Can an Odometer Measure Relativistic Distance?

[PAllen]

In a number of threads over the years the idea has been discussed that if odometers were as well defined as (and as nearly realizable) as clocks, arguments we sometimes get that distance/length contraction 'disappears' when you stop, so it is meaningless, would have less weight.

Now, nothing can get around the fact that given manifold with metric, proper time interval is defined along any arbitrary world line, while something more is needed for any concept of an odometer. The minimal extra thing needed to imagine any world line as having an odometer as well as clock is congruence of reference world lines, which would normally be desired to be some flavor of co-moving congruence (but this is not necessary). Then, the congruence defines an imaginary, space filling history of reference markers. We have no interest in their separation with respect to any foliation - my proposal needs only the congruence, not a foliation.

Then, the motivating concept is that a traveler can watch markers go by, unrolling imaginary tape measure matching the speed of markers as they go by. Then distance traveled for any path with respect to the markers is the amount of tape unrolled. Putting this into a formula is easy. At any moment along a curve parametrized by tau, you have the orthonormal local frame or tretrad defined by the 4 -velocity (really, many of them, but it won't matter which you choose). In this frame, the congruence world line at this event has some speed (we don't care about direction, which is why we don't care about which orthonormal frame you choose at any point of the travel world line). The odometer reading is defined as:

∫v(\tau) d\tau

Then, when a traveler reaches some destination, they see that local clocks have advanced far more than their clocks, but they know that their path through spacetime elapsed less time, and this is a characteristic of their path. Similarly, when they stop, they see that observations by local astronomers say they traveled a long distance, but their odometer, characterizing travel relative to regional markers, is much less, and this is also a function of their path (for a given family of markers). In general, the higher the average v for a path, between two chosen world lines of the congruence, the lower the proper time and the shorter the distance traveled. But, for any average v << c, the distance is essentially constant (while proper time = ∫ d\tau obviously grows without bound as average v decreases).

Opinions on value, pitfalls, or any other responses?

 

[ghwellsjr]

I'm confused. Is the odometer supposed to be measuring the distance an observer is traveling as defined by a single Inertial Reference Frame or by multiple IRF's? In other words, if we consider an observer at rest on the Earth where his trip odometer is reading zero and not accumulating any distance and he departs to a star four light-years distant where he comes to rest, is his odometer supposed to read four-light years when he gets there and remain at that number or does it switch to the IRF while he is traveling such that only during his "trip" it is accumulating a distance that is less than four-light years?

And if you want to consider a non-inertial rest frame, then the odometer is pretty simple--it just reads zero (or any other constant you want).

 

[PAllen]

The aim is not to involve frames per se, at all. Instead, generalize the idea of a 'road' going by. You reel out tape measure matching the speed of road as it passes. When done, you have a measure of how much road has gone by. A congruence of world lines formalizes and greatly generalized the concept of 'road', to a allow a stretchy, twisting road if desired, and arbitrary motion by the 'traveler' (traveler only in the sense that the road is going by them; otherwise, the traveler may consider themselves at rest).

The definition is coordinate and frame independent. The only thing specified is a congruence of world lines of 'reference objects'. This is much less than a coordinate system. Given this, only invariants are computed.

 

[ghwellsjr]

The "traveler" has a worldline and maybe you are considering his source and destination but what other worldlines are you talking about?

 

[WannabeNewton]

The "traveler" has a worldline and maybe you are considering his source and destination but what other worldlines are you talking about?

Hi George. Imagine we have a family of observers that may be undergoing all kinds of instantaneous radial and rotational motion relative to one another; furthermore imagine that no two observers in the family ever bump into one another. Then since each observer in the family has his/her own worldline, and the observers never bump into one another, we get a unique worldline passing through each point of space-time. The entire family of worldlines is what we call a time-like congruence.

The worldlines can be doing all sorts of things like twisting around, expanding, contracting (all of which corresponding to the observers having instantaneous radial and rotational velocities relative to one another). But, in principle and sometimes in practice, we don't need any kind of coordinate system or frame field to describe the aforementioned kinematics of this congruence. We just compute tensor quantities (the most common being the shear tensor, expansion scalar, and vorticity tensor) and use their invariant physical interpretations to describe the physics of the congruence. This is what PAllen was referring to.

 

[ghwellsjr]

Your description sounds like the odometer could get different readings for a trip depending on how these "observers" are dancing around. I thought the whole idea of PAllen's odometer was to relate the reading to a specific Length Contraction.

 

[PAllen]

Your description sounds like the odometer could get different readings for a trip depending on how these "observers" are dancing around. I thought the whole idea of PAllen's odometer was to relate the reading to a specific Length Contraction.

The most useful congruence for my odometer concept would be some family of comoving world lines, but several interesting cases arise:

- A family of Rindler observers representing Born rigid uniform acceleration

- A family of comoving observers in an FLRW spacetime

- as well as the obvious case of mutually stationary inertial world lines

However, having fixed any valid congruence, my odometer reading is defined for any path. At least the following property seems true in the most wildly general case:

Given an event P, and a world line L in the congruence not containing P, then a sequence of geodesic world lines connecting P to L such that min(v) is increasing, and min(v) approaches c, then the odometer reading for the sequence approaches zero. [I'd like to find a stronger statement. A problem is ruling out increasing zigzag in the world line from P to L as min(v) increases.]

Other sensible properties would require restriction to something closer to a comoving congruence. In particular, the idea that for a wide range of v << c, the odometer reading for a geodesic world line between some P an L is 'nearly' constant seems to require something close to a co-moving family.

 

[yuiop]

... Similarly, when they stop, they see that observations by local astronomers say they traveled a long distance, but their odometer, characterizing travel relative to regional markers, is much less, and this is also a function of their path (for a given family of markers). In general, the higher the average v for a path, between two chosen world lines of the congruence, the lower the proper time and the shorter the distance traveled.

It might be worth noting that a real odometer that basically counts the revolutions of a wheel that is rolling along a road without slipping, will measure less distance between two fixed points on the road as velocity increases.

The aim is not to involve frames per se, at all. Instead, generalize the idea of a 'road' going by. You reel out tape measure matching the speed of road as it passes. When done, you have a measure of how much road has gone by.

