Can Gravity Exist in an Inertial Frame?

[analyst5]

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?

Thanks

 

[WannabeNewton]

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.

 

[WannabeNewton]

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.

 

[A.T.]

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.

 

[sardar]

for your question! An inertial frame can be defined as a reference frame in which Newton's first law of motion holds true - that an object will remain at rest or in uniform motion unless acted upon by an external force. So, in essence, an inertial frame is one where there are no external forces acting on objects within that frame.

However, you are correct in thinking that gravity can complicate this definition. In the case of your bed, it is at rest relative to the Earth because the force of gravity is counteracting any potential acceleration. This means that, in a way, the bed is still experiencing an external force - the force of gravity.

So, can something be considered an inertial frame even if gravity acts upon it? The answer is yes, but with some caveats. Inertial frames are often thought of as being free from any external forces, but in reality, it is more accurate to say that they are free from any net external forces. In other words, the forces acting on an object within an inertial frame may not necessarily cancel out, but the resulting acceleration is still zero.

To conceptualize the correlation between inertial frames and gravity, it may be helpful to think about it in terms of relative motion. In an inertial frame, objects will appear to move in a straight line at a constant speed, while in a non-inertial frame (such as one experiencing the force of gravity), objects will appear to be accelerating. This is because gravity is a non-inertial force, meaning it causes objects to deviate from their natural, straight-line motion.

I hope this helps clarify the relationship between inertial frames and gravity. Keep asking questions and exploring the fascinating world of science!