Undergrad Inertial & non-inertial frames & the principle of equivalence

[PeterDonis]

I thought that the Riemann normal coordinate system was an exponential mapping from the tangent space at a point to the manifold at that point?

And I asked why you think that. Just repeating it doesn't answer the question. Where are you getting it from? What textbook? What peer-reviewed paper?

is it the case that when one uses a coordinate basis induced by a coordinate system on the spacetime manifold the component form of the mathematical representations of the laws don't take on their special relativistic form exactly because the manifold is curved?

You shouldn't be thinking about coordinates until you understand things in coordinate-independent terms. SR assumes spacetime is flat, so obviously whatever the laws of physics are in a curved spacetime, they can't be the exact laws of SR. That is true regardless of what coordinates you choose.

If the metric is defined on the tangent spaces of the spacetime manifold, how does it describe the curved geometry on the manifold?

The metric on the tangent space is additional structure on the tangent space. The metric on the tangent space is not the same as the metrc on the actual manifold. That should be obvious since the tangent space metric is flat and the metric on the actual manifold is curved.

 

[Frank Castle]

And I asked why you think that. Just repeating it doesn't answer the question. Where are you getting it from? What textbook? What peer-reviewed paper?

Apologies, I misunderstood what you were getting at here. I’ve been reading Nakahara’s book “Geometry, Topology and Physics” . In section 7.4.4 he discusses normal coordinate systems in terms of a map $EXP: T_{p}M\rightarrow M$, $X_{q}\mapsto q$ where $X_{q}\in T_{p}M$. I realize now that the notation might be misleading and he’s not referring to an exponential map given how he’s defined the actual mapping of tangent vectors to points.

The metric on the tangent space is additional structure on the tangent space. The metric on the tangent space is not the same as the metrc on the actual manifold. That should be obvious since the tangent space metric is flat and the metric on the actual manifold is curved.

But I thought the metric tensor was defined as $g:T_{p}M\times T_{p}M\rightarrow\mathbb{R}$, i.e. it acts on tangent vectors? In Nakahara’s book he makes no mention of there being to different metrics defined, one on the manifold and one on the tangent spaces.

 

[PeterDonis]

I thought the metric tensor was defined as $g:T_{p}M\times T_{p}M\rightarrow\mathbb{R}$, i.e. it acts on tangent vectors?

All tensors act on tangent vectors. But there are still two metrics and you have to avoid confusing them. The tangent space itself is flat, and has its own flat metric, distinct from the metric of the actual curved manifold. In other words, in addition to the tensor $g$, the metric of the curved manifold, there is another tensor, usually called $\eta$, which is the metric of the flat tangent space. Both of these are tensors and so they act on tangent vectors; but they are different tensors.

 

[Frank Castle]

All tensors act on tangent vectors. But there are still two metrics and you have to avoid confusing them. The tangent space itself is flat, and has its own flat metric, distinct from the metric of the actual curved manifold. In other words, in addition to the tensor $g$, the metric of the curved manifold, there is another tensor, usually called $\eta$, which is the metric of the flat tangent space. Both of these are tensors and so they act on tangent vectors; but they are different tensors.

Ah ok. I’ve never seen this distinction being explicitly made before. Do you know of any good textbooks that discuss this?

In practice, how does one make the distinction between them? Is it simply that one chooses to evaluate $g$ on coordinate basis vectors that are tangent to coordinate curves on the actual manifold, thus describing the geometry on the manifold

 

[PeterDonis]

I’ve never seen this distinction being explicitly made before.

Probably because textbooks generally have no need to discuss the actual metric of the tangent space, as distinct from the metric of the manifold, in any detail; just saying that the tangent space is flat is enough.

In practice, how does one make the distinction between them?

If you see the metric written in the form $g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$, or something similar, that's how. But that form is usually used only in the weak field approximation, where it is assumed that $h_{\mu \nu} \ll 1$.

Is it simply that one chooses to evaluate $g$ on coordinate basis vectors that are tangent to coordinate curves on the actual manifold

No; the distinction between any two tensors is independent of any choice of coordinates.

 

[Frank Castle]

Probably because textbooks generally have no need to discuss the actual metric of the tangent space, as distinct from the metric of the manifold, in any detail; just saying that the tangent space is flat is enough.

Ah ok, so it’s implicitly assumed then.

If you see the metric written in the form gμν=ημν+hμνgμν=ημν+hμνg_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}, or something similar, that's how. But that form is usually used only in the weak field approximation, where it is assumed that hμν≪1hμν≪1h_{\mu \nu} \ll 1.

So in this case is $g_{\mu\nu}$ the pullback metric from the tangent space?

No; the distinction between any two tensors is independent of any choice of coordinates.

