The Metric Tensor

A smooth manifold, on its own, is a forgetful object. It knows which functions are differentiable and what a tangent vector is — but it has no idea how long a vector is, whether two directions are perpendicular, or how far apart two points lie. Those are exactly the data a ruler and a protractor supply, and the metric tensor is the field of infinitesimal rulers-and-protractors that supplies them. At each point $p$ it is a bilinear form $g_p:T_pM\times T_pM\to\mathbb{R}$ that is symmetric and nondegenerate, varying smoothly from point to point; in a chart it is a symmetric matrix-valued field $g_{\mu\nu}(x)$ transforming as a $(0,2)$ tensor. From this one gadget the whole of geometry unrolls. Contract it with a displacement twice and you get the line element $\mathrm{d}s^2 = g_{\mu\nu}\,\mathrm{d}x^\mu\,\mathrm{d}x^\nu$ , the squared length of an infinitesimal step; integrate its root for arc length, extremize that for geodesics. Its inverse $g^{\mu\nu}$ raises and lowers indices, welding the tangent space to its dual. Demand a connection that is torsion-free and leaves $g$ constant and you get a unique one — Levi-Civita — whose coefficients, the Christoffel symbols, are built from first derivatives of $g$ ; push to second derivatives and out falls the Riemann tensor that decides whether the space is genuinely curved. Two warnings frame everything below. A metric that varies from point to point is not by itself curved — flat space in spherical coordinates has a position-dependent $g$ and zero curvature. And the signature of $g$ is an invariant: a Riemannian $(+,+,\dots)$ metric measures ordinary lengths, while a Lorentzian $(-,+,+,+)$ metric measures spacetime intervals and carves out the causal cones of relativity. In general relativity the metric stops being a fixed backdrop and becomes the dynamical field itself, so geometry and gravity are one.

Key takeaways

The metric tensor $g_{\mu\nu}$ is a symmetric, nondegenerate rank-two covariant tensor field assigning an inner product to every tangent space, fixing lengths, angles, and volumes.
The line element $\mathrm{d}s^2 = g_{\mu\nu}\,\mathrm{d}x^\mu\,\mathrm{d}x^\nu$ packages all local geometry; its components are the dot products $g_{\mu\nu}=\mathbf{e}_\mu\cdot\mathbf{e}_\nu$ of the basis vectors.
The inverse $g^{\mu\nu}$ (defined by $g^{\mu\alpha}g_{\alpha\nu}=\delta^\mu_{\ \nu}$ ) gives the musical isomorphism — raising and lowering indices.
Torsion-free plus metric-compatible uniquely fixes the Levi-Civita connection $\Gamma^\lambda_{\ \mu\nu}=\tfrac12 g^{\lambda\sigma}(\partial_\mu g_{\sigma\nu}+\partial_\nu g_{\sigma\mu}-\partial_\sigma g_{\mu\nu})$ , hence geodesics and curvature.
Signature is invariant: $(+,+,+)$ Riemannian length versus $(-,+,+,+)$ Lorentzian interval; the determinant gives the measure $\sqrt{|g|}\,\mathrm{d}^n x$ .
A position-dependent $g$ is not curvature: flat $\mathbb{R}^3$ in spherical coordinates has $g=\mathrm{diag}(1,r^2,r^2\sin^2\theta)$ yet Riemann tensor zero; a true 2-sphere has $K=1/a^2$ .
Key idea: A metric is a smoothly varying, symmetric, nondegenerate inner product on tangent spaces; it turns a bare differentiable manifold into a measurable geometry — lengths, angles, volumes, the vector-covector isomorphism, and, through metric compatibility with zero torsion, a unique connection and its curvature.

The core insight

A metric is a smoothly varying, symmetric, nondegenerate inner product on tangent spaces; it converts the purely topological-differentiable manifold into a measurable geometry, fixing lengths, angles, volumes, the vector-covector isomorphism, and — by metric compatibility plus zero torsion — a unique connection and its curvature.

Where this lives today

The metric is measured in laboratories and read off telescopes every day. GPS is the everyday case: each satellite's clock runs fast against the ground by about 38 microseconds per day once the Schwarzschild $g_{tt}$ redshift and the orbital-speed term are combined, and without that curved-spacetime correction positioning would drift by kilometres within a day. At the opposite extreme, LIGO and Virgo detect gravitational waves as oscillations of the spatial metric itself — a passing wave stretches one arm and squeezes the other by a strain $h\sim10^{-21}$ , less than a thousandth of a proton's width over a four-kilometre arm, first caught in GW150914 in 2015. The Event Horizon Telescope images the photon ring predicted by the Schwarzschild and Kerr metrics around M87 and Sgr A*, and numerical relativity solves the Einstein equations for $g_{\mu\nu}$ to predict merger waveforms. Far from gravity the same mathematics reappears as information geometry: the Fisher-Rao metric turns a family of probability distributions into a Riemannian manifold whose geodesics and volume element give coordinate-invariant notions of statistical distinguishability, now standard in machine learning and quantum information.

