Schwarzschild Metric

Equivalent forms

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

The first and most important exact solution of Einstein's equations, containing black holes, gravitational time dilation, light bending and Mercury's precession in one line element.

Symbol	Name	SI unit	Dimension	Range
$gtt$	Time Metric Componentoutput The factor -(1 - r_s/r) multiplying c^2 dt^2	dimensionless	1	-5 – 1
$r$	Radial Coordinate Areal radius from the central mass	m	L	1 – 30
$rs$	Schwarzschild Radius Horizon radius r_s = 2GM/c^2 (in same units as r)	m	L	0.1 – 10

Dimensional analysis

ds^2 -> [L^2]

(1-r_s/r) c^2 dt^2 ->

[1][L^2\,\mathrm{T}^-2][T^2] = [L^2]

; r^2 dOmega^2 -> [L^2]

All terms area; r_s/r is dimensionless.

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Discovery

Karl Schwarzschild · 1916

Within weeks of Einstein publishing the field equations, Schwarzschild found this exact solution from the trenches of WWI, months before dying of illness contracted at the front.

Research status: stable

Real-world applications

Black-hole shadow modelling (EHT)
GPS and precision-clock gravitational corrections (weak-field limit)
Accretion-disk and orbit dynamics around compact objects

Common misconceptions

The horizon $r=r_s$ is a coordinate singularity, not a physical one — a falling observer crosses it uneventfully
r is an 'areal' radius (area $= 4\,\mathrm{pi} r^2)$ , not the measured distance to the centre
The metric is static outside the horizon but t and r swap roles inside it

Experimental verification

Mercury's perihelion precession (43''/century), light deflection at the Sun (1919 eclipse), Shapiro delay, and gravitational redshift are all Schwarzschild-metric predictions confirmed to high precision; the Event Horizon Telescope images match its photon ring.

Derivation

Assume a static, spherically symmetric vacuum:

ds^2 =

-A(r)c^2 dt^2 + B(r)dr^2 + r^2 dOmega^2.

Plug into the vacuum field equations

R_{\mu \nu }=0

; they force A

(r) = 1/B(r) = 1 - K/r

, and matching the Newtonian potential -GM/r at large r fixes

K = 2GM/c^2 = r_s

.

Birkhoff's theorem shows this is the UNIQUE spherically symmetric vacuum solution.

Limiting cases

r >> r_s⟶ metric -> MinkowskiFar from the mass, spacetime is flat and special relativity is recovered.

r -> r_s⟶ g_tt -> 0, g_rr -> infinityThe event horizon; the coordinate singularity is removable but the horizon is real.

r -> 0⟶ true curvature singularityTidal forces diverge — a genuine singularity, not a coordinate artefact.

What if…

What if the mass rotates?

Use the Kerr metric; frame-dragging adds cross terms and the horizon shrinks with spin.

What if you use isotropic coordinates?

The horizon coordinate singularity moves but the physics is identical; coordinates are just labels.

1

Time-dilation factor at r = 3 r_s

Given ·

r:: 3
rs:: 1

Find ·

\sqrt{-g_tt} =

rate of a clock vs infinity

Steps

$1 - r_s/r = 1 - 1/3 = 2/3$
$\sqrt{2/3} = 0.816$

Result ·

\sqrt{1 - r_s/r} = \sqrt{1 - 1/3} = \sqrt{2/3} = 0.816

. A clock there ticks at 82% of the rate of a distant clock.

Schwarzschild Radius→Event Horizon→Gravitational Time Dilation→The Metric Tensor→Geodesic Equation→