Can Special Relativity Predict Gravitational Effects?

[atyy]

I see. I guess that's basically the argument made in this paper? http://arxiv.org/abs/gr-qc/0603087

Yes. Just to clarify what I meant, I would say GR is distinguished by general covariance of the field that is varied in the action, although not by general covariance of the equations of motion.

Although I might agree in principle, in reality general covariance seems to be very, very hard for most people to accept. Even smart people who have studied a lot of GR often tie themselves up in horrible knots because they keep thinking that coordinates have some direct physical interpretation. For instance, the end of this paper

Davis and Lineweaver, Publications of the Astronomical Society of Australia, 21 (2004) 97, msowww.anu.edu.au/~charley/papers/DavisLineweaver04.pdf

has a long list of statements by authoritative people (including Feynman) about how you can never receive a photon from a galaxy that's receding from you at >c. All those people made that mistake, but it's not a mistake you'd make if you'd truly accepted coordinate-independence.

Well, Feynman even made mistakes with Gauss's law in his celebrated Lectures on Physics (http://www.feynmanlectures.info/flp_errata.html, see Thorne's preface at the bottom), which I am sure he understood most of the time, so I think the list is entertaining, but I wouldn't read much more into it. Actually, the fact that Feynman got some bits of classical electrodynamics wrong based on the equivalence principle is further reason not to take Einstein's motivating ideas (general covariance, EP) for GR too seriously - modern textbooks such as MTW, Rindler and Carroll essentially concur - d'Inverno is the odd one out, I think.

 

[Fredrik]

the rulers would be contracted when aligned to measure the cicumference but wouldn't they be their proper rest length when aligned with the diameter ??

That's right.

 

[DrGreg]

Some people want to describe this scenario as circumference/diameter≠pi so desperately that they define a new manifold where there is such a circle. I have no idea why anyone would want to do this (other than a desire to change an incorrect claim into a correct one), but I know that you can do it by taking the world lines of the points on the disc (the spirals in the spacetime diagram) to be the points of a new 3-dimensional manifold, and use the Minkowski metric to define a Riemannian metric on this new manifold. This manifold is curved, but I haven't found a reason to think that this is relevant (and for that reason, I also haven't studied the details).

This is one of the rare occasions I have to disagree with Fredrik, not on the technical details but on his opinion that this is irrelevant. The manifold being referred to here (technically a "quotient manifold" I believe), represents the intuitive notion of a geometry being represented by a rigid grid of rulers glued together.

For instance, if you set out to measure the disk using rulers, you might think that that would eliminate time as a factor. But real rulers are dynamical objects that stretch and bend due to centrifugal and Coriolis forces, and relativity places absolute limits on the rigidity of materials, so you can't just say you want to make them infinitely rigid. There is a notion of Born rigidity, but it doesn't have all the properties you'd like in a rigid object (e.g., a Born-rigid object with finite area can't have a nonzero angular acceleration), and in order to make something act Born-rigid, you have to decide on a notion of simultaneity (which is frame-dependent), and then arrange to have forces applied simultaneously to various points on the object in order to keep it from deforming.

Imagine a giant wheel of rulers, one set of rulers as radial spokes and another set connected to form concentric circles. If you glued the rulers together at rest and then tried to rotate the wheel, it would fall apart as the circumferential rulers would contract and the radial rulers would not. Nevertheless, you could imagine gluing the structure together while it was already spinning. It might be a severe engineering challenge to achieve this in practice, but we can envisage it happening without any laws of physics being broken. Once constructed, the spinning structure is perfectly stable so long as it continues to rotate with constant angular velocity and we don't try to move any pieces to a new relative location (provided the centripetal forces are not large enough for the structure to fall apart). This wheel of rulers is therefore a physical realisation of the space-metric Fredrik and bcrowell referred to. On that basis I think the metric does have some relevance.

Of course, if we're considering making simultaneous physical measurements of the circumference and diameter, why bother with the time dimension at all?

Because a length measurement requires two simultaneous readings.

