Schwarzschild Outer Metric

In 1915, the same year as the publication of Einstein’s Field Equations, the German physicist Karl Schwarzschild produced the first exact solution. He did this while in hospital having developed a fatal skin condition while serving in the German army on the Russian front in World War I.

The Schwarzschild Outer Metric is an exact solution to Einstein’s Field Equations for the vacuum of space around a static and symmetric body. The metric must be independent of time and the spherical $$\theta$$ and $$\phi$$ coordinates.

Define the metric

The metric can be written as:

$ds^2 = A(r)dt^2 - B(r)dr^2 - r^2d\theta^2 - r^2 \sin^2\theta d\phi^2$

Where $$A$$ and $$B$$ are unknown functions of $$r$$. The solution must become the Minkowski metric as $$r \rightarrow \infty$$ and as $$M \rightarrow 0$$.

The metric and its inverse are now defined to be:

$g_{\alpha\beta} = \begin{bmatrix} A & 0 & 0 & 0\\ 0 & -B & 0 & 0\\ 0 & 0 & -r^2 & 0\\ 0 & 0 & 0 & -r^2 \sin^2 \theta \end{bmatrix}$ $g^{\alpha\beta} = \begin{bmatrix} \frac{1}{A} & 0 & 0 & 0\\ 0 & -\frac{1}{B} & 0 & 0\\ 0 & 0 & -\frac{1}{r^2} & 0\\ 0 & 0 & 0 & -\frac{1}{r^2 \sin^2 \theta} \end{bmatrix}$

Calculate the Christoffel Symbols

Using the comma notation, where a comma subscript means a partial derivative of that variable, the Christoffel symbols are:

$\Gamma^{\lambda}_{\mu\nu} = \frac{1}{2}g^{\lambda\rho}(g_{\mu\rho,\nu} + g_{\nu\rho,\mu} - g_{\mu\nu,\rho})$

There are four sets of Christoffel symbols, one set for each of $$t, r, \theta, \phi$$.

For the $$t$$ component, $$\lambda = t$$, and for all non-zero symbols $$\rho = t$$.

$\Gamma^{t}_{\mu\nu} = \frac{1}{2}g^{tt}(g_{\mu t,\nu} + g_{\nu t,\mu} - g_{\mu\nu,t})$

The term $$g_{\mu\nu,t} = 0$$ as nothing is time dependent. Substitute the value of $$g^{tt}$$, gives:

$\Gamma^{t}_{\mu\nu} = \frac{1}{2A}(g_{\mu t,\nu} + g_{\nu t,\mu})$

This leaves two symbols:

$\Gamma^{t}_{rt} = \Gamma^{t}_{tr} = \frac{A'}{2A}$

For the $$r$$ component, $$\lambda = r$$, and for all non-zero symbols $$\rho = r$$.

$\Gamma^{r}_{\mu\nu} = \frac{1}{2}g^{rr}(g_{\mu r,\nu} + g_{\nu r,\mu} - g_{\mu\nu,r})$

This gives four symbols:

\begin{align*} \Gamma^r_{tt} &= -\frac{1}{2}g^{rr}g_{tt,r} = \frac{A'}{2B}\\ \Gamma^r_{rr} &= \frac{1}{2}g^{rr}g_{rr,r} = \frac{B'}{2B}\\ \Gamma^r_{\theta\theta} &= -\frac{1}{2}g^{rr}g_{\theta\theta,r} = -\frac{r}{B}\\ \Gamma^r_{\phi\phi} &= -\frac{1}{2}g^{rr}g_{\phi\phi,r} = -\frac{r\sin^2\theta}{B} \end{align*}

For the $$\theta$$ component, $$\lambda = \theta$$, and for all non-zero symbols $$\rho = \theta$$.

$\Gamma^{\theta}_{\mu\nu} = \frac{1}{2}g^{\theta\theta}(g_{\mu\theta,\nu} + g_{\nu\theta,\mu} - g_{\mu\nu,\theta})$

This gives three symbols:

\begin{align*} \Gamma^\theta_{\theta r} &= \Gamma^\theta_{r\theta} = \frac{1}{2}g^{\theta\theta}g_{\theta\theta, r} = \frac{1}{2r^2}2r = \frac{1}{r}\\ \Gamma^\theta_{\phi\phi} &= -\frac{1}{2}g^{\theta\theta}g_{\phi\phi,\theta} = \frac{1}{2r^2}2r^2\sin\theta\cos\theta = -\sin\theta\cos\theta \end{align*}

For the $$\phi$$ component, $$\lambda = \phi$$, and for all non-zero symbols $$\rho = \phi$$.

