Second postulate of SR quiz question

[stevendaryl]

I think the last paragraph can be misleading. The geodesics of a space-time depend only on the space-time and they are what they happen to be, for example the geodesics of Minkowski space-time are stright lines (in the usual sense), and that's that. They are not geodesics with respect to one frame and non-geodesics wrt another.

vanhees71 didn't say anything to suggest that something was geodesic with respect to one frame and not another. He was saying that the geodesics are "straight" relative to one frame versus the other. I'm not sure if there is a standard definition of what "straight" means, but relative to a coordinate system, one could say that a path is straight or not depending on whether there is a parametrization x^\mu(s) such that \dfrac{d^2 x^\mu}{ds^2} = 0

 

[martinbn]

Well, that's why it can be misleading. For me geodesic and straight are the same thing, and it is a reference independent property of lines.

 

[Dale]

However, one can also overemphasize the geometrical aspects and forget that besides the elegant formulations in terms of geometric objects (differentiable manifolds, affine Euclidean and pseudo-Euclidean, Riemann- and pseudo-Riemann, varous fiber bundles,...) we still do physics, and physics is about what you can really observe (in a very broad sense, from naive looking at things with our senses to high-precision quantitative measurements with very tricky technology).

I don't think that this is a real problem. I think that physicists are well aware that the purpose of the math is to analyze and predict the outcome of measurements.

I think that you are making the opposite mistake, which is overly identifying the math with the measurements.

Usually we don't realize it anymore, but any measurement (more or less tacitly) uses and introduces a reference frame.

This is simply false. No specific reference frame is required to analyze any given measurement. Many measurements are more conveniently or simply analyzed in the rest frame of the measurement apparatus, but it is not a requirement. The physicist retains complete freedom in which mathematical reference frame they prefer to use for the analysis of any measurement. The measurement itself does not introduce the frame, the physicist does.

 

[vanhees71]

Well, but at least there is a "restframe of the measurement apparatus", and you statement is not always true. E.g., the temperature of a flowing fluid is measured in its local restframe, i.e., with a co-moving thermometer. That's why in the modern definition of temperature for relativistic fluids temperature is a scalar (no Lorentz-$\gamma$ factors as in the older definition)!

Also cross sections are defined in the lab frame and then written covariantly. Of course, physicists measure it in the rest frame of the detectors and then recalculate it to the invariant cross section they like to. But to be able to do so you need clearly defined reference frames (here the lab-frame and the center-of-momentum frame in a collider experiment).

So still, I think that the frame-free formulation of RT is a theoretical construct while to address observations you always have to introduce a reference frame, of course you can just calculate from one frame to the other, but you need to define a frame (or the different frames you want to analyze your data in), to make sense of your measurement results.

 

[vanhees71]

I think the last paragraph can be misleading. The geodesics of a space-time depend only on the space-time and they are what they happen to be, for example the geodesics of Minkowski space-time are stright lines (in the usual sense), and that's that. They are not geodesics with respect to one frame and non-geodesics wrt another.

Yes, maybe this was an unfortunate formulation. Still, for Rindler observers the geodesics appear not as straight lines, but for any inertial observer they appear as such. As an invariant object the geodesics are of course always straight lines.

 

[Demystifier]

For me geodesic and straight are the same thing

If you accept that language, then you must say, e.g., that trajectory of the planet Mercury around the Sun is straight. I think it is quite obvious that such a language would not satisfy an astronomer, for example. Thus for physicists (who are supposed to be able to speak with both astronomers and mathematicians) it makes a lot of sense not to treat geodesic and straight as the same thing.

Or are we talking only about special relativity? In that case my remark may be irrelevant.

 

[martinbn]

If you accept that language, then you must say, e.g., that trajectory of the planet Mercury around the Sun is straight. I think it is quite obvious that such a language would not satisfy an astronomer, for example. Thus for physicists (who are supposed to be able to speak with both astronomers and mathematicians) it makes a lot of sense not to treat geodesic and straight as the same thing.

Or are we talking only about special relativity? In that case my remark may be irrelevant.

Well, the world line of Mercury is a straight line in space-time, yes. The trajectory, which is a projection in space, of some choice of space and time split of the space-time need not be straight.

 

[Demystifier]

Well, the world line of Mercury is a straight line in space-time, yes. The trajectory, which is a projection in space, of some choice of space and time split of the space-time need not be straight.

OK, but how about the following problem? Consider non-relativistic physics, where everything happens in the 3-dimensional Euclidean space. (The concept of time, according to such a non-relativistic theory, is not related to the geometry of space.) In such a space, consider observers A and B. A is an inertial observer, so his trajectory is straight. B moves along a circle, so his trajectory is not straight. However, using a standard folk english, one can say that from the point of view of B, it looks as if the trajectory of A is not straight. How would you express this fact in a more precise language?

