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

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One particular form of the equivalence principle states that

The laws of physics for freely falling particles in a gravitational field are locally indistinguishable from those in a uniformly accelerating frame in Minkowski spacetime

My question is, does one arrive at this conclusion from a Newtonian perspective, i.e. before positing that gravity is the manifestation of spacetime curvature?

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If this is the case, then this form of the EP makes sense, since according to an observer on the surface of a planet, within the gravitational field (which we assume to be uniform), particles in free-fall are in a uniformly accelerating reference frame. Viewing gravity as a real force in this context, the observer on the planet would see that according to the freely falling reference frame, the physical laws are those of special relativity. Equivalently, an observer at rest in a uniformly accelerating rocket in deep space (far from an gravitational fields) would observe that freely falling particles are accelerating uniformly (in exactly the same manner as in the gravitational field), and again, the rocket observer would conclude that the laws of physics in this free fall frame are those of special relativity. The two cases are indistinguishable. Thus, the laws of physics of freely falling particles in a gravitational field and a uniformly accelerating reference frame in Minkowski spacetime are locally indistinguishable.

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PeterDonis
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One particular form of the equivalence principle states
Where are you getting this from? Please give a reference.

PeterDonis
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(page 23)
This appears to contain more or less the formulation you quoted in the OP. I would add some clarification as follows:

Locally in spacetime, the laws of physics for freely falling particles in a gravitational field in a frame at rest with respect to the source of the field are the same as those in a uniformly accelerating frame in Minkowski spacetime.

(Note that the first clarification is actually in the source, but you abbreviated it to "locally" in your formulation in the OP; I think the longer form is better. The second clarification is critical, since without it your statement is false.)

does one arrive at this conclusion from a Newtonian perspective
How can one, since Newtonian physics does not even contain the concept of "spacetime", let alone "Minkowski spacetime"?

How can one, since Newtonian physics does not even contain the concept of "spacetime", let alone "Minkowski spacetime"?
Sorry, this is a bad phrasing on my part. What I meant by this, is does this argument originate from initially considering gravity as a real force? I’m a little confused by this formulation of the equivalence principle since I thought that free-fall frames were locally inertial frames, since objects in free-fall follow the laws of physics. However, in this formulation seems to state that the laws of physics in a freely falling laboratory in a gravitational field are locally indistinguishable from those in a uniformly accelerating reference frame in Minkowski spacetime. Especially the part on page 23 of the second reference that say “locally an observer can always eliminate gravity by moving to an accelerated frame of reference” (I’ve paraphrased a bit here). Maybe I’m misinterpreting this?!

Is the point that, locally in spacetime, the laws of physics for an observer at rest in an arbitrary gravitational field are indistinguishable from those in a uniformly accelerating reference frame in Minkowski spacetime. Furthermore, locally in spacetime, the laws of physics in a freely falling frame of reference are those of special relativity.

PAllen
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No, you misunderstand the source as described by Peter. A frame at rest with respect to a gravitation source is NOT a free fall frame. Free falling particles in this frame will be accelerating, just like inertial particles in an SR accelerating frame.

A different aspect of the equivalence principle is local equivalence of SR inertial frames an free fall frames near a gravity source.

PeterDonis
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does this argument originate from initially considering gravity as a real force?
No. That would be inconsistent, since the conclusion of the argument is that gravity is not a real force.

I thought that free-fall frames were locally inertial frames
They are. But the source you quoted from is not considering that aspect of the equivalence principle. It's considering a different aspect: the local equivalence of uniformly accelerated frames in Minkowski spacetime with frames at rest in a gravitational field.

No, you misunderstand the source as described by Peter. A frame at rest with respect to a gravitation source is NOT a free fall frame. Free falling particles in this frame will be accelerating, just like inertial particles in an SR accelerating frame.

A different aspect of the equivalence principle is local equivalence of SR inertial frames an free fall frames near a gravity source.
No. That would be inconsistent, since the conclusion of the argument is that gravity is not a real force.

They are. But the source you quoted from is not considering that aspect of the equivalence principle. It's considering a different aspect: the local equivalence of uniformly accelerated frames in Minkowski spacetime with frames at rest in a gravitational field.
You’re right, I’ve got myself completely mixed up on this one.

So is it correct to say that the equivalence principle states that locally, it is impossible for an observer to distinguish whether they are at rest in an arbitrary gravitational field, or in a uniformly accelerating frame of reference in Minkowski spacetime, by carrying out any kind of experiment. That is, locally the laws of physics in a frame at rest in an arbitrary gravitational field are identical to those in a uniformly accelerating reference frame in Minkowski spacetime.

