Laws in freely falling elevator

[beacon]

It seems that when talking about the GR, the textbooks usually give the example of a freely falling elevator. They say in such an elevator the laws take the same form as in SR.

Why in such a elevator the laws take the same form as in SR? How is it related to equivalance theorem?

 

[Severian596]

I am summarizing "A Short Course in General Relativity," by creating a lab that is free falling, that takes up a small region in spacetime, and that is nonrotating with respect to distant cosmic matter, you create an environment that has the effect of eliminating gravity. That way you can emulate the conditions necessary for SR to apply, and the equivalence of inertial mass and gravitiational mass is an essential aspect of these conditions.

 

[EL]

Originally posted by beacon
It seems that when talking about the GR, the textbooks usually give the example of a freely falling elevator. They say in such an elevator the laws take the same form as in SR.

Why in such a elevator the laws take the same form as in SR? How is it related to equivalance theorem?

Imagine you are in a small elevator without knowing what is happening outside it, and holding a ball in your hand. When dropping the ball you can see it hoover in the elevator right at the place where you let it go. Then you have two ways of interpreting this situation:
Either you are falling freely in a gravitational field (the ball has the same acceleration as the elevator according to Newton), or the elevator is situated somewhere without being affected by gravity (i.e. you are in a region of flat spacetime = SR holds). You really can´t tell which one is true.
Note that this only holds for elevators of infinitesimal size. In a large elevator the shape of the gravitational field will affect the motion of the ball (e.g. a sperical field will make the ball move towards the centre of the elevator).
This is a motivation for the equivalence theorem, which says that localy (i.e. at a point) you can always choose coordinates so you get flat spacetime.

 

[beacon]

I think there is a hypothesis in your argument:
One sees the laws with the same form as in SR when one can not feel the exsistence of gravity.

Is this it?

 

[Severian596]

That's the biggest point, but there are other conditions that must be met, specifically that the lab is very small (because otherwise whatever gravity you are attempting to transform away shows itself through converging falling vectors), and that the lab be nonrotating.

But you've nailed the most important idea: transforming away gravity.

 

[DW]

Originally posted by beacon
It seems that when talking about the GR, the textbooks usually give the example of a freely falling elevator. They say in such an elevator the laws take the same form as in SR.

Why in such a elevator the laws take the same form as in SR? How is it related to equivalance theorem?

These kinds of mind experiments were what led to general relativity, but in hindsight it is obvious in terms of the general relativistic equation of motion. The general relativistic equation of motion is
$$F^\lambda = \frac{DP^\lambda}{d\tau} = \frac{dP^\lambda}{d\tau} + \Gamma ^{\lambda}_{\mu}_{\nu}U^{\mu}P^{\nu}$$
In free fall frames the affine connections $$\Gamma ^{\lambda}_{\mu}_{\nu}$$ vanish and what is left is
$$F^\lambda = \frac{dP^\lambda}{d\tau}$$
which is the Minkowski force equation which is the special relativistic law of motion.

 

[beacon]

I see.
Thank you all. £º£_£©