# Galilean equivalence for extended objects

• I
The Galilean equivalence principle (or weak equivalence principle) is the statement of the universality of free-fall under gravity. For example, according to Wikipedia, it can be stated as follows
The trajectory of a point mass in a gravitational field depends only on its initial position and velocity, and is independent of its composition and structure.
My question regards the limitation of the principle to point masses. Does universality of free-fall not hold for extended objects? It seems to me that in Newtonian gravity if one focuses on the center of mass of a body universality of free-fall is satisfied.

What about GR? If the particle is point-like, it is well-known that universality of free-fall is embodied in the fact that the particle trajectories are the geodesics of the spacetime manifold. Do extended objects move geodetically?

I assume the object is extended but still has a small mass, so that one can neglect the gravitational influence of the object.

## Answers and Replies

PeterDonis
Mentor
2020 Award
Do extended objects move geodetically?

Theoretically in GR, the general answer is no, they don't. More precisely, the center of mass of an extended object in GR does not, theoretically, move on a geodesic.

However, in practical terms, deviations from geodesic motion for extended objects are way too small to be observable in most cases. For example, the Sun and all of the planets in the solar system move on geodesics as best we can tell with our most accurate measurements. (Note that these objects are large enough that they don't have negligible gravity, yet we still can't measure deviations from geodesic motion. This is one manifestation of how weak gravity is as an interaction compared to the other known interactions.)

A.T.
Science Advisor
It seems to me that in Newtonian gravity if one focuses on the center of mass of a body universality of free-fall is satisfied.
What exactly is meant by this?

Theoretically in GR, the general answer is no, they don't. More precisely, the center of mass of an extended object in GR does not, theoretically, move on a geodesic.
Is this due to the fact that the extended object may be spinning? I vaguely recall that spinning objects in GR do not move on geodesics. I am more interested in understanding whether the fact that the object is extended, and therefore can probe inhomogeneities in the field, invalidates the GEP (or not).

What exactly is meant by this?

Extended objects of different shapes and different masses generally will be spinning in different ways while falling, but their centers of mass move along the same trajectory, which is the same as that of any point-mass in free-fall in the same gravitational field and with the same initial conditions.

PeterDonis
Mentor
2020 Award
Is this due to the fact that the extended object may be spinning?

No. Even a non-spinning extended object, theoretically, will not move on a geodesic (more precisely, its center of mass won't). The general reason is interaction between the internal structure of the object and the curvature of spacetime.

PeterDonis
Mentor
2020 Award
I am more interested in understanding whether the fact that the object is extended, and therefore can probe inhomogeneities in the field, invalidates the GEP (or not)

What is the GEP? I'm aware of the weak, Einstein, and strong equivalence principles. All of them contain the qualification that they only apply to "test objects" and only apply in a small enough patch of spacetime that the effects of spacetime curvature is negligible. So the failure of extended objects to move on geodesics does not invalidate any of the EPs, since it is due to the effects of spacetime curvature.

A.T.
Science Advisor
Extended objects of different shapes and different masses generally will be spinning in different ways while falling, but their centers of mass move along the same trajectory, which is the same as that of any point-mass in free-fall in the same gravitational field and with the same initial conditions.
Is that really always true in Newtonian gravity, if the field is non-uniform?

What is the GEP? I'm aware of the weak, Einstein, and strong equivalence principles.
GEP=Galilean equivalence principle. It is the same as the WEP=weak equivalence principle.

Is that really always true in Newtonian gravity, if the field is non-uniform?

Actually, now I think it isn't. If ##g(x)## is the position-varying gravitational field, then we want to find some ##\bar x## such that ##\ddot{\bar x}=g(\bar x)##. The center of mass does the trick for a uniform gravitational field, but this doesn't carry over to the nonuniform case and probably there is no ##\bar x## that satisfies in general the above condition.