Furthermore, locally a freely falling reference frame in an arbitrary gravitational field constitutes a local inertial reference frame and the laws of physics will reduce to those of special relativity in such frames.

PeterDonis
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is it correct to say that the equivalence principle states that locally, it is impossible for an observer to distinguish whether they are at rest in an arbitrary gravitational field, or in a uniformly accelerating frame of reference in Minkowski spacetime, by carrying out any kind of experiment
Yes.

locally the laws of physics in a frame at rest in an arbitrary gravitational field are identical to those in a uniformly accelerating reference frame in Minkowski spacetime
Yes. And these laws of physics are just the laws of special relativity, expressed in a uniformly accelerating frame.

locally a freely falling reference frame in an arbitrary gravitational field constitutes a local inertial reference frame
Yes.

and the laws of physics will reduce to those of special relativity in such frames
More precisely, they will reduce to the laws of SR, expressed in an inertial frame. These are the same laws as above, just expressed in a different frame.

Yes. And these laws of physics are just the laws of special relativity, expressed in a uniformly accelerating frame.
Is this because an observer cannot locally distinguish between an arbitrary gravitational field and a uniformly accelerating reference frame in Minkowksi spacetime by performing any kind of experiment. The laws of physics in Minkowski spacetime are those of SR and so the laws of physics for the observer should be SR, but expressed in a uniformly accelerating reference frame?

Would it be correct then to say that locally within an arbitrary gravitational field, the laws of physics reduce to those of SR for all reference frames (inertial and non-inertial)?

PeterDonis
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Is this because an observer cannot locally distinguish between an arbitrary gravitational field and a uniformly accelerating reference frame in Minkowksi spacetime by performing any kind of experiment.
Isn't that what you already said in post #9, and I agreed to in post #10?

The laws of physics in Minkowski spacetime are those of SR and so the laws of physics for the observer should be SR, but expressed in a uniformly accelerating reference frame?
Isn't that what I already said?

Would it be correct then to say that locally within an arbitrary gravitational field, the laws of physics reduce to those of SR for all reference frames (inertial and non-inertial)?
Statements about the laws of physics are independent of any choice of reference frame. At least, that's the proper way to consider such statements in relativity. You can deduce particular consequences of that for how the laws will appear when expressed in a particular frame, but the laws themselves don't depend on which frame you choose.

Isn't that what you already said in post #9, and I agreed to in post #10?
Isn't that what I already said?
Sorry, yes I'm repeating things :/

Statements about the laws of physics are independent of any choice of reference frame. At least, that's the proper way to consider such statements in relativity. You can deduce particular consequences of that for how the laws will appear when expressed in a particular frame, but the laws themselves don't depend on which frame you choose.
If one is starting from a basic heuristic argument though, would the chain of arguments go like this:

1. The laws of physics reduce to those of SR in a local free-fall frame;

2. One then recasts the mathematical form of these laws into tensorial form;

3. The equations hold in one (set of) reference frame (a local free-fall frame), and hence, since they are in tensorial form, they hold in all reference frames.

PeterDonis
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would the chain of arguments go like this
No. Logically speaking (though not historically), putting the laws in tensorial form comes first. You don't have to pick a frame at all to do that.

No. Logically speaking (though not historically), putting the laws in tensorial form comes first. You don't have to pick a frame at all to do that.
Ok. I was trying to understand how it was done historically, and how Einstein arrived at these conclusions.

Given the requirement that the laws of physics should reduce to those of SR in local reference frames, are the generalisations of these equations to curved spacetime unique?

PeterDonis
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I was trying to understand how it was done historically, and how Einstein arrived at these conclusions.
Historically, Einstein's recognition of the equivalence principle as crucial for extending relativity to cover gravity predates the expression of the laws of physics in tensorial form. So Einstein's original understanding of the EP didn't involve the "tensorial form" stuff at all.

In fact, Einstein's recognition of the EP as crucial predates even the Minkowski spacetime formulation of SR (Einstein's recognition of the EP as crucial was in 1907; Minkowski came up with his formulation in 1908). So if you want to understand Einstein's original thought process about the EP, you have to go back to his 1905 formulation of SR.

Given the requirement that the laws of physics should reduce to those of SR in local reference frames, are the generalisations of these equations to curved spacetime unique?
If you mean, are the expressions of the laws in tensorial form unique, yes.

Historically, Einstein's recognition of the equivalence principle as crucial for extending relativity to cover gravity predates the expression of the laws of physics in tensorial form. So Einstein's original understanding of the EP didn't involve the "tensorial form" stuff at all.