In general that's true, but it's not true when the points being measured are permanently at rest in your coordinate system. In that case you can take your measurements any time you like. So when you measure your circumference, it doesn't matter whether each infinitesimal measurement you make is synchronised to any other measurement or not. You just start at one worldline, and work your way round the circle until you get back the the worldline you started at, taking as much time as you want. The length measurement you make does not need to be associated with a curve in space-time. It is a space measurement, not a spacetime measurement. In particular there's no need to identify a spacelike surface within which to make measurements (and in the case of rotating coordinates, you can't find any such surface orthogonal to the "at rest" spiralling worldlines).

 

[Fredrik]

This is one of the rare occasions I have to disagree with Fredrik, not on the technical details but on his opinion that this is irrelevant. The manifold being referred to here (technically a "quotient manifold" I believe), represents the intuitive notion of a geometry being represented by a rigid grid of rulers glued together.

I don't mind when you disagree with me, because I usually learn something new when you do. I think I expressed myself a bit too strongly in #19. #28 is a better representation of what I've actually been thinking:

Maybe there is a good reason do those fancy definitions and complicated calculations, but I still haven't seen one.

I actually started writing an addition to #19, where I said that the fancy stuff might be useful if we're trying to develop an understanding of non-local measurements, perhaps with the intention of replacing the axioms of SR and GR (which refer to local measurements) with axioms that refer to measurements of a more general kind. But I never finished it, because I felt that it might just lead to 20 questions about what I meant, and then I'd have to spend too much time explaining it.

If this approach does lead to a more general notion of measurement, and especially if it gives us what we need to replace the length measurement axiom of GR with a prettier one, I might just have to study those details.

Imagine a giant wheel of rulers,
...you could imagine gluing the structure together while it was already spinning.
It might be a severe engineering challenge to achieve this in practice,

I'm not convinced that it's just an engineering challenge, because we have to consider the elastic deformation of the rulers (and spokes) under centrifugal forces. Or are we just assuming that they're rigid in some specific sense?

This wheel of rulers is therefore a physical realisation of the space-metric Fredrik and bcrowell referred to. On that basis I think the metric does have some relevance.

I'm trying to come up with a sentence that describes how it's relevant, something like the axiom I use to describe how clocks are relevant: "A clock measures the proper time of the curve in spacetime that represents its motion". What would the corresponding statement be for this wheel? How about this?

"Let Q be the quotient manifold defined by the congruence of curves in spacetime that represents the wheel's motion, and let p be the unique point in Q whose projection onto spacetime is a timelike geodesic. The wheel measures proper lengths in Q of lines through p and circles around p".

In general that's true, but it's not true when the points being measured are permanently at rest in your coordinate system. In that case you can take your measurements any time you like. So when you measure your circumference, it doesn't matter whether each infinitesimal measurement you make is synchronised to any other measurement or not.

OK, so you're thinking of rulers attached to the disc in a very complicated way, rather than inertial rulers momentarily comoving with a small segment of the edge. The thing is, they give us the same results as inertial comoving rulers, and each of those rulers involve two simultaneous readings, and it's clear that they're measuring the length of a discontinous curve in spacetime, or the length of a continuous spacelike spiral.

I don't know what constraints you would impose on the attached rulers to prevent them from deforming under all the forces involved, or rather to get them to deform in exactly the right way. I also don't know how you would justify the claim that what you're measuring is "the circumference".

The length measurement you make does not need to be associated with a curve in space-time.

Then why would we call it a "length" measurement? Because there exists a manifold that has nothing to do with what we previously called "space", in which there is a circle with that circumference? I think we need a much better motivation than that.

It is a space measurement, not a spacetime measurement.

But the result is also not the proper length of any (relevant) continuous curve in space, so it doesn't make sense to call it a space measurement either...

...until we have defined "space" to mean something very different from what it meant to us before we started working on this problem!

