Equivalence principle and local inertial reference frames

Question

Local inertial reference frames are defined as follows:

Pick a set coordinates for the manifold (assuming the manifold can be described by global coordinates) $\{x^i\}$ such that $g(p) = \eta$ is the metric at a point $p$ at the manifold and the first derivative of the metric coefficients will vanish at the point $p$.

But the problem that I have is not this formal definition, which is mathematically clear, but the physics is difficult. Since in special relativity we defined inertial frames as frames that have no acceleration (which has to be absolute). But if we look for example how the equivalence principle is described, then people always tend to use accelerating elevators and then conclude that locally the system should be inertial, but I do not get this statement. How can an object be accelerated and describe an inertial frame locally at the same time? The statement "we can apply SR" locally does not make quite sense. Mathematically it does, since we showed that the metric is the Minkowski metric at a point $p$ so that in the tangent space (isomorphic to Minkowski space, only at $p$) we can define 4-vectors and so on, but physically it is not obvious for me.

Could somebody elaborate on this notion?

Answer 1

Local inertial frames (LIF) are simply frames , or observers, for whom the laws of inertia hold. This means that freely falling objects either stay put or move with constant velocity in the frame. Locally means in a small area of space and for a short period of time, that is in a small patch of space-time. Locally inertial frames are therefore simply freely falling ,non rotating, observers who observe freely falling neutral particles move in straight lines with constant speed (or stay put). In LIF the laws of SR hold, so charged particles will follow Maxwell eqs. A typical Earth bound lab is not LIF:grab a PC and let it go! will it float? No it will crash on the floor. The best known example of LIF is the ISS . Everything is freely floating there and it is in fact used for experiments in microgravity. MTW provides nice descriptions of LIF in the first chapters.

The OP is is confused because in SR inertial frames are defined as frames which are not accelerated. But in fact this definition is carried over to general relativity in the following way : inertial frames are frames with no 4-acceleration. If you have a frame with null 4-acceleration it means that you are freely falling. It also means that you might have a non null 3-acceleration when the components of a 4-acceleration are calculated in a non inertial frame {$x^\mu$}: $$a^\mu=u^\mu,_\nu u^\nu+\Gamma^\mu_{\nu\rho}u^\nu u^\rho$$

On RHS the first term is closely related to the 3-acceleration (it is a derivative of a velocity) while the second term contains correction terms representing forces and they cancel the first term so that the left-hand side is 0. In short

$$u^\mu,_\nu u^\nu= -\Gamma^\mu_{\nu\rho}u^\nu u^\rho$$

so the acceleration on the LHS is "explained" by the non null gamma terms on the RHS.

Please note that the 4-acceleration is 0 for all frames (obviously since it is a 4-vector).

Answer 2

In general relativity we are studying gravity. One interesting observation that we can make about a uniform gravitational field is that when you are freely falling under its influence, and you are closed in some sort of a box, you can not tell that you are freely falling AND you can not tell that you are under the influence of any sort of gravitational field. Effects of gravity are canceled buy your free fall and you are actually, in your freely falling system, in the same situation as some guy that is floating in space far away from any source of gravity. Since there is no way to experimentaly confirm that you are freely falling in some gravitational field because everything arround you is falling at the same rate, this is actually an inertial frame. But keep in mind, this is only if you are freely falling (in a gravitational field). Of course, all of this is possible because everything in your box is falling at the same rate: the apple, glass marble, loaf of bread etc, regardless of their mass. But, regarding the equivalence principle, we can also state it in this way: standing still in a grav field is same as accelerating upwards somewhere where there is no field. Now, all of this is OK if we are talking about uniform gravitational field, but in real life, you have for example, grav field of the Earth which is non-uniform, so you could actually, if you experiment carefuly, detect this non-uniformity. And you would know that you are in a grav field. But, locally, in a small amount of space-time arround some point, you could not tell. Nevertheless observation is still amazing and tells us something deep about the space-time even if we are talking only locally.