Sorry, what I meant by this is, if I represent both metrics in a coordinate basis (induced by a coordinate chart on the manifold) how do I differentiate between the two? Does one simply calculate the Riemann tensor for both?

 

[PeterDonis]

if I represent both metrics in a coordinate basis (induced by a coordinate chart on the manifold) how do I differentiate between the two?

Um, by their components? [Edit: actually, strictly speaking, this won't work--see my post #69.]

Does one simply calculate the Riemann tensor for both?

To do that you need their components anyway, so you need to know those first.

 

[PeterDonis]

in this case is $g_{\mu\nu}$ the pullback metric from the tangent space?

No. Why would you think that?

 

[PeterDonis]

if I represent both metrics in a coordinate basis (induced by a coordinate chart on the manifold)

Actually, strictly speaking, you can't even do this, because the actual spacetime and its tangent space, considered as manifolds, are different manifolds, so you would need two different coordinate charts (one for the spacetime and one for the manifold), and each chart can only represent one of the two metrics.

 

[PeterDonis]

I’ve been reading Nakahara’s book “Geometry, Topology and Physics” .

I should note that I am not an expert on this topic, or on this textbook. Also, it is a very advanced textbook, and I'm not sure you have the background for it. We seem to be increasing your confusion in this thread instead of reducing it.

It might be better at this point to go back to your original question about the EP. Has it been answered? If not, what has not been answered? To be clear, answering your original question about the EP should not require all of this advanced differential geometry and topology. The fact that we are getting deeper into those topics indicates, to me, that we have gotten off the track.

 

[Frank Castle]

Um, by their components?

I was meaning in terms of which one is the flat metric and which one corresponds to the curved manifold, but I guess this question has been answered now.

I've been re-reading Sean Carroll's notes and he talks about the exponential map as a local mapping of the tangent space to the manifold via $exp_{p}:T_{p}M\rightarrow M$, $exp_{p}(k^{\mu})=x^{\mu}(\lambda =1)$ where $x^{\mu}(\lambda)$ is a solution to the geodesic equation subject to $\frac{dx^{\mu}(0)}{d\lambda}=k^{\mu}$. Is this what you were referring to on being able to approximate the manifold locally with the tangent space near a given point?

No. Why would you think that?

Sorry, ignore me on this one. I was mis-remembering a section I'd read in Sean Carroll's notes.

 

[Frank Castle]

I should note that I am not an expert on this topic, or on this textbook. Also, it is a very advanced textbook, and I'm not sure you have the background for it. We seem to be increasing your confusion in this thread instead of reducing it.

It might be better at this point to go back to your original question about the EP. Has it been answered? If not, what has not been answered? To be clear, answering your original question about the EP should not require all of this advanced differential geometry and topology. The fact that we are getting deeper into those topics indicates, to me, that we have gotten off the track.

The original question has been answered, and I think I understood it. Apologies for getting so side-tracked, I managed to get myself muddled with some concepts that I thought I understood. I guess I'm going to have to go back and read/re-read some things. Are there any particularly good textbooks or notes that you've read and found helpful?

 

[PeterDonis]

he talks about the exponential map as a local mapping of the tangent space to the manifold via $exp_{p}:T_{p}M\rightarrow M$, $exp_{p}(k^{\mu})=x^{\mu}(\lambda =1)$ where $x^{\mu}(\lambda)$ is a solution to the geodesic equation

Note that last qualifier: a solution to the geodesic equation. That's crucial. What Carroll is saying here is that a point in spacetime and a particular tangent vector at that point, taken together, determine a unique geodesic throughout the spacetime (or at least throughout some open connected region of the spacetime). We can then construct a map from the tangent space at the chosen point to the spacetime by mapping each tangent vector to the point in the spacetime that lies exactly one unit (of affine parameter) along the unique geodesic in the spacetime determined by that tangent vector.

Is this what you were referring to on being able to approximate the manifold locally with the tangent space near a given point?

No. The mapping I described above, heuristically, is a mapping from the tangent space at a point to a "unit circle" in the spacetime centered on that point. It is not a mapping from the tangent space, considered as a flat manifold, to an open neighborhood of the chosen point in the actual curved manifold.

Once again, we seem to be getting further away from the actual topic of this thread; this discussion is not supposed to be a general discussion about differential geometry. Is there anything specifically about the equivalence principle that still needs to be clarified?

 

[PeterDonis]

Are there any particularly good textbooks or notes that you've read and found helpful?

I personally think Carroll's lecture notes are enough of a treatment of differential geometry for GR unless you are actually doing active research in the field. MTW give a more detailed treatment, but it can be hard to follow. Wald also gives a more detailed treatment, but it's more abstract and I'm not sure the physical meaning comes through as clearly.