This is what really bugs me about these claims (when they're made in books). The authors seem to start out thinking that we can measure the circumference of a disc in a certain way, and then when they realize that this is just wrong, they redefine the meaning of the words to make their first claim right. It's like saying that an obviously discontinuous function is continuous...and then go "oops...uh...because every subset of the real numbers is open". I don't doubt that some of them have a better understanding of the issues than that, but they don't always show it.

I find it very strange that some authors try to slip in a redefinition of the term "space" without even mentioning that it is a redefinition.

The claim (not made by you) that I find the most bizarre is that the quotient manifold is space in the rotating coordinate system. It clearly doesn't have anything to do with that coordinate system, or any other.

 

[DrGreg]

I'm not convinced that it's just an engineering challenge, because we have to consider the elastic deformation of the rulers (and spokes) under centrifugal forces. Or are we just assuming that they're rigid in some specific sense?

If you like, we can wait until the rotating structure has settled down to an equilibrium state, i.e. all points are relatively at rest in the rotating frame, and then calibrate it (i.e. engrave the markings on the rulers) with the help of some comoving inertial rulers. Or, even better, by using local (infinitesimal) radar.

How about this?

"Let Q be the quotient manifold defined by the congruence of curves in spacetime that represents the wheel's motion, and let p be the unique point in Q whose projection onto spacetime is a timelike geodesic. The wheel measures proper lengths in Q of lines through p and circles around p".

Points in Q do not correspond to timelike geodesics in spacetime. They correspond to spiral worldlines (at rest in the rotating coordinates) which are timelike but certainly not geodesic. I don't know whether that was just a typo from you, or whether you have misunderstood the quotient construction.

And although I talked of radial and tangential rulers, in principle you could have rulers in any direction you like (again assembled and radar-calibrated while already rotating).

OK, so you're thinking of rulers attached to the disc in a very complicated way, rather than inertial rulers momentarily comoving with a small segment of the edge. The thing is, they give us the same results as inertial comoving rulers, and each of those rulers involve two simultaneous readings, and it's clear that they're measuring the length of a discontinous curve in spacetime, or the length of a continuous spacelike spiral.

Any distance is a "sum of infinitesimal distances" (to use somewhat imprecise language, but you know what I mean) and each individual infinitesimal distance coincides with a comoving inertial measurement, and the two ends of that infinitesimal measurement are indeed simultaneous in the comoving frame (i.e. the infinitesimal segment in spacetime is orthogonal to the spiral worldlines at that point). But when you stitch all these measurements together, it doesn't matter whether whether they are synchronised to each other. You could, if you want, synchronise them to each other and find that when you complete the circle (reaching the same spiral worldline where you started), the two ends aren't simultaneous. It doesn't matter. When you project each infinitesimal segment onto the quotient manifold, the time gets lost and you get the same projected segment no matter how each segment is synced to its neighbours.

Then why would we call it a "length" measurement? Because there exists a manifold that has nothing to do with what we previously called "space", in which there is a circle with that circumference?

I'm not sure why you have such difficulty in accepting this as "space". It takes a bit of getting used to (and non-mathematical souls who haven't previously come across other examples of "quotient spaces" in other maths contexts may have more difficulty) but to my mind this seems to be the correct definition of space. It just happens to be isomorphic to a spacetime surface of simultaneity in other cases such as Minkowski coords or Rindler coords where there is no torsion.

But the result is also not the proper length of any (relevant) continuous curve in space, so it doesn't make sense to call it a space measurement either...

It is the length of a continuous curve in the quotient manifold, a circle in fact. It's just not the length of a closed curve in spacetime.

The claim (not made by you) that I find the most bizarre is that the quotient manifold is space in the rotating coordinate system. It clearly doesn't have anything to do with that coordinate system, or any other.

I don't really understand why you think that last sentence is true.

