6.3.1. Schwarzschild metrics#

Spatially spherically symmetric and static vacuum solution of the Einstein equation, representing the exterior gravitational field of a non-rotating, uncharged massive body.

Schwarzschild metrics gives

(6.3)#\[ ds^2 = - \left(1 - \frac{r_s}{r} \right) c^2 dt^2 + \left(1 - \frac{r_s}{r} \right)^{-1} dr^2 + r^2 d \theta^2 + r^2 \sin^2 \theta \, d\phi^2 \ . \]
Vacuum equation

Without mass density and electric charge and current, \(\mathsf{T} = \mathsf{0}\). From the expression of (6.2), it immediately follows, \(\mathsf{R} = \mathsf{0}\).

6.3.1.1. Metric Ansatz & Symmetry Assumptions#

The four coordinates used to parametrize the space-time are \((t, r, \theta, \phi)\). The two physical symmetry constraints in Schwarzschild solution of EFE are

  1. Spherical Symmetry: the spatial geometry is invariant under rotation, s.t. the angular part of the metric is given by the standard round metric on \(S^2\), \(d\Omega^2 = d \theta^2 + \sin^2 \theta \, d\phi^2\)

  2. Staticity: the metric coefficients are independent of the time coordinate \(t\) (\(\partial_t g_{\mu u} = 0\)); the further assumption that the line element is invariant under time reversal eliminates cross-terms such as \(dt\,dr\).

Under these assumptions, the general line element can be parameterized using two undetermined functions of radius, \(A(r)\) and \(B(r)\):

\[ds^2 = -e^{2A(r)} c^2 dt^2 + e^{2B(r)} dr^2 + r^2 \left( d \theta^2 + \sin^2 \theta \, d\phi^2 \right)\]

Remark. Using exponential parameterizations \(e^{2A(r)}\) and \(e^{2B(r)}\) ensures that \(g_{tt} < 0\) and \(g_{rr} > 0\) outside any horizon. todo Add some remark/examples about the role of time, and its consequence on metrics

From the choosen parametrization, the non-zero covariant components of the metric tensors are

\[\begin{split}\begin{aligned} g_{tt} & = - e^{- 2 A(r)} c^2 \\ g_{rr} & = e^{ 2 B(r)} \\ g_{\theta \theta} & = r^2 \\ g_{\phi \phi} & = r^2 \sin^2 \theta \\ \end{aligned}\end{split}\]

As the metric tensor is diagonal, it’s contravariant components are just the inverse of its covariant components.

6.3.1.2. Christoffel Symbols#

The Christoffel symbols of the second kind are computed via the expression (6.1),

\[\Gamma_{ac}^{b} = \frac{1}{2} g^{bd} \left( \partial_c g_{ad} + \partial_a g_{dc} - \partial_{d} g_{ac} \right) \ .\]

Writing the derivative w.r.t. \(r\) as \(\partial_r A(r) = A'(r)\)

  • \(\Gamma^t_{tr} = \Gamma^t_{rt} = \frac{1}{2} g^{t d} \left( \partial_r g_{t d} + \partial_t g_{dr} - \partial_d g_{tr} \right) = - \frac{1}{2} e^{2 A} c^{-2} \cdot ( - 2 A' ) e^{-2A} c^2 = A'\)

  • \(\Gamma^r_{tt} = A' e^{2(A-B)} c^2\)

  • \(\Gamma^r_{rr} = B'\)

  • \(\Gamma^r_{\theta \theta} = -r e^{-2B}\)

  • \(\Gamma^r_{\phi\phi} = -r \sin^2 \theta \, e^{-2B}\)

  • \(\Gamma^\theta_{r \theta} = \Gamma^\theta_{\theta r} = \frac{1}{r}\)

  • \(\Gamma^\theta_{\phi\phi} = -\sin\theta\cos \theta\)

  • \(\Gamma^\phi_{r\phi} = \Gamma^\phi_{\phi r} = \frac{1}{r}\)

  • \(\Gamma^\phi_{\theta\phi} = \Gamma^\phi_{\phi\theta} = \cot \theta\)

6.3.1.3. Components of the Ricci Tensor#

The Ricci tensor is defined by contracting the Riemann curvature tensor:

\[R_{\mu u} = \partial_{\lambda}\Gamma^{\lambda}_{\mu u} - \partial_{ u}\Gamma^{\lambda}_{\mu\lambda} + \Gamma^{\lambda}_{\mu u}\Gamma^{\sigma}_{\lambda\sigma} - \Gamma^{\sigma}_{\mu\lambda}\Gamma^{\lambda}_{ u\sigma}\]

Evaluating the non-zero independent components yields:

  • Time-Time Component (\(R_{tt}\))

    \[R_{tt} = e^{2(A-B)} c^2 \left[ A'' + (A')^2 - A'B' + \frac{2A'}{r} \right]\]

  • Radial-Radial Component (\(R_{rr}\))

    \[R_{rr} = -A'' - (A')^2 + A'B' + \frac{2B'}{r}\]

  • Angular Component (\(R_{\theta \theta}\))

    \[R_{\theta \theta} = 1 - e^{-2B} \left[ 1 + r(A' - B') \right]\]

Remarks.

