A basic test of general relativity is to check that it reduces to
good old Newtonian gravity in the limit where gravitational effects are
weak and velocities are small compared to the speed of light.
To do this, we can use our formulation of Einstein's equation
to derive Newton's inverse-square force law for a planet with mass
and radius . Since we can only do this when gravitational effects
are weak, we must assume that the planet's radius is much greater than
its Schwarzschild radius: in units where
.
Then the curvature of *space* - as opposed to spacetime -
is small. To keep things simple, we make a couple of additional
assumptions: the planet has uniform density , and the pressure
is negligible.

We want to derive the familiar Newtonian result

giving the radial gravitational acceleration of a test particle at distance from the planet's center, with of course.

To do this, let be a sphere of radius centered on the planet. Fill the interior of the sphere with test particles, all of which are initially at rest relative to the planet. At first, this might seem like an illegal thing to do: we know that notions like `at rest' only make sense in an infinitesimal neighborhood, and is not infinitesimal. But because space is nearly flat for weak gravitational fields, we can get away with this.

We can't apply our formulation of Einstein's equation directly to , but we can apply it to any infinitesimal sphere within . In the picture below, the solid black circle represents the planet, and the dashed circle is . The interior of has been divided up into many tiny spheres filled with test particles. Green spheres are initially inside the planet, and red spheres are outside.

Suppose we pick a tiny green sphere that lies within the planet's
volume, a distance less than from the center. Our formulation
of Einstein's equation
(2)
tells us that the fractional change in volume
of this sphere will be

On the other hand, as we saw in our discussion of tidal effects, spheres outside the planet's volume will be distorted in shape by tidal effects, but remain unchanged in volume. So, any little red sphere that lies outside the planet will undergo no change in volume at all:

Thus, after a short time has elapsed, the test particles will be distributed like this:

Because the volume occupied by the spheres of particles within
the planet went down, the whole big sphere of test particles had
to shrink. Now consider the following question:
what is the change in volume of the large sphere of test
particles? All the little green spheres shrank by the same fractional
amount:

to leading order in , while all the little red spheres underwent no change in volume. So, if stands for the volume occupied by the test particles that were initially inside the planet, and stands for the volume of the big sphere , then the overall change in volume is

We can also express in terms of the change in
the sphere's radius, :

Setting the two expressions for equal, we see

But the change in radius is just the radial displacement of a test particle on the outer edge of the sphere, so where is the radial acceleration of this particle. Plugging this into the above equation, we obtain

We've been working all along in units in which . Switching back to conventional units, we obtain

just as Newton said.

* © 2006 John Baez and Emory Bunn*