The physical picture

Picture the metric as a local ruler-and-protractor bolted to each point. Its whole job is to eat two infinitesimal arrows and return a number, $g(\mathbf{u},\mathbf{v})$ . From that single bilinear operation you extract every metric quantity there is: lengths $\|\mathbf{u}\|=\sqrt{|g(\mathbf{u},\mathbf{u})|}$ , angles $\cos\theta = g(\mathbf{u},\mathbf{v})/(\|\mathbf{u}\|\,\|\mathbf{v}\|)$ , and volumes through the determinant factor $\sqrt{|g|}$ . The components are just the pairwise dot products of the coordinate basis vectors, $g_{\mu\nu}=\mathbf{e}_\mu\cdot\mathbf{e}_\nu$ : where coordinate lines cross at right angles the off-diagonal entries vanish, and where a basis vector stretches — like the azimuthal $\partial_\phi$ far from the axis — the matching diagonal entry swells, as the first figure shows for the plane in polar coordinates.

Figure

The metric reads the geometry of a coordinate grid: in the plane the polar basis vectors are e_r of fixed length and e_φ of length r, so ds² = dr² + r² dφ² and g = diag(1, r²) — the off-diagonal entry is zero because the grid is orthogonal, and g_φφ grows as r² outward

At a single point you can always tidy the metric up: symmetry diagonalizes it and nondegeneracy rescales it, so by Sylvester's law of inertia there is a local orthonormal frame with $g=\mathrm{diag}(\pm1,\dots)$ . What you generally cannot do is make $g$ constant across a neighbourhood — the obstruction is precisely curvature, measured invariantly by whether triangles close the way Euclid promised. The sharpest demonstration is a triangle on a sphere: its angles overshoot $180^\circ$ , and the overshoot equals the enclosed area divided by $a^2$ . That is a real, calculable number carried entirely by the metric $g=a^2\,\mathrm{diag}(1,\sin^2\theta)$ , and the second figure pins it down for the cleanest case.

Figure

A geodesic triangle on the 2-sphere of radius a, metric ds² = a²(dθ² + sin²θ dφ²) with area element √g = a² sinθ: the octant bounded by three great-circle quarter-arcs has three right angles, so its angle sum is 270° — a spherical excess 90° = π/2 = Area divided by a², giving Gauss curvature K = 1/a² exactly

In a Lorentzian metric the inner product is indefinite, so $g(\mathbf{u},\mathbf{u})$ can be positive, negative, or zero, and that sign — not any coordinate label — sorts vectors into spacelike, timelike, and null. The causal skeleton of spacetime, light cones and all, is read straight off the signature of $g$ : one tensor field carries the entire local geometry — lengths, angles, volumes, the split of space from time, and the seed of curvature.

The core equations

The one indispensable formula is the line element: contract the metric with a displacement twice for the squared length of an infinitesimal step. Everything else here specializes, differentiates, or inverts it.

\mathrm{d}s^2 = g_{\mu\nu}\,\mathrm{d}x^\mu\,\mathrm{d}x^\nu \qquad (1)

where: $\mathrm{d}s^2$ — squared length of the infinitesimal displacement $\mathrm{d}x^\mu$ ; $g_{\mu\nu}$ — covariant metric components (a symmetric matrix field); $\mathrm{d}x^\mu$ — coordinate differentials; repeated indices summed (Einstein convention). Two algebraic properties make $g$ a metric rather than a generic tensor: symmetry is free (an antisymmetric part could never contribute to $\mathrm{d}s^2$ ), while nondegeneracy is the load-bearing assumption that guarantees an inverse.

g_{\mu\nu}=g_{\nu\mu},\qquad \det(g_{\mu\nu})\neq 0 \qquad (2)

where: $g_{\mu\nu}=g_{\nu\mu}$ — symmetry, so the matrix is diagonalizable with real eigenvalues; $\det(g)\neq0$ — nondegeneracy, so no nonzero vector is invisible to the inner product. The inverse metric $g^{\mu\nu}$ is defined purely as the matrix inverse — never by "raising indices on $g$ ", which would just reproduce the identity.

g^{\mu\alpha}g_{\alpha\nu}=\delta^{\mu}_{\ \nu} \qquad (3)

where: $g^{\mu\nu}$ — contravariant metric, the matrix inverse of $g_{\mu\nu}$ ; $\delta^\mu_{\ \nu}$ — Kronecker delta, whose existence is exactly what nondegeneracy buys. With both $g$ and $g^{-1}$ in hand the tangent space and its dual become interchangeable: lowering an index with $g$ turns a vector into a covector, raising with $g^{-1}$ undoes it. This is the musical isomorphism.