If the tape measure is reeled out at a rate that matches the speed of the road, it will only measure the proper length of the road, because the reeled out part of the tape measure is at rest with the road. In other words, if the proper length of the road is 4 light years, the tape measure will always measure 4 light years, independent of velocity, using this method. On the other hand, the mechanical odometer attached to a wheel rolling along the road will measure much less than 4 light years as velocity increases. (I am of course ignoring the practical difficulties of keeping the radius of the wheel constant.) Is that the sort of result you are looking for?

 

[PAllen]

If the tape measure is reeled out at a rate that matches the speed of the road, it will only measure the proper length of the road, because the reeled out part of the tape measure is at rest with the road. In other words, if the proper length of the road is 4 light years, the tape measure will always measure 4 light years, independent of velocity, using this method. On the other hand, the mechanical odometer attached to a wheel rolling along the road will measure much less than 4 light years as velocity increases. (I am of course ignoring the practical difficulties of keeping the radius of the wheel constant.) Is that the sort of result you are looking for?

No, my definition disagrees with this. My mathematical model does not measure the length of the emitted tape in its rest frame. While I used tape reeling as a motivational analogy, my formal definition is better described as simply defining that if the 'road' is going by at v relative to me at some moment, then the amount of road that went by is v d\tau. This would be equivalent to measuring each piece of tape I emit in my local frame, and adding these up, rather than measuring the tape in its rest frame. But my definition really doesn't involve modelling tape at all.

 

[ghwellsjr]

Your description sounds like the odometer could get different readings for a trip depending on how these "observers" are dancing around. I thought the whole idea of PAllen's odometer was to relate the reading to a specific Length Contraction.

The most useful congruence for my odometer concept would be some family of comoving world lines, but several interesting cases arise:

- A family of Rindler observers representing Born rigid uniform acceleration

- A family of comoving observers in an FLRW spacetime

- as well as the obvious case of mutually stationary inertial world lines

However, having fixed any valid congruence, my odometer reading is defined for any path. At least the following property seems true in the most wildly general case:

Given an event P, and a world line L in the congruence not containing P, then a sequence of geodesic world lines connecting P to L such that min(v) is increasing, and min(v) approaches c, then the odometer reading for the sequence approaches zero. [I'd like to find a stronger statement. A problem is ruling out increasing zigzag in the world line from P to L as min(v) increases.]

Other sensible properties would require restriction to something closer to a comoving congruence. In particular, the idea that for a wide range of v << c, the odometer reading for a geodesic world line between some P an L is 'nearly' constant seems to require something close to a co-moving family.

If you addressed my concerns, I can't see it.

Let me restate my question:

Suppose we have an astronaut at rest on the Earth who then departs at 0.8c for a star 4 light-years away where he comes to rest. Does his odometer accumulate 2.4 light-years?

 

[PAllen]

If you addressed my concerns, I can't see it.

Let me restate my question:

Suppose we have an astronaut at rest on the Earth who then departs at 0.8c for a star 4 light-years away where he comes to rest. Does his odometer accumulate 2.4 light-years?

Yes, if the congruence used is the family of all world lines representing rest in an earth/star inertial frame. That has (in my mind) been answered in the OP and in the response you quote. However, what matters is what is communicated to the reader - so I guess I failed to present it successfully for you.

 

[ghwellsjr]

If you addressed my concerns, I can't see it.

Let me restate my question:

Suppose we have an astronaut at rest on the Earth who then departs at 0.8c for a star 4 light-years away where he comes to rest. Does his odometer accumulate 2.4 light-years?

Yes, if the congruence used is the family of all world lines representing rest in an earth/star inertial frame. That has (in my mind) been answered in the OP and in the response you quote. However, what matters is what is communicated to the reader - so I guess I failed to present it successfully for you.

Then it seems to me that an odometer can be implemented trivially by simply looking at the Doppler signals coming from the star destination point and/or the Earth departure point and/or any other objects also at rest with respect to those bodies. Of course, this only works for line-of-sight distances. I don't know if it would be possible for several targets all at rest in three dimensions to permit a 3D odometer.

If we call the observed Doppler Ratio "R", then we can calculate the speed β (in the rest frame of the observed inertial object) as:

β = (R²-1)/(R²+1)

Then for an inertial trip, the "odometer" distance reading "O" is simply βΔτ.

Here is a spacetime diagram depicting the above quoted scenario. The Earth is the thick blue line, the star is the thick red line and the astronaut is shown in black. The dots mark off 1-year increments of time for each object. The thin lines show the signals going from the different objects toward the astronaut:

Since at 0.8c the Doppler Ratio is 3 and the astronaut's odometer will calculate this as,

β = (3²-1)/(3²+1) = (9-1)/(9+1) = (8)/(10) = 0.8

And for the trip the odometer will accumulate,

O = βΔτ = 0.8*3 = 2.4 light-years.

The astronaut could have another odometer that looks back at the Earth where the Doppler ratio is 1/3 or .333 and its calculations would be:

β = (0.333²-1)/(0.333²+1) = (0.111-1)/(0.111+1) = (-0.888)/(1.111) = -0.8

= βΔτ = -0.8*3 = -2.4 light-years.

The negative sign means that the astronaut is moving away from the Earth departure point.

A real odometer would be making continuous measurements and calculations and accumulating distance continuously.

Here is a spacetime diagram showing the rest frame of the astronaut during his trip:

You can see that at the start of the trip, the star is 2.4 light-years away from the astronaut and at the end the Earth is 2.4 light-years away from the astronaut.

As I mentioned earlier, the line-of-sight objects must be inertial. For example, if the star were to accelerate toward the earth, traveling 2.4 light-years in its traveling rest frame, the Earth's odometer looking at the star would measure a distance of 0.8*1 or 0.8 light-years which doesn't correspond to anything:

Do you think that his would give less weight to the arguments that length contraction is meaningless because it 'disappears' when you stop?

 

[yuiop]

... my formal definition is better described as simply defining that if the 'road' is going by at v relative to me at some moment, then the amount of road that went by is v d\tau.

Yes, this works, but I am still mulling over whether we can call this an invariant quantity or not, because it relies on a relative velocity.