 

[bcrowell]

For instance, if you set out to measure the disk using rulers, you might think that that would eliminate time as a factor. But real rulers are dynamical objects that stretch and bend due to centrifugal and Coriolis forces, and relativity places absolute limits on the rigidity of materials, so you can't just say you want to make them infinitely rigid. There is a notion of Born rigidity, but it doesn't have all the properties you'd like in a rigid object (e.g., a Born-rigid object with finite area can't have a nonzero angular acceleration), and in order to make something act Born-rigid, you have to decide on a notion of simultaneity (which is frame-dependent), and then arrange to have forces applied simultaneously to various points on the object in order to keep it from deforming.

Imagine a giant wheel of rulers, one set of rulers as radial spokes and another set connected to form concentric circles. If you glued the rulers together at rest and then tried to rotate the wheel, it would fall apart as the circumferential rulers would contract and the radial rulers would not. Nevertheless, you could imagine gluing the structure together while it was already spinning. It might be a severe engineering challenge to achieve this in practice, but we can envisage it happening without any laws of physics being broken. Once constructed, the spinning structure is perfectly stable so long as it continues to rotate with constant angular velocity and we don't try to move any pieces to a new relative location (provided the centripetal forces are not large enough for the structure to fall apart). This wheel of rulers is therefore a physical realisation of the space-metric Fredrik and bcrowell referred to. On that basis I think the metric does have some relevance.

I don't see anything to disagree with here. Is this in reply to the text you quoted from my #27? I think what I said and what you said are consistent with one another. I agree with you that the rotating spatial metric is well defined, interesting, and non-Euclidean.

 

[DrGreg]

I don't see anything to disagree with here. Is this in reply to the text you quoted from my #27? I think what I said and what you said are consistent with one another. I agree with you that the rotating spatial metric is well defined, interesting, and non-Euclidean.

I wasn't disagreeing with you, I was quoting this as it seemed relevant to the point I was making in my post, and I wanted to address the issues of rigidity etc. My post was really aimed at Fredrik, Austin0 & snoopies622. Sorry for any confusion.

 

[jason12345]

I just had a strange thought:

In section 23 of his popular book, "Relativity - The Special and General Theory", Einstein explains why a clock on the edge of a rotating disk will run more slowly than one at the center, and then says,

"thus on our circular disk, or, to make the case more general, in every gravitational field, a clock will go more quickly or less quickly, according to the position in which the clock is situated (at rest)."

In the next paragraph he uses the same kind of argument - applying a prediction of special relativity to the rotating disk - to conclude that,

"the propositions of Euclidean geometry cannot hold exactly on the rotating disk, nor in general in a gravitational field.."

Since an observer on the disk will also notice a Coriolis force (if, say, he has a little Foucault pendulum and sets it in motion) why does it not then follow that every gravitational field is accompanied by a Coriolis acceleration? Are Einstein's arguments in this section simply disingenuous?

Maybe you should do a search on the The Ehrenfest Paradox as well, since the above is related:

http://en.wikipedia.org/wiki/Ehrenfest_paradox

I find it curious that Einstein jumps to the conclusion that the rods around the periphery of the circle are Lorentz contracted even though they're being accelerated. Worse still, all the points of the rotating frame have unique comoving frames and so there is no comoving frame for an infinitesimal rod on the circumference. Of course, since I know little about GR, the error must lie with my lack of understaning ;)

 

[Fredrik]

If you like, we can wait until the rotating structure has settled down to an equilibrium state, i.e. all points are relatively at rest in the rotating frame, and then calibrate it (i.e. engrave the markings on the rulers) with the help of some comoving inertial rulers. Or, even better, by using local (infinitesimal) radar.

OK, that makes sense.

Points in Q do not correspond to timelike geodesics in spacetime. They correspond to spiral worldlines (at rest in the rotating coordinates) which are timelike but certainly not geodesic. I don't know whether that was just a typo from you, or whether you have misunderstood the quotient construction.

It wasn't a typo, and I don't think I have misunderstood it. Actually, it looks like you just misread what I said. There is exactly one point in the quotient manifold that corresponds to a timelike geodesic. That's the point I called p. The only thing I want to change right away about what I said is the word "projection". (To be consistent with standard terminology, e.g. in the context of fiber bundles, it's the map that takes a point in spacetime to the spiral in spacetime that it belongs to, i.e. to a point in Q, that should be called a projection). So let's replace that with "corresponds to".

