Interior Schwarzschild metric


In Einstein's theory of general relativity, the interior Schwarzschild metric is an exact solution for the gravitational field in the interior of a non-rotating spherical body which consists of an incompressible fluid and has zero pressure at the surface. This is a static solution, meaning that it does not change over time. It was discovered by Karl Schwarzschild in 1916, who earlier had found the exterior Schwarzschild metric.

Mathematics

The interior Schwarzschild metric is framed in a spherical coordinate system with the body's centre located at the origin, plus the time coordinate. Its line element is
where
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.

Other formulations

Defining a parameter, we get
We can also define an alternative radial coordinate and a corresponding parameter, yielding

Properties

Volume

With and the area
the integral for the proper volume is
which is larger than the volume of an euclidean reference shell.

Density

The fluid has a constant density by definition. It is given by
were 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, 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, 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.

Redshift

for radiation from the sphere's surface is
From the stability condition follows.

Visualization

The spatial curvature of the interior Schwarzschild metric can be visualized by taking a slice with constant time and 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, 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.

Examples

Here are the relevant parameters for some astronomical objects, disregarding rotation and inhomogeneities such as deviation from the spherical shape and variation in density.
Object
Earth6,370 km8.87 mm170,000,000 km
9.5 light-minutes		7.7″7×10−10
Sun696,000 km2.95 km338,000,000 km
19 light-minutes		7.0′2×10−6
White dwarf with 1 solar mass5000 km2.95 km200,000 km1.4°3×10−4
Neutron star with 2 solar masses20 km6 km37 km30°0.15

History

The interior Schwarzschild solution was the first static spherically symmetric perfect fluid solution that was found. It was published on 24 February 1916, only three months after Einstein's field equations and one month after Schwarzschild's exterior solution.