# The equivalence principle and identifying free-fall as a locally inertial frame

As far as I understand it the equivalence principle in general relativity is the statement that a frame at rest in a uniform gravitational field is equivalent (indistinguishable) to an accelerated (non-inertial) frame in the absence of gravity (i.e. in Minkowski spacetime).

I am trying to reason to myself why free-fall (i.e. a frame in which gravity is the only force acting) defines a locally inertial frame.

Consider an elevator with an observer inside it in free-fall in a uniform gravitational field. Is the reasoning simply that in free-fall, due to the equivalence principle, both the elevator and the observer inside the elevator will accelerate towards the centre of the massive body (that is the source of the gravitational field) at the same rate, hence there will be no relative acceleration between the elevator and the observer inside the elevator. As this is the case for any other objects inside the lift also, assuming that the observer cannot see outside of the lift, then there is no experiment that they can perform that will determine whether they are in free-fall or following uniform unaccelerated motion in deep space (in the absence of gravity), the two scenarios are indistinguishable. Hence, the observer inside the elevator must conclude that they are in an inertial frame (of course, this only holds locally due to tidal forces).

Would this be a correct way to argue the situation?

• An observer on the surface of the Earth is not in an inertial system. Newton's laws simply don't apply. Objects at rest, for instance, don't stay at rest and objects in motion aren't moving in straight lines but on ballistic curves (parabolas). Did I misunderstand what you are trying to say about observer A? Commented Feb 22, 2016 at 14:22
• @CuriousOne Yes you're right, I accidentally left that in after rewriting the question. I'll remove that part now. Do you have a good way of wording an argument as to why free-fall defines an inertial frame in GR?
– Will
Commented Feb 22, 2016 at 14:55

Your argument is true classically also. You see, the effect of Gravitation close to earth and that of acceleration in flat spacetime is same. But this equivalence goes way past just the elevator experience, where all the observer feeling is a state of weightlessness or equal weight. While introducing GR Einstein took this equivalence farther and said that there are no laws of physics that can distinguish between an accelerated frame and a stationary frame in gravitational field. This goes for all laws including Electro-magnetism. So yes, if you see a charged particle in accelerated frame it should radiate. This can then be extrapolated and we can say that a charged particle which is stationary in a gravitational field should also radiate.

That may not be the correct explanation. The first assumption of Newton that the non accelerating frames with respect to the Earth's surface is inertial. This assumption is what is not taken in the first place in general relativity. Moreover the comparison of the two reference frames that are a frame which is far away in space and freely falling frames in earth. To solve this question Newton assumed a new force to counter the effect of 'acceleration' on Earth's surface. Einstein questions this assumption and says that we need to assume a new force whose cause is not visible as we are in a non inertial reference frame. This brings us the geometry of space and time and curvatures produces in it due to mass. Now according to the above assumption we can say that we are in a non inertial and the freely falling reference frame as inertial.

• If I have a charged body or even a charged particele in a free-fall reference frame, will it radiate? Commented Feb 22, 2016 at 19:00
• I have no clue on that. Commented Feb 22, 2016 at 19:08
• The equivalence priciple doesn't work globally because of the self force of the electric charge. It will radiate. Commented Feb 22, 2016 at 19:51
• @lattitude I haven't assumed that the Earth represents an inertial frame though (I'm not using Newton's assumptions), I was purely stating that the elevator is accelerating relative to an Earthbound observer, but due to the equivalence principle, the observer (and all other objects) inside the elevator will also accelerate at the same rate as the elevator. Therefore there is no relative acceleration between the elevator and the observer (and all other objects) in the elevator, so according to the observer in the elevator they are in a local inertial frame, since it is impossible...
– Will
Commented Feb 22, 2016 at 20:35
• ...for them to determine whether they're in free-fall or in uniform (unaccelerated) motion in absence of any gravitational sources (assuming the elevator has no windows). Is there a better way to put this then? How should it be correctly argued?
– Will
Commented Feb 22, 2016 at 20:37