantaneous velocity" by taking two position measurements arbitrarily close together will yield a velocity arbitrarily close to c, by some application of the uncertainty principle that I don't really understand (probably because I haven't studied relativistic quantum mechanics).

Anyway, I don't understand what connection you're trying to draw between Dirac's point about the instantaneous velocity of particles and relativity.

I'm not trying to draw any connection between Dirac's point and relativity per se. Relativity is a coping strategy for life in a universe without a locally preferred frame. The connection that I'm drawing is between Dirac's point and Lorentz symmetry. If you take an instantaneous measurement of an electron's velocity and you get c in the direction of the constellation Leo, then that's pretty interesting to me if I understand it correctly.

If you're suggesting this might somehow imply a preferred frame, remember that his comments are based on theoretical calculations from relativistic QM, a Lorentz-symmetric theory.

OK. It only caught my eye because I was already looking for this particular phenomena based on a prediction from somewhere else. A successive approximation process of "propose new symmetry", "formulate new QM", "make new theoretical calculations", and "repeat" is to be expected. I'm not yet able to complete this particular cycle on my own though.

This is still totally convoluted...when you said "acknowledged", you mean you weren't acknowledging the thing I said I thought you should acknowledge ('that you are making a general point that all speeds are coordinate-dependent'), but rather that you were acknowledging a different point of your own ('that no experiment can directly measure a dimensionful quantity') which you didn't spell out? Usually if one person says "you should acknowledge X" and the other person says "acknowleged", it's assumed that the second person is still referring to X.

I acknowledged what you said because I recognized that all velocities were dimensionful quantities and therefore that they can't possibly be directly measurable, and "coordinate-dependent" seemed consistent with that concept although I didn't understand the implications of that then as well as I do now. Then I added to it. Take this in context; I'm still sorting out exactly what "coordinate-dependent" really means.

And like I said, the issue of dimensionful vs. dimensionless quantities is entirely separate from the issue of coordinate-dependent vs. coordinate-independent quantities, as there are plenty of dimensionful quantities that are not coordinate-dependent, like the proper time of a particular path through spacetime. Likewise, it's possible to come up with dimensionless quantities that are coordinate-dependent, like the ratio of times between one tick of my clock and one tick of your clock (if we use the coordinate systems specified by the Lorentz transform, each of us will say it's the other guy's clock ticks which are longer).

OK.

OK, so as long as SR is described in a way that makes it clear the two postulates only apply in coordinate systems built to Einstein's specifications, then maybe you don't have a problem with SR at all. I think physicists all have this implicit understanding of what is meant by SR even if they don't always state it with complete clarity.

I would never have noticed how "SR is described" in the first place if it wasn't being misrepresented.

 

[JesseM]

That vector doesn't really have four components. I'll rewrite it this way: d \bold{s}=\bold {c_0}dt-d \bold{v}.

That equation doesn't make sense to me. First off it isn't even correct according to the standard definition of ds, which would be ds = \sqrt{c^2 dt^2 - dx^2 - dy^2 - dz^2}. Also, \bold {c_0}dt-d \bold{v} seems to be subtracting a vector from a scalar, it is not the same as cdt - dx - dy - dz, because it's not correct to write d \bold{v} = dx + dy + dz. Or if you mean to create a new meaning for \bold {c_0}dt where it is a vector rather than a scalar, it would still need to have the same number of components as the velocity vector in order to subtract the velocity vector from it, so what are its three components?

I'm not trying to draw any connection between Dirac's point and relativity per se. Relativity is a coping strategy for life in a universe without a locally preferred frame. The connection that I'm drawing is between Dirac's point and Lorentz symmetry. If you take an instantaneous measurement of an electron's velocity and you get c in the direction of the constellation Leo, then that's pretty interesting to me if I understand it correctly.

Since Dirac says that the velocity of c comes about because uncertainty in momentum approaches infinity, presumably the direction would be entirely random.

I acknowledged what you said because I recognized that all velocities were dimensionful quantities and therefore that they can't possibly be directly measurable, and "coordinate-dependent" seemed consistent with that concept although I didn't understand the implications of that then as well as I do now. Then I added to it. Take this in context; I'm still sorting out exactly what "coordinate-dependent" really means.

OK, if you were thinking at the time that coordinate-dependent and dimensionful were the same thing, presumably that means you thought you were acknowledging the same point I was making even though the point you had in mind was not the one I had in mind.

