The Metric Tensor: Measuring Curved Spacetime
How do you measure distance on a curved surface? On a flat plane, Pythagoras works: $ds^2 = dx^2 + dy^2$. On a sphere, the formula changes — it depends on where you are. The metric tensor $g_{\mu\nu}$ encodes this position-dependent notion of distance. It is the central object of general relativity.
The line element
An infinitesimal displacement in spacetime is described by the line element:
$$ds^2 = g_{\mu\nu} \, dx^\mu \, dx^\nu$$
(summing over $\mu$ and $\nu$ from 0 to 3). This is a generalisation of both Pythagoras and the Minkowski interval. The metric tensor $g_{\mu\nu}(x)$ is a symmetric (0,2) tensor field — at each point, it’s a $4 \times 4$ symmetric matrix with 10 independent components.
Three geometries
Draw a short path on each surface. The metric determines how the coordinate displacement maps to actual distance.
| Geometry | Metric $g_{\mu\nu}$ | Signature | Curvature |
|---|---|---|---|
| Flat plane | $\text{diag}(1, 1)$ | (+,+) | Zero |
| Sphere of radius $R$ | $\text{diag}(R^2, R^2 \sin^2\theta)$ | (+,+) | Positive (constant) |
| Minkowski spacetime | $\text{diag}(-1, 1, 1, 1)$ | (−,+,+,+) | Zero |
| Schwarzschild (outside) | Position-dependent | (−,+,+,+) | Non-zero, varies |
What the metric does
The metric tensor serves multiple roles:
- Measures distances: $ds^2 = g_{\mu\nu} dx^\mu dx^\nu$
- Measures angles: The angle between vectors $u$ and $v$ comes from $\cos\theta = g_{\mu\nu} u^\mu v^\nu / (|u||v|)$
- Raises and lowers indices: $v_\mu = g_{\mu\nu} v^\nu$ converts contravariant to covariant
- Determines volumes: The volume element is $\sqrt{|g|} \, d^4x$, where $g = \det(g_{\mu\nu})$
- Defines geodesics: The “straightest paths” are determined entirely by $g_{\mu\nu}$ and its derivatives
- Encodes gravity: In GR, the metric is the gravitational field
Signature
The signature of a metric is the pattern of signs in its diagonal form. Riemannian metrics have all positive signs — they measure ordinary distance on curved spaces. Lorentzian (or pseudo-Riemannian) metrics have one negative sign — the time direction. All spacetime metrics in GR are Lorentzian with signature $(-,+,+,+)$.
The negative sign means that the “distance” between two events can be negative (timelike), zero (lightlike), or positive (spacelike). This isn’t a mathematical quirk — it’s the structure that enforces causality.
From flat to curved
Minkowski spacetime has $g_{\mu\nu} = \eta_{\mu\nu} = \text{diag}(-1, 1, 1, 1)$ everywhere — constant, the same at every point. In general relativity, mass and energy cause $g_{\mu\nu}$ to vary from point to point. The difference between the actual metric and the flat Minkowski metric is the gravitational field.
Newton’s gravitational potential $\Phi$ corresponds roughly to the $(0,0)$ component: $g_{00} \approx -(1 + 2\Phi/c^2)$. Where $\Phi$ is large and negative (near a massive body), clocks run slower — gravitational time dilation falls out of the metric.