 

[PAllen]

Yes, this works, but I am still mulling over whether we can call this an invariant quantity or not, because it relies on a relative velocity.

Given a congruence, the result as I carefully defined it, is independent of coordinates. In fact, it formally requires no coordinates at all to define.

 

[PAllen]

Then it seems to me that an odometer can be implemented trivially by simply looking at the Doppler signals coming from the star destination point and/or the Earth departure point and/or any other objects also at rest with respect to those bodies. Of course, this only works for line-of-sight distances. I don't know if it would be possible for several targets all at rest in three dimensions to permit a 3D odometer.

If we call the observed Doppler Ratio "R", then we can calculate the speed β (in the rest frame of the observed inertial object) as:

β = (R²-1)/(R²+1)

Then for an inertial trip, the "odometer" distance reading "O" is simply βΔτ.

Here is a spacetime diagram depicting the above quoted scenario. The Earth is the thick blue line, the star is the thick red line and the astronaut is shown in black. The dots mark off 1-year increments of time for each object. The thin lines show the signals going from the different objects toward the astronaut:

Since at 0.8c the Doppler Ratio is 3 and the astronaut's odometer will calculate this as,

β = (3²-1)/(3²+1) = (9-1)/(9+1) = (8)/(10) = 0.8

And for the trip the odometer will accumulate,

O = βΔτ = 0.8*3 = 2.4 light-years.

The astronaut could have another odometer that looks back at the Earth where the Doppler ratio is 1/3 or .333 and its calculations would be:

β = (0.333²-1)/(0.333²+1) = (0.111-1)/(0.111+1) = (-0.888)/(1.111) = -0.8

= βΔτ = -0.8*3 = -2.4 light-years.

The negative sign means that the astronaut is moving away from the Earth departure point.

A real odometer would be making continuous measurements and calculations and accumulating distance continuously.

Here is a spacetime diagram showing the rest frame of the astronaut during his trip:

You can see that at the start of the trip, the star is 2.4 light-years away from the astronaut and at the end the Earth is 2.4 light-years away from the astronaut.

As I mentioned earlier, the line-of-sight objects must be inertial. For example, if the star were to accelerate toward the earth, traveling 2.4 light-years in its traveling rest frame, the Earth's odometer looking at the star would measure a distance of 0.8*1 or 0.8 light-years which doesn't correspond to anything:

Do you think that his would give less weight to the arguments that length contraction is meaningless because it 'disappears' when you stop?

For the special case of inertial co-moving congruence, this is a nice, practical shortcut. However, the advantage in the abstraction to congruences is that you last case is handles equally well. The answer depends on the congruence, but if in your last diagram, you imagine:

- a familly of world lines displaced downward and left from your star elbow word line, such that they don't intersect,

then that last year of Earth history corresponds correctly to an odometer reading of .8 light years. Instead of the Earth world line, if you pick a world line almost comoving with this congruence, passing through the slant left portions that intersect this last year of Earth history, it would measure 4/3+ε light years. And (.8/ (4/3)) = .6, as expected.

I like your suggestion, and it is equivalent to mine if you assume an appropriate coming congruence (and that such exists - it may not exist).

 

[ghwellsjr]

For the special case of inertial co-moving congruence, this is a nice, practical shortcut.

Co-moving with the earth/star or the astronaut?

However, the advantage in the abstraction to congruences is that you last case is handles equally well. The answer depends on the congruence, but if in your last diagram, you imagine:

- a familly of world lines displaced downward and left from your star elbow word line, such that they don't intersect,

Is this what you mean?

The green lines are worldlines at 0.99c and each is a one-year interval.

But I'm going to need some help with the rest of this:

then that last year of Earth history corresponds correctly to an odometer reading of .8 light years. Instead of the Earth world line, if you pick a world line almost comoving with this congruence, passing through the slant left portions that intersect this last year of Earth history, it would measure 4/3+ε light years. And (.8/ (4/3)) = .6, as expected.

If I did the first step correctly, can you copy the diagram and draw in the next step or all the remaining steps would be even better. I just am not grasping what your are saying.

I like your suggestion, and it is equivalent to mine if you assume an appropriate coming congruence (and that such exists - it may not exist).

Are you saying that even for my above example you don't know if your method will always work? And what do you mean by "coming" congruence?

 

[PAllen]

Obviously 'coming' congruence was a typo. I meant co-moving as used earlier.

By a co-moving congruence I mean any of the following:

- in SR, a congruence where every world line of the congruence would see see bodies following nearby world lines as at rest by some criteria (e.g. Doppler, or Fermi-Normal distance). A co-moving congruence need not inertial. I would consider a Rindler congruence to be comoving (by the Born rigidity criterion).

- in GR, various flavors of the closest you can come the SR concept, plus, for cosmology, I would include co-moving with the expansion.

A co-moving congruence is a special case of congruence

As to the question of co-moving with what, that is a choice. The idea is for any choice of a congruence, there is a corresponding odometer that can measure travel distance for an arbitrary world line against the chosen congruence. For an astronaut traveling from Earth to a star, I might choose to use a co-moving congruence in which the star's world line is in the congruence (and the Earth moving slowly - its max v(tau) as defined in my first post would be <<<c). But I could choose to define a congruence, comoving with the rocket and discuss travel distance of the star or Earth in this rocket centric congruence.

Your picture is not what I meant for the congruence. I meant to take your red world line, and displace it a tiny bit down and to the left for each new world line of the congruence (that we show in a diagram; one assumes there is a mathematical description of the continuous infinity of non-intersecting world lines).

As to your last question, my method always 'works' because it is a definition: Given a manifold, metric, and a congruence, I define coordinate independent method for associating an odometer reading with any interval of any world line. Since sometimes people include null paths in world lines, let me say that I am ruling that out - the congruence must be timelike, and the world lines for which the odometer reading is defined by my procedure are timelike world lines. A different sense of 'work' is whether it has desired properties. For any congruence, my definition has some desired properties. For an arbitrary, non-comoving, congruence it lacks other properties one might expect - but those are properties that can't exist for such a congruence in the first place.