I would never have noticed how "SR is described" in the first place if it wasn't being misrepresented.

By who, in what instances? Like I said, I think when people say things like "relativity says light always moves at c" there's an implicit assumption that we're using the most physically natural system of coordinates. And again, this is true of any statement about velocities, including ones like "that car is moving at 55 mph relative to me".

 

[Aether]

That equation doesn't make sense to me. First off it isn't even correct according to the standard definition of ds, which would be ds = \sqrt{c^2 dt^2 - dx^2 - dy^2 - dz^2}. Also, \bold {c_0}dt-d \bold{v} seems to be subtracting a vector from a scalar, it is not the same as cdt - dx - dy - dz, because it's not correct to write d \bold{v} = dx + dy + dz. Or if you mean to create a new meaning for \bold {c_0}dt where it is a vector rather than a scalar, it would still need to have the same number of components as the velocity vector in order to subtract the velocity vector from it, so what are its three components?

The \bold{c_0} is in bold, has three components, and they represent the instantaneous velocity of an electron which has magnitude c_0 and whatever direction. Sorry, I meant: \textbf{v} dt = dx + dy + dz.

Since Dirac says that the velocity of c comes about because uncertainty in momentum approaches infinity, presumably the direction would be entirely random.

I don't think that the momentum approaches infinity, and have a precise finite prediction for what it approaches. Like I said, a successive approximation process seems more reasonable than infinite just-about-anything to me.

OK, if you were thinking at the time that coordinate-dependent and dimensionful were the same thing, presumably that means you thought you were acknowledging the same point I was making even though the point you had in mind was not the one I had in mind.

Seems that way, though my statement didn't even capture completely what I did mean, so apologies for that.

By who, in what instances? Like I said, I think when people say things like "relativity says light always moves at c" there's an implicit assumption that we're using the most physically natural system of coordinates. And again, this is true of any statement about velocities, including ones like "that car is moving at 55 mph relative to me".

OK. I'm not really that interested in comparing coordinate systems per se anyway. I'm looking for actual symmetry violations, and need to be able to distinguish empirical facts from coordinate system induced hallucinations.

 

[JesseM]

The \bold{c_0} is in bold, has three components, and they represent the instantaneous velocity of an electron which has magnitude c_0 and whatever direction.

What electron? Normally ds just refers to the distance between two points (perhaps infinitesimally close points) in spacetime. Can you show how your new notation would work in practice? Like, if I want to know the interval ds between two points along a straight path through flat spacetime, and the first point is t=5,x=8,y=10,z=1 and the second is t=21,x=11,y=7,z=-5, then dt would just be 21-5, dx would be 11-8, and so on. Does your notation only apply to paths taken by actual particles like electrons rather than arbitrary paths through spacetime? If so it can't be used to make a full metric, it seems. And does it depend on assuming particles have a well-defined position and velocity at every instant?

Why isn't it correct to write d \bold{v} = dx + dy + dz, because if there is something wrong with that notation then that needs to be cleaned up first.

Because one is a vector and the other is just the sum of the vector's components, a scalar quantity. You can write d \bold{v} = (dx,dy,dz) though.

I don't think that the momentum approaches infinity, and have a precise finite prediction for what it approaches. Like I said, a successive approximation process seems more reasonable than infinite just-about-anything to me.

I'm just going by what Dirac says--in the paper you posted, he writes:

It may easily be verified that a measurement of a component of the velocity must lead to the result ±c in relativity theory, simply from an elementary application of the principle of uncertainty of (24). To measure the velocity we must measure the position at two slightly different times and then divide the change of position by the time interval. (It will not do to measure the momentum and apply a formula, as the ordinary connexion between velocity and momentum is not valid.) In order that our measured velocity may approximate to the instantaneous velocity, the time interval between the two measurements must be very accurate. The great accuracy with which the position of the electron is known during the time-interval must give rise, according to the principle of uncertainty, to an almost complete indeterminacy in its momentum. This means that almost all values of the momentum are equally probable, so that the momentum is almost certain to be infinite. An infinite value for a component of momentum corresponds to the value of ±c for the corresponding component of velocity.

So it seems Dirac is saying that this is an unescapable consequence of the uncertainty principle, that as the accuracy of your position measurement approaches 100%, the uncertainty in momentum increases without bound, and I take it that in relativistic quantum theory this means that the likelihood that the velocity will be found to be as close to c as your measurement can resolve will approach 100%. If you don't think the uncertainty in momentum becomes arbitrarily large and so the expectation value of the momentum approaches infinity in this case, then I think your ideas must violate the uncertainty principle.

