Curved Spacetime and the Metric

The equivalence principle told us that gravity can be transformed away at a point but leaves a residue — tidal forces — that no change of frame can erase. That residue is the clue Einstein followed to its stunning conclusion: gravity is not a force at all, but the geometry of spacetime. Mass and energy warp the four-dimensional fabric of space and time, and objects with no forces on them simply glide along the straightest paths available in that warped geometry. A thrown ball and an orbiting planet are both coasting — through a spacetime bent by the Earth and the Sun.

To make "warped fabric" into physics we need one mathematical object that measures distances and times in a curved geometry: the metric tensor g_{\mu\nu}. You already met it as the thing that turns coordinate steps into real distances; here it graduates to the central character of general relativity. This page assembles three ideas — the spacetime interval, the metric as the field that encodes gravity, and intrinsic curvature — into a single picture of what "curved spacetime" actually means.

The interval: the one number everyone agrees on

In flat, gravity-free spacetime, special relativity already taught us that different observers disagree about distances and about time intervals — but they all agree on one combination, the spacetime interval between two nearby events:

ds^2 = -c^2\,dt^2 + dx^2 + dy^2 + dz^2.

This is Pythagoras with one crucial minus sign — the mark of time. The sign of ds^2 sorts every pair of events into three kinds: timelike (ds^2 < 0, close enough in space that a sub-light worldline connects them — cause can reach effect), null (ds^2 = 0, connected by a light ray), and spacelike (ds^2 > 0, so far apart that not even light can bridge them — "elsewhere"). The boundary is the light cone, and it fences off cause from effect.

Compactly, using the Einstein summation convention, the interval is ds^2 = \eta_{\mu\nu}\,dx^\mu\,dx^\nu, where \eta_{\mu\nu} = \operatorname{diag}(-1, +1, +1, +1) is the Minkowski metric — flat spacetime's rulebook for distance. Everything special about gravity is contained in what happens when we replace this constant \eta with something that varies from place to place.

From η to g: the metric becomes a field

Einstein's leap is to promote the flat metric \eta_{\mu\nu} to a varying field g_{\mu\nu}(x) — a different symmetric rank-2 tensor at every event. The interval in a general spacetime is

The proper time actually ticked by a clock along its worldline is read straight off the metric: c\,d\tau = \sqrt{-ds^2} for a timelike step. In flat space with a clock at rest, ds^2 = -c^2 dt^2 and d\tau = dt, as it must. But bend g_{00} away from -1 — as a mass does — and the same coordinate time dt yields a different proper time d\tau: this is gravitational time dilation, now baked directly into the geometry.

A vital subtlety: a varying metric does not by itself mean curvature. Recall flat space in polar coordinates, ds^2 = dr^2 + r^2\,d\theta^2 — the metric component r^2 varies, yet the plane is perfectly flat. You can always find coordinates that make gravity look present when it isn't (an accelerating frame) or absent when it is (a free-falling frame, locally). Real, physical curvature is what cannot be coordinate-changed away — and to detect it we need an honest geometric test.

Intrinsic curvature: the surveyor's test

How would a flat ant living on a surface, unable to peek into a third dimension, discover that its world is curved? It surveys. Draw a triangle from three straight (geodesic) sides and add up the interior angles. On a flat plane they sum to exactly 180^\circ. On a sphere — think of a triangle made of two lines of longitude and a stretch of the equator — the angles bulge out and sum to more than 180^\circ. On a saddle they sum to less. The excess is proportional to the enclosed area and to the curvature, and the ant needs no outside view to measure it. This is intrinsic curvature — Gauss's Theorema Egregium — and it is exactly the kind of curvature spacetime has.

Other intrinsic signatures agree: on a curved surface the circumference of a circle is no longer 2\pi r, parallel-transporting a vector around a loop rotates it, and — the physical heart of it — initially parallel geodesics fail to stay parallel. Two travellers setting off "straight" and side by side on a sphere draw together; on a saddle they spread apart. Translate that into spacetime and "geodesics that start parallel and then converge" is precisely the tidal drawing-together of two nearby free-falling apples. Tidal force is curvature. All of this is packaged into the Riemann curvature tensor built from the second derivatives of g_{\mu\nu} — the true, coordinate-proof measure of gravity, and the star of the next pages.

This is the deep question, and the answer is beautifully clean: you take derivatives. At any single event you can always choose local inertial coordinates in which the metric equals \eta_{\mu\nu} and its first derivatives vanish — that is the equivalence principle, the free-falling frame where gravity disappears. But you cannot, in a genuine gravitational field, also make the second derivatives vanish. Those irremovable second derivatives are the components of the Riemann tensor, and if any of them are non-zero the spacetime is truly curved — no coordinate trick can hide it. So the test for "real gravity versus a funny frame" is exactly the surveyor's test one level up: does the metric's second-order structure show a residue? Flat spacetime in exotic coordinates has a Riemann tensor that is identically zero; the spacetime around the Sun does not.

The rubber-sheet picture — a bowling ball denting a trampoline — is a helpful cartoon and a misleading one. Three cautions. First, that picture needs an outside third dimension for the sheet to sag into (extrinsic curvature); real spacetime curvature is intrinsic, measured entirely from within, with no "outside" required. Second, the trampoline seems to say balls roll inward because they slide "downhill" — but that sneaks gravity back in to explain gravity, and worse, a purely spatial dent cannot explain why a ball dropped from rest starts to move at all. Third, and most important: it is the time part of the metric, g_{00}, that does most of the work for everyday gravity — objects fall because their worldlines bend toward the region where time runs slow, not because space is dented. Curved spacetime, not curved space, is the point.