Schwarzschild Solution: Derivation

Introduction

The Schwarzschild solution, discovered by Karl Schwarzschild in 1916 (just months after Einstein published general relativity), describes the spacetime geometry outside a spherically symmetric, non-rotating mass. It's the first exact solution to Einstein's field equations and describes black holes.

We'll derive this solution step by step from Einstein's field equations using spherical symmetry.

Step 1: Symmetry Assumptions

Key Assumptions

Spherical symmetry: The metric depends only on radial coordinate r
Static: The metric is time-independent (no time derivatives)
Asymptotically flat: At large distances, spacetime approaches Minkowski space
Vacuum solution: We solve Einstein's equations in vacuum (outside the mass)

The most general spherically symmetric, static metric in spherical coordinates (t, r, θ, φ) can be written as:

$$ds^2 = -e^{2\alpha(r)} c^2 dt^2 + e^{2\beta(r)} dr^2 + r^2 d\Omega^2$$

where $d\Omega^2 = d\theta^2 + \sin^2\theta \, d\phi^2$ is the metric on a unit 2-sphere, and α(r) and β(r) are unknown functions we must determine.

Step 2: Einstein Field Equations in Vacuum

In vacuum (no matter or energy), Einstein's field equations reduce to:

$$R_{\mu\nu} = 0$$

This says the Ricci tensor must vanish everywhere outside the mass. The Ricci tensor is computed from the Christoffel symbols, which are derived from the metric.

Step 3: Christoffel Symbols

The Christoffel symbols are given by:

$$\Gamma^\rho_{\mu\nu} = \frac{1}{2}g^{\rho\sigma}\left(\frac{\partial g_{\sigma\mu}}{\partial x^\nu} + \frac{\partial g_{\sigma\nu}}{\partial x^\mu} - \frac{\partial g_{\mu\nu}}{\partial x^\sigma}\right)$$

For our metric, the non-zero Christoffel symbols include:

$$\Gamma^t_{tr} = \alpha'(r), \quad \Gamma^r_{tt} = \alpha'(r) e^{2(\alpha - \beta)}, \quad \Gamma^r_{rr} = \beta'(r)$$

$$\Gamma^r_{\theta\theta} = -r e^{-2\beta}, \quad \Gamma^r_{\phi\phi} = -r\sin^2\theta \, e^{-2\beta}$$

$$\Gamma^\theta_{r\theta} = \Gamma^\phi_{r\phi} = \frac{1}{r}, \quad \Gamma^\theta_{\phi\phi} = -\sin\theta\cos\theta$$

$$\Gamma^\phi_{\theta\phi} = \cot\theta$$

Here, primes denote derivatives with respect to r: α'(r) = dα/dr.

Step 4: Ricci Tensor Components

The Ricci tensor is computed from:

$$R_{\mu\nu} = \partial_\rho \Gamma^\rho_{\mu\nu} - \partial_\nu \Gamma^\rho_{\mu\rho} + \Gamma^\rho_{\mu\nu}\Gamma^\sigma_{\rho\sigma} - \Gamma^\rho_{\mu\sigma}\Gamma^\sigma_{\nu\rho}$$

After substantial calculation (which I'll spare you the details of!), the independent components are:

$$R_{tt} = e^{2(\alpha-\beta)}\left[\alpha'' + \alpha'^2 - \alpha'\beta' + \frac{2\alpha'}{r}\right]$$

$$R_{rr} = -\alpha'' - \alpha'^2 + \alpha'\beta' + \frac{2\beta'}{r}$$

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

$$R_{\phi\phi} = \sin^2\theta \, R_{\theta\theta}$$

Step 5: Solving R_μν = 0

Setting each component to zero and solving the coupled differential equations:

From R_θθ = 0:

$$e^{-2\beta}\left[r(\beta' - \alpha') - 1\right] + 1 = 0$$

$$\Rightarrow \frac{d}{dr}\left(r e^{-2\beta}\right) = 1$$

$$\Rightarrow r e^{-2\beta} = r - r_s$$

$$\Rightarrow e^{2\beta} = \frac{1}{1 - r_s/r}$$

where r_s is an integration constant (the Schwarzschild radius).

From R_tt + R_rr = 0:

$$\alpha' + \beta' = 0 \quad \Rightarrow \quad \alpha = -\beta + \text{const}$$

Choosing the constant so that the metric is asymptotically flat (approaches Minkowski as r → ∞):

$$e^{2\alpha} = 1 - \frac{r_s}{r}$$

Step 6: The Schwarzschild Metric

Substituting our solutions for α and β back into the metric:

Schwarzschild Metric

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

where the Schwarzschild radius is:

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

This is determined by matching to the Newtonian limit at large distances.

Birkhoff's Theorem

Theorem:

The Schwarzschild solution is the unique spherically symmetric vacuum solution to Einstein's field equations.