$\Gamma^{\phi}_{\mu\nu} = \frac{1}{2}g^{\phi\phi}(g_{\mu\phi,\nu} + g_{\nu\phi,\mu} - g_{\mu\nu,\phi})$

This gives four symbols:

\begin{align*} \Gamma^\phi_{\phi r} &= \Gamma^\phi_{r\phi} = \frac{1}{2}g^{\phi\phi}g_{\phi\phi,r} = \frac{1}{2r^2\sin^2\theta}2r\sin^2\theta = \frac{1}{r}\\ \Gamma^\phi_{\theta\phi} &= \Gamma^\phi_{\phi\theta} = \frac{1}{2}g^{\phi\phi}g_{\phi\phi,\theta} = \frac{1}{2r^2\sin^2\theta}2r^2\sin\theta\cos\theta = \cot\theta \end{align*}

So, the Christoffel symbols are:

\begin{align*} \Gamma^{t}_{rt} &= \Gamma^{t}_{tr} = \frac{A'}{2A}\\ \Gamma^r_{tt} &= \frac{A'}{2B}\\ \Gamma^r_{rr} &= \frac{B'}{2B}\\ \Gamma^r_{\theta\theta} &= -\frac{r}{B}\\ \Gamma^r_{\phi\phi} &= -\frac{r\sin^2\theta}{B}\\ \Gamma^\theta_{\theta r} &= \Gamma^\theta_{r\theta} = \frac{1}{r}\\ \Gamma^\theta_{\phi\phi} &= -\sin\theta\cos\theta\\ \Gamma^\phi_{\phi r} &= \Gamma^\phi_{r\phi} = \frac{1}{r}\\ \Gamma^\phi_{\theta\phi} &= \Gamma^\phi_{\phi\theta} = \cot\theta \end{align*}

Construct the Ricci Tensor

The Ricci tensor is defined to be:

$R_{\mu\nu} = \Gamma^\lambda_{\mu\nu,\lambda} - \Gamma^\lambda_{\lambda\mu,\nu} + \Gamma^\lambda_{\lambda\eta}\Gamma^\eta_{\mu\nu} - \Gamma^\lambda_{\mu\eta}\Gamma^\eta_{\lambda\nu}$

Due to symmetry the Ricci tensor has just the four trace emements:

$R_{\mu\mu} = \Gamma^\lambda_{\mu\mu,\lambda} - \Gamma^\lambda_{\lambda\mu,\mu} + \Gamma^\lambda_{\lambda\eta}\Gamma^\eta_{\mu\mu} - \Gamma^\lambda_{\mu\eta}\Gamma^\eta_{\lambda\mu}$

To calculate $$R_{tt}$$, $$\mu = t$$ expand each term and eliminate zero symbols. Only derivatives of $$r$$ exist.

\begin{align*} R_{tt} &= \Gamma^\lambda_{tt,\lambda} - \Gamma^\lambda_{t\lambda,t} + \Gamma^\lambda_{\lambda\eta}\Gamma^\eta_{tt} - \Gamma^\lambda_{t\eta}\Gamma^\eta_{\lambda t}\\ &= \Gamma^r_{tt,r} + \bigg(\Gamma^t_{tr} + \Gamma^r_{rr} + \Gamma^\theta_{\theta r} + \Gamma^\phi_{\phi r}\bigg)\Gamma^r_{tt} - 2\Gamma^t_{tr}\Gamma^r_{tt}\\ &= \bigg(\frac{A'}{2B}\bigg)' + \bigg(\frac{A'}{2A} + \frac{B'}{2B} + \frac{1}{r} + \frac{1}{r}\bigg)\frac{A'}{2B} - 2\frac{A'}{2A}\frac{A'}{2B}\\ &= \frac{A''}{2B} - \frac{A'B'}{2B^2} + \frac{A'^2}{4AB} + \frac{A'B'}{4B^2} + \frac{A'}{rB} -\frac{A'^2}{2AB}\\ &= \frac{A''}{2B} - \frac{A'}{4B}\bigg(\frac{A'}{A} + \frac{B'}{B}\bigg) + \frac{A'}{rB} \end{align*}