Seems that way, though my statement didn't even capture what I did mean, so apologies for that.

No problem, glad we got it cleared up.

 

[Aether]

What electron? Normally ds just refers to the distance between two points (perhaps infinitesimally close points) in spacetime. Can you show how your new notation would work in practice? Like, if I want to know the interval ds between two points along a straight path through flat spacetime, and the first point is t=5,x=8,y=10,z=1 and the second is t=21,x=11,y=7,z=-5, then dt would just be 21-5, dx would be 11-8, and so on. Does your notation only apply to paths taken by actual particles like electrons rather than arbitrary paths through spacetime? If so it can't be used to make a full metric, it seems. And does it depend on assuming particles have a well-defined position and velocity at every instant?

An electron is needed to define the instantaneous direction of \bold{c_0} using Dirac's analysis, though I suppose that many other types of particles would do just as well. To know whether this notation would apply to paths taken by particles as well as individual particles one would need to know if two particles some distance apart both always prefer the same direction simultaneously (or any other predictable correlation). If it is a random orientation then we probably couldn't interpolate between particles. The particle doesn't have any instantaneous position and velocity in terms of (dx,dy,dz) at all as these are only averages of (dA,dB,dC) over relatively long periods of time. Assuming that flat spacetime also means that all particles point in the same direction at the same time, then you can recover the familiar concept of a spacetime interval by taking the magnitude of the time vectors. The actual vector intervals...need to know if there is a predictable function for Dirac's rapid oscillation of the electron to answer that.

Because one is a vector and the other is just the sum of the vector's components, a scalar quantity. You can write d \bold{v} = (dx,dy,dz) though.

I changed that notation after I posted it: \textbf{v} dt = dx+dy+dz.

I'm just going by what Dirac says--in the paper you posted, he writes: So it seems Dirac is saying that this is an unescapable consequence of the uncertainty principle, that as the accuracy of your position measurement approaches 100%, the uncertainty in momentum increases without bound, and I take it that in relativistic quantum theory this means that the likelihood that the velocity will be found to be as close to c as your measurement can resolve will approach 100%. If you don't think the uncertainty in momentum becomes arbitrarily large and so the expectation value of the momentum approaches infinity in this case, then I think your ideas must violate the uncertainty principle. No problem, glad we got it cleared up.

Imagine Dirac doing an immitation of Don Adams ("Get Smart"): I said "almost certain to be infinite". My number is so large that you would double over laughing, but it isn't infinite. Which do you like better: infinite momentum, or a deterministic explanation for the uncertainty principle?

 

[JesseM]

An electron is needed to define the instantaneous direction of \bold{c_0} using Dirac's analysis, though I suppose that many other types of particles would do just as well. To know whether this notation would apply to paths taken by particles as well as individual particles

What would it mean for it to apply to "individual particles" rather than their paths? What would dx, dy and dz represent if not the incremental distance traveled by the particle in the incremental time dt?

one would need to know if two particles some distance apart both always prefer the same direction simultaneously (or any other predictable correlation). If it is a random orientation then we probably couldn't interpolate between particles.

So if two particles take the same path you wouldn't have a single value for the integral of ds along that path, right? And like I said, you have no way to integrate along paths that don't happen to be taken by any actual particles. So whatever you're doing here, you aren't defining a new type of metric, since a metric is supposed to give a single distance for an arbitrary path in your spacetime, and the ds you use appears to have no relation to the ds used in relativity (for example, note that the integral of ds along a path in relativity is just c times the proper time along that path--is anything like that true of your ds?) So what is the purpose of the ds you're calculating? What do you do with it? How do you use it to make predictions about measurable things?

The particle doesn't have any instantaneous position and velocity in terms of (dx,dy,dz) at all as these are only averages of (dA,dB,dC) over relatively long periods of time.

Again, what are dA, dB, and dC? You seem to have introduced these symbols without defining them.

