Gravitational force equation derived from GR

[PAllen]

I was talking about the proper acceleration of a hovering object which is discussed in previous answers by Ibix and others.

a_p = $\frac{GM}{r^2\sqrt{1-2GM/r}}$

So now the question is if we can take this proper acceleration and put it into the equation
F = γ³ma_p ?

Ok, but originally you seemed to be asking about the simplest useful approximate generalization of Newton's law of gravitation that capture GR effects. Einstein-Infeld-Hoffman equations are exactly that, containing Newtons law of gravitation as the first term.

As for relating force needed to maintain hovering to proper acceleration of a hovering body, note that in a momentarily comoving local inertial frame (in this case, a momentarily stationary free fall frame coinciding with some hovering body), you have that coordinate acceleration equals proper acceleration, and also that simply F=ma is true (m and F measured in the local, momentarily stationary, free fall frame). Recall, as well, that proper acceleration magnitude is a scalar invariant, thus it can be computed in non-inertial Schwarzschild coordinates, then used as described in a local momentaritly stationary free fall frame.

 

[sha1000]

That is not the gravitational force. There is no gravitational force in GR. It is the force needed to be provided by other means to keep the object stationary.

Also note that you make explicit reference to a gamma factor. It is unclear what you intend this gamma factor to be.

Yes, you are right. This is indeed a force needed to keep the object stationary.

A.T. who indicated the proper acceleration equation also wrote: "The force needed to hover it would be the proper acceleration times its mass." F = ma

But the relativistic Newtonian second law is F=dP/dT = F = γ³ma.

So I thought that it would be more convenient to use this relativistic equation.

 

[PeterDonis]

the relativistic Newtonian second law is F=dP/dT = F = γ^3*ma.

This is not a fully general law. It is only valid in an inertial frame, for acceleration parallel to velocity. Neither of those applies to the situation in which you are trying to apply it.

 

[PAllen]

The general form of F=ma in special relativity, using 3-vectors in an inertial frame is:
$\mathbf F = m\gamma^3(\mathbf v \cdot \mathbf a)\mathbf v + m\gamma \mathbf a$

 

[sha1000]

This is not a fully general law. It is only valid in an inertial frame, for acceleration parallel to velocity. Neither of those applies to the situation in which you are trying to apply it.

PAllen wrote: " As for relating force needed to maintain hovering to proper acceleration of a hovering body, note that in a momentarily comoving local inertial frame (in this case, a momentarily stationary free fall frame coinciding with some hovering body), you have that coordinate acceleration equals proper acceleration..."

I'm really not an expert. Does this mean that we can define a co-moving local inertial frame (stationary free fall frame coinciding with hovering body)? From this point would it be possible to apply F = γ³ma?
Since we are dealing with co-moving local inertial frame?

 

[PAllen]

PAllen wrote: " As for relating force needed to maintain hovering to proper acceleration of a hovering body, note that in a momentarily comoving local inertial frame (in this case, a momentarily stationary free fall frame coinciding with some hovering body), you have that coordinate acceleration equals proper acceleration..."

I'm really not an expert. Does this mean that we can define a co-moving local inertial frame (stationary free fall frame coinciding with hovering body)? From this point would it be possible to apply F = γ³ma?
Since we are dealing with co-moving local inertial frame?

NO, you would just apply F=ma in that frame. $\gamma$ would be 1 by definition in such a frame.

 

[Ibix]

Ok I got it. Thanks

Just last question if you don't mind. Can we apply the same reasoning to the object in circular oribit around massive body. Is it possible to use naively the same proper acceleration equation obtained for a hovering object?

No. Circular orbits can be powered or unpowered, and the necessary proper acceleration can be anywhere between $\pm\infty$. However, the typical use of the term "circular orbit" refers to an unpowered orbit. This is zero proper acceleration.