"Gravitational field" vs. "acceleration due to gravity"

thecommexokid

So I'm pretty sure the following paragraph is all true. Do the citizens of PhysicsForums agree? Please confirm and/or correct and/or clarify, as needed.

In electostatics, the electric field E is a completely different quantity than the acceleration due to an electric force a_E, which you could calculate as a_E = q_E E / m (where q_E E is the electric charge). But the gravitational field Φ and the acceleration due to gravitational force a_G are the same thing. By analogy with the electric case, a_G = q_G Φ / m, but q_G, the "gravitational charge", is itself just m. So it all comes down to the familiar fact that inertial mass and gravitational mass are the same thing...which is an unexplained coincidence in the context of Newtonian mechanics (though it possibly has firmer footing in general relativity).

cmos

Completely correct (excpet for the slight typo in the first parentheses ). This happy coincidence you speak of is commonly called the equivalence of inertial and gravitational mass; i.e. that the m appearing in Newton's Second Law is the same m that appears in Newton's Law of Universal Gravitation. In the electrostatic analogy, we could think of the "electric charge" as an "electric mass," which is completely unrelated the inertial/gravitational mass. Of course, this is completely equivalent to your explanation.

One word of warning, the above is true when you formulate gravity as a non-relativistic field theory in analogy to electrostatics. There might be some subtleties when you go into proper General Relativity (I'm truthfully not sure); but never mind that since we're not posting in the Relativity Forums.

vanhees71

Yep, it's all fine within the non-relativistic (Newtonian) approximation.

Concerning General Relativity (GR) (which is a classical theory after all and thus belongs also to this subforum although there is a more specialized one on relativity), one should emphasize that here the sources of the gravitational field is not the mass distribution but all kinds of energy and momentum, to which according to GR the gravitational field couples universally. That comes directly out of the application of the here discussed equivalence principle, leading to a geometrical reinterpretation of the gravtiational field as the curvature of the four-dimensional pseudo-Riemannian space-time manifold.