Lorentz factor equal to infinity + lots of questions

[Austin0]

I'm just saying you use your own clock measuring your own personal time, not a "Rindler coordinate clock" that deliberately runs at the wrong speed to keep itself synced to the observer's "master clock". Time how long it takes for light to reflected back to you from a mirror and calculate the average round-trip speed, and take the limit as the distance to the mirror drops to zero. You'll always get c.

OK I misread correct as proper.
When you say deliberately runs at the wrong speed wrt "Rindler coordinate clock" are you talking about a natural Rindler clock that is differentially dilated due to acceleration or to a clock we have deliberately recalibrated to compensate for the Born dilation?

 

[Austin0]

He didn't say "locally proper", he said locally "correct". Think of it as clocks that are small enough for curvature and acceleration to be irrelevant, and rulers with distance markings that on small scales agree with radar measurements. Proper time is defined here. A clock measures the proper time of the curve in spacetime that represents its motion.

I know what proper time is.
I also understand your definition of a clock; as measuring the proper time of the worldline.
In practical application this definition is obviously quite workable but I would not agree with it in principle.
What a clock actually measures is relative motion , whether it is a photon or the internal occillation of an atom, so its output is a function of a completely different worldline or worldlines. This does not keep it from being applicable to the frame in which the complete mechanism is at rest, of course.

 

[Austin0]

Yes but the limit is not the only thing that is interesting, when we increase the distance to the mirror we get different values and the direction of measurement is also a factor.

For instance if we have a Born rigid spaceship with proper constant acceleration and we want to measure the roundtrip time between the front and the back then we find that the roundtrip time measured from the front is longer than measured from the back of the spaceship.

We can even, in principle, test this with massive objects as well (were we assume a fully elastic reflection), then if the initial proper velocity is $> 1/\sqrt{2}$ the result if similar as for light but if the velocity is $< 1/\sqrt{2}$ the roundtrip time measured from the front is shorter than measured from the back of the spaceship. When the velocity is exactly $1/\sqrt{2}$ the two directions are of equal duration, this speed seems to 'ignore' gravitation.

All this is independent of the actual rate of acceleration. And of course, by the equivalence principle, the same would happen if we measure this in a tower.

Have you actually done calculations to arrive at these figures?
I can see that the reflected velocity would be greater coming off the back than coming off the front due to closing and reccession of the reflector but is this enough to over compansate for the dilation differential between front and back and measure a shorter trip from the front /back/front?

 

[Fredrik]

I also understand your definition of a clock;

It's not my definition of a clock. It's a part of my definition of special relativity. A clock is defined by instructions that tell you how to build one.

Have you actually done calculations to arrive at these figures?

Hurkyl has, and I have checked his calculations and found them to be correct. See post #21 in this thread and the link in it.

 

[Austin0]

It's not my definition of a clock. It's a part of my definition of special relativity. A clock is defined by instructions that tell you how to build one.

It is pretty clearly your definition of a clock by function, which is independant of any specific mechanical design, no?

Have you actually done calculations to arrive at these figures?
I can see that the reflected velocity would be greater coming off the back than coming off the front due to closing and reccession of the reflector but is this enough to over compansate for the dilation differential between front and back and measure a shorter trip from the front /back/front?

Hurkyl has, and I have checked his calculations and found them to be correct. See post #21 in this thread and the link in it.

Hi Fredrik Very interesting thread you linked in Post #21 ,,,I am going to have to go back and spend more time.
But it is a completely different question from my post above to PassionFlower which is regarding a massive particle reflecting from back /front/back and vice versa.

Regarding the linked thread I do have a couple of questions:
1) In that thread it seemed to be implied that any arbitrary acceleration differential could be assigned between front and back as long as it was greater in the back and would then , still comply with the Born rigid hypothesis requirements.I.e. Constant proper length.
Is this the case?
2) It did not factor in a dynamic dilation factor between reflections as a consequence of acceleration. Wouldn't this be assumed to change the measurements of distance/c , over time??

 

[George Jones]

Have you actually done calculations to arrive at these figures?

I did the calculations a few months ago. After doing the calculations, I searched for similar calculations, and I found the references that I gave in posts #36 and #38 in the thread

https://www.physicsforums.com/showthread.php?p=2839878#post2839878.

If I get time later today, maybe I'll post my calculations for the constant acceleration case.

 

[Fredrik]

It is pretty clearly your definition of a clock by function, which is independant of any specific mechanical design, no?

It's definitely not my definition of a clock. If we take that sentence to define a clock, it doesn't work as a definition of special relativity. The whole thing would be circular. It would be like defining left and right by saying that on the left hand, the thumb is to the right.

There's more than one way to design an adequate clock, but I would take one of the design specifications as "the" definition, and then extend it by saying that any other design specification that gives us a device that produces the same results is good too.

But it is a completely different question from my post above to PassionFlower which is regarding a massive particle reflecting from back /front/back and vice versa.

My mistake. I thought you guys were talking about light, but now I see that the word "massive" is in there too.

1) In that thread it seemed to be implied that any arbitrary acceleration differential could be assigned between front and back as long as it was greater in the back and would then , still comply with the Born rigid hypothesis requirements.I.e. Constant proper length.
Is this the case?