Assuming that flat spacetime also means that all particles point in the same direction at the same time

What possible reason is there to make this assumption, since the theory that predicts the instantaneous velocity of a particle is c in the first place would definitely not lead to this conclusion? I don't even know if the prediction about the instantaneous velocities has been experimentally tested to any great degree of precision. It seems like you're just picking and choosing predictions of quantum theory you think are neat and throwing out ones that don't fit with your personal intuitions, with very little understanding of the underlying theory and why it makes these predictions, and why they go together. Same with your picking symbols used in relativity and redefining them in ways that seem to have no relation to their original use...a lot of what you are doing looks like a kind of http://www.physics.brocku.ca/etc/cargo_cult_science.html to me.

then you can recover the familiar concept of a spacetime interval by taking the magnitude of the time vectors. The actual vector intervals...need to know if there is a predictable function for Dirac's rapid oscillation of the electron to answer that.

Since the prediction is based on total uncertainty in the momentum due to extremely precise position measurements, it's a safe bet that the direction of the momentum would be randomized along with its magnitude.

I changed that notation after I posted it. \bold{v} = dx+dy+dz.

Why did you get rid of the "d"? That makes even less sense, how do infinitesimal displacements along the x,y, and z lead to a non-infinitesimal velocity vector? And you didn't address my point about one side of your equation being a vector and the other being a scalar.

He says "almost certain to be infinite".

I'd guess that what he means is that the expectation value of the momentum approaches infinity as the uncertainty in position approaches zero.

My number is so large that you would double over laughing, but it isn't infinite. Which do you like better: infinite momentum, or a deterministic explanation for the uncertainty principle?

If it's an "explanation" for the uncertainty principle, then it should make the same empirical predictions as the uncertainty principle--if you're saying there's an upper limit to the momentum no matter how much you reduce the uncertainty in the position, that would seem to be a violation of the uncertainty principle.

Shouldn't this go in the Independent Research forum, anyway?

 

[Aether]

Again, what are dA, dB, and dC? You seem to have introduced these symbols without defining them.

I defined them in posts #79 & #83. However, in #79 I used dv where I meant \textbf{v} dt, and didn't make the vectors bold. So, my apologies for causing confusion in that way.

Why did you get rid of the "d"? That makes even less sense, how do infinitesimal displacements along the x,y, and z lead to a non-infinitesimal velocity vector? And you didn't address my point about one side of your equation being a vector and the other being a scalar.

You're right about the notation. I changed it to: \textbf{v} dt=dx+dy+dt. Both sides of the equation are vectors.

Shouldn't this go in the Independent Research forum, anyway?

This forum looks like a great idea, but I wasn't looking to get into an open discussion of my own personal theories at this time. I was merely attempting to answer all of your questions as I have been doing all along. Your questions and comments have been generally helpful, and I appreciate that.

 

[DrGreg]

Technical note on vectors in TEX

I noticed in some posts in this thread that on this site's TEX system \bold text is sometimes hard to distinguish from normal italic TEX.

I'd suggest using \textbf instead.

\textbf r = (x, y)

Or, if you prefer, \vec

\vec r = (x, y)

 

[DrGreg]

If I understand correctly what Paul Dirac is saying (and I think I do because I have several of his subsequently published papers and articles in Nature where he says explicitly that "an ether is rather forced upon us" or words to that effect) then I don't see why we can't represent \textbf c_0 dt as a vector, and d \textbf s =\textbf c_0 dt-dx-dy-dz as a vector. I understand that for many practical purposes this would be a meaningless complication, but for my purposes I would like to figure out how to do it correctly. On second thought, it may be the vector \textbf c_0 dt that should be held invariant in the transformation that I'm trying to develop.

I preface my remarks by saying I don't know a lot about quantum theory. This website has a separate forum devoted to the subject.

In quantum theory particles do not have an exact position or momentum. There is always some error in any measurment you make. The more accurately you measure one, the less accurately you measure the other.

So if you try to measure a quantum particle's velocity by measuring \delta x / \delta t, for very, very, very, small \delta x and \delta tyou are doomed to failure because the errors will overwhelm the tiny difference you are trying to measure. So "instantaneous velocity" calculated this way is pretty meaningless. Your best bet is to measure a whole sequence of distances and times, plot them on a graph and perform a straight line curve fit. This averages out all the measurement errors.

In this forum I assume that all particles are "classical" particles postulated to have a precise position and momentum.

Mansouri-Sexl add three arbitrary synchronization parameters to the LET transformation equation for the t coordinate; one for each direction in space. Making \textbf c_0 dt a vector may imply three time coordinates; one for each direction in space.

No they are not saying anything of the sort. The three parameters are something you, the observer, decide when you choose how to synchronize your clocks, they are not extra dimensions of anything.