To calculate $$R_{rr}$$, $$\mu = r$$ expand each term and eliminate zero symbols. Only derivatives of $$r$$ exist.

\begin{align*} R_{rr} &= \Gamma^\lambda_{rr,\lambda} - \Gamma^\lambda_{r\lambda,r} + \Gamma^\lambda_{\lambda\eta}\Gamma^\eta_{rr} - \Gamma^\lambda_{r\eta}\Gamma^\eta_{\lambda r}\\ &= \Gamma^r_{rr,r} - (\Gamma^t_{rt,r} + \Gamma^r_{rr,r} + \Gamma^\theta_{r\theta,r} + \Gamma^\phi_{r\phi,r}) + (\Gamma^t_{tr}\Gamma^r_{rr} + \Gamma^r_{rr}\Gamma^r_{rr} + \Gamma^\theta_{\theta r}\Gamma^r_{rr} + \Gamma^\phi_{\phi r}\Gamma^r_{rr}) - (\Gamma^t_{rt}\Gamma^t_{rt} + \Gamma^r_{rr}\Gamma^r_{rr} + \Gamma^\theta_{r\theta}\Gamma^\theta_{r\theta} + \Gamma^\phi_{r\phi}\Gamma^\phi_{\phi r})\\ &= - (\Gamma^t_{rt,r} + \Gamma^\theta_{r\theta,r} + \Gamma^\phi_{r\phi,r}) + (\Gamma^t_{tr}\Gamma^r_{rr} + \Gamma^\theta_{\theta r}\Gamma^r_{rr} + \Gamma^\phi_{\phi r}\Gamma^r_{rr}) - (\Gamma^t_{rt}\Gamma^t_{rt} + \Gamma^\theta_{r\theta}\Gamma^\theta_{r\theta} + \Gamma^\phi_{r\phi}\Gamma^\phi_{\phi r})\\ &= -\bigg(\frac{A'}{2A}\bigg)' + \frac{2}{r^2} + \frac{A'B'}{4AB} + \frac{B'}{rB} - \bigg(\frac{A'}{2A}\bigg)^2 - \frac{2}{r^2}\\ &= -\frac{A''}{2A} + \frac{A'^2}{2A^2} - \frac{A'^2}{4A^2} + \frac{A'B'}{4AB} + \frac{B'}{rB}\\ &= -\frac{A''}{2A} + \frac{A'}{4A}\bigg(\frac{A'}{A} + \frac{B'}{B}\bigg) + \frac{B'}{rB} \end{align*}

To calculate $$R_{\theta\theta}$$, $$\mu = \theta$$ expand each term and eliminate zero symbols. Only derivatives of $$r$$ and $$\theta$$ exist.

\begin{align*} R_{\theta\theta} &= \Gamma^\lambda_{\theta\theta,\lambda} - \Gamma^\lambda_{\theta\lambda,\theta} + \Gamma^\lambda_{\lambda\eta}\Gamma^\eta_{\theta\theta} - \Gamma^\lambda_{\theta\eta}\Gamma^\eta_{\lambda\theta}\\ &= \Gamma^r_{\theta\theta,r} - \Gamma^\phi_{\theta\phi,\theta} + (\Gamma^t_{tr} + \Gamma^r_{rr} + \Gamma^\theta_{\theta r} + \Gamma^\phi_{\phi r})\Gamma^r_{\theta\theta} - 2\Gamma^\theta_{\theta r}\Gamma^r_{\theta\theta} - \Gamma^\phi_{\theta\phi}\Gamma^\phi_{\theta\phi}\\ &= - \bigg(\frac{r}{B}\bigg)' + \csc^2\theta - \frac{r}{B}\bigg(\frac{A'}{2A} + \frac{B'}{2B} + \frac{1}{r} + \frac{1}{r}\bigg) + 2\frac{1}{r}\frac{r}{B} - \cot^2\theta\\ &= -\frac{1}{B} + \frac{rB'}{B^2} + \frac{2}{B} - \frac{r}{2B}\bigg(\frac{A'}{A} + \frac{B'}{B}\bigg) - \frac{2}{B} + 1\\ &= -\frac{1}{B} - \frac{r}{2B}\bigg(\frac{A'}{A} - \frac{B'}{B}\bigg) + 1 \end{align*}

