Gravity and inertial frames

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I was just wondering what would the definition of the inertial frame be, since as I've understood the description it seems that an inertial frame seems to be 'free' from any external forces, correct me please if I'm wrong. Can something be an inertial frame even if gravity acts upon it? For instance my bed is at rest wrt to Earth (because of gravity), it seems that it isn't accelerating but gravity still acts upon it, can it be considered an inertial frame? How do conceptualize the correlation between inertial frames and gravity?

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WannabeNewton
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In the presence of arbitrary gravitational fields (curved space-times), there are no global inertial frames. There exist only local inertial frames. They are defined in the exact same way as in flat space-time except they only extend infinitesimally at a single event in the curved space-time whereas in flat space-time we have global inertial frames extended throughout space-time. These local inertial frames will, at any given event, be freely falling non-rotating frames.
 
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WannabeNewton
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How do conceptualize the correlation between inertial frames and gravity?

Sorry I forgot to answer this part but it's quite simple. It's a consequence of the equivalence principle. Imagine you're in a non-rotating freely falling elevator.

*Why non-rotating? Well if it's rotating you can easily experimentally verify the presence of centrifugal forces but we want to find a relationship with an inertial frame, for which there are no inertial forces*

Now we must take the dimensions of the elevator to be much smaller than the characteristic length scales of the space-time over which space-time curvature varies. This is to eliminate tidal forces as potential measurements. For arbitrary curved space-times this will amount to taking the limit as the freely falling elevator's dimensions become infinitesimally small at a single event.

Now say you're inside this elevator and you drop a ball. Well you and the ball (and the elevator) will all be falling at the exact same rate because of the equivalence principle. So no matter what experimental apparatus you have at hand, you will invariably conclude that you, the ball, and the elevator are simply floating in free space i.e. inertial.

So locally (meaning at a single event), a freely falling non-rotating elevator in curved space-time will correspond to an inertial elevator in flat space-time. This can be made more mathematically precise using the language of general relativity which I can delve into if you wish.
 
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A.T.
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I was just wondering what would the definition of the inertial frame be, since as I've understood the description it seems that an inertial frame seems to be 'free' from any external forces, correct me please if I'm wrong. Can something be an inertial frame even if gravity acts upon it?
- Newtonian Gravity is a real force, so objects in free fall under gravity are not inertial, but accelerated.

- In General Relativity gravitational acceleration is just a coordinate effect in non-inertial frames, so objects in free fall under gravity are inertial.

See the clip below for a comparison. Note that in Einsteins model there are no forces acting on the falling red apple. This fits well with a free falling accelerometer, which measures zero proper acceleration:

https://www.youtube.com/watch?v=DdC0QN6f3G4

For instance my bed is at rest wrt to Earth (because of gravity), it seems that it isn't accelerating but gravity still acts upon it, can it be considered an inertial frame?

The bed is like the green apple still hanging on the tree, in the clip above.

- In Newtonian Gravity the bed is inertial, because gravity and ground reaction force cancel, so the net force is zero.

- In General Relativity there is no force of gravity, just the ground reaction accelerating the bed upwards. So the bed is not inertial. This fits well with an accelerometer resting on the bed, which measures 1g proper acceleration upwards.
 
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