No. When the rocket is doing Born rigid motion, every part of it has a world line that's mapped to a hyperbola by an inertial frame, and these hyperbolas must all approach the same straight line asymptotically. So given the proper acceleration of the rear, and the length of the rocket, we can calculate the proper acceleration of the front.

2) It did not factor in a dynamic dilation factor between reflections as a consequence of acceleration. Wouldn't this be assumed to change the measurements of distance/c , over time??

Hurkyl calculated the proper time of the world line of the rear between an event where it emits light, and an event where it receives the same light after a reflection in the front. This calculation doesn't involve any coordinate changes from one inertial frame to another, so the dilation factor doesn't enter into it. I guess another way of looking at it is to say that this dilation factor is already included implicitly in the specification of the two hyperbolas.

 

[DrGreg]

OK I misread correct as proper.
When you say deliberately runs at the wrong speed wrt "Rindler coordinate clock" are you talking about a natural Rindler clock that is differentially dilated due to acceleration or to a clock we have deliberately recalibrated to compensate for the Born dilation?

There are two kinds of clocks (in this context)

• a natural clock measuring proper time (which exhibits "gravitational dilation" when compared with other proper-time clocks at different heights)
• a "coordinate clock" measuring Rindler coordinate time that deliberately runs fast or slow compared with proper time, and which is synchronised to all the other Rindler clocks according to the current comoving inertial observer

You use a coordinate clock to measure coordinate speed (and get an answer that usually isn't c). You use a proper-time clock to make a "local measurement" (and get an answer that tends to c as the distance shrinks to zero).

In practice all real-world clocks measure proper time and coordinate time is just something you calculate; you don't actually need to implement a physical clock displaying coordinate time.

(All "clocks" referred to here are idealised clocks -- actual physical clocks might only approximate proper time on the assumption that acceleration or tidal effects have negligible effect on the practical clock mechanism.)

Footnote: strictly speaking, the metre is now defined in terms of the speed of light, so any experiment to "measure the speed of light" using a ruler does no such thing: it is really an experiment to measure the accuracy of your ruler.

 

[Austin0]

I did the calculations a few months ago. After doing the calculations, I searched for similar calculations, and I found the references that I gave in posts #36 and #38 in the thread

https://www.physicsforums.com/showthread.php?p=2839878#post2839878.

If I get time later today, maybe I'll post my calculations for the constant acceleration case.

Thanks for the reference George Jones unfortunately for some reason I can't access those archives.
Is it possible to give a simple conceptual explanation of the reason for a velocity threshold which would result in a coordinate reversal from slowing down to speeding up?
Thanks

 

[Austin0]

A clock measures the proper time of the curve in spacetime that represents its motion.

It's definitely not my definition of a clock. If we take that sentence to define a clock, it doesn't work as a definition of special relativity. The whole thing would be circular. It would be like defining left and right by saying that on the left hand, the thumb is to the right.

Sure seems to me to be both a functional definition of a clock and as part of an accurate definition of relativity and spacetime.

There's more than one way to design an adequate clock, but I would take one of the design specifications as "the" definition, and then extend it by saying that any other design specification that gives us a device that produces the same results is good too.

I always assumed that the mechanical design of a clock was irrelevant , that virtualy all clocks and physical processes would be equally affected and measure time along the trajectory through spacetime.

No. When the rocket is doing Born rigid motion, every part of it has a world line that's mapped to a hyperbola by an inertial frame, and these hyperbolas must all approach the same straight line asymptotically. So given the proper acceleration of the rear, and the length of the rocket, we can calculate the proper acceleration of the front.

So the acceleration at the front in Hurkyl's calculations, which was half the acceleration in the rear with a length of 1 , was the correct Born differential?
Then kev suggested different front and rear accelerations,, were they also in compliance with the requisite factor??
How do you calculate that from acceleration and length?

Hurkyl calculated the proper time of the world line of the rear between an event where it emits light, and an event where it receives the same light after a reflection in the front. This calculation doesn't involve any coordinate changes from one inertial frame to another, so the dilation factor doesn't enter into it. I guess another way of looking at it is to say that this dilation factor is already included implicitly in the specification of the two hyperbolas.

But doesn't the dilation factor increase with the slope of the hyperbolas over time??

WHy would it be relevant whether or not an inertial frame was involved??
If the natural clocks are being used to measure the speed of light locally or the distance between the front and back, then the relevant comparison is between those clocks, with themselves at different points on the worldlines , yes?
So if a local transverse measurement of c is conducted at a later point in time, when the clock is running relatively slower, then the conclusion would seem to be that light was faster, or if you assume constant c, then that the ruler was shorter ,,do you agree??
Then measurements from the back, to the front and back , would not only appear shorter due to increased dilation over time, but would also have the additional factor of increased contraction over time , yes?
So what factor would make it possible to either measure a constant c or measure a constant Born rigid spatial relationship with the accelerated clocks without a specific artificial recalibration of the whole clock systems individual rates?

