Can an Odometer Measure Relativistic Distance?

[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.

 

[ghwellsjr]

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.

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).

I still have no idea what you are talking about. Remember, I'm trying to understand your statement from post #15 that the inertial Earth's measurement of the accelerated star as having traveled 0.8 light-years has some meaning. I'm wondering specifically if it means something like what I illustrated in my previous post where I can transform to a different rest frame and show a distance of 2.4 or 4 light-years but in this case it would be 0.8 light-years. Of course I understand that we can have a conveyer belt traveling at some speed that makes an odometer read 0.8 light-years during the time that the star is traveling to the Earth but that seems contrived to me.

 

[PAllen]

Thanks, great diagrams, as always. One further point obvious from the drawings, is that for the officers world line portion from Earth to the star (the part where both odometers are functioning), as of passing the star:

- the prisoner progress odometer reads: 2.02 light years
- the Earth - star odometer reads 2.4 light years

both applying to same officer spacetime path.

 

[PAllen]

I still have no idea what you are talking about. Remember, I'm trying to understand your statement from post #15 that the inertial Earth's measurement of the accelerated star as having traveled 0.8 light-years has some meaning. I'm wondering specifically if it means something like what I illustrated in my previous post where I can transform to a different rest frame and show a distance of 2.4 or 4 light-years but in this case it would be 0.8 light-years. Of course I understand that we can have a conveyer belt traveling at some speed that makes an odometer read 0.8 light-years during the time that the star is traveling to the Earth but that seems contrived to me.

Imagine a second star (B) whose world line goes vertically in your diagram upwards to the point (3,-3.75). Then this world line turns left going .8c, reaching Earth world line at (0,0). Between these two world lines, you can fit in many similar 'marker' world lines. That is an example of a non-inertial congruence.

Now simply imagine that the passage of B is is when Earth turns on their odometer to measure travel distance from B to the original star. They find .8 light years. Note that in the Earth frame, the coordinate distance between B and A (original star) after both are moving, is, indeed, .8.

You could also draw the frame in which B and the original star are at rest after their sudden acceleration. They would be 4/3 light years apart in this frame, would see Earth take 5/3 years to get from B to A, during which 1 Earth year would go by. And if you compute the Earth odometer in this frame, of course you still get .8 for measure of B to A travel by earth.

 

[ghwellsjr]

Thanks, great diagrams, as always. One further point obvious from the drawings, is that for the officers world line portion from Earth to the star (the part where both odometers are functioning), as of passing the star:

- the prisoner progress odometer reads: 2.02 light years
- the Earth - star odometer reads 2.4 light years

both applying to same officer spacetime path.

Thanks, that was actually my intended point but I neglected to make it.