General Relativity: Definition of Locally Inertial Frame

Question

The Einstein Equivalence Principle states that in a sufficiently small frame of reference is impossible to know if we are into a gravitational field or not. Equivalently we cannot say if we are in an accelerating frame or not. This is because gravity and inertia are equivalent (hence the name of the principle).

So because in a local frame we indeed can't establish if we are accelerating or not, makes more sense to define a local inertial frame of reference (synonym of not accelerating frame) as a "free falling frame". In fact into a local free falling frame things behave as if in a perfectly inertial special relativity frame. ($\mathbb{M}^4$)

Perfect, but seems to me that this should imply that a local observer standing on the earth (so not free falling at all) should be considered as an accelerating, non inertial frame.

Ok, this seems also fine. But we know that there is another, more geometrical, equivalent formulation of EEP:

Locally spacetime looks like $\mathbb{M}^4$

This is not the precise formulation of the geometrical formulation, but it's good enough. This means that in every sufficiently small region of spacetime it's like being into a inertial special relativity frame, so no accelerating, no gravity, no shenanigans.

But: earlier we said that me, writing this question on the surface of the earth, should be considered as an accelerating frame! But the geometrical formulation states that every sufficiently small reference frame, myself included, should be like an inertial SR frame!

So, in the context of GR, am I accelerating? Or on the contrary am I into a local inertial SR frame? And most important of all: why this two formulation of EEP seems to contradict each other?

Answer 1

seems to me that this should imply that a local observer standing on the earth (so not free falling at all) should be considered as an accelerating, non inertial frame.

Yes, an observer standing on the earth is not inertial in relativity. The definitive test is to have the observer carry a good accelerometer. In this case it will indicate an acceleration of 1 g upwards, conclusively showing that the observer is non-inertial.

Just a nitpick on language: an observer isn’t a reference frame, he or she has a reference frame, or even better there is a reference frame where he or she is at rest.

there is another, more geometrical, equivalent formulation of EEP: Locally spacetime looks like 𝕄4 This is not the precise formulation of the geometrical formulation, but it's good enough.

Agreed, it is good enough for present purposes.

This means that in every sufficiently small region of spacetime it's like being into a inertial special relativity frame, so no accelerating, no gravity, no shenanigans.

It does not mean that at all. You can certainly have accelerating reference frames with pseudo-gravitational forces in 𝕄4. All 𝕄4 means is that you cannot have any tidal effects.

𝕄4 is a flat spacetime manifold and can be equipped with an endless number of coordinate systems, including non-inertial ones. What “locally spacetime looks like 𝕄4 means is that all of the curvature tensors on spacetime are 0 over the local region, but it does not restrict you to a certain class of coordinate systems.

But the geometrical formulation states that every sufficiently small reference frame, myself included, should be like an inertial SR frame!

No, the observer is unambiguously non-inertial. The geometrical formulation does not contradict that at all. The geometrical formulation merely says that in a small region spacetime is flat, not that an observer is inertial. It is perfectly consistent to have non-inertial observers and reference frames in flat spacetime. Only tidal effects are forbidden.

Answer 2

Standing on the surface of the Earth, the reference frame at rest relative to yourself is certainly not a frame with Minkowski metric. Here is the proof: release an object, so that it is in free fall. There is relative acceleration between the object and the chosen frame. Hence the frame is not inertial and its metric is not Minkowskian.

To define a tangent space in general relativity it is not sufficient that the metric be Minkowskian just at one event. It must be Minkowskian AND have no first-order dependence on distance or time near that event. In other words the Christoffel symbols must all vanish. But since the released object is accelerating relative to the frame at rest on Earth, at least one of the Christoffel symbols is not zero.

Answer 3

Great question. If I could somewhat reframe your question, I believe you confused by the apparent contradiction between these two statement about the equivalent principle:

1. Any manifold in GR locally looks like Minkowski space.
2. Even (very small) local frames can demonstrate gravitational effects (e.g. you can feel yourself "accelerating upward" if you are standing on the surface of the Earth).

Your objection is that the statement 1 seems to imply that there can't be any observable gravitational effects over very small regions of spacetime, while statement 2 seems to imply that there can be.

The resolution to this apparent contradiction is that statements 1 and 2 and using different quantitative notions of the word "local", and statement 1 is restricting the word "local" to smaller regions than statement 2 is.

More precisely: statement 1 can be more precisely rephrased as:

For any point $p$ on any pseudo-Riemannian manifold (i.e. spacetime), there exists a local coordinate system around $p$ in which the Taylor expansion of metric tensor agrees with the Minkowski metric $\eta$ to first order about $p$.

In other words, $g(p) = \eta$ and $\partial_\mu g(p) \equiv 0$ in these particular coordinates (which are known as Riemann normal coordinates). So if you define "local" to mean "so small that only first-order variations are non-negligeable," which is the implicit assumption in statement 1, then indeed no gravitational effects can be detected locally.

But it turns out that the effects of curvature (or the acceleration of the "owner" of a local coordinate system) necessarily come in at second order in the metric. More precisely:

A manifold has intrinsic curvature at a point $p$ iff the second-order Taylor expansion of the metric tensor about $p$ deviates from the Minkowski metric.

Or even more precisely:

At any point $p$ on any pseudo-Riemannian manifold, the second-order partial derivatives of the metric tensor $\partial_\mu \partial_\nu g(p)$ are either identically zero in every coordinate system, or have some nonzero components in every coordinate system. Therefore, the proposition $\partial_\mu \partial_\nu g(p) \equiv 0$ is coordinate independent. The Riemann curvature tensor vanishes at $p$ iff $\partial_\mu \partial_\nu g(p) \equiv 0$ in some coordinate system (and therefore in all of them).

Therefore, you can always make the Taylor expansion of the metric about a point agree with the Minkowski metric to first order (by using Riemann normal coordinates), but you can't make it agree to second order if the manifold is curved at point $p$. Since gravitational effects are a physical manifestation of the curvature of the spacetime manifold, you can detect them if your local frame is big enough to capture second-order deviations about the point $p$. This slightly weaker sense of "local" is the sense being used in statement 2. If your region of spacetime is only "first-order big" in the time direction, then you won't have time to measure any relative acceleration of a nearby test particle.

(By the way, you can't actually forces that induce an acceleration, but only forces that induce a tidal acceleration - defined broadly as any spatial variation in the acceleration field. The only reason that you can feel the Earth accelerating you upward is because your body is big enough that the second-order terms in the metric (which are proportional to the acceleration constant $g$) are non-negligeable. You many not be used to thinking of the electrostatic repulsion acceleration of the Earth pushing up on you as a "tidal" acceleration, but it is: the only reason you can feel it is that it's applied at the soles of your feel but not elsewhere on your body, which induces internal compression forces within your body that you feel. If it were somehow distributed in such a way as to induce a uniform acceleration over your entire body, then it would work just like gravity and you wouldn't be able to feel it.)

Answer 4

General relativity is quite clear. As an observer on the surface of the Earth, you are in an accelerating (non-inertial) frame. Your other formulation is consequently not good enough. Only inertial frames look like Minkowski frames as used in special relativity. This is the essence of the equivalence principle, and it should be clear that only one of your formulations is correct.

The standard test of an inertial frame is to use an accelerometer (you may even have one in and app for your mobile phone). You can therefore tell whether you are in an inertial frame without looking outside of your immediate locality.