So to clarify my last comment with the 'coming congruence' typo, I meant:

For a many (not sure about all) cases of a comoving congruence in SR, your procedure would give the same odometer reading as mine. It clearly does for an inertial comoving congruence. It also agrees in at least some cases for non-inertial comoving congruences.

My comment about non-existence is that if you want to include some world line as part of your congruence (e.g. one moving in a circle in some inertial frame, but with varying speed), you can always (in SR) build such congruence, but you will not be able to construct a co-moving congruence - the world lines will have to appear in relative motion to each other (this is a consequence of the Herglotz-Noether theorem). My odometer definition will work just fine for such a congruence, but your 'destination Doppler' method will not agree with it. The equivalence of 'destination Doppler' to my method depends critically on the congruence being co-moving.

 

[phyti]

In a number of threads over the years the idea has been discussed that if odometers were as well defined as (and as nearly realizable) as clocks, arguments we sometimes get that distance/length contraction 'disappears' when you stop, so it is meaningless, would have less weight.

Now, nothing can get around the fact that given manifold with metric, proper time interval is defined along any arbitrary world line, while something more is needed for any concept of an odometer. The minimal extra thing needed to imagine any world line as having an odometer as well as clock is congruence of reference world lines, which would normally be desired to be some flavor of co-moving congruence (but this is not necessary). Then, the congruence defines an imaginary, space filling history of reference markers. We have no interest in their separation with respect to any foliation - my proposal needs only the congruence, not a foliation.

Then, the motivating concept is that a traveler can watch markers go by, unrolling imaginary tape measure matching the speed of markers as they go by. Then distance traveled for any path with respect to the markers is the amount of tape unrolled. Putting this into a formula is easy. At any moment along a curve parametrized by tau, you have the orthonormal local frame or tretrad defined by the 4 -velocity (really, many of them, but it won't matter which you choose). In this frame, the congruence world line at this event has some speed (we don't care about direction, which is why we don't care about which orthonormal frame you choose at any point of the travel world line). The odometer reading is defined as:

∫v(\tau) d\tau

Then, when a traveler reaches some destination, they see that local clocks have advanced far more than their clocks, but they know that their path through spacetime elapsed less time, and this is a characteristic of their path. Similarly, when they stop, they see that observations by local astronomers say they traveled a long distance, but their odometer, characterizing travel relative to regional markers, is much less, and this is also a function of their path (for a given family of markers). In general, the higher the average v for a path, between two chosen world lines of the congruence, the lower the proper time and the shorter the distance traveled. But, for any average v << c, the distance is essentially constant (while proper time = ∫ d\tau obviously grows without bound as average v decreases).

Opinions on value, pitfalls, or any other responses?

In the example of the anaut traveling at .8c to the 4 lyr destination, his clock reads 2.4 yr. Since he interprets his own time dilation as a universal length contraction, his perception is that of traveling 2.4 lyr.
A clock works as a mechanical integrator.
I suggest your odometer is a clock.

 

[ghwellsjr]

In the example of the anaut traveling at .8c to the 4 lyr destination, his clock reads 2.4 yr. Since he interprets his own time dilation as a universal length contraction, his perception is that of traveling 2.4 lyr.
A clock works as a mechanical integrator.
I suggest your odometer is a clock.

I don't understand what you are saying. As you can see by my first diagram in post #12, his clock accumulates 3 years, not 2.4.

Besides, the issue with an odometer is that you need a separate odometer for every source or destination. One clock just won't fit the bill.

 

[PAllen]

In the example of the anaut traveling at .8c to the 4 lyr destination, his clock reads 2.4 yr. Since he interprets his own time dilation as a universal length contraction, his perception is that of traveling 2.4 lyr.
A clock works as a mechanical integrator.
I suggest your odometer is a clock.

gwellsjr already made the main points, but I'll just say: no way.

A clock reads a value for path given a metric. An odometer, as I've defined, will read differently given a world line and a metric, for every congruence defining a 'road way' through the universe. In particular, without affecting the clock reading of my own world line, if I chose a congruence that includes my own world line (co-moving or wildly general - as long as it include my world line), my odometer reading is zero. This simply states I don't travel relative to myself.

 

[nitsuj]

^ Zing!

 

[nitsuj]

Opinions on value, pitfalls, or any other responses?

I can only offer my opinion, since the details of this goes above my understanding. But I love the idea of length being a measure similarly reported as time. Anything that puts the relativity of length on equal footing with time.

The only pitfall is I can't understand how you have proposed to do this measurement Oh and not sure if the measure you propose is the same as the definition of proper length.

 

[ghwellsjr]

I can only offer my opinion, since the details of this goes above my understanding. But I love the idea of length being a measure similarly reported as time. Anything that puts the relativity of length on equal footing with time.

The only pitfall is I can't understand how you have proposed to do this measurement Oh and not sure if the measure you propose is the same as the definition of proper length.

Proper Length? I thought the whole idea was to measure contracted length (or distance).

 

[PAllen]

Proper Length? I thought the whole idea was to measure contracted length (or distance).

Yes, you are right. Given a congruence (my mathematical implementation of a universe filling roadway), the distance traveled will be be function of moment to moment speed relative to the roadway. With my construction (like a real odometer) you can't even talk about distance traveled if you don't move relative to the roadway - your reading will always be zero. However, for a congruence without shear or expansion (at least), whenever speed relative to the congruence is <<c, but > 0, the distance is nearly constant. This gets at the non-relativistic expectation that odometer reading does not depend on speed.

 

[ghwellsjr]

Yes, you are right. Given a congruence (my mathematical implementation of a universe filling roadway), the distance traveled will be be function of moment to moment speed relative to the roadway. With my construction (like a real odometer) you can't even talk about distance traveled if you don't move relative to the roadway - your reading will always be zero. However, for a congruence without shear or expansion (at least), whenever speed relative to the congruence is <<c, but > 0, the distance is nearly constant. This gets at the non-relativistic expectation that odometer reading does not depend on speed.

I'm still trying to figure out your congruence scheme but doesn't it work at all speeds? Why do you keep mentioning v<<c?

 

[PAllen]

I'm still trying to figure out your congruence scheme but doesn't it work at all speeds? Why do you keep mentioning v<<c?