V_\mu = g_{\mu\nu}V^\nu,\qquad V^\mu = g^{\mu\nu}V_\nu \qquad (4)

where: $V^\nu$ — contravariant (tangent-vector) components; $V_\mu$ — covariant (covector / one-form) components. In an orthonormal frame $g=\mathbb{1}$ , so upper and lower components coincide — which is why elementary vector calculus never needs the distinction. Because $\mathrm{d}s^2$ is a coordinate-independent scalar, the components must transform with two lower Jacobian factors — the defining law of a $(0,2)$ tensor.

g'_{\alpha\beta} = \frac{\partial x^\mu}{\partial x'^\alpha}\frac{\partial x^\nu}{\partial x'^\beta}\,g_{\mu\nu} \qquad (5)

where: $x'^\alpha$ — new coordinates; $\partial x^\mu/\partial x'^\alpha$ — Jacobian of the change of coordinates. Substitute $\mathrm{d}x^\mu=(\partial x^\mu/\partial x'^\alpha)\mathrm{d}x'^\alpha$ into (1) and match, and the two factors are forced. Differentiating $g$ once produces the Christoffel symbols — the unique connection coefficients that are metric-compatible and torsion-free, the hinge between algebra and calculus on a manifold.

\Gamma^{\lambda}_{\ \mu\nu} = \tfrac{1}{2}g^{\lambda\sigma}\!\left(\partial_\mu g_{\sigma\nu} + \partial_\nu g_{\sigma\mu} - \partial_\sigma g_{\mu\nu}\right) \qquad (6)

where: $\Gamma^\lambda_{\ \mu\nu}$ — Levi-Civita connection coefficients, symmetric in $\mu\nu$ ; $\partial_\mu g_{\sigma\nu}$ — first partial derivatives of the metric. $\Gamma$ vanishes wherever $g$ is constant. The property that singles out this connection is metric compatibility: the covariant derivative of $g$ is zero.

\nabla_\lambda g_{\mu\nu} = 0 \qquad (7)

where: $\nabla_\lambda$ — covariant derivative with respect to the Levi-Civita connection. It guarantees lengths and angles survive parallel transport, so a unit vector stays a unit vector as you carry it around. The determinant of $g$ supplies the scalar density every covariant integral needs — the invariant volume element.

\mathrm{d}V = \sqrt{|g|}\,\mathrm{d}^n x \qquad (8)

where: $g$ — determinant $\det(g_{\mu\nu})$ ; $|g|$ — absolute value (Lorentzian metrics have $g<0$ ); $\mathrm{d}^n x$ — coordinate volume. The factor $\sqrt{|g|}$ makes $\int f\,\sqrt{|g|}\,\mathrm{d}^n x$ the same number in every chart. Angles come from the same machine: $\cos\theta = g_{\mu\nu}U^\mu V^\nu/(\|U\|\,\|V\|)$ in the Riemannian case, with Cauchy-Schwarz keeping the ratio in $[-1,1]$ .

Derivation AUniqueness of the Levi-Civita connection

The goal is to prove a cornerstone of Riemannian geometry: on a manifold with a metric there is exactly one connection that is both torsion-free and metric-compatible, with coefficients fixed entirely by the metric through formula (6). We do not assume the answer; we impose the two conditions on an unknown $\Gamma$ and solve the resulting linear system, watching four of the six terms cancel by symmetry.

Step 1. Write out metric compatibility $\nabla_\lambda g_{\mu\nu}=0$ using the covariant-derivative rule for a $(0,2)$ tensor: $\partial_\lambda g_{\mu\nu} = \Gamma^{\sigma}_{\ \lambda\mu}g_{\sigma\nu} + \Gamma^{\sigma}_{\ \lambda\nu}g_{\mu\sigma}$ . Why: this turns the geometric demand into an algebraic relation between derivatives of $g$ and the unknown connection.