 

[Fredrik]

Sure seems to me to be both a functional definition of a clock and as part of an accurate definition of relativity and spacetime.

Again, that would make the definitions circular.

I always assumed that the mechanical design of a clock was irrelevant , that virtualy all clocks and physical processes would be equally affected and measure time along the trajectory through spacetime.

It is, and they are, but that doesn't mean that we can allow circular definitions.

So the acceleration at the front in Hurkyl's calculations, which was half the acceleration in the rear with a length of 1 , was the correct Born differential?

Yes.

Then kev suggested different front and rear accelerations,, were they also in compliance with the requisite factor??
How do you calculate that from acceleration and length?

I haven't read his calculations yet. For constant, Born rigid, proper acceleration, you need to make sure that the world lines of each particle in the object is approacing the same asymptote, i.e. that the hyperbolas they're on when they're described in an inertial frame have the same center.

But doesn't the dilation factor increase with the slope of the hyperbolas over time??

I thought that was exactly what I was answering with the text you quoted.

WHy would it be relevant whether or not an inertial frame was involved??

Because you're talking about time dilation, which is what we call the fact that the time coordinate of a point on the time axis of one inertial frame is different by a factor of gamma in another inertial frame with the same origin.

If the natural clocks are being used to measure the speed of light locally or the distance between the front and back, then the relevant comparison is between those clocks, with themselves at different points on the worldlines , yes?

Yes, and that's precisely why you shouldn't expect to see the time dilation formula, Lorentz transformations, or anything like that.

the clock is running relatively slower,

Only relative to the time coordinate of some specific inertial coordinate system. In Hurkyl's calculation, such coordinates are only used to specify events, not to describe ticking rates. He's just calculating the proper time of a segment of a curve. The inertial frame is only used to determine what events to use as the endpoints of the curve.

then the conclusion would seem to be that light was faster, or if you assume constant c, then that the ruler was shorter ,,do you agree??
Then measurements from the back, to the front and back , would not only appear shorter due to increased dilation over time, but would also have the additional factor of increased contraction over time , yes?

I don't follow your reasoning, but this rocket is doing Born rigid motion, so its lengths in the comoving inertial frames are all =1.

So what factor would make it possible to either measure a constant c or measure a constant Born rigid spatial relationship with the accelerated clocks without a specific artificial recalibration of the whole clock systems individual rates?

If the first question is about what the acceleration, length and so on need to be for the velocity measurement to give us the result c, I don't see why I should spend my time calculating that. And I don't understand the other question.

 

[Austin0]

. So the acceleration at the front in Hurkyl's calculations, which was half the acceleration in the rear with a length of 1 , was the correct Born differential?

Yes.

I thought the acceleration differential was supposed to be such that it resulted in the same length contraction as the CMIF's. Just looking at it doesn't it seem like a rear acceleration twice that of the front would result in a much greater instantaneous velocities differential than that represented by the motion required to produce contraction?

Originally Posted by Fredrik
Hurkyl calculated the proper time of the world line of the rear between an event where it emits light, and an event where it receives the same light after a reflection in the front. This calculation doesn't involve any coordinate changes from one inertial frame to another, so the dilation factor doesn't enter into it. I guess another way of looking at it is to say that this dilation factor is already included implicitly in the specification of the two hyperbolas. .

But doesn't the dilation factor increase with the slope of the hyperbolas over time?? .

I thought that was exactly what I was answering with the text you quoted. .

Well yes but you confused the issue by also saying "so the dilation factor doesn't enter into it"

WHy would it be relevant whether or not an inertial frame was involved?? .

Because you're talking about time dilation, which is what we call the fact that the time coordinate of a point on the time axis of one inertial frame is different by a factor of gamma in another inertial frame with the same origin.

Yes, and that's precisely why you shouldn't expect to see the time dilation formula, Lorentz transformations, or anything like that. .

Well the term time dilation seems to be used wrt clocks at different Schwarzschild radii , and also clocks at different Rindler R's so I 'm not sure what your point is here.

So if a local transverse measurement of c is conducted at a later point in time, when the clock is running relatively slower, then the conclusion would seem to be that light was faster, or if you assume constant c, then that the ruler was shorter ,,do you agree?? .

Only relative to the time coordinate of some specific inertial coordinate system. In Hurkyl's calculation, such coordinates are only used to specify events, not to describe ticking rates. He's just calculating the proper time of a segment of a curve. The inertial frame is only used to determine what events to use as the endpoints of the curve.

Well if successive reflections are the events that determine the endpoints of succeeding line segments with a different slope [i.e. greater dilation], it appears the end result would be the same Yes?

So what factor would make it possible to either measure a constant c or measure a constant Born rigid spatial relationship with the accelerated clocks without a specific artificial recalibration of the whole clock systems individual rates?

I am not talking about specific accelerations or lengths but of the general conditions as described for Born acceleration.
Both relative to the front of the system of whatever length and the comparative inertial frame, the clocks at the back are dilated, this would seem to have to result in a faster measured speed of light no matter how short the distance of transit. Or comparably; with the assumption of constant c it would n infer a shorter ruler for that measurement. Is this somehow innaccurate or incomprehensible?