 

[stevendaryl]

Well, the world line of Mercury is a straight line in space-time, yes. The trajectory, which is a projection in space, of some choice of space and time split of the space-time need not be straight.

I would say that "straight" is a concept from Euclidean geometry, and that "geodesic" is a generalization to non-Euclidean geometry. They're not synonyms--one is a generalization of the other.

 

[vanhees71]

With straight line I mean a straight line in an affine space. It's the geodesic of the affine space, but there cannot be a straight line in the Schwarzschild metric, describing spacetime around the sun. Mercury is on a timelike geodesic in this spactime, but it's not in a straight line. In curved space there are no straight lines but geodesics. I think, one should not confuse the issue by making "straight line" a synonym with geodesic.

 

[martinbn]

OK, but how about the following problem? Consider non-relativistic physics, where everything happens in the 3-dimensional Euclidean space. (The concept of time, according to such a non-relativistic theory, is not related to the geometry of space.) In such a space, consider observers A and B. A is an inertial observer, so his trajectory is straight. B moves along a circle, so his trajectory is not straight. However, using a standard folk english, one can say that from the point of view of B, it looks as if the trajectory of A is not straight. How would you express this fact in a more precise language?

Well, that seems fine to me, but trajectories are frame dependent, while geodesics are not.

I would say that "straight" is a concept from Euclidean geometry, and that "geodesic" is a generalization to non-Euclidean geometry. They're not synonyms--one is a generalization of the other.

That is a possible and acceptable convention, but why the distinction. If you consider a seven dimensional Euclidean space, then are the geodesics straight lines or not? After all higher dimensional spaces are generalizations of Euclidean geometry too.

 

[loislane]

You're mixing up observers and coordinates. An observer is inertial if he has zero proper acceleration; that is a direct physical observable that can be measured with an accelerometer. A Rindler observer is not inertial because he has nonzero proper acceleration; his accelerometer does not read zero. That's true regardless of what coordinates you are using to describe the Rindler observer's motion.

You seem to be using the word observer as if it was something physical instead of a mathematical abstraction, according to what Dalespam wrote above , apparently a distinction can be made between an inertial observer and an inertial frame, I was not aware of that distinction, I'm only used to talk in terms of coordinates that is something well defined mathematically.

What if we make the distinction between an observer at rest in the rindler coordinates, that is noninertial because it has to accelerate to counter the "acceleration" of the curvilinear coordinates, and an observer following the hyperbolic motion ("slidng" along the coordinate hyperbolas), would you say the latter is an inertial or a noninertial observer?

Similarly a Minkowski observer in Rindler coordinates, is it inertial or noninertial?

I think there is a problem with observers and frames as objects with motion, so it is best to stick to coordinates. If a frame or observer is something physical rather than an abstract labeling the conclusions derived from them are different.

 

[stevendaryl]

That is a possible and acceptable convention, but why the distinction

Because "straight" has connotations that don't apply to geodesics in general.

If you consider a seven dimensional Euclidean space, then are the geodesics straight lines or not?

Yes.

After all higher dimensional spaces are generalizations of Euclidean geometry too.

Yes, and the concept of "straight" applies equally well to higher dimensional spaces, but does not apply equally well to curved spaces.

 

[martinbn]

Because "straight" has connotations that don't apply to geodesics in general.

Why not?

Yes, and the concept of "straight" applies equally well to higher dimensional spaces, but does not apply equally well to curved spaces.

Why not?

 

[stevendaryl]

Why not?

Left as an exercise.

 

[martinbn]

A line is straight if its tangent vector is parallel transported along the line. It seems to apply equally well to Euclidean and curved spaces, no?

 

[stevendaryl]

A line is straight if its tangent vector is parallel transported along the line. It seems to apply equally well to Euclidean and curved spaces, no?

No. I would say that the concept of a "straight line" is described by Euclid's geometry, which includes several properties that don't hold of geodesics (or autoparallels) including:

1. Between any two points, there is a unique line connecting them.
2. If two distinct lines can intersect at at most one point.

Geodesics (or autoparallels) in curved space don't satisfy the notion of "straight line". The clue is the word "curved". A space with a connection is curved if its autoparallels are not straight lines.

Look, nobody uses the word straight in any context other than Euclidean space, as far as I know. Unless it's in quotes, where it's meant to remind you that geodesics are analogous to the straight lines of Euclidean geometry. But I don't really care what terminology you use, I'm just telling you that other people don't use that terminology.

 

[vanhees71]

I'd say for a straight line an arbitrary affine space is sufficient. You don't need a metric. It also makes sense in Minkowski space, which is only pseudo-Euclidean. Also in curved manifolds it's sufficient to have an affine connection to define geodesics. It doesn't need to be Riemannian.