Step 2. Cyclically permute the free indices $\lambda\to\mu\to\nu\to\lambda$ to get a second copy, $\partial_\mu g_{\nu\lambda} = \Gamma^{\sigma}_{\ \mu\nu}g_{\sigma\lambda} + \Gamma^{\sigma}_{\ \mu\lambda}g_{\nu\sigma}$ . Why: the identity holds for any ordering, and a second one gives an independent equation.

Step 3. Permute once more for a third copy, $\partial_\nu g_{\lambda\mu} = \Gamma^{\sigma}_{\ \nu\lambda}g_{\sigma\mu} + \Gamma^{\sigma}_{\ \nu\mu}g_{\lambda\sigma}$ . Why: three equations in the symmetric derivative combinations are exactly enough to isolate a single $\Gamma$ .

Step 4. Form (Step 2) + (Step 3) − (Step 1). Using the torsion-free symmetry $\Gamma^\sigma_{\ \mu\nu}=\Gamma^\sigma_{\ \nu\mu}$ , four of the six $\Gamma g$ terms cancel in pairs, leaving $\partial_\mu g_{\nu\lambda} + \partial_\nu g_{\lambda\mu} - \partial_\lambda g_{\mu\nu} = 2\,\Gamma^{\sigma}_{\ \mu\nu}g_{\sigma\lambda}$ . Why: torsion-freeness is exactly the symmetry that annihilates the unwanted terms — both conditions are needed for uniqueness.

Step 5. Contract with the inverse metric $g^{\lambda\rho}$ to isolate $\Gamma^{\rho}_{\ \mu\nu} = \tfrac12 g^{\rho\lambda}(\partial_\mu g_{\nu\lambda}+\partial_\nu g_{\lambda\mu}-\partial_\lambda g_{\mu\nu})$ , the boxed Christoffel formula. Why: nondegeneracy guarantees $g^{\rho\lambda}$ exists, so the solution is unique — one and only one Levi-Civita connection. *The result is symmetric in the lower pair $\mu\nu$ and vanishes identically whenever $g_{\mu\nu}$ is constant.*

Go deeper · Derivation B · 1 minThe geodesic equation from extremizing arc length

Derivation BThe geodesic equation from extremizing arc length

The aim is to show that the straightest curves — stationary paths of the length functional — obey an equation whose coefficients are exactly the Christoffel symbols, so geodesics are determined by $g$ alone. We vary the quadratic energy $L=\tfrac12 g_{\mu\nu}\dot x^\mu\dot x^\nu$ (equivalent to arc length for an affine parameter), push it through Euler-Lagrange, and watch the connection assemble itself from the metric derivatives.

Step 1. Apply the Euler-Lagrange equation $\dfrac{\mathrm{d}}{\mathrm{d}\lambda}\dfrac{\partial L}{\partial \dot x^\lambda} - \dfrac{\partial L}{\partial x^\lambda}=0$ to $L=\tfrac12 g_{\mu\nu}(x)\dot x^\mu\dot x^\nu$ . Why: extremizing length is a variational problem, and Euler-Lagrange turns "stationary path" into a differential equation.

Step 2. Compute the pieces: $\dfrac{\partial L}{\partial \dot x^\lambda} = g_{\lambda\nu}\dot x^\nu$ and $\dfrac{\partial L}{\partial x^\lambda}=\tfrac12 \partial_\lambda g_{\mu\nu}\,\dot x^\mu\dot x^\nu$ . Why: the second term survives only because $g$ depends on position — the source of all the connection terms.

Step 3. Differentiate the momentum along the curve: $\dfrac{\mathrm{d}}{\mathrm{d}\lambda}(g_{\lambda\nu}\dot x^\nu) = g_{\lambda\nu}\ddot x^\nu + \partial_\mu g_{\lambda\nu}\,\dot x^\mu\dot x^\nu$ . Why: the chain rule splits it into an acceleration piece and a metric-variation piece.

Step 4. Substitute and symmetrize $\partial_\mu g_{\lambda\nu}\dot x^\mu\dot x^\nu$ over $\mu\nu$ (legal, since it multiplies the symmetric $\dot x^\mu\dot x^\nu$ ): $g_{\lambda\nu}\ddot x^\nu + \tfrac12(\partial_\mu g_{\lambda\nu}+\partial_\nu g_{\lambda\mu}-\partial_\lambda g_{\mu\nu})\dot x^\mu\dot x^\nu = 0$ . Why: symmetrization is what conjures the three-term Christoffel combination from a single derivative.