It works at all speeds. But it has the property that for "well behaved congruences" the distance traveled is essentially independent of speed for speeds <<c relative to the congruence. For speeds approaching c relative to the congruence, the distance traveled approaches zero (this is true for any congruence, not just well behaved ones).

 

[ghwellsjr]

It works at all speeds. But it has the property that for "well behaved congruences" the distance traveled is essentially independent of speed for speeds <<c relative to the congruence. For speeds approaching c relative to the congruence, the distance traveled approaches zero (this is true for any congruence, not just well behaved ones).

You keep piling on confusion. What does the behavior of a congruence have to do with anything? Are you saying that some congruences yield incorrect answers? How do you know if a congruence is well behaved or not? Why not just always use well behaved ones?

Maybe it would help if you would explain the realization of your odometer. How do you actually make one and use one?

 

[PAllen]

You keep piling on confusion. What does the behavior of a congruence have to do with anything? Are you saying that some congruences yield incorrect answers? How do you know if a congruence is well behaved or not? Why not just always use well behaved ones?

Maybe it would help if you would explain the realization of your odometer. How do you actually make one and use one?

There is no wrong or right answer. Well behaved means the congruence behaves like our intuitive explanations of a road. For any congruence, odometer is 'right' for that congruence. Well behaved would mean that the expansion and shear tensor of the congruence are zero (that the road doesn't stretch or twist). But an odometer (implemented e.g. by a wheel contacting the road) can give a valid reading for a stretching and twisting road.

An example of a well behaved congruence would be the family of world lines x=.8t+α, where α is the constant specifying each world line. An less well behaved congruence, but perfectly usable for an odometer, would be: x= .9 sin(t)+α. For either of these, you can compute, using the recipe I gave in the OP, a distance traveled along any world line e.g. x=0, or x=.1t, or x = ln (t) for t > 1.

 

[nitsuj]

Proper Length? I thought the whole idea was to measure contracted length (or distance).

Sure, but all clocks measure proper time, yet some have "ticked" more slowly then others. And this can be seen from the cumulative value of "ticks". I though this was the whole idea of it being referred to as an "odometer". Is this supposed to be an invariant measure?

 

[PAllen]

Sure, but all clocks measure proper time, yet some have "ticked" more slowly then others. And this can be seen from the cumulative value of "ticks". I though this was the whole idea of it being referred to as an "odometer". Is this supposed to be an invariant measure?

My definition is an invariant given a congruence. Proper length would be a terrible choice of term, because proper length has a meaning in SR - length measured in an object's rest frame. That is not even applicable to this discussion.

 

[nitsuj]

My definition is an invariant given a congruence. Proper length would be a terrible choice of term, because proper length has a meaning in SR - length measured in an object's rest frame. That is not even applicable to this discussion.

Huh, so I have no clue what you are talking about. and am disinterested in looking up what given a congruence means /

 

[ghwellsjr]

There is no wrong or right answer. Well behaved means the congruence behaves like our intuitive explanations of a road. For any congruence, odometer is 'right' for that congruence. Well behaved would mean that the expansion and shear tensor of the congruence are zero (that the road doesn't stretch or twist). But an odometer (implemented e.g. by a wheel contacting the road) can give a valid reading for a stretching and twisting road.

An example of a well behaved congruence would be the family of world lines x=.8t+α, where α is the constant specifying each world line. An less well behaved congruence, but perfectly usable for an odometer, would be: x= .9 sin(t)+α. For either of these, you can compute, using the recipe I gave in the OP, a distance traveled along any world line e.g. x=0, or x=.1t, or x = ln (t) for t > 1.

Is your odometer purely a mathematical construct--no actual hardware? In other words, is it something that can be built into a vehicle and have a continually updating real-time display (like the trip odometers in your car)? Or is it like Time Dilation and Length Contraction in that a traveler only has access to those parameters after he collects a lot of radar and other data and after the trip is over, he can go back and calculate what his "traveled" distance versus his Proper Time was?

 

[PAllen]

Huh, so I have no clue what you are talking about. and am disinterested in looking up what given a congruence means /

Suppose you have an ordinary road from Toledo to Segovia, and also a conveyor belt from Toledo to Segovia. A car traveling from Toledo to Segovia will get a different reading on its odometer depending on whether it took the ordinary road or the conveyor belt. Thus the distance is function of the congruence (= road; the conveyor belt is a road in a different frame). However, for speed << c, the distance on each 'road' will be independent of speed [edit: for the conveyor belt, what would be independent of speed for low speeds is the reading between two marks on the conveyor belt]. For high speeds relative to either road, the distance on either road will decrease as speed increases.

Note, I started this thread with the idea that if someone didn't know about congruences and tetrads, they probably shouldn't participate, because I didn't want to define everything. Oh well.

 

[PAllen]

Is your odometer purely a mathematical construct--no actual hardware? In other words, is it something that can be built into a vehicle and have a continually updating real-time display (like the trip odometers in your car)? Or is it like Time Dilation and Length Contraction in that a traveler only has access to those parameters after he collects a lot of radar and other data and after the trip is over, he can go back and calculate what his "traveled" distance versus his Proper Time was?

Well, implementing it in the real world for interstellar travel could not really be done. However, if it could, it would give a reading moment to moment as you travel, just like a car's odometer.

 

[ghwellsjr]

Proper Length? I thought the whole idea was to measure contracted length (or distance).

Sure, but all clocks measure proper time, yet some have "ticked" more slowly then others. And this can be seen from the cumulative value of "ticks". I though this was the whole idea of it being referred to as an "odometer". Is this supposed to be an invariant measure?

I can't say for PAllen's scheme (since I don't understand it yet) but for my scheme (presented in post #12), two travelers, starting from the same location and going in straight lines to the same inertial destination but traveling with different arbitrary speed profiles will get different answers, as they should, and since my scheme is based on Doppler signals and Proper Times, both of which are invariants, their results will be invariant, meaning, of course, that it doesn't matter which frame we analyze the scenario in, as I show in the first two diagrams in post #12.

The same thing applies to a single traveler, going to and/or passing by several inline inertial destinations that have relative inline motion. He would have multiple trip odometers which he can reset and accumulate distances between pairs of objects.