 

[martinbn]

@stevendaryl, ok, no need for me to argue and derail the thread over terminology. All I wanted to say is that it is a matter of convention. You seem to insist on your preferred choice, but it is still a choice. It is not true that it is as universally accepted that straight line is used only in the context of Euclidean geometry as you suggest. For example non-Euclidean geometry (Lobachevsky).

 

[stevendaryl]

@stevendaryl, ok, no need for me to argue and derail the thread over terminology. All I wanted to say is that it is a matter of convention. You seem to insist on your preferred choice, but it is still a choice.

Yes, since the words "autoparallel' and "geodesic" are already used for the concept, I don't see any reason to use a different word, such as "straight"

 

[PeterDonis]

You seem to be using the word observer as if it was something physical instead of a mathematical abstraction

Exactly. An "observer" as I'm using the term is a physical thing that can carry an accelerometer and observe and record its readings. It's certainly not a mathematical abstraction. You and I can be observers.

apparently a distinction can be made between an inertial observer and an inertial frame

Yes, certainly. A physical thing that can carry an accelerometer has some physical reading on its accelerometer, independent of how we describe the thing and its motion mathematically. Without such a distinction, our physical theories would have no meaning, because we would have no way of linking them up with actual observations.

What if we make the distinction between an observer at rest in the rindler coordinates, that is noninertial because it has to accelerate to counter the "acceleration" of the curvilinear coordinates, and an observer following the hyperbolic motion

These two are the same. An observer at rest in Rindler coordinates is non-inertial, and is following hyperbolic motion in Minkowski coordinates.

a Minkowski observer in Rindler coordinates, is it inertial or noninertial?

If you mean an observer at rest in Minkowski coordinates, such an observer is inertial. This observer won't be at rest in Rindler coordinates, but that is irrelevant to the question of whether the observer (as opposed to the coordinates) is inertial or not.

I think there is a problem with observers and frames as objects with motion, so it is best to stick to coordinates. If a frame or observer is something physical rather than an abstract labeling the conclusions derived from them are different.

I'm not sure why you would think this. The fact that observers are physical things doesn't mean we can't describe them mathematically. One way is to describe observers by the worldlines they follow; then we can write equations for those worldlines in different coordinate charts. Another way is to describe observers by frame fields (mappings of sets of orthonormal basis vectors to points in spacetime). I don't see that mathematically modeling observers poses any particular problem; it just requires a clear understanding of what you're trying to model.

 

[loislane]

If you mean an observer at rest in Minkowski coordinates, such an observer is inertial. This observer won't be at rest in Rindler coordinates, but that is irrelevant to the question of whether the observer (as opposed to the coordinates) is inertial or not.

And yet it is possible to Lorentz transform from Minkowski coordinates to Rindler coordinates(in the wedge area of Minkowski space where both coordinates exist), what happens to the Minkowski observer when he tries to keep at rest in the new coordinates? Does he have to accelerate uniformly?

 

[stevendaryl]

And yet it is possible to Lorentz transform from Minkowski coordinates to Rindler coordinates(in the wedge area of Minkowski space where both coordinates exist), what happens to the Minkowski observer when he tries to keep at rest in the new coordinates? Does he have to accelerate uniformly?

I'm not sure what you're asking. Are you saying that an observer is initially at rest relative to Minkowsky coordinate system, and then changes his motion so that he is at rest relative to Rindler coordinates? If that's the case, that means that the observer at some point starts undergoing constant proper acceleration.

 

[PeterDonis]

what happens to the Minkowski observer when he tries to keep at rest in the new coordinates?

Um, what? A Minkowski observer, by definition, is inertial. He can't "try to keep at rest in the new coordinates". His state of motion is already specified. When you transform to Rindler coordinates, his worldline looks like whatever it looks like.

 

[Dale]

Well, but at least there is a "restframe of the measurement apparatus",

There is a center of momentum frame for all measurement devices, and many devices have a rest frame. But that frame is not uniquely privileged from a physics standpoint and any other frame can be used.

and you statement is not always true.

My point is that YOUR statement is not always true. In order to correct what you see as a mistake in physics (overemphasis of geometry) you go way too far by asserting that it is the measurement which introduces the reference frame, and further asserting that it applies to "any measurement".

Of course, physicists measure it in the rest frame of the detectors

No, this is simply false. The device functions according to the laws of physics, the laws of physics are the same in all frames, s the physicist can use any frame they like to analyze the measurement.

It often is convenient to use the rest frame of the detector, but sometimes it is easier to use a different frame (eg the GPS ECI). If a different frame is simpler then a good physicist will choose that one. The choice of frame is not forced on the physicist by the physics or the measurement device.

Physicists do not perform an experiment in a reference frame, they perform an analysis in a reference frame.

So still, I think that the frame-free formulation of RT is a theoretical construct while to address observations you always have to introduce a reference frame

I am not disagreeing with you on this. I understand this point, but you are going way too far beyond this claim in making your case.