"Let Q be the quotient manifold defined by the congruence of curves in spacetime that represents the wheel's motion, and let p be the unique point in Q that corresponds to a timelike geodesic in spacetime. The wheel measures proper lengths in Q of lines through p and circles around p".

If we do what you suggested in the first quote above, and engrave distance markings on e.g. a rotating solid disc, at locations determined by radar, we might be able to use it to measure the proper length of other curves in the quotient manifold. I'm saying "might", because I still haven't thought about how to define the metric on Q.

Any distance is a "sum of infinitesimal distances" (to use somewhat imprecise language, but you know what I mean) and each individual infinitesimal distance coincides with a comoving inertial measurement, and the two ends of that infinitesimal measurement are indeed simultaneous in the comoving frame (i.e. the infinitesimal segment in spacetime is orthogonal to the spiral worldlines at that point). But when you stitch all these measurements together, it doesn't matter whether whether they are synchronised to each other. You could, if you want, synchronise them to each other and find that when you complete the circle (reaching the same spiral worldline where you started), the two ends aren't simultaneous. It doesn't matter. When you project each infinitesimal segment onto the quotient manifold, the time gets lost and you get the same projected segment no matter how each segment is synced to its neighbours.

Agreed. But this isn't relevant until we have redefined "space" to mean a quotient manifold defined by a congruence instead of a submanifold defined by a coordinate system.

I'm not sure why you have such difficulty in accepting this as "space". It takes a bit of getting used to (and non-mathematical souls who haven't previously come across other examples of "quotient spaces" in other maths contexts may have more difficulty) but to my mind this seems to be the correct definition of space. It just happens to be isomorphic to a spacetime surface of simultaneity in other cases such as Minkowski coords or Rindler coords where there is no torsion.

My strongest objection isn't against the claim that it makes sense to call this "space". It's against the suggestion that this simply is space, and no redefinition is necessary, even though the term already means something else.

I think I understand what you said about how the distance markings we engrave on the disc will assign lengths to curves in Q in agreement with the metric on Q, but I don't know if I can be convinced to think that this is a good enough reason to redefine the word "space" to not be a submanifold of spacetime.

One thing I need to know before I start thinking that this is a good idea is if this procedure makes sense for an arbitrary congruence. Let X be an arbitrary, smooth, timelike, local vector field, defined on an open subset U of spacetime, and consider its integral curves. Should we define "space" as a quotient manifold here too? I'm guessing that this wouldn't even make sense.

I don't really understand why you think that last sentence is true.

Perhaps that statement was too strong. I meant that the quotient manifold is defined by the congruence, not by the coordinate system. The coordinate system defines something completely different as "space". The only relationship between the rotating coordinate system and the quotient manifold is that the congruence that defines the quotient manifold is easier to describe in those coordinates than any other.

 

[DrGreg]

It wasn't a typo, and I don't think I have misunderstood it. Actually, it looks like you just misread what I said. There is exactly one point in the quotient manifold that corresponds to a timelike geodesic. That's the point I called p.

Oops, my mistake. I get it now.

If we do what you suggested in the first quote above, and engrave distance markings on e.g. a rotating solid disc, at locations determined by radar, we might be able to use it to measure the proper length of other curves in the quotient manifold. I'm saying "might", because I still haven't thought about how to define the metric on Q.

Have you seen bcrowell's own website that he linked to in an earlier post? http://www.lightandmatter.com/html_books/genrel/ch03/ch03.html#Section3.4

I already talked about how to measure each infinitesimal "ds" between two spirals in a direction orthogonal to them. That's more-or-less it.

One thing I need to know before I start thinking that this is a good idea is if this procedure makes sense for an arbitrary congruence. Let X be an arbitrary, smooth, timelike, local vector field, defined on an open subset U of spacetime, and consider its integral curves. Should we define "space" as a quotient manifold here too? I'm guessing that this wouldn't even make sense.