The important thing to remember is that the purpose of this exercise is to come up with an instrument that can accumulate a reading of distance "traveled" in the rest frame of the "traveler" but the distance we are concerned with is not really the distance that the "traveler" is traveling because it's his rest frame, but rather it is the contracted distance that the destination object has traveled toward the "traveler" (or for the contracted distance that the source object has traveled away from the "traveler").

So I think your question about whether it's an invariant measure is really asking does it give the contracted distance in the rest frame of the traveler, correct? My scheme does that. I don't know about PAllen's.

 

[PAllen]

So I think your question about whether it's an invariant measure is really asking does it give the contracted distance in the rest frame of the traveler, correct? My scheme does that. I don't know about PAllen's.

It does give a contracted distance when the 'road' is moving by 'fast'. This is formally described in the property I gave in post #7:

Given an event P, and a world line L in the congruence not containing P, then a sequence of geodesic world lines connecting P to L such that min(v) is increasing, and min(v) approaches c, then the odometer reading for the sequence approaches zero.

 

[ghwellsjr]

Note, I started this thread with the idea that if someone didn't know about congruences and tetrads, they probably shouldn't participate, because I didn't want to define everything. Oh well.

That's surprising since you started this thread with:

In a number of threads over the years the idea has been discussed that if odometers were as well defined as (and as nearly realizable) as clocks, arguments we sometimes get that distance/length contraction 'disappears' when you stop, so it is meaningless, would have less weight.

...and it sounded to me like you wanted to come up with a way to counter those "arguments we sometimes get that distance/length contraction 'disappears' when you stop". I seriously doubt that anyone who knows about congruences and tetrads is putting forth those arguments.

 

[PAllen]

That's surprising since you started this thread with:

...and it sounded to me like you wanted to come up with a way to counter those "arguments we sometimes get that distance/length contraction 'disappears' when you stop". I seriously doubt that anyone who knows about congruences and tetrads is putting forth those arguments.

You've got a point. I was thinking after getting feed back, maybe it could be explained in pictures. I did try to give some intuitive descriptions, and I'll try another:

Imagine the universe is filled with a distribution of markers, reasonably densely, but travelers magically never hit them. In general, the markers may move around relative to each other, but they are not allowed to collide. Obviously, each marker has a world line. This is a congruence. If no marker sees its neighbors move, then we say this congruence has no expansion or shear. This is what I was calling a co-moving congruence for the SR case (lets leave out GR and cosmology for now).

Now (magically) we allow there to be different congruences as we feel like: Earth and a star have one such that Earth and the star are motionless relative to the congruence (markers). A high speed rocket could set up its own congruence relative to which it is motionless (which really means the rocket world line is a member of a congruence it sets up; the Earth and sun world lines are members of the congruence they set up).

A traveler can set their odometer to read a congruence. In general, the traveler can be accelerating wildly, and the markers need not be following inertial paths (but they must never collide, and it is 'nicer' if they have no expansion or shear). If the traveler picks a congruence that contains their world line [there is a marker right next to them sitting motionless ], the odometer will always read zero because their speed relative to themselves is always zero. When the traveler sets their odometer to read a congruence, it watches markers of that congruence go by (caring only about the very closest). As a marker goes by with speed v, the odometer increments by v* Δtau, with Δtau being their clock time till the next marker goes by. That one may have different speed and/ or direction. Whatever its speed, say v2, the odometer increments v2 Δtau, etc.

This is obviously very crude, but approximates the formal definition I gave in the OP.

 

[ghwellsjr]

As a marker goes by with speed v, the odometer increments by v* Δtau, with Δtau being their clock time till the next marker goes by. That one may have different speed and/ or direction. Whatever its speed, say v2, the odometer increments v2 Δtau, etc.

How does the traveler determine the speed?

 

[PAllen]

How does the traveler determine the speed?

Well, I know you don't seem to like geometric answers, but I'll give that first, then discuss physical implementation. Answering this question clarified that in my OP, tetrads (local frame of traveler) are completely superfluous. Only the congruence is needed. So, at any event on the traveler world line, you have a congruence (marker) world line intersecting it. We call the 4-velocity of the traveler at that event U1 - it is just the unit tangent vector of the traveler world line in spacetime. We call U2 the 4 velocity if the intersecting congruence world line at that event. Then speed is simply:

v = || U2 - (U1 \bullet U2) U1 || / (U1 \bullet U2)

and the odometer reading is ∫ v d\tau. Note, this definition carries unchanged into GR, and totally arbitrary congruences and travel paths.

There are many ways to measure the speed of object locally going past you, e.g. radar equivalent of echolocation. However, since you like Doppler, just imagine each marker emits at a standard frequency in its own rest frame. As it is coming towards you, from whatever direction, on near collision course, measure its Doppler and use the formula you gave earlier. This fact explains the equivalence between your odometer and mine for special cases: if the congruence is comoving, and you are traveling geodesically toward a destination described by a specified, initially distant, marker, then the Doppler of passing markers will be the same as that of the destination marker.

 

[ghwellsjr]

...This fact explains the equivalence between your odometer and mine for special cases: if the congruence is comoving, and you are traveling geodesically toward a destination described by a specified, initially distant, marker, then the Doppler of passing markers will be the same as that of the destination marker.

But if you aren't doing the equivalent of what my example does (a traveler going between two mutually at rest objects at some speed), then in what sense are you showing Distance Contraction? I thought that was the whole point of this exercise.

 

[PAllen]

But if you aren't doing the equivalent of what my example does (a traveler going between two mutually at rest objects at some speed), then in what sense are you showing Distance Contraction? I thought that was the whole point of this exercise.

Distance contraction as described by a traveler relative to distances measured in some inertial frame is a very special case. I want to include many other cases in the most equivalent way. For example, suppose someone travels in a big circle, and varying speed, from earth, to a star and back. Using an inertial comoving congrurence including the Earth (idealized not to be moving relative t the star) and star, my approach gives (I believe) the only reasonable answer to what the rocket would consider their distance traveled within this framework. Further, such a rocket, using my odometer, would find that whatever their speed variations, if the max(v) stayed < .01c, (however long the circle took to travel), they would find about the same distance traveled. However, as speed increased, e.g. min(v) approached c, they would get smaller and smaller odometer readings. (And I don't need any coordinates or frame for the rocket, and the calculation would come out the same no matter what coordinates I used for the whole thing).