 

[atyy]

vanhees71 is probably thinking along the lines of MTW 13.6: "A physicist .. may use several different coordinates systems at once. But a coordinate system of special utility is one at rest relative to all the apparatus bolted into the floor and walls ... This proper reference frame ..."

 

[atyy]

I think this is the main difference between mathematicians, who are very much emphasizing the geometric aspects of spacetime.

A reference frame is somehow defined by real physical objects, be it a human being with his senses looking at a phenomenon or any fancy measurement device invented to discover accurate quantitative facts about nature that are not directly "detectable" by our senses.

I largely agree with DaleSpam's comments. But I would like to add that in a way, what you are proposing is more "geometric", not less. In your view, the laboratory walls serve as a physical manifestation of a proper reference frame and ideal clocks measure proper time, test particles and light rays follow geodesics. All those things are test objects or external rulers in the sense that they are assumed not to contribute to spacetime curvature. In conventional geometry, our rulers are rigid external objects, so I would say your view is more "geometric".

The "mathematical" view is not necessarily more "geometric". One can think of it as more correct than what you are proposing, because what they really mean is gauge-invariant (which is geometric in their sense, but not in the sense of having an external physical ruler or test particle). Of course it is not always physical in the sense of strict GR because there are neither observers nor measuring instruments in any vacuum spacetime. "It is the theory that says what can be observed"

So when martinbn says mercury follows a geodesic - it is not correct in the sense that it fails to account for mercury's perturbation of the background spacetime, but it is physical in the sense that he is talking about a gauge-invariant and hence physical object.

 

[vanhees71]

And yet it is possible to Lorentz transform from Minkowski coordinates to Rindler coordinates(in the wedge area of Minkowski space where both coordinates exist), what happens to the Minkowski observer when he tries to keep at rest in the new coordinates? Does he have to accelerate uniformly?

The transformation from Minkowski (i.e. pseudo-Caratesian coordinates in an inertial frame) to Rindler coordinates are clearly not Lorentz transformations. They are not even linear. I've given the formulae somewhere in this thread yesterday.

 

[vanhees71]

There is a center of momentum frame for all measurement devices, and many devices have a rest frame. But that frame is not uniquely privileged from a physics standpoint and any other frame can be used.

My point is that YOUR statement is not always true. In order to correct what you see as a mistake in physics (overemphasis of geometry) you go way too far by asserting that it is the measurement which introduces the reference frame, and further asserting that it applies to "any measurement".

I think it's semantics, but can you give me an example for a measurement that can be made without a clear specification of a reference frame? This is impossible, because measuring something means to have a reference you can compare the measured quantity to. Of course, measurement apparati are part of the physical system and thus follow the generally valid physical laws. Otherwise you couldn't define such a reference. Many quantities are defined in a certain reference frame by convention. E.g., cross sections in relativistic collisions are defined as if they were made in an fixed-target experiment and then expressed in a manifestly covariant way such that I don't need to think much anymore when doing a measurement and make my histograms in quantities everybody is used to.

Another convention is that phase-space distrubtion functions in relativistic kinetics are by definition Lorentz scalars. In (local) equilibrium intrinsic quantities like temperature and chemical potentials are, by convention, scalar (fields) defined in the (local) rest frame of the fluid. For details, see

http://fias.uni-frankfurt.de/~hees/publ/kolkata.pdf

All these definitions use "natural" reference frames, which are distinguished in the one or the other way by the physical situation discribed ("lab" frame of a two-body collision, (local) rest frame(s) of a fluid, etc.). All these quantities can written in manifestly covariant form and thus directly and conveniently measured in any reference frame.

 

[vanhees71]

I largely agree with DaleSpam's comments. But I would like to add that in a way, what you are proposing is more "geometric", not less. In your view, the laboratory walls serve as a physical manifestation of a proper reference frame and ideal clocks measure proper time, test particles and light rays follow geodesics. All those things are test objects or external rulers in the sense that they are assumed not to contribute to spacetime curvature. In conventional geometry, our rulers are rigid external objects, so I would say your view is more "geometric".

The "mathematical" view is not necessarily more "geometric". One can think of it as more correct than what you are proposing, because what they really mean is gauge-invariant (which is geometric in their sense, but not in the sense of having an external physical ruler or test particle). Of course it is not always physical in the sense of strict GR because there are neither observers nor measuring instruments in any vacuum spacetime. "It is the theory that says what can be observed"

So when martinbn says mercury follows a geodesic - it is not correct in the sense that it fails to account for mercury's perturbation of the background spacetime, but it is physical in the sense that he is talking about a gauge-invariant and hence physical object.

This I don't understand. Of course, you can only measure gauge-invariant properties, but I think, we are really drifting too much apart from the original topic of this thread.