 

[Frank Castle]

Is there anything specifically about the equivalence principle that still needs to be clarified?

Just a (hopefully) quick clarification. By the way, thanks for answering my further questions despite going of on a massive tangent (pardon the pun), sorry it ended up being so long.

So, if I've understood correctly, the equivalence principle corresponds to our ability to construct a RNC system on the spacetime manifold, within which the laws of physics take their SR form (mathematically), however, the metric is only Minkowski to first-order (apart from at the origin of the coordinate system). Since this is a coordinate system on the actually manifold, physics is only approximately that of SR within the local neighbourhood of the origin of this coordinate system. Alternatively, one can work in the tangent space to a given point in which the laws of physics are exactly those of SR - this will approximate an infinitesimal neighbourhood of the manifold around a given point precisely because the manifold is (pseudo-) Riemannian.

The will (probably) be the last thing I wanted to ask related to this is, does a RNC system generally cover a smaller patch of the manifold than a more general coordinate system, in which the laws of physics do not reduce to their SR form?

I personally think Carroll's lecture notes are enough of a treatment of differential geometry for GR unless you are actually doing active research in the field. MTW give a more detailed treatment, but it can be hard to follow. Wald also gives a more detailed treatment, but it's more abstract and I'm not sure the physical meaning comes through as clearly.

Thanks very much for the recommendations.

 

[PeterDonis]

if I've understood correctly, the equivalence principle corresponds to our ability to construct a RNC system on the spacetime manifold, within which the laws of physics take their SR form

The EP is independent of any choice of coordinates. And the ability to construct RNC centered on a point is a property of any manifold, as a matter of mathematics, independent of any physical interpretation. So I don't know if what you say here is a useful way of looking at it.

 

[Frank Castle]

The EP is independent of any choice of coordinates. And the ability to construct RNC centered on a point is a property of any manifold, as a matter of mathematics, independent of any physical interpretation. So I don't know if what you say here is a useful way of looking at it.

I think this is still a bit of a sticking point for me. I get that the EP also requires that the laws of physics are those of SR for a sufficiently small patch of spacetime in uniformly accelerating reference frames as well as free-fall frames, but I’m unsure how this is realized in practice? I mean, if one is in a non-inertial reference frame, how does one know how local a region around a given point one has to be for the EP to hold?
In RNCs these is more explicitly obvious, since the derivative of the Christoffel symbols, $\frac{\partial\Gamma^{\mu}_{\;\alpha\beta}}{\partial x^{\nu}}=-\frac{1}{3}\left(R^{\mu}_{\;\alpha\beta\nu}+R^{\mu}_{\;\beta\alpha\nu}\right)$, determines how far one can move from the origin of the coordinate system before curvature becomes non-negligible (explicitly one uses the Jacobi equation to calculate the geodesic deviation).

 

[PeterDonis]

I get that the EP also requires that the laws of physics are those of SR for a sufficiently small patch of spacetime in uniformly accelerating reference frames as well as free-fall frames

That is because the EP is independent of your choice of coordinates, and all you are doing when you use an accelerating frame instead of a free-fall frame is to choose different coordinates (Rindler coordinates vs. Minkowski coordinates).

how does one know how local a region around a given point one has to be for the EP to hold?

This has nothing to do with your choice of coordinates. It has to do with how curved the spacetime is as compared to how accurate your measurements are.

In RNCs these is more explicitly obvious

Yes, but that's a calculational convenience, not a necessity.

 

[Frank Castle]

That is because the EP is independent of your choice of coordinates, and all you are doing when you use an accelerating frame instead of a free-fall frame is to choose different coordinates (Rindler coordinates vs. Minkowski coordinates).

So does one simply exploit the EP by noting that one can calculate a quantity using SR in either an inertial or non-inertial frame and this calculation will be valid for a sufficiently small region in curved spacetime?

This has nothing to do with your choice of coordinates. It has to do with how curved the spacetime is as compared to how accurate your measurements are.

Can one not calculate the geodesic deviation of test particles to determine the range of validity of ones chosen local inertial coordinates (i.e. the range at which curvature causes the geodesics to intersect)?

 

[PeterDonis]

does one simply exploit the EP by noting that one can calculate a quantity using SR in either an inertial or non-inertial frame and this calculation will be valid for a sufficiently small region in curved spacetime?

You can do that, yes, and the EP says it will work.

Can one not calculate the geodesic deviation of test particles to determine the range of validity of ones chosen local inertial coordinates (i.e. the range at which curvature causes the geodesics to intersect)?

You can do that for any coordinates. Geodesic deviation is independent of coordinates.