In fact, Einstein's recognition of the EP as crucial predates even the Minkowski spacetime formulation of SR (Einstein's recognition of the EP as crucial was in 1907; Minkowski came up with his formulation in 1908). So if you want to understand Einstein's original thought process about the EP, you have to go back to his 1905 formulation of SR.
Ah ok. Maybe I'll stick with the modern interpretation then

If you mean, are the expressions of the laws in tensorial form unique, yes.
Ok cool, that's what I thought.

Thanks for your time and patience.

Historically, Einstein's recognition of the equivalence principle as crucial for extending relativity to cover gravity predates the expression of the laws of physics in tensorial form. So Einstein's original understanding of the EP didn't involve the "tensorial form" stuff at all.

In fact, Einstein's recognition of the EP as crucial predates even the Minkowski spacetime formulation of SR (Einstein's recognition of the EP as crucial was in 1907; Minkowski came up with his formulation in 1908). So if you want to understand Einstein's original thought process about the EP, you have to go back to his 1905 formulation of SR.

If you mean, are the expressions of the laws in tensorial form unique, yes.

Sorry to resurrect this post, but I have a further question that I'm hoping you can help me with.

By the equivalence principle, within a sufficiently small neighbourhood of a given spacetime point the laws of physics are those of SR in both inertial and non-inertial reference frames, so what is the difference when one considers a larger region of a given coordinate chart?

The reason I ask is because in a non-inertial frame, Christoffel symbols will naturally be present in the equations due to fact of being in an accelerated reference frame. The same is true in a coordinate system in the presence of a gravitational field, however, if one considers a large enough region within a given coordinate chart one should be able to detect the effects of curvature. Shouldn't one be able to distinguish between the laws of SR expressed in non-inertial coordinates, and the case in which curvature has to be taken into account?

Is the point that, if we are in a non-inertial reference frame in which the laws of SR apply we can always find a coordinate transformation that reduces the metric to the Minkowksi metric. However, for larger regions around a spacetime point it is impossible to choose a coordinate system in which the metric reduces to the the Minkowski metric and curvature effects have to be taken into account.

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PeterDonis
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what is the difference when one considers a larger region of a given coordinate chart?
That you can no longer ignore the effects of spacetime curvature, if it is present (see below).

Shouldn't one be able to distinguish between the laws of SR expressed in non-inertial coordinates, and the case in which curvature has to be taken into account?
Yes, by computing the Riemann tensor and seeing whether it is zero or nonzero. If it's zero, you are in flat spacetime and any stuff like Christoffel symbols, etc. is just due to choosing non-inertial coordinates. If the Riemann tensor is nonzero, spacetime curvature is present and there is no such thing as global inertial coordinates--any inertial chart only works in a small enough patch of spacetime that curvature can be ignored.

Yes, by computing the Riemann tensor and seeing whether it is zero or nonzero. If it's zero, you are in flat spacetime and any stuff like Christoffel symbols, etc. is just due to choosing non-inertial coordinates. If the Riemann tensor is nonzero, spacetime curvature is present and there is no such thing as global inertial coordinates--any inertial chart only works in a small enough patch of spacetime that curvature can be ignored.
So when one says that within a small enough patch of spacetime the laws of physics reduce to SR, is it the case that one could start in non-inertial coordinates within this sufficiently small patch, such that the Christoffel symbols don't vanish, but it would be possible to find a coordinate transformation such that they vanish and one can explicitly see that the laws are those of SR? If the patch is larger, then such a transformation does not exist as a consequence of spacetime curvature.

Is it correct to say that the equivalence principle implies that within a sufficiently small neighbourhood of each spacetime point the laws of physics are those of SR, regardless of ones choice of coordinates (i.e. in both inertial and non-inertial coordinates)?

If this is true though, then I'm confused about the fact that non-inertial frames are included since the Riemann tensor will not vanish (since the metric will only be Minkowski to second order).

PAllen
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So when one says that within a small enough patch of spacetime the laws of physics reduce to SR, is it the case that one could start in non-inertial coordinates within this sufficiently small patch, such that the Christoffel symbols don't vanish, but it would be possible to find a coordinate transformation such that they vanish and one can explicitly see that the laws are those of SR? If the patch is larger, then such a transformation does not exist as a consequence of spacetime curvature.
You don’t need a coordinate transform to verify flatness. The curvature tensor is an expression based on the Christoffel symbols. If it is zero, this fact is true in all coordinates. For example, polar coordinates for a plane have nonzero christoffel symbols, but the curvature tensor constructed from them vanishes. Similarly, Rindler coordinates in SR have non vanishing christoffel symbols, but vanishing curvature tensor constructed from. You can never make the curvature tensor vanish at a point unless there is no curvature there.

