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
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
tells us that the fractional change in volume
of this sphere will be
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
We can also express in terms of the change in
the sphere's radius, :
© 2006 John Baez and Emory Bunn