To calculate $$R_{\phi\phi}$$, $$\mu = \phi$$ expand each term and eliminate zero symbols. Only derivatives of $$r$$ and $$\theta$$ exist.

\begin{align*} R_{\phi\phi} &= \Gamma^\lambda_{\phi\phi,\lambda} - \Gamma^\lambda_{\phi\lambda,\phi} + \Gamma^\lambda_{\lambda\eta}\Gamma^\eta_{\phi\phi} - \Gamma^\lambda_{\phi\eta}\Gamma^\eta_{\lambda\phi}\\ &= \Gamma^r_{\phi\phi,r} + \Gamma^\theta_{\phi\phi,\theta} + \bigg(\Gamma^t_{tr} + \Gamma^r_{rr} + \Gamma^\theta_{\theta r} + \Gamma^\phi_{\phi r}\bigg)\Gamma^r_{\phi\phi} - 2\Gamma^\phi_{\phi r}\Gamma^r_{\phi\phi} - \Gamma^\phi_{\phi\theta}\Gamma^\theta_{\phi\phi}\\ &= - \frac{\sin^2\theta}{B} + \frac{r\sin^2\theta B'}{B^2} - \cos^2\theta + \sin^2\theta -\frac{r\sin^2\theta}{B}\bigg(\frac{A'}{2A} + \frac{B'}{2B} + \frac{1}{r} + \frac{1}{r}\bigg) + 2\frac{1}{r}\frac{r\sin^2\theta}{B} + \cos^2\theta\\ &= \sin^2\theta\bigg(-\frac{1}{B} + \frac{rB'}{B^2} - \frac{r}{2B}\bigg(\frac{A'}{A} - \frac{B'}{B}\bigg) + 1\bigg)&&\\ &= \sin^2\theta R_{\theta\theta} \end{align*}

The Ricci tensor is:

\begin{align*} R_{tt} &= \frac{A''}{2B} - \frac{A'}{4B}\bigg(\frac{A'}{A} + \frac{B'}{B}\bigg) + \frac{A'}{rB}\\ R_{rr} &= -\frac{A''}{2A} + \frac{A'}{4A}\bigg(\frac{A'}{A} + \frac{B'}{B}\bigg) + \frac{B'}{rB}\\ R_{\theta\theta} &= -\frac{1}{B} - \frac{r}{2B}\bigg(\frac{A'}{A} - \frac{B'}{B}\bigg) + 1\\ R_{\phi\phi} &= \sin^2\theta R_{\theta\theta} \end{align*}

Construct the Ricci Scalar

The Ricci scalar is:

$R = g^{\mu\nu}R_{\mu\nu}$

\begin{align*} R &= \frac{R_{tt}}{A} - \frac{R_{rr}}{B} - \frac{R_{\theta\theta}}{r^2} - \frac{R_{\phi\phi}}{r^2\sin^2\theta}\\ &= \frac{A''}{AB} - \frac{A'}{2AB}\bigg(\frac{A'}{A} + \frac{B'}{B}\bigg) + \frac{2}{r^2B}(1 + \frac{rA'}{A} - \frac{rB'}{B} - B) \end{align*}

Construct the Einstein Tensor

The Einstein Tensor can now be constructed.

$G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu}$

\begin{align*} G_{tt} &= -\frac{A}{r^2B}(1 - r\frac{B'}{B} - B)\\ G_{rr} &= \frac{1}{r^2}(1 + r\frac{A'}{A} - B)\\ G_{\theta\theta} &= \frac{r^2}{2B}\bigg(\frac{A''}{A} + \frac{A'}{rA} - \frac{B'}{rB} - \frac{A'(AB)'}{2A^2B}\bigg)\\ G_{\phi\phi} &= G_{\theta\theta}\sin^2\theta \end{align*}

Construct the Field Equations

The Einstein tensor is zero because $$T_{\mu\nu} = 0$$ in a vacuum.