You don’t need a coordinate transform to verify flatness. The curvature tensor is an expression based on the Christoffel symbols. If it is zero, this fact is true in all coordinates. For example, polar coordinates for a plane have nonzero christoffel symbols, but the curvature tensor constructed from them vanishes. Similarly, Rindler coordinates in SR have non vanishing christoffel symbols, but vanishing curvature tensor constructed from. You can never make the curvature tensor vanish at a point unless there is no curvature there.
This is what I'm confused about though. The equivalence principle (EP) states that in a local free-fall frame the laws of physics should reduce to those of SR, which mathematically, means that within a sufficiently small region around a spacetime point, a local coordinate transformation exists such that the metric is Minkowksi (up to second order), with the Christoffel symbols vanishing within this region. However, the EP also states that locally, the effects of gravitation should be indistinguishable from that of a uniformly accelerating reference frame in Minkowski spacetime. So the laws of physics should reduce to those of SR in non-inertial reference frames within a sufficiently small patch of spacetime too, right?

PAllen
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This is what I'm confused about though. The equivalence principle (EP) states that in a local free-fall frame the laws of physics should reduce to those of SR, which mathematically, means that within a sufficiently small region around a spacetime point, a local coordinate transformation exists such that the metric is Minkowksi (up to second order), with the Christoffel symbols vanishing within this region. However, the EP also states that locally, the effects of gravitation should be indistinguishable from that of a uniformly accelerating reference frame in Minkowski spacetime. So the laws of physics should reduce to those of SR in non-inertial reference frames within a sufficiently small patch of spacetime too, right?
The EP is effectively a first order statement. The curvature tensor measures second and higher order deviations, and physics attributable to curvature generally vanishes in the limit at a point, while curvature does not. Consider a trivial calculus analog. To first order, the tangent to a curve approximates the curve near the point of tangency. But the second derivative does not vanish at a point.

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Mister T
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is it the case that one could start in non-inertial coordinates within this sufficiently small patch, such that the Christoffel symbols don't vanish, but it would be possible to find a coordinate transformation such that they vanish and one can explicitly see that the laws are those of SR?
Yes. More precisely, you can explicitly see that the laws are those of SR as expressed in an inertial frame. But you don't have to express the laws of SR in an inertial frame to see that they're the laws of SR. You can write those laws in tensorial form, independent of any choice of coordinates.

Is it correct to say that the equivalence principle implies that within a sufficiently small neighbourhood of each spacetime point the laws of physics are those of SR, regardless of ones choice of coordinates (i.e. in both inertial and non-inertial coordinates)?
Go back and read what I said in post #12 (the last part).

I'm confused about the fact that non-inertial frames are included since the Riemann tensor will not vanish
Yes, it will. Don't make assumptions. Try it. For example, try computing the Riemann tensor for Rindler coordinates on Minkowski spacetime.

https://en.wikipedia.org/wiki/Rindler_coordinates

Yes. More precisely, you can explicitly see that the laws are those of SR as expressed in an inertial frame. But you don't have to express the laws of SR in an inertial frame to see that they're the laws of SR. You can write those laws in tensorial form, independent of any choice of coordinates.
Does this mean that in an arbitrary frame of reference the laws of physics are those of SR, but in coordinate form include terms taking into account curvature?

I think what I've got myself confused about is what happens when one isn't considering a sufficiently small patch of spacetime. Are the non-gravitational laws of physics still those of SR (since they are in tensorial form and thus coordinate independent) when curvature cannot be neglected?

Is the point that the EP principle requires that within a sufficiently small neighbourhood of each spacetime point the laws of physics are those of SR, and this amounts to being able to find a local coordinate transformation in which the metric reduces to the Minkowski metric (to second order), where one can explicitly see that the laws of physics take on their SR form. Thus, any observer (for example a Rindler coordinate frame) can find such local coordinates. However, this coordinate transformation will only hold for a sufficiently small region, after which it breaks down and the effects of curvature become apparent and one must take into account the Christoffel symbols. One can furthermore calculate the Riemann tensor and find that it is non-vanishing and hence the presence of the Christoffel symbols is due to the curvature of spacetime and not the effects of an accelerated reference frame.

This is distinguished from SR, in which spacetime is globally Minkowski. An observer who starts of in Rindler coordinates will be able to find a global coordinate transformation that transforms the metric in its Minkowski form. Furthermore, if they compute the Riemann tensor in their (initial) frame of reference they will find that it vanishes and hence they can conclude that they are in flat spacetime.

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