It only makes sense with a field associated with a stationary coordinate system, so that each pair of timelike worldlines are a constant distance from each other e.g. as determined by two-way doppler shift. (I'm still a beginner in this area but I think this all means that stationary coordinates are characterised by time-independent spacetime metric components or equivalently that your vector field X must be a Killing vector.)

If you have a copy of Rindler's Relativity: Sp, Gen & Cosm, it's covered there in Chap 9 of the 2nd edition.

(Haven't time to say more now, it is way past bedtime in my part of the world.)

 

[bcrowell]

One thing I need to know before I start thinking that this is a good idea is if this procedure makes sense for an arbitrary congruence. Let X be an arbitrary, smooth, timelike, local vector field, defined on an open subset U of spacetime, and consider its integral curves. Should we define "space" as a quotient manifold here too? I'm guessing that this wouldn't even make sense.

It only makes sense with a field associated with a stationary coordinate system, so that each pair of timelike worldlines are a constant distance from each other e.g. as determined by two-way doppler shift. (I'm still a beginner in this area but I think this all means that stationary coordinates are characterised by time-independent spacetime metric components or equivalently that your vector field X must be a Killing vector.)

If you have a copy of Rindler's Relativity: Sp, Gen & Cosm, it's covered there in Chap 9 of the 2nd edition.

My own current understanding of this is written up here: http://www.lightandmatter.com/html_books/genrel/ch07/ch07.html#Section7.2 The distinction I make there between stationary spacetimes with only one (timelike) Killing vector and those with additional symmetries may be one that can be eliminated, but for me at my current point of understanding it seems helpful. If there's only one Killing vector, which is timelike, then points in space have permanent identities that are defined in a natural way, and defining a spatial metric becomes a pretty natural thing to do.

I found Rindler's treatment helpful, but a little opaque in places.

 

[Fredrik]

It only makes sense with a field associated with a stationary coordinate system,

Minkowski spacetime is a stationary spacetime no matter what curves we choose to consider. Perhaps what you have in mind is that there's a coordinate system in which all points that belong to the image of the same curve in the congruence have the same spatial coordinates? We can use my almost arbitrary vector field X to define such a coordinate system, e.g. like this: Choose some inertial frame x. For each event p, follow the integral curve of X through p to the event q, where it intersects the t=0 plane. Assign spatial coordinates (x¹(q),x²(q),x³(q)) to p.

I don't know if this can be used for anything interesting. I'm just saying that (almost) every congruence seems to be associated with such a coordinate system.

For me to feel that it's appropriate to call something "space", I think it should at least give meaning to "space, at time t". A quotient manifold doesn't seem to do that.

 

[DrGreg]

Perhaps what you have in mind is that there's a coordinate system in which all points that belong to the image of the same curve in the congruence have the same spatial coordinates?

That is indeed what I meant (and why I said "stationary coordinates" rather than "stationary spacetime"). Your vector field needs to be a timelike non-vanishing Killing vector field for the coordinates to be "stationary", i.e. a constant distance from each other over time.

For me to feel that it's appropriate to call something "space", I think it should at least give meaning to "space, at time t". A quotient manifold doesn't seem to do that.

But space applies to all possible times. You are thinking of a bundle of spaces, one for each time, but I'm thinking of a single space.

 

[DrGreg]

For each event p, follow the integral curve of X through p to the event q, where it intersects the t=0 plane. Assign spatial coordinates (x¹(q),x²(q),x³(q)) to p.

You can use that to define the coordinates, but what I'm saying is you don't want to use the t=0 plane to measure distance. The distance between nearby points in the plane does not necessarily equal the orthogonal distance between the curves, so it doesn't represent distance as measured by a comoving inertial observer (a tangent line to the curve).

 

[Fredrik]

You are thinking of a bundle of spaces, one for each time, but I'm thinking of a single space.