Step 5. Contract with $g^{\rho\lambda}$ to free the acceleration: $\ddot x^\rho + \Gamma^{\rho}_{\ \mu\nu}\dot x^\mu\dot x^\nu=0$ . Why: the bracket is precisely $2g_{\rho\lambda}\Gamma^\rho_{\ \mu\nu}$ from the previous derivation — geodesics and Levi-Civita are one content. *As a check: when $g$ is constant all Christoffel symbols vanish and the equation reduces to $\ddot x^\rho=0$ , the straight lines of flat space.*

Go deeper · Derivation C · 1 minFlat space in spherical coordinates has zero curvature

Derivation CFlat space in spherical coordinates has zero curvature

The purpose is to demolish the most common beginner's error — reading curvature off the appearance of the metric. We take flat Euclidean $\mathbb{R}^3$ in spherical coordinates, where $g$ is manifestly position-dependent, build its Christoffel symbols and Riemann tensor, and watch every curvature component cancel to exactly zero — confirming that curvature lives in the second-order structure of $g$ , not its coordinate dress.

Step 1. Start from $\mathrm{d}s^2=\mathrm{d}r^2+r^2\mathrm{d}\theta^2+r^2\sin^2\theta\,\mathrm{d}\phi^2$ and feed it into formula (6). The nonzero Christoffels are $\Gamma^{r}_{\ \theta\theta}=-r$ , $\Gamma^{r}_{\ \phi\phi}=-r\sin^2\theta$ , $\Gamma^{\theta}_{\ r\theta}=1/r$ , $\Gamma^{\theta}_{\ \phi\phi}=-\sin\theta\cos\theta$ , $\Gamma^{\phi}_{\ r\phi}=1/r$ , $\Gamma^{\phi}_{\ \theta\phi}=\cot\theta$ . Why: these are nonzero purely because the coordinate basis rotates and stretches — a coordinate effect, not yet curvature.

Step 2. Assemble $R^{\rho}_{\ \sigma\mu\nu}=\partial_\mu\Gamma^{\rho}_{\ \nu\sigma}-\partial_\nu\Gamma^{\rho}_{\ \mu\sigma}+\Gamma^{\rho}_{\ \mu\lambda}\Gamma^{\lambda}_{\ \nu\sigma}-\Gamma^{\rho}_{\ \nu\lambda}\Gamma^{\lambda}_{\ \mu\sigma}$ and test the component $R^{\theta}_{\ r\theta r}$ . Why: if any second-order component survived, the space would be curved.

Step 3. Its two surviving pieces are the derivative term $-\partial_r\Gamma^{\theta}_{\ r\theta}=-\partial_r(1/r)=1/r^2$ and the quadratic term $\Gamma^{\theta}_{\ r\theta}\Gamma^{\theta}_{\ r\theta}=1/r^2$ , entering with opposite sign. Why: both are individually nonzero, so the vanishing is a genuine cancellation, not an absence of terms.

Step 4. Add them: $1/r^2 - 1/r^2 = 0$ , and the same cancellation occurs for every component, so $R^{\rho}_{\ \sigma\mu\nu}=0$ everywhere. Why: a nonconstant $g$ can hide but never create curvature — always check the Riemann tensor, not $g$ itself. *This confirms $\mathbb{R}^3$ is globally flat. By contrast, on the genuine 2-sphere the analogous component is $R^{\theta}_{\ \phi\theta\phi}=\sin^2\theta\neq0$ , giving $K=1/a^2$ .*

Limiting cases

The one metric formalism collapses onto every familiar geometry when fed the right components; the table maps abstract $g_{\mu\nu}$ back to cases you already know.

Case	Result	Meaning
Flat Euclidean (Cartesian)	$g_{\mu\nu}=\delta_{\mu\nu}$	ordinary Pythagorean space, $\mathrm{d}s^2=\mathrm{d}x^2+\mathrm{d}y^2+\mathrm{d}z^2$ ; all Christoffel symbols vanish
Minkowski (Lorentzian)	$\eta_{\mu\nu}=\mathrm{diag}(-1,+1,+1,+1)$	flat spacetime; the signature splits time from space and gives the invariant interval $\mathrm{d}s^2=-c^2\mathrm{d}t^2+\mathrm{d}\mathbf{x}^2$
Flat space, spherical coordinates	$g_{\mu\nu}=\mathrm{diag}(1,\,r^2,\,r^2\sin^2\theta)$	position-dependent yet zero curvature — nonconstant $g$ alone is not curvature
Local inertial (Riemann normal) frame	$g_{\mu\nu}(p)=\eta_{\mu\nu},\ \partial_\lambda g_{\mu\nu}(p)=0$	at any point one can flatten $g$ to first order; the irreducible second-order remainder is the Riemann tensor
2-sphere of radius $a$	$\mathrm{d}s^2 = a^2(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2)$	a genuinely curved Riemannian metric, constant Gauss curvature $K=1/a^2$ , Ricci scalar $R=2/a^2$

