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InEinstein's theory ofgeneral relativity, theinterior Schwarzschild metric (alsointerior Schwarzschild solution orSchwarzschild fluid solution) is anexact solution for thegravitational field in the interior of a non-rotating spherical body which consists of anincompressible fluid (implying thatdensity is constant throughout the body) and has zeropressure at the surface. This is a static solution, meaning that it does not change over time. It was discovered byKarl Schwarzschild in 1916, who earlier had found theexterior Schwarzschild metric.^[1]

The interior Schwarzschild metric is framed in aspherical coordinate system with the body's centre located at the origin, plus the time coordinate. Itsline element is^[2][3]

where

${\displaystyle \tau }$ is theproper time (time measured by a clock moving along the sameworld line with thetest particle),

${\displaystyle c}$ is thespeed of light,

${\displaystyle t}$ is the time coordinate (measured by a stationary clock located infinitely far from the spherical body),

${\displaystyle r}$ is the Schwarzschild radial coordinate; each surface of constant${\displaystyle t}$ and${\displaystyle r}$ has the geometry of a sphere with measurable (proper) circumference${\displaystyle 2\pi r}$ and area${\displaystyle 4\pi r^2}$ (as by the usual formulas), but the warping of space means that the proper distance from each shell to the center of the body is greater than${\displaystyle r}$,

${\displaystyle \theta }$ is thecolatitude (angle from north, in units ofradians),

${\displaystyle \phi }$ is thelongitude (also in radians),

${\displaystyle r_s}$ is theSchwarzschild radius of the body, which is related to its mass${\displaystyle M}$ by${\displaystyle r_s=2GM/c^2}$, where${\displaystyle G}$ is thegravitational constant (for ordinary stars and planets, this is much less than their proper radius),

${\displaystyle r_g}$ is the value of the${\displaystyle r}$ 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${\displaystyle r\le r_g}$. For a complete metric of the sphere's gravitational field, the interior Schwarzschild metric has to be matched with the exterior one,

${\displaystyle -c^{2}d{\tau }^2=\left(1-\frac{r_s}{r}\right)c^{2}d{t}^2-{\left(1-\frac{r_s}{r}\right)}^{-1}d{r}^2-r^2(d{\theta }^2+{\sin }^2\theta d{\phi }^2),}$

at the surface. It can easily be seen that the two have the same value at the surface, i.e., at${\displaystyle r=r_g}$.

Defining a parameter${\displaystyle {\mathcal{R}}^2=r_g^3/r_s}$, we get

We can also define an alternative radial coordinate${\displaystyle \eta =\arcsin \frac{r}{\mathcal{R}}}$ and a corresponding parameter${\displaystyle {\eta }_g=\arcsin \frac{r_g}{\mathcal{R}}=\arcsin \sqrt{\frac{r_s}{r_g}}}$, yielding^[4]

${\displaystyle c^{2}d{\tau }^2={\left(\frac{3\cos {\eta }_g-\cos \eta }{2}\right)}^2{c}^{2}d{t}^2-\frac{d{r}^2}{{\cos }^2\eta }-r^2(d{\theta }^2+{\sin }^2\theta d{\phi }^2).}$

With${\displaystyle g_{rr}=(1-r_s{r}^2/r_g^3{)}^{-1}}$ and the area${\displaystyle A=4\pi r^2}$, the integral for the proper volume is

${\displaystyle V={\int }_0^{r_g}A\sqrt{g_{rr}}dr=2\pi \left(\frac{r_g^{9/2}\arcsin \sqrt{{\displaystyle \frac{r_s}{r_g}}}}{r_s^{3/2}}-\frac{r_g^4\sqrt{1-{\displaystyle \frac{r_s}{r_g}}}}{r_s}\right),}$

which is larger than the volume of a euclidean reference shell.

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

${\displaystyle \rho =\frac{M}{\frac{4\pi }{3}{r}_g^3}=\frac{3}{\kappa {\mathcal{R}}^2},}$

where${\displaystyle \kappa =8\pi G/c^2}$ is theEinstein gravitational constant.^[3][5] It may be counterintuitive that the density is the mass divided by the volume of a sphere with radius${\displaystyle r_g}$, 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,${\displaystyle M}$ 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.

The pressure of the incompressible fluid can be found by calculating theEinstein tensor${\displaystyle G_{\mu \nu }}$ from the metric. The Einstein tensor isdiagonal (i.e., all off-diagonal elements are zero), meaning there are noshear stresses, and has equal values for the three spatial diagonal components, meaning pressure isisotropic. Its value is

${\displaystyle p=\rho c^2\frac{\cos \eta -\cos {\eta }_g}{3\cos {\eta }_g-\cos \eta }.}$

As expected, the pressure is zero at the surface of the sphere and increases towards the centre. It becomes infinite at the centre if${\displaystyle \cos {\eta }_g=1/3}$, which corresponds to${\displaystyle r_s=\frac{8}{9}{r}_g}$ or${\displaystyle {\eta }_g\approx {70.5}^{\circ }}$, which is true for a body that is extremely dense or large. Such a body suffersgravitational collapse into ablack hole. As this is a time dependent process, the Schwarzschild solution does not hold any longer.^[2][3]

Gravitational redshift for radiation from the sphere's surface (for example, light from a star) is

${\displaystyle z=\frac{1}{\cos {\eta }_g}-1.}$

From the stability condition${\displaystyle \cos {\eta }_g>1/3}$ follows.^[3]

The spatialcurvature 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.${\displaystyle t=const.,\theta =\pi /2}$. This two-dimensional slice can beembedded in a three-dimensional Euclidean space and then takes the shape of aspherical cap with radius${\displaystyle \mathcal{R}}$ and half opening angle${\displaystyle {\eta }_g}$. ItsGaussian curvature${\displaystyle K}$ is proportional to the fluid's density and equals${\displaystyle {\mathcal{R}}^{-2}=r_s/r_g^3=\rho \kappa /3}$. As the exterior metric can be embedded in the same way (yieldingFlamm's paraboloid), a slice of the complete solution can be drawn like this:^[5][6]

In this graphic, the blue circular arc represents the interior metric, and the blackparabolic arcs with the equation${\displaystyle w=2\sqrt{r_s(r-r_s)}}$ represent the exterior metric, or Flamm's paraboloid. The${\displaystyle \eta }$-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${\displaystyle {\eta }_g\mathcal{R}}$.

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 implyextrinsic curvature.)

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	${\displaystyle r_g}$	${\displaystyle r_s}$	${\displaystyle \mathcal{R}}$	${\displaystyle {\eta }_g}$	${\displaystyle z}$ (redshift)
Earth	6,370 km	8.87 mm	170,000,000 km 9.5 light-minutes	7.7″	7×10−10
Sun	696,000 km	2.95 km	338,000,000 km 19 light-minutes	7.0′	2×10−6
White dwarf with 1 solar mass	5000 km	2.95 km	200,000 km	1.4°	3×10−4
Neutron star with 2 solar masses	20 km	6 km	37 km	30°	0.15

The interior Schwarzschild solution was the firststatic spherically symmetric perfect fluid solution that was found. It was published on 24 February 1916, only three months afterEinstein's field equations and one month after Schwarzschild's exterior solution.^[1][2]

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