The Metric Tensor

Equivalent forms

ds^2 = g_{\mu\nu}\,dx^\mu dx^\nu

g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}\ (\text{weak field})

A single symmetric rank-2 tensor encodes all local geometry: distances, angles, causal structure, time dilation, and (through its derivatives) the entire gravitational field.

Symbol	Name	SI unit	Dimension	Range
$ds2$	Interval Squaredoutput Measured spacetime interval squared	m^2	L^2	-100 – 100
$gtt$	Time-Time Component g_00 metric component (sets clock rate)	dimensionless	1	-2 – -0.05
$gxx$	Space-Space Component g_11 metric component (sets ruler scale)	dimensionless	1	0.2 – 3

Dimensional analysis

ds^2 -> [L^2]

g_{mu nu} dx^mu dx^nu ->

[1][L][L] = [L^2]

(with

x^0 = ct)

Interval is an area.

\checkmark

Discovery

Bernhard Riemann / Albert Einstein · 1915

Riemann built the geometry of curved spaces in 1854; sixty years later Einstein promoted the metric from a passive backdrop to a dynamical field sourced by matter.

Research status: stable

Real-world applications

GPS relativistic timing (weak-field metric)
Numerical relativity of black-hole mergers
Cosmological FLRW modelling

Common misconceptions

The metric is not a single number — it is 10 independent functions of spacetime (symmetric 4x4)
Coordinates by themselves mean nothing; only g_{mu nu} dx^mu dx^nu is physical
A non-trivial metric does not always mean curvature — flat space in odd coordinates has a non-diagonal g

Experimental verification

Every GR test is a measurement of a metric component: GPS clock rates measure g_tt in Earth's field; Shapiro delay measures g_rr; frame-dragging (Gravity Probe B, LAGEOS) measures off-diagonal g_{t phi}.

Derivation

Distance on any manifold is defined by a symmetric bilinear form:

ds^2 = g_{\mu \nu } dx^\mu dx^\nu

.

Requiring the interval be coordinate-independent forces g to transform as a rank-2 tensor:

g'_{\alpha \beta } =

(partial x^mu/partial x'^alpha)(partial x^nu/partial x'^beta) g_{mu nu}.

Its inverse g^{mu nu} raises indices, and its derivatives build the Christoffel symbols and curvature.

Limiting cases

g =

diag(-1,1,1,1)⟶ MinkowskiFlat spacetime; special relativity holds globally.

g = \eta + h

, |h| << 1⟶ weak-field/linearized gravityThe regime of gravitational waves and Newtonian gravity.

g_tt -> 0⟶ horizon / infinite redshiftClocks freeze relative to infinity, as at a black-hole horizon.

What if…

What if the metric has off-diagonal g_{t phi}?

Space is 'dragged' around the mass — frame dragging, as in the rotating Kerr metric.

What if g is degenerate

(det g = 0)

?

The geometry breaks down — a signal of a horizon or a genuine singularity depending on the case.

1

Interval from a diagonal metric

Given ·

gtt:: -1
gxx:: 1
dt(ct):: 3
dx:: 4

Find · ds^2

Steps

$ds^2 = g_tt (cdt)^2 + g_xx dx^2$
$= -(3)^2 + (4)^2$
$= 7$ -> spacelike

Result ·

ds^2 = g_tt (c dt)^2 + g_xx dx^2 = -1*(3)^2 + 1*(4)^2 = -9 + 16 = 7

(spacelike).

Minkowski Metric→Schwarzschild Metric→Geodesic Equation→christoffel symbolseinstein field equations