 

[Aether]

In this forum I assume that all particles are "classical" particles postulated to have a precise position and momentum.

OK. What is the right term (Minkowski geometry? Topology?) for that science which emcompasses all possible coordinate systems (e.g., from which new coordinate systems may be contructed), all types of particles, which directly addresses all questions of Lorentz symmetry and violations thereof, and of which relativity is an infinitesimally thin slice (e.g., that it explicitly applies only to inertial frames and classical particles)?

No they are not saying anything of the sort. The three parameters are something you, the observer, decide when you choose how to synchronize your clocks, they are not extra dimensions of anything.

That's right. Only the first sentence describes what Mansouri-Sexl are saying. The second sentence follows it so that the two may be compared to show that there is a precedent for parameterizing the time coordinate with three arbitrary components, one for each direction in space.

 

[Jimmy Snyder]

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.

Do I have this right? These guys are trying to prove that a postulate is true? By the way, it's the second postulate. Who are these guys?

 

[Aether]

Do I have this right? These guys are trying to prove that a postulate is true?

Mansouri-Sexl do not try to prove, by experiment, that the postulate is true. On page 499 of their first paper they say "When clocks are synchronized according to the Einstein procedure the equality of the velocity of light in two opposite directions is trivial and cannot be the subject of an experiment."

By the way, it's the second postulate.

One of my GR textbooks, A Short Course in General Relativity, by Foster & Nightingale, lists the speed of light postulate first.

Who are these guys?

They developed a popular test theory of special relativity and published it in 1977. Since then, most published experiments testing for violations of local Lorentz invariance have referenced their work.

After lengthy discussion, I see more clearly now that the key limitation of the constancy of the speed of light postulate is that it is only true "in all inertial frames". Inertial frames are coordinate systems in which the speed of light is defined to be constant in all directions, and that is the end of it. There can be no such thing as an experiment to verify that this is true. If someone claims that there is, then they are not observing this explicit limitation that is built into the postulate. That would be like saying that "experiments have proven that the inches on an english ruler really are inches, so there".

 

[Aether]

The primary reason to synchronize clocks is to be able to measure velocities. When we demand that an object of mass m and velocity v moving north have an equal and opposite momentum to an object of mass m and velocity v moving south, we require Einsteinan clock synchronization.

Empirically, this means that we require an two objects of equal masses moving at the same speed in opposite directions to stop when they collide inelastically.

It is indeed *possible* to use non-Einsteinain clock synchronizations, and under some circumstances it is more-or-less forced on us. In such circumstances, one must not remember that momentum is not isotropic.

Note that Newton's laws assume that momentum is isotropic (an isotropic function of velocity). Therfore Newton's laws (with the definition of momentum as p=mv) cannot be used unless Einstein's clock synchronization is used. Some other definition of momentum other than p=mv must be used if it is to remain a conserved quantity when non-standard clock synchronizations are used.

The ability to use Newton's laws at low velocities was what motivated Einstein to define his method of clock synchronization.

pervect, I agree with you now. I did not fully appreciate that it is the very definition of what an inertial reference frame represents which causes momentum to be conserved as p=mv rather than a law of nature per se. That is as interesting to me as is the fact that the constancy of the speed of light is an artifact of the definition of an inertial frame.

I interpreted "Empirically, this means that we require an two objects of equal masses moving at the same speed in opposite directions to stop when they collide inelastically" to imply that SR and LET were not empirically equivalent, but I see now that by this you were correctly defining how "at the same speed in opposite directions" is used to define an intertial frame.

 

[Jimmy Snyder]

One of my GR textbooks, A Short Course in General Relativity, by Foster & Nightingale, lists the speed of light postulate first.

One of Einstein's papers, 'On the Electrodynamics of Moving Bodies', lists it second.

There can be no such thing as an experiment to verify that this [postulate] is true.

Do you know of a postulate that has been proven true?

 

[Aether]

One of Einstein's papers, 'On the Electrodynamics of Moving Bodies', lists it second.

OK.

Do you know of a postulate that has been proven true?

If you have a point to make, then please make it.

 

[DrGreg]

In this forum I assume that all particles are "classical" particles postulated to have a precise position and momentum.

Aether, I said this simply to suggest that if you want to discuss the subtleties of quantum theory, it would be a good idea to do that in the Quantum Theory forum of this website.