I also want to handle cases where e.g. Earth and star both have proper accelerations. Given a reasonable choice of markers moving as best as they can in tandem with accelerating Earth and star, my odometer still works, and still shows distance contraction with increasing speed relative to the markers. Further, I can do all this analysis without even producing a non-inertial coordinate system. I can specify the congruence in any inertial coordinates I feel like and do the computation there.

My method also works in GR at cosmological scales. All with the same simple mathematical formulation.

Any case your odometer can work for, mine will give the same answer (if I chose an equivalent congruence). Is there some reason you don't like generalization?

 

[ghwellsjr]

Any case your odometer can work for, mine will give the same answer (if I chose an equivalent congruence). Is there some reason you don't like generalization?

I don't like it when it confuses the issue and if the issue is that someone argues that Length Contraction is meaningless because it 'disappears' when you stop, then I think my odometer might be a whole lot more effective at persuasion with that person than yours.

I am getting the impression that you are pioneering new territory (pun intended) by investigating Length Contraction, which is a coordinate effect in Special Relativity with Inertial Reference Frames, in other areas where it is not defined. But I don't know much about General Relativity, so I don't really know.

 

[PAllen]

I don't like it when it confuses the issue and if the issue is that someone argues that Length Contraction is meaningless because it 'disappears' when you stop, then I think my odometer might be a whole lot more effective at persuasion with that person than yours.

I am getting the impression that you are pioneering new territory (pun intended) by investigating Length Contraction, which is a coordinate effect in Special Relativity with Inertial Reference Frames, in other areas where it is not defined. But I don't know much about General Relativity, so I don't really know.

I will go over the value I see in my odometer concept (though I have a hard time believing someone else has not done something similar, over history). As to 'original research', I would call this more like routine exercise in the type of college relativity course that introduces congruences, straightforward application of well established techniques.

1) This concept of odometer is a direct, intuitive, generalization of every day experience, that can be presented so it seems almost inescapably correct. If I see space buoys (laid out on flight path - any flight path - from Earth to star) going by my rocket at a certain speed each, computing distance traveled relative to them as I outline it seems hard to argue with. It is completely equivalent to what actual odometers do, as noted by yuiop (thank you for pointing out that reeled out tape as an image presents unnecessary complications).

2) Where the Lorentz transform between inertial frames can be used, or where watching Doppler of a single destination you are traveling straight toward, works, this agrees, acting as another motivator for these.

3) Each more general case handled by my odometer proposal has well defined use cases that are not covered by the methods of (2), showing the concept of contracted distance is none-the less useful for explaining observations. The first case is simply a rocket following an irregular flight path against the stellar background. Doppler from destination and/or departure point will give a meaningless result. Trying to come up with the 'right' non-inertial frame for the rocket is basically impossible (which one is 'right'?). However, for discussion of travel relative to stellar background, my approach (using the unique inertial congruence for the stellar background) gives a well defined answer with exactly the desired properties. Distance contraction is applicable and observable in principle in such a case. Yet we have no need to leave a single inertial frame to do the analysis because the congruence method is invariant - all that matters is the travel world line and the congruence.

4) Generalization to congruences that have expansion and shear is motivated by a somewhat fanciful example. Suppose you have a rapidly spinning disk, that changes its spin from one speed to another, then settles down. Herglotz-Noether theorem says that there must be expansion and/or shear in the disk during this process (that can settle out after the final speed is reached). The way in SR for describing such a disk is with a congruence; choices you have in the congruence model different cases of expansion and shear. Now, imagine a relativistic roach scampering at near c across the disk as it is spinning up. My approach says the problem of distance traveled in the roach's experience is perfectly well defined for any given congruence (that is, any specific model of the spinning up disk). Further, it has the key attributes of distance contraction for faster and faster roaches. In fact, the really difficult question would be asking the distance traveled by the roach in disk's frame, because there is no defensible implementation of this. Yet the roach's travel experience is perfectly well defined using only local computation. Again, I can compute this in any inertial frame, even one where the disk as a whole is moving, and get the same results (as long as I am using the same physical model of the disk - the congruence).

5) The case for GR and cosmology is to answer: suppose ship were traveling such as get between two galaxies hundreds of millions of light years apart, while the crew still lived. Expansion would come into play, and I would claim the most useful answer to distance traveled by the rocket 'against the cosmic flow', would be to use the FLRW comoving congruence.

I see it as a big advantage that one simple formula, expressed in terms of the now heavily used concept of congruence, will cover all these cases.

 

[phyti]

ghwellsjr:
Sorry George, do you ever have days when you can't multiply?

PAllen:
A (light) clock accumulates "time" dependent on the path it takes. By design it integrates the "time" which is a function of speed. Since the "time" is actually light motion within the clock, it can be converted to distance using c. This is the so called "proper time", and by SR definition, "proper distance".
Anyone leaving you and returning can have their clock recorded the same as an odometer on a vehicle. If you left and returned following the same speed profile, your clock would show the same elapsed interval. If you chose a different path, a different interval.
I see no difference between clock and odometer.
You would have to subtract your time between readings for the traveling observer since you didn't go anywhere.

 

[PAllen]

PAllen:
A (light) clock accumulates "time" dependent on the path it takes. By design it integrates the "time" which is a function of speed. Since the "time" is actually light motion within the clock, it can be converted to distance using c. This is the so called "proper time", and by SR definition, "proper distance".

This is incorrect. Proper distance is the invariant interval along a spacelike geodesic connecting two events with spacelike separation. I have never seen any other definition, and this has nothing to do with what you are describing or what I was describing. Your c * proper time along a path produces something with units of distance, but it is not what my odometer measures nor is it proper distance. One other quantity with a similar name, but also irrelevant, is proper length which is the length of a rigid body in its rest frame.

Anyone leaving you and returning can have their clock recorded the same as an odometer on a vehicle. If you left and returned following the same speed profile, your clock would show the same elapsed interval. If you chose a different path, a different interval.
I see no difference between clock and odometer.
You would have to subtract your time between readings for the traveling observer since you didn't go anywhere.

