The Metric Tensor

What is a Metric?

A metric defines the notion of distance on a space. In flat 3D space, the distance between two nearby points is given by the Pythagorean theorem:

But in curved spaces or spacetime, the relationship between coordinate differences and actual distances is more complex.

The Line Element

In general coordinates, the infinitesimal spacetime interval is:

Here $g_{\mu\nu}$ is the metric tensor, a symmetric rank-2 tensor that encodes all geometric information about the spacetime.

Being symmetric ($g_{\mu\nu} = g_{\nu\mu}$), the metric has 10 independent components in 4D spacetime.

Minkowski Metric

In flat spacetime (special relativity), the metric is the Minkowski metric:

This gives the spacetime interval:

The minus sign on the time component distinguishes time from space and is what makes spacetime Lorentzian rather than Euclidean.

Raising and Lowering Indices

The metric tensor converts between contravariant and covariant components:

Here $g^{\mu\nu}$ is the inverse metric, defined by:

where $\delta^\mu_\nu$ is the Kronecker delta (identity matrix).

Schwarzschild Metric

The simplest curved spacetime solution is the Schwarzschild metric, describing the spacetime outside a spherically symmetric, non-rotating mass:

where $d\Omega^2 = d\theta^2 + \sin^2\theta \, d\phi^2$ is the angular part, and the Schwarzschild radius is $r_s = 2GM/c^2$.

Physical Meaning

The metric determines:

• Proper time: $d\tau = \sqrt{-ds^2}/c$ for timelike intervals
• Proper distance: $d\ell = \sqrt{ds^2}$ for spacelike intervals
• Angles between vectors via inner products
• Null geodesics: paths where $ds^2 = 0$ (light rays)

Signature

The signature of a metric refers to the signs of its eigenvalues. In GR, we use a Lorentzian signature with one time and three space dimensions:

• $(-,+,+,+)$ - "mostly plus" convention (used here)
• $(+,-,-,-)$ - "mostly minus" convention

Both conventions are equivalent; just be consistent!