Black Holes

The Schwarzschild solution describes the curved spacetime outside a spherically symmetric mass. At the Schwarzschild radius r_s = 2GM/c², the event horizon forms — a surface from which nothing, not even light, can escape. Stephen Hawking showed that quantum effects cause black holes to emit thermal radiation.

Key Concepts

Schwarzschild metric: ds² = -(1-r_s/r)c²dt² + dr²/(1-r_s/r) + r²dΩ²
Event horizon at r = r_s = 2GM/c²
Gravitational time dilation diverges at r_s
Hawking temperature: T_H = ℏc³/(8πGMk_B)
Photon sphere at r = 3GM/c² = 3r_s/2

Key Equations

Schwarzschild metric

ds^2 = -\left(1-\frac{r_s}{r}\right)c^2dt^2 + \frac{dr^2}{1-r_s/r} + r^2d\Omega^2

Schwarzschild radius

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

Hawking temperature

T_H = \frac{\hbar c^3}{8\pi G M k_B}

Photon sphere

r_{ph} = \frac{3GM}{c^2} = \frac{3r_s}{2}

Worked Example

Example Problem

Problem

Find the Hawking temperature of a black hole with M = 10 M_sun.

Solution

T_H = ℏc³/(8πGMk_B). With M=2×10³¹ kg: T_H = 1.055×10⁻³⁴×2.7×10²⁵/(8π×6.674×10⁻¹¹×2×10³¹×1.38×10⁻²³) = 2.84×10⁻⁹/1.46×10⁻¹ = 1.94×10⁻⁸ K ≈ 6×10⁻⁹ K.

Practice

Exercises

7 problems

1 of 7

Find r_s for a black hole of M=10 M_sun (M_sun=2×10³⁰ kg) in km.

Your answer

2 of 7

Find the Hawking temperature of a solar-mass black hole (M=2×10³⁰ kg) in nK.

Your answer

3 of 7

The photon sphere for a Schwarzschild BH with r_s=3 km is at r_ph = 3r_s/2 in km.

Unlock Exercise 3