The field equations are:

\begin{align*} -\frac{A}{r^2B}(1 - \frac{rB'}{B} - B) &= 0\\ \frac{1}{r^2}(1 + r\frac{A'}{A} - B) &= 0\\ \frac{r^2}{2B}\bigg(\frac{A''}{A} + \frac{A'}{rA} - \frac{B'}{rB} - \frac{A'(AB)'}{2A^2B}\bigg) &= 0\\ G_{\theta\theta}\sin^2\theta &= 0 \end{align*}

Solve the Equations

The first two field equations reduce to:

\begin{align*} 1 - \frac{rB'}{B} - B &= 0\\ 1 + \frac{rA'}{A} - B &= 0 \end{align*}

Subtracting gives:

$\frac{A'}{A} +\frac{B'}{B} = \frac{BA' + AB'}{AB} = \frac{(AB)'}{AB} = 0$

Integrating gives:

$AB = k_1$

Where $$k_1$$ is a constant of integration. As the metric must conform to the Minkowski metric $$k_1 = c^2$$, $$AB = c^2$$.

The first equation can be written as:

$\frac{1}{B} - \frac{rB'}{B^2} = \frac{d}{dr}\bigg(\frac{r}{B}\bigg) = 1$

Integrating gives:

$\frac{r}{B} = r + k_2$

Where $$k_2$$ is a constant of integration.

This gives.

$B = \bigg(1 + \frac{k_2}{r}\bigg)^{-1}$

Therefore.

$A = c^2\bigg(1 + \frac{k_2}{r}\bigg)$

As the solution must become the Minkowski metric as $$M \rightarrow \infty$$ the constant must be a multiple of the mass $$k_2 = kM$$. The $$A$$ and $$B$$ terms are in fact the Lorentz transformations.

$1 + \frac{kM}{r} = 1 - \frac{v^2}{c^2}$

$\frac{kM}{r} = -\frac{v^2}{c^2}$

This must agree with Newton’s law of gravity for a small object of mass $$m$$ falling from an infinite distance. In this case the kinetic energy of the object is equal to the gravitational potential energy.

$\frac{1}{2}mv^2 = \frac{GMm}{r}$

$v^2 = \frac{2GM}{r}$

Now substitute the value of $$v^2$$.

$\frac{kM}{r} = -\frac{2GM}{c^2r}$

$k = -\frac{2G}{c^2}$

$r_s = \frac{2GM}{c^2}, c^2r_s = 2GM$

The Christoffel Symbols

Now, the Christoffel symbols are:

\begin{align*} \Gamma^{t}_{rt} &= \Gamma^{t}_{tr} = \frac{r_s}{2r(r - r_s)}\\ \Gamma^r_{tt} &= \frac{GM(r - r_s)}{r^3}\\ \Gamma^r_{rr} &= \frac{-r_s}{2r(r - r_s)}\\ \Gamma^r_{\theta\theta} &= -(r - r_s)\\ \Gamma^r_{\phi\phi} &= -\sin^2\theta(r - r_s)\\ \Gamma^\theta_{\theta r} &= \Gamma^\theta_{r\theta} = \frac{1}{r}\\ \Gamma^\theta_{\phi\phi} &= -\sin\theta\cos\theta\\ \Gamma^\phi_{\phi r} &= \Gamma^\phi_{r\phi} = \frac{1}{r}\\ \Gamma^\phi_{\theta\phi} &= \Gamma^\phi_{\phi\theta} = \cot\theta \end{align*}

The Schwarzschild Solution

The Schwarzschild outer solution becomes:

$$$ds^2 = c^2d \tau^2 = \bigg(1 - \frac{2GM}{c^2r}\bigg)c^2dt^2 - \bigg(1 - \frac{2GM}{c^2r}\bigg)^{-1}dr^2 - r^2d\theta^2 - r^2 \sin^2\theta d\phi^2$$$ $$$ds^2 = c^2d \tau^2 = \bigg(1 - \frac{r_s}{r}\bigg)c^2dt^2 - \bigg(1 - \frac{r_s}{r}\bigg)^{-1}dr^2 - r^2d\theta^2 - r^2 \sin^2\theta d\phi^2$$$

The solution has two singularities. The singularity ar $$r = 0$$ is not an issue as the solution is only valid in the vacuum of space around a massive object.

The singularity at $$r = r_s$$ has interesting consequences. The radius $$r_s$$ is normally well inside the object and hence out of scope of the solution. If however, all of the mass is compressed into a volume less than the Schwarzschild radius, the Schwarzschild radius defines the event horizon of a black hole.