is the time coordinate (measured by a stationary clock located infinitely far from the spherical body).
is the Schwarzschild radial coordinate. Each surface of constant and has the geometry of a sphere with measurable (proper) circumference and area (as by the usual formulas), but the warping of space means the proper distance from each shell to the center of the body is greater than .
is the value of the -coordinate at the body's surface. (This is less than its proper (measurable interior) radius, although for the Earth the difference is only about 1.4 millimetres.)
This solution is valid for . For a complete metric of the sphere's gravitational field, the interior Schwarzschild metric has to be matched with the exterior one,
at the surface. It can easily be seen that the two have the same value at the surface, i.e., at .
Defining a parameter , we get
We can also define an alternative radial coordinate and a corresponding parameter , yielding
With and the area
the integral for the proper volume is
which is larger than the volume of a euclidean reference shell.
The fluid has a constant density by definition. It is given by
where is the
Einstein gravitational constant. It may be counterintuitive that the density is the mass divided by the volume of a sphere with radius , which seems to disregard that this is less than the proper radius, and that space inside the body is curved so that the volume formula for a "flat" sphere shouldn't hold at all. However, is the mass measured from the outside, for example by observing a test particle orbiting the gravitating body (the "
Kepler mass"), which in general relativity is not necessarily equal to the proper mass. This mass difference exactly cancels out the difference of the volumes.
Pressure and stability
The pressure of the incompressible fluid can be found by calculating the
Einstein tensor from the metric. The Einstein tensor is
diagonal (i.e., all off-diagonal elements are zero), meaning there are no
shear stresses, and has equal values for the three spatial diagonal components, meaning pressure is
isotropic. Its value is
As expected, the pressure is zero at the surface of the sphere and increases towards the centre. It becomes infinite at the centre if , which corresponds to or , which is true for a body that is extremely dense or large. Such a body suffers
gravitational collapse into a
black hole. As this is a time dependent process, the Schwarzschild solution does not hold any longer.
Embedding of a Schwarzschild metric's slice in three-dimensional Euclidean space. The interior solution is the darker cap at the bottom. This embedding should not be confused with the unrelated concept of a
curvature of the interior Schwarzschild metric can be visualized by taking a slice (1) with constant time and (2) through the sphere's equator, i.e. . This two-dimensional slice can be
embedded in a three-dimensional Euclidean space and then takes the shape of a
spherical cap with radius and half opening angle . Its
Gaussian curvature is proportional to the fluid's density and equals . As the exterior metric can be embedded in the same way (yielding
Flamm's paraboloid), a slice of the complete solution can be drawn like this:
In this graphic, the blue circular arc represents the interior metric, and the black
parabolic arcs with the equation represent the exterior metric, or Flamm's paraboloid. The -coordinate is the angle measured from the centre of the cap, that is, from "above" the slice. The proper radius of the sphere – intuitively, the length of a measuring rod spanning from its centre to a point on its surface – is half the length of the circular arc, or .
This is a purely geometric visualization and does not imply a physical "fourth spatial dimension" into which space would be curved. (Intrinsic curvature does not imply
Here are the relevant parameters for some astronomical objects, disregarding rotation and inhomogeneities such as deviation from the spherical shape and variation in density.