Common trapa position-dependent metric means the space is curved

This is the most common misreading of the metric, and it is false. Flat $\mathbb{R}^3$ in spherical coordinates has $g_{\mu\nu}=\mathrm{diag}(1,r^2,r^2\sin^2\theta)$ , blatantly position-dependent, yet its Riemann tensor is identically zero (see the third derivation). Curvature lives in the second derivatives of $g$ assembled into $R^{\rho}_{\ \sigma\mu\nu}$ , not in $g$ itself, and it is coordinate-invariant: it cannot be conjured by bad coordinates nor removed by clever ones. A second trap rides along: $g^{\mu\nu}$ is the matrix inverse of $g_{\mu\nu}$ , never the result of "raising indices" on $g$ — that would just hand you back $\delta^\mu_{\ \nu}$ . The reliable test that two differently-written metrics are the same geometry is agreement of their curvature invariants (Ricci scalar, Kretschmann scalar), not the look of their components.

Worked examples

1 · The round 2-sphere, end to end (fully worked)

Take the 2-sphere of radius $a$ with $\mathrm{d}s^2 = a^2(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2)$ . Compute the metric, its inverse, the area element, the Christoffel symbols, the curvature, and the Gauss curvature — then read off the octant area.

Step 1. Read the components off the line element: $g_{\theta\theta}=a^2$ , $g_{\phi\phi}=a^2\sin^2\theta$ , off-diagonal zero. The determinant is $g=a^4\sin^2\theta$ , so $\sqrt{g}=a^2\sin\theta$ and the area element is $\mathrm{d}A=a^2\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\phi$ ; integrating over $\theta\in[0,\pi],\ \phi\in[0,2\pi]$ recovers the total area $a^2\cdot 2\cdot 2\pi = \mathbf{4\pi a^2}$ , as it must.

Step 2. Invert the diagonal metric — reciprocate each diagonal entry — to get $g^{\theta\theta}=1/a^2$ and $g^{\phi\phi}=1/(a^2\sin^2\theta)$ . This is the matrix inverse, not any index-raising trick.

Step 3. Feed $g$ into the Christoffel formula (6). Only two independent symbols survive: $\Gamma^{\theta}_{\ \phi\phi}=-\sin\theta\cos\theta$ and $\Gamma^{\phi}_{\ \theta\phi}=\cot\theta$ . Everything carrying an $r$ index is gone (no $r$ direction), and the constant radius $a$ never differentiates.

Step 4. Compute the one independent Riemann component $R^{\theta}_{\ \phi\theta\phi}=\partial_\theta\Gamma^{\theta}_{\ \phi\phi}-\Gamma^{\theta}_{\ \phi\phi}\Gamma^{\phi}_{\ \theta\phi}$ . The pieces are $\partial_\theta(-\sin\theta\cos\theta)=\sin^2\theta-\cos^2\theta$ and $-(-\sin\theta\cos\theta)(\cot\theta)=+\cos^2\theta$ ; summing gives $\mathbf{\sin^2\theta}\neq0$ — the sphere is genuinely curved.

Step 5. Contract to the scalars. The Ricci tensor is $R_{ij}=(1/a^2)g_{ij}$ , so $R=g^{ij}R_{ij}=\mathbf{2/a^2}$ and the Gauss curvature is $K=R/2=\mathbf{1/a^2}$ — constant, positive, shrinking as the sphere grows. As a payoff, Gauss-Bonnet gives the octant (three right angles) a spherical excess $E=\pi/2=\text{Area}/a^2$ , so its area is $(\pi/2)a^2=\tfrac18(4\pi a^2)$ — one-eighth of the sphere, with no integration needed.

2 · Schwarzschild time dilation from a single metric component

The Schwarzschild metric outside a mass $M$ has $g_{tt}=-(1-r_s/r)$ with $r_s=2GM/c^2$ . Estimate $r_s$ for the Earth and the Sun, and the rate at which a clock on Earth's surface runs slow relative to one infinitely far away.

Step 1. The Schwarzschild radius is $r_s=2GM/c^2$ . For Earth, $GM_\oplus=3.986\times10^{14}\ \mathrm{m^3 s^{-2}}$ and $c^2=8.988\times10^{16}\ \mathrm{m^2 s^{-2}}$ , so $r_s=\mathbf{8.87\times10^{-3}\ m}$ — Earth's mass squeezed inside about 8.9 mm would become a black hole.

