# A relativistic odometer

1. Jan 5, 2014

### 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?

2. Jan 5, 2014

### 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).

3. Jan 5, 2014

### 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.

4. Jan 5, 2014

### ghwellsjr

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

5. Jan 5, 2014

### WannabeNewton

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.

6. Jan 5, 2014

### 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.

7. Jan 5, 2014

### PAllen

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.

8. Jan 5, 2014

### yuiop

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.

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?

9. Jan 5, 2014

### PAllen

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.

Last edited: Jan 5, 2014
10. Jan 6, 2014

### 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?

11. Jan 6, 2014

### PAllen

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.

12. Jan 6, 2014

### ghwellsjr

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?

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13. Jan 6, 2014

### yuiop

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.

14. Jan 6, 2014

### PAllen

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.

15. Jan 6, 2014

### PAllen

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).

16. Jan 7, 2014

### ghwellsjr

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

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:
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.

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?

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17. Jan 7, 2014

### 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.

Last edited: Jan 7, 2014
18. Jan 7, 2014

### phyti

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.

19. Jan 7, 2014

### ghwellsjr

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.

20. Jan 7, 2014

### PAllen

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.