Implications:

Any spherically symmetric mass distribution has Schwarzschild geometry outside it
A pulsating spherical star doesn't produce gravitational waves (spherical symmetry prevents it)
The exterior geometry doesn't depend on the star's internal structure—only its total mass M

Physical Interpretation

For the Sun:

$$r_s = \frac{2GM_\odot}{c^2} \approx 3 \text{ km}$$

The Sun's actual radius is 696,000 km, so we're far from the Schwarzschild radius.

For Earth:

$$r_s = \frac{2GM_\oplus}{c^2} \approx 9 \text{ mm}$$

Earth would need to be compressed to 9 millimeters to become a black hole!

What is a Black Hole?

A black hole forms when mass M is compressed within its Schwarzschild radius r_s. At r = r_s, the metric component g_tt vanishes and g_rr diverges—this is the event horizon, the point of no return.

Interactive Simulations

Schwarzschild Radius Calculator for Astrophysical Objects

Python

script.py47 lines

import numpy as np

# Constants
G = 6.674e-11      # m^3 kg^-1 s^-2
c = 2.998e8         # m/s
M_sun = 1.989e30    # kg

print("=" * 60)
print("SCHWARZSCHILD RADIUS CALCULATOR")
print("r_s = 2GM/c^2")
print("=" * 60)

objects = {
    "Earth":          5.972e24,
    "Sun":            M_sun,
    "White Dwarf":    0.6 * M_sun,
    "Neutron Star":   1.4 * M_sun,
    "Sgr A*":         4.0e6 * M_sun,
    "M87*":           6.5e9 * M_sun,
    "TON 618":        6.6e10 * M_sun,
}

print(f"{'Object':<18} {'Mass (kg)':<14} {'r_s':<18} {'Density at r_s'}")
print("-" * 60)

for name, M in objects.items():
    r_s = 2 * G * M / c**2
    vol = (4.0/3.0) * np.pi * r_s**3
    rho = M / vol if vol > 0 else np.inf

if r_s < 1e-2:
        r_str = f"{r_s*1000:.3f} mm"
    elif r_s < 1e3:
        r_str = f"{r_s:.2f} m"
    elif r_s < 1e9:
        r_str = f"{r_s/1000:.2f} km"
    else:
        r_str = f"{r_s/1.496e11:.3f} AU"

print(f"{name:<18} {M:.3e}    {r_str:<18} {rho:.2e} kg/m^3")

print()
print("Key insight: larger BHs have LOWER density at the horizon!")
print(f"Sun density:   {M_sun / ((4/3)*np.pi*(2*G*M_sun/c**2)**3):.2e} kg/m^3")
print(f"M87* density:  {6.5e9*M_sun / ((4/3)*np.pi*(2*G*6.5e9*M_sun/c**2)**3):.2e} kg/m^3")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Schwarzschild Metric Components vs Radial Distance

Fortran

program.f9045 lines

program schwarzschild_metric
  implicit none
  integer, parameter :: n = 12
  double precision :: r_rs_vals(n), r_ratio, g_tt, g_rr
  double precision :: G, c, M_sun, r_s
  integer :: i

G = 6.674d-11
  c = 2.998d8
  M_sun = 1.989d30
  r_s = 2.0d0 * G * M_sun / (c * c)

data r_rs_vals /0.5d0, 0.8d0, 0.95d0, 0.99d0, 1.0d0, &
                  1.01d0, 1.05d0, 1.5d0, 2.0d0, 3.0d0, &
                  5.0d0, 10.0d0/

write(*,'(A)') '======================================================'
  write(*,'(A)') ' Schwarzschild Metric Components vs r/r_s'
  write(*,'(A)') ' g_tt = -(1 - r_s/r),  g_rr = 1/(1 - r_s/r)'
  write(*,'(A)') '======================================================'
  write(*,'(A)') ' r/r_s      g_tt          g_rr          Region'
  write(*,'(A)') '------------------------------------------------------'

do i = 1, n
    r_ratio = r_rs_vals(i)
    if (abs(r_ratio - 1.0d0) < 1.0d-10) then
      write(*,'(F6.2,A)') r_ratio, '      0.000         INFINITY      HORIZON'
    else
      g_tt = -(1.0d0 - 1.0d0/r_ratio)
      g_rr = 1.0d0 / (1.0d0 - 1.0d0/r_ratio)
      if (r_ratio < 1.0d0) then
        write(*,'(F6.2,2X,F12.5,2X,F12.5,2X,A)') r_ratio, g_tt, g_rr, 'INTERIOR'
      else
        write(*,'(F6.2,2X,F12.5,2X,F12.5,2X,A)') r_ratio, g_tt, g_rr, 'EXTERIOR'
      end if
    end if
  end do

write(*,'(A)') ''
  write(*,'(A)') 'Note: Inside horizon (r < r_s), g_tt > 0 and g_rr < 0'
  write(*,'(A)') 'The roles of time and space coordinates swap!'
  write(*,'(A,F8.3,A)') 'Solar mass r_s = ', r_s, ' m (about 3 km)'
end program schwarzschild_metric

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

← Back to Part I Overview Next: Event Horizon →