Step 2. For the Sun, $GM_\odot=1.327\times10^{20}\ \mathrm{m^3 s^{-2}}$ gives $r_s=\mathbf{2.95\times10^{3}\ m}$ , the familiar 2.95 km Schwarzschild radius.

Step 3. A static clock ticks at the rate $\sqrt{-g_{tt}}=\sqrt{1-r_s/r}$ . At Earth's surface $r=6.371\times10^{6}\ \mathrm{m}$ , so $r_s/r=\mathbf{1.39\times10^{-9}}$ and $\sqrt{1-r_s/r}\approx 1-6.96\times10^{-10}$ .

Step 4. So a surface clock loses about $6.96\times10^{-10}\times 86\,400\ \mathrm{s}\approx\mathbf{60\ \mu s\ per\ day}$ against a clock at infinity — the same $g_{tt}$ effect (opposite sign for higher, orbiting clocks) that forces GPS to correct its atomic clocks by tens of microseconds a day. Geometry from one metric component, measured routinely in orbit.

Physics of the portrait

The metric tensor has no single inventor, but it has a decisive one: the idea that geometry is carried by an intrinsic quadratic form, freed from any embedding, belongs to Bernhard Riemann.

Origins · 1 minBernhard Riemann and how this was discovered

Bernhard Riemann

1826-1866 · the invention of Riemannian geometry and the intrinsic metric

Bernhard Riemann held his 1854 habilitation before the Göttingen faculty — with the aging Gauss, who had proposed the topic, in the audience — and in a lecture almost devoid of formulas ("Über die Hypothesen, welche der Geometrie zu Grunde liegen") he generalized Gauss's theory of curved surfaces to any dimension. His radical move was to define distance intrinsically, by a quadratic form $\mathrm{d}s^2=\sum g_{ij}\,\mathrm{d}x^i\mathrm{d}x^j$ living on the manifold itself with no ambient space, and from that single form he extracted a notion of curvature — anticipating by six decades the mathematics Einstein would need. Riemann could not yet compute freely with $g_{ij}$ as a tensor: that language was forged by Christoffel, who in 1869 introduced the three-index symbols, and by Ricci-Curbastro and his student Levi-Civita, whose 1900 "absolute differential calculus" turned the vision into a working machinery of covariant derivatives and parallel transport. When Einstein reached for it in 1915, $g_{\mu\nu}$ stopped describing a fixed space and became the gravitational field itself. Riemann died at thirty-nine, but the geometry that bears his name is now the language of gravitation, cosmology, and the shape of spacetime.

History

1827

— Gauss's Disquisitiones generales defines the first fundamental form of a curved surface

1854

— Riemann's habilitation lecture generalizes the metric to n-dimensional manifolds

1869

— Christoffel introduces the three-index symbols and covariant differentiation of quadratic forms

1900

— Ricci and Levi-Civita publish the absolute differential calculus, the tensor calculus proper

1915

— Einstein makes the metric g the dynamical gravitational field in general relativity

Going further

Texts. Stone & Goldbart, Mathematics for Physics, Ch. 11 (differential geometry and tensors) · Arfken & Weber, Mathematical Methods for Physicists, tensors and curvilinear-coordinates chapters · Hassani, Mathematical Physics, Part on differential geometry · Nakahara, Geometry, Topology and Physics, Ch. 7 · Byron & Fuller, Mathematics of Classical and Quantum Physics, tensor chapter · Misner, Thorne & Wheeler, Gravitation, Ch. 8 · Wald, General Relativity, Ch. 3. Key results to keep in mind: the Christoffel formula follows from the (2)+(3)−(1) cyclic permutation of metric compatibility; the geodesic equation is recovered by Euler-Lagrange on $L=\tfrac12 g_{\mu\nu}\dot x^\mu\dot x^\nu$ ; spherical coordinates in $\mathbb{R}^3$ give nonzero $\Gamma$ but $R^\rho{}_{ \sigma\mu\nu}=0$ ; on the 2-sphere $\sqrt{g}=a^2\sin\theta$ , $R^\theta{}_{ \phi\theta\phi}=\sin^2\theta$ , $K=1/a^2$ , $R=2/a^2$ ; and $r_s=2GM/c^2=8.87\ \mathrm{mm}$ (Earth), $2.95\ \mathrm{km}$ (Sun), with gravitational surface slowing $\approx6.96\times10^{-10}$ .

