Here is a question so basic it sounds silly: how do you measure a distance? On a flat sheet of graph paper the answer is Pythagoras — a tiny step (dx, dy) has length ds with ds^2 = dx^2 + dy^2. But now describe the same flat plane using polar coordinates (r,\theta). A step d\theta in angle covers a bigger arc the further out you are — its length is r\,d\theta, not d\theta. So now

Where did that r^2 come from? It is the metric tensor doing its job. The metric is the object that tells you how to turn coordinate differences into real, physical distances — and once distances change from place to place, or space itself curves, it becomes the single most important tensor in physics. This page is about that one object, g_{ij}.

What the metric is

The metric tensor is a symmetric rank-2 tensor g_{ij} whose job is to eat two small displacement vectors and return their inner product. Fed the same displacement twice, it returns the squared length — the master formula (with the Einstein summation convention) is:

Read off the metric by matching the line element. Flat Cartesian space, ds^2 = dx^2 + dy^2, has g_{ij} = \delta_{ij} — the identity, ones on the diagonal. Polar coordinates, ds^2 = dr^2 + r^2 d\theta^2, have

Same flat plane, different coordinates, different-looking metric — but it measures the very same distances. That is the whole point: the metric carries the geometry, whatever coordinates you dress it in.

Seeing the metric: why the r² is there

The figure makes the polar metric visible. A fixed wedge of angle \Delta\theta is drawn from the origin; slide the radius r outward and watch the arc it subtends. The coordinate step \Delta\theta never changes, but the physical arc length r\,\Delta\theta grows in proportion to r.

That growth is exactly the factor g_{\theta\theta} = r^2 in the line element: length-per-unit-\theta is r, so length-squared-per-unit-\theta^2 is r^2. The metric is the dictionary between coordinate steps and true distances, and here it says "one radian of angle is worth r units of length."

The metric's second job: raising and lowering indices

In the tensor world there are two flavours of index — contravariant (upstairs, v^i) and covariant (downstairs, v_i). The metric is the machine that converts between them. Lowering an index means contracting with g_{ij}; raising it means contracting with the inverse metric g^{ij}:

The inverse metric is literally the matrix inverse of g_{ij}; together they multiply to the Kronecker delta. This raising and lowering is not bookkeeping fluff — it is how you form the dot product \mathbf{u}\cdot\mathbf{v} = u_i v^i = g_{ij}u^i v^j, and in curved space it is the only correct way to compare vectors with covectors. A symmetric n\times n metric has \tfrac{n(n+1)}{2} independent components — the count that matters when you tally the unknowns in a field theory.

Flat versus curved — the deepest use

Here is the payoff. In genuinely flat space you can always find coordinates in which the metric is the plain identity \delta_{ij} everywhere at once — the polar r^2 is just a coordinate artefact you can transform away. In a curved space you cannot: on the surface of a sphere of radius a,

and no clever change of coordinates can make this constant everywhere — the honest curvature of the sphere is locked inside how g_{ij} varies from point to point. This is the idea Einstein built general relativity on: gravity is not a force but the curvature of a four-dimensional spacetime metric g_{\mu\nu}. Even flat spacetime has a non-trivial metric — the Minkowski metric \eta_{\mu\nu} = \mathrm{diag}(-1, 1, 1, 1), whose crucial minus sign is what separates time from space and makes the whole of special relativity work.

Worked examples

Example 1 — read off a metric. Cylindrical coordinates have ds^2 = dr^2 + r^2 d\theta^2 + dz^2. Matching ds^2 = g_{ij}dx^i dx^j, the metric is diagonal: g_{ij} = \mathrm{diag}(1, r^2, 1).

Example 2 — the inverse metric. For the polar metric g_{ij} = \mathrm{diag}(1, r^2), the inverse is g^{ij} = \mathrm{diag}(1, 1/r^2) — invert each diagonal entry. Check: g^{\theta\theta}g_{\theta\theta} = \tfrac{1}{r^2}\cdot r^2 = 1 = \delta^\theta_{\,\theta}.

Example 3 — a distance on a sphere. On a unit sphere (a=1), a small step of d\theta = 0.1 at the equator (\theta = 90^\circ) and d\varphi = 0.1 has ds^2 = 1\cdot(0.1)^2 + 1\cdot\sin^2(90^\circ)\cdot(0.1)^2 = 0.02, so ds = \sqrt{0.02}\approx 0.141.

Example 4 — counting components. In 4-dimensional spacetime the metric is a symmetric 4\times 4 tensor, so it has \tfrac{n(n+1)}{2} = \tfrac{4\cdot5}{2} = 10 independent components — the ten functions Einstein's equations solve for.