Yes, but Minkowski spacetime is a bundle of spaces, and it's quite useful. I still don't really get why we would want a single space.

You can use that to define the coordinates, but what I'm saying is you don't want to use the t=0 plane to measure distance. The distance between nearby points in the plane does not necessarily equal the orthogonal distance between the curves, so it doesn't represent distance as measured by a comoving inertial observer (a tangent line to the curve).

Good point.

I've been thinking some more about what sort of congruence of curves that can define a quotient manifold in a useful way, and I think I get it. I just don't have an elegant way of saying it yet. I'm thinking that if we consider two small neighborhoods of two points on the same curve in the congruence, they should "look approximately the same" in their comoving inertial frames. The approximation must become exact in the limit where the size of the region goes to zero, and this must hold for any two points on any single curve in the congruence. To "look approximately the same" here means that any other curve that passes through the two neighborhoods must be at approximately the same spatial coordinates in the two comoving inertial frames.

(There must be a better way of saying that).

This seems to be the requirement that must be met for the "object" whose motion is represented by the congruence of curves to be useful as a generalized "ruler", because it allows us to calibrate it once and for all with local radar measurements, and engrave it with distance markings (as you described above) that will not "move around relative to each other".

 

[DrGreg]

I've been thinking some more about what sort of congruence of curves that can define a quotient manifold in a useful way, and I think I get it. I just don't have an elegant way of saying it yet. I'm thinking that if we consider two small neighborhoods of two points on the same curve in the congruence, they should "look approximately the same" in their comoving inertial frames. The approximation must become exact in the limit where the size of the region goes to zero, and this must hold for any two points on any single curve in the congruence. To "look approximately the same" here means that any other curve that passes through the two neighborhoods must be at approximately the same spatial coordinates in the two comoving inertial frames.

As I understand it, and if I understand your point, the Killing vector condition ensures that your 2 neighbourhoods don't just look approximately the same, they are exactly the same (well, isomorphic).
In the original spiral example, the spiral has a symmetry: if you rotate about and translate along the time axis, all the spirals look the same as before. I'm not an expert with Killing vectors, but I believe the same thing happens there too.

 

[Fredrik]

As I understand it, and if I understand your point, the Killing vector condition ensures that your 2 neighbourhoods don't just look approximately the same, they are exactly the same (well, isomorphic).

I thought that might be the case. I'll have to think about this some more, but not today.

 

[DrGreg]

In another thread, a member Anamitra has defined concepts of "physical length", "physical time", "physical speed", all relative to a frame, which I have attempted to summarise by the following post in that thread:

I haven't come across Anamitra's technique before, but I think this is what is happening.

Given a metric of the form

$$ds^2 = g_{00}\,dt^2 + g_{ij}\,dx^i\,dx^j$$​

(where i, j take values 1,2,3 only) define two new metrics:

$$dT^2 = -g_{00}\,dt^2$$
$$dL^2 = g_{ij}\,dx^i\,dx^j$$​

T is being called "physical time" and L is being called "physical length". Both are evaluated by integrating along the same spacetime worldline that you would integrate ds along. And both are dependent on your choice of coordinate system.

It should be clear that if you were to evaluate T along a worldline of constant x¹,x²,x³ it would equal proper time. If you were to evaluate L along a curve of constant t it would equal proper length. But for an arbitrary worldline you evaluate both along the same worldline.

If you calculate |dL/dT| along a null worldline you get 1.

That seems to be closely related to what we're discussing here. In our case we have a more general metric of the form

$$ds^2 = g_{00}\,(dt - w_i\,dx^i)^2 + k_{ij}\,dx^i\,dx^j$$​

Rindler shows any stationary metric can be written in this form. So we can define metrics

$$dT^2 = -g_{00}\,(dt - w_i\,dx^i)^2$$
$$dL^2 = k_{ij}\,dx^i\,dx^j$$​

to define "physical time" and "physical length". We are effectively decomposing a 4-vector ds as sum of dT parallel to the Killing vector field and dL orthogonal to dT.