Prerequisites. Tensor Algebra and the Tensor Product · Covariant and Contravariant Tensors · Manifolds and Tangent Spaces. Leads to. Raising and Lowering Indices · Christoffel Symbols and Affine Connections · Parallel Transport and Geodesics · The Riemann Curvature Tensor · Hodge Star and the Codifferential.

Ask the tutor: Ask me to work any metric you like — polar, cylindrical, the 2-sphere, Schwarzschild, or FRW cosmology — all the way from $g_{\mu\nu}$ to its Christoffel symbols, geodesics, and curvature, step by step.

Is the metric tensor the same thing as the distance between two points?

Not quite — it is the infinitesimal ingredient from which distance is built. The metric gives $\mathrm{d}s^2=g_{\mu\nu}\mathrm{d}x^\mu\mathrm{d}x^\nu$ , the squared length of a tiny step, at every point. To get the distance between two finite points you integrate $\sqrt{|\mathrm{d}s^2|}$ along a curve joining them and minimize over all such curves; the minimizer is a geodesic and its length is the geodesic distance. So the metric is local and the distance is global — the metric alone does not pick a curve until you extremize. On a sphere, the same metric yields great-circle distances only after that minimization.

What is the difference between g_μν and g^μν, really?

They are matrix inverses of each other, and that is the whole story: $g^{\mu\alpha}g_{\alpha\nu}=\delta^\mu_{\ \nu}$ . The lower-index $g_{\mu\nu}$ turns a vector into a covector, and the upper-index $g^{\mu\nu}$ raises one back. A frequent error is to think $g^{\mu\nu}$ comes from "raising the indices on $g_{\mu\nu}$ " — but raising both indices of $g_{\mu\nu}$ using itself just returns $\delta$ . Compute $g^{\mu\nu}$ the honest way: write $g_{\mu\nu}$ as a matrix and invert it, which for a diagonal metric means reciprocating each diagonal entry.

If a nonconstant metric is not curvature, how do I actually detect curvature?

Compute the Riemann tensor $R^{\rho}_{\ \sigma\mu\nu}$ from second derivatives of the metric (via the Christoffel symbols) and ask whether it is zero. If every component vanishes the space is flat, no matter how baroque $g$ looks; if any is nonzero the space is genuinely curved and no coordinate change can hide it. Because comparing full tensors is awkward, one usually checks coordinate-invariant scalars — the Ricci scalar $R$ and the Kretschmann scalar $R_{\rho\sigma\mu\nu}R^{\rho\sigma\mu\nu}$ . Flat space in spherical coordinates has all of these zero; a 2-sphere of radius $a$ has $R=2/a^2\neq0$ , settling the question invariantly.

Why does metric compatibility ∇g = 0 matter physically?

It keeps geometry and calculus talking to each other. Metric compatibility means the covariant derivative of $g$ vanishes, so when you parallel-transport two vectors along any curve their inner product — hence their lengths and angle — stays constant. Without it, a connection could stretch or twist vectors as it slid them, and "unit vector" or "constant speed along a geodesic" would lose meaning. Combined with zero torsion it singles out a unique connection (Levi-Civita), which is why we can speak of "the" Christoffel symbols of a metric rather than choosing among many connections. It is what makes the affine and metric structures one geometry.

What does the signature of the metric tell me?

The signature is the pattern of plus and minus signs when the metric is diagonalized to $\pm1$ s, and by Sylvester's law of inertia it is an invariant — no coordinate transformation can change it. A Riemannian signature $(+,+,\dots,+)$ makes every nonzero vector have positive squared length, so the metric measures ordinary distances. A Lorentzian signature $(-,+,+,+)$ lets $g(\mathbf{u},\mathbf{u})$ be negative, positive, or zero, sorting vectors into timelike, spacelike, and null and building the light cones of relativity. That single minus sign is the entire difference between a length and a spacetime interval — why time is not just a fourth spatial direction.

Can I always make the metric look flat at a point?

At a single point, yes — always. Symmetry plus nondegeneracy let you diagonalize and rescale, so there exist coordinates (Riemann normal coordinates) with $g_{\mu\nu}(p)=\eta_{\mu\nu}$ and even $\partial_\lambda g_{\mu\nu}(p)=0$ . This is the mathematical face of Einstein's equivalence principle: a freely falling observer sees locally flat spacetime, gravity "transformed away". What you cannot kill is the second derivatives — the irreducible remainder is the Riemann tensor, so curvature is exactly the obstruction to extending a local inertial frame over a finite region. Flat at a point and to first order, yes; true curvature always shows up at second order.