 

[Frank Castle]

You can do that for any coordinates. Geodesic deviation is independent of coordinates.

By this do you mean that one can use any coordinate system you want (inertial or non-inertial) such that the laws of physics are those of SR for a sufficiently small neighbourhood - one can calculate the geodesic deviation in any of these coordinate systems to determine how small this neighbourhood has to be in order for the approximation to of SR to hold?

 

[PeterDonis]

By this do you mean that one can use any coordinate system you want (inertial or non-inertial) such that the laws of physics are those of SR for a sufficiently small neighbourhood - one can calculate the geodesic deviation in any of these coordinate systems to determine how small this neighbourhood has to be in order for the approximation to of SR to hold?

Yes.

 

[Frank Castle]

Yes.

Ok great, I think I'm getting it now. So is the point that if one considers larger regions of a given coordinate system the approximation breaks down and one has to take into account the effects of the gravitational field? The equations (for the non-gravitational laws of physics) will look the same as they do in a non-inertial reference frame (i.e. including connection terms), however, the Riemann tensor will be non-zero indicating that the spacetime is curved (this is true in an infinitesimal neighbourhood of a point too, but the point is the tidal effects are too small to be observable for small enough regions). Furthermore, the geodesic deviation for finite patches of the coordinate system will be non-negligible meaning that full GR is required in order to correctly describe physical experiments.

 

[PeterDonis]

is the point that if one considers larger regions of a given coordinate system the approximation breaks down and one has to take into account the effects of the gravitational field?

Not larger regions of a given coordinate system. Larger regions of the spacetime. All of this is independent of any choice of coordinates. As I said, it depends on how curved the spacetime is and how accurate your measurements are. Those are independent of coordinates.

 

[Frank Castle]

Not larger regions of a given coordinate system. Larger regions of the spacetime. All of this is independent of any choice of coordinates. As I said, it depends on how curved the spacetime is and how accurate your measurements are. Those are independent of coordinates.

Sorry, by larger region I was assuming this corresponded to covering a larger region of spacetime.

So depending on how accurate one's measurements are and how curved spacetime actually is will determine the size of the region of spacetime around each point in which the laws of SR (approximately) hold, and this will be true for any coordinate system?

If one can always choose a RNC system, and furthermore, because the laws of physics are in tensorial form, one can choose any coordinate system in which the laws of physics are those of SR for sufficiently small neighbourhoods of each point, is it the case that the only real point where GR comes in is determining the geodesics of spacetime such that a RNC can be constucted, and working out the geodesic deviation such that one can determine how small the region around each point has to be in order for curvature to be negligible?

 

[PeterDonis]

depending on how accurate one's measurements are and how curved spacetime actually is will determine the size of the region of spacetime around each point in which the laws of SR (approximately) hold

Yes.

and this will be true for any coordinate system?

It is true independently of coordinates. You seem to have the logic backwards. You don't first choose coordinates and then figure out the size of the region. You first figure out the size of the region, using coordinate-independent facts (the accuracy of your measurements and the curvature of spacetime are both coordinate-independent), and then, if you must, you choose coordinates and calculate what the coordinate-independent facts translate to in those coordinates.

is it the case that the only real point where GR comes in is determining the geodesics of spacetime such that a RNC can be constucted, and working out the geodesic deviation such that one can determine how small the region around each point has to be in order for curvature to be negligible?

You make it sound like this isn't very much. In fact it's everything. "GR comes in" in determining the actual curved geometry of the spacetime. That is everything. It's not just a small thing added on.

 

[Frank Castle]

You make it sound like this isn't very much. In fact it's everything. "GR comes in" in determining the actual curved geometry of the spacetime. That is everything. It's not just a small thing added on.

Sorry, I realize it's a much bigger deal than I make it sound. I was just wondering how this enters the non-gravitational laws of physics - since they are in tensorial form they "look" the same whether or not spacetime is curved, it's just in coordinate form that they differ, i.e. partial derivatives becoming covariant derivatives and the metric becoming non-Minkowski, however, this would be true in a non-inertial frame in flat spacetime too. Can differences be seen, for example, from the EM wave equation, in which a term proportional to curvature appears in curved spacetime?

 

[PeterDonis]

partial derivatives becoming covariant derivatives and the metric becoming non-Minkowski

Both of these statements are independent of coordinates.

 

[Frank Castle]

Both of these statements are independent of coordinates.

Ah ok. So this is the key point - the fact that the derivatives become covariant derivatives and the metric non-Minkowski is a due to the manifold being curved, which is a coordinate independent statement.

 

[PeterDonis]

the fact that the derivatives become covariant derivatives and the metric non-Minkowski is a due to the manifold being curved, which is a coordinate independent statement

Yes.