  • The \(R_{\phi\phi}\) component yields \(R_{\phi\phi} = R_{\theta \theta} \sin^2 \theta\), providing no independent equation).

  • The non-diagonal components are identically zero (todo prove it!)

6.3.1.4. Solving the Differential Equations#

Combining \(R_{tt}\) and \(R_{rr}\) (linear combination \(e^{-2(A-B)} \frac{R_{tt}}{c^2} + R_{rr} = 0\)) immediately gives

\[A'(r) = -B'(r) \ .\]

and integrating with respect to \(r\),

\[A(r) + B(r) = C = \text{const.}\]

Boundary conditions at infinity. To satisfy the boundary condition of asymptotic flatness—that spacetime approaches flat Minkowski space as \(r \rightarrow \infty\), then

\[\begin{split}\begin{aligned} g_{tt} & = - e^{2A(r)} \rightarrow - 1 && A(r) \rightarrow 0 \\ g_{rr} & = e^{2B(r)} \rightarrow 1 && B(r) \rightarrow 0 \end{aligned}\end{split}\]

and thus \(C = 0\), and \(A(r) = - B(r)\).

Solving for \(A(r)\) via \(R_{\theta \theta}\). Substituting \(B' = -A'\) and \(e^{-2B} = e^{2A}\) into \(R_{\theta \theta} = 0\):

\[\begin{split}\begin{aligned} 0 & = R_{\theta \theta} = \\ & = 1 - e^{2A} \left( 1 + 2rA' \right) = \\ & = 1 - \dfrac{d}{dr} \left( r e^{2 A} \right) \ , \end{aligned}\end{split}\]

and thus, integrating in \(r\),

\[r e^{2 A(r)} = r + C \ .\]

or

\[e^{2A(r)} = 1 + \frac{C}{r}\]

where \(C\) is a constant of integration. Since \(e^{2B(r)} = e^{-2A(r)}\), it follows \(e^{2B(r)} = \left(1 + \frac{C}{r} \right)^{-1}\).

6.3.1.5. Determining the Integration Constant \(C\) (The Newtonian Limit)#

To determine \(C\), we examine the metric in the weak-field, low-velocity limit (\(r \to \infty\)). In this limit, general relativity recovers Newtonian gravity, as shown in classical limit,

\[g_{tt} = \eta_{tt} + h_{tt} \approx - 1 - \frac{2}{c^2} \Phi \ ,\]

where \(\Phi(r) = - \frac{GM}{r}\) is the Newtonian gravitational potential of a central mass \(M\). Equating the two expression of the coefficient \(g_{tt}\)

\[g_{tt} = -e^{2A(r)} = - 1 - \frac{C}{r} = - 1 + \frac{2}{c^2} \frac{G M}{r} \]

and thus

\[C = - \frac{2GM}{c^2} \ .\]

Defining the Schwarzschild radius \(r_s := \frac{2GM}{c^2}\), the expression of Schwarzschild metric becomes

\[ ds^2 = - \left(1 - \frac{r_s}{r} \right) c^2 dt^2 + \left(1 - \frac{r_s}{r} \right)^{-1} dr^2 + r^2 d \theta^2 + r^2 \sin^2 \theta \, d\phi^2 \ . \]

6.3.1.6. Special Case: Complete Vacuum (\(M = 0\))#

If the spacetime contains no mass (\(M = 0\)), the Minkowski flat spacetime follows

\[ds^2 = -c^2 dt^2 + d r^2 + r^2 d \theta^2 + r^2 \sin^2 \theta \, d\phi^2 = - c^2 dt^2 + |d \vec{r}|^2 \ , \]

or, using the common transformation between spherical and Cartesian space coordinates,

\[\begin{split}\begin{cases} t = t \\ x = r \sin \theta \cos \phi \\ y = r \sin \theta \sin \phi \\ z = r \cos \theta \ , \end{cases}\end{split}\]
\[ds^2 = - c^2 dt^2 + dx^2 + dy^2 + dz^2 \ .\]