You don't seem to understand my proposal at all. By definition, my odometer depends not only on space time path (which is the sole determinant of proper time) but also on an abstraction of a 'road'. What road you reference makes a huge difference in the odometer reading. I gave what I thought was simple, non-relativistic analogy. I pick two events representing leaving Paris and arriving at Bern. Imagine there is a road connecting them and also a conveyor belt. A real odometer would give a completely different reading depending on whether it was responding to the road or the conveyor belt. Mine also does. The congruence is the mathematical abstraction of universe filling road that the odometer reads.

If you choose to respond, please try to be specific about what part of this you don't understand or disagree with rather than repeating incorrect statements.

 

[phyti]

You don't seem to understand my proposal at all. By definition, my odometer depends not only on space time path (which is the sole determinant of proper time) but also on an abstraction of a 'road'. What road you reference makes a huge difference in the odometer reading. I gave what I thought was simple, non-relativistic analogy. I pick two events representing leaving Paris and arriving at Bern. Imagine there is a road connecting them and also a conveyor belt. A real odometer would give a completely different reading depending on whether it was responding to the road or the conveyor belt. Mine also does. The congruence is the mathematical abstraction of universe filling road that the odometer reads.

There are an unlimited range of paths between two locations, so yes the time would depend on the path. Beyond that I don't see a benefit of reading road signs. It works in the everyday experience only because the signs are static. If you are proposing a similar idea in a dynamic universe, it is questionable.

The term you used is:
the odometer increments by v* Δtau. This is exactly what the clock does as it moves along a geodesic/spacetime path, with its rate varing with v. If it doesn’t, it’s not measuring time correctly.
In the example d=4ly, v=.8c, the anaut upon arrival will think he has traveled 3*.8 = 2.4 ly, which is his odometer reading, i.e. how far he has traveled.
If it isn’t, why not, and what should it be.

 

[PAllen]

There are an unlimited range of paths between two locations

I said events, not locations, and there is one spacetime path but two roads. They can even be read simultaneously: imagine the conveyor is on it sided, and the car has wheels against both the road and coming out its side against the conveyor. Each set of wheels is connected to an odometer. You will have two readings for one spacetime path.

, so yes the time would depend on the path. Beyond that I don't see a benefit of reading road signs. It works in the everyday experience only because the signs are static. If you are proposing a similar idea in a dynamic universe, it is questionable.

The term you used is:
the odometer increments by v* Δtau. This is exactly what the clock does as it moves along a geodesic/spacetime path, with its rate varing with v. If it doesn’t, it’s not measuring time correctly.
In the example d=4ly, v=.8c, the anaut upon arrival will think he has traveled 3*.8 = 2.4 ly, which is his odometer reading, i.e. how far he has traveled.
If it isn’t, why not, and what should it be.

No clock reads v Δ tau. It reads Δ tau (tau is universally used for proper time along a world line). The v is the momentary speed of the road/buoy/marker/congruence relative the travel world line. Please read, especially, post #40. There is some given world line we integrate along. A congruence gives a field of 4-velocities throughout spacetime. In post 40 I give the precise integration measured by the odometer.

The last part is correct, but the point of view is that 2.4 light years is a function of the worldline and the congruence. In this case a congruence of world lines that have no expansion or shear (are mutually stationary) and in which the Earth and star are members of the congruence. Thus what determines the 2.4 light years is that this 'road' is going by the traveler at .8c. A different road going by would give a different answer - for the same spacetime path.

 

[phyti]

I said events, not locations, and there is one spacetime path but two roads. They can even be read simultaneously: imagine the conveyor is on it sided, and the car has wheels against both the road and coming out its side against the conveyor. Each set of wheels is connected to an odometer. You will have two readings for one spacetime path.


No clock reads v Δ tau. It reads Δ tau (tau is universally used for proper time along a world line). The v is the momentary speed of the road/buoy/marker/congruence relative the travel world line. Please read, especially, post #40. There is some given world line we integrate along. A congruence gives a field of 4-velocities throughout spacetime. In post 40 I give the precise integration measured by the odometer.

The last part is correct, but the point of view is that 2.4 light years is a function of the worldline and the congruence. In this case a congruence of world lines that have no expansion or shear (are mutually stationary) and in which the Earth and star are members of the congruence. Thus what determines the 2.4 light years is that this 'road' is going by the traveler at .8c. A different road going by would give a different answer - for the same spacetime path.

Thanks for your patient effort. This would require additional study for me to understand your idea.
Will return to more basic topics.

 

[ghwellsjr]

I want to provide another example involving two odometers in use by the same observer as he is traveling towards two other objects/observers that are not at mutual rest. This will be a follow-on to my example in post #12 where we had an astronaut leaving Earth for a star 4 light-years away but since he was traveling at 0.8c, his odometer measured 2.4 light-years for the trip.

Let's imagine that a prisoner has been exiled to a planet on the star system four light-years from Earth with the requirement that he stay there. The authorities on Earth monitor his position but eventually see that he has departed at a speed of 27.47%c. They can tell this from the Doppler signal coming from his tracking device. So after a short time they send an officer to intercept him. The officer, shown in black has two odometers, one that is looking at the Doppler signal coming from the star system and one looking at the Doppler signal coming from the prisoner's tracking device. In this first diagram, I show the Doppler signals coming from the prisoner:

As you can see, there are 13.6 annual tracking signals coming from the prisoner during the officer's six-year trip. This is a Doppler ratio of 13.6/6 = 2.267. Plugging this into the formula for extracting speed from Doppler:

β = (R²-1)/(R²+1) = (2.267²-1)/(2.267²+1) = (5.137-1)/(5.137+1) = 4.137/6.137 = 0.674

Now we multiply this by the officer's time of travel, 6 and get 4.04 light-years as the odometer reading when he apprehends the prisoner.

Meanwhile, his other odometer has measured a Doppler ratio of 3 yielding a speed of 0.8 (as shown in post #12) and a distance of 2.4 light-years:

To confirm these two measurements, we transform to the officer's rest frame during his trip:

In summary, we see that a traveler's one clock in conjunction with his real-time observation of Doppler signals from two different sources can enact two odometers and that they accumulate distances differently, as they should.