Manifolds, Vectors, and the Metric

The equivalence principle guarantees that spacetime is Minkowskian in a small neighbourhood of every event but says nothing about how those neighbourhoods fit together. The mathematical object that is flat locally and possibly curved globally is the manifold. This lesson assembles the apparatus general relativity runs on: charts, tangent vectors as differential operators, one-forms, and the metric tensor that recovers lengths, times, and angles from raw coordinates. The four-vectors of special relativity reappear here as the tangent-space objects at a single event, with the new feature that the tangent spaces at different events are distinct and must be connected before they can be compared.

Manifolds and coordinate charts

An $n$-dimensional manifold is a set that is locally homeomorphic to ${\mathbb{R}}^n$: each point has a neighbourhood that can be mapped smoothly and invertibly onto an open region of ${\mathbb{R}}^n$. Such a map is a coordinate chart, assigning to each point in its patch a set of $n$ real numbers $x^{\mu }$. A single chart rarely covers the whole manifold — the sphere is the standard example, since no one chart can cover it without a singular point — so a manifold is described by an atlas of charts whose patches overlap.

Where two charts overlap, a point carries two sets of coordinates related by an invertible smooth transformation $x^{\prime \mu }(x)$, with a nonsingular Jacobian matrix $\partial x^{\prime \mu }/\partial x^{\nu }$. The whole tensor calculus of general relativity is the statement of how geometric objects transform between charts under these Jacobians. Nothing physical depends on the choice of chart; coordinates are labels, and a coordinate that misbehaves (the longitude at the pole) need not signal anything wrong with the manifold.

Tangent vectors as directional derivatives

In flat space a vector can be pictured as an arrow joining two points, but on a curved manifold there is no such thing as a straight arrow between distant points, and no canonical way to subtract the coordinates of separated points. The definition of a vector must be intrinsic and local. The device is to identify a vector at a point $p$ with the directional derivative it defines along curves through $p$.

Let $x^{\mu }(\lambda )$ be a curve through $p$ and $f$ a smooth function on the manifold. The rate of change of $f$ along the curve is

using the summation convention. The operator in parentheses acts on any function and depends only on the curve's tangent at $p$. The tangent vector to the curve is that operator,

and the set of all such operators at $p$ forms an $n$-dimensional vector space, the tangent space $T_{p}M$. The coordinate partial derivatives $\{{\partial }_{\mu }\}=\{{\partial }_1,\dots ,{\partial }_n\}$ are a basis for it, the coordinate basis. Reading a vector as a differential operator makes its transformation law automatic: under a change of chart, the chain rule gives

Components transform with the Jacobian $\partial x^{\prime }/\partial x$ (contravariantly), basis vectors with its inverse, and the vector $V=V^{\mu }{\partial }_{\mu }$ itself is invariant. This reproduces the contravariant transformation of four-vectors, now with a general Jacobian in place of the constant Lorentz matrix.

One-forms and the dual space

Alongside each tangent space sits its dual, the space of linear maps from vectors to real numbers. Its elements are one-forms (covariant vectors). The prototypical one-form is the gradient of a function, $\mathbf{d}f$, whose action on a vector returns the directional derivative:

The coordinate differentials $\{\mathbf{d}{x}^{\mu }\}$ are the basis of the dual space, the dual basis, defined by the pairing

A general one-form is $\omega ={\omega }_{\mu }\mathbf{d}{x}^{\mu }$ with a lower index, and its components transform with the inverse Jacobian, opposite to a vector:

The pairing $\omega (V)={\omega }_{\mu }{V}^{\mu }$ is a number independent of chart, because the two Jacobians cancel. Vectors and one-forms are distinct species — one cannot be added to the other — until a metric is supplied to convert between them. A tensor of type $(k,l)$ is a multilinear map taking $k$ one-forms and $l$ vectors to a number, with components $T^{{\mu }_1\cdots {\mu }_k}{}_{{\nu }_1\cdots {\nu }_l}$ carrying $k$ upper and $l$ lower indices, each transforming with its own Jacobian factor. The metric is a $(0,2)$ tensor.

The metric tensor and the line element

Coordinates alone carry no notion of length: the numbers $x^{\mu }$ could be any smooth relabelling. The metric tensor $g_{\mu \nu }$ supplies the geometry, assigning an invariant squared length to every infinitesimal displacement through the line element

The metric is a symmetric $(0,2)$ tensor, $g_{\mu \nu }=g_{\nu \mu }$, nondegenerate (so its matrix is invertible), and in general relativity it has Lorentzian signature $(-,+,+,+)$: at any point a basis can be chosen in which $g_{\mu \nu }$ reduces to ${\eta }_{\mu \nu }=\mathrm{diag}(-1,+1,+1,+1)$. This is the metric statement of the equivalence principle — a local inertial frame is a chart in which $g_{\mu \nu }={\eta }_{\mu \nu }$ at the point and its first derivatives vanish. The inverse metric $g^{\mu \nu }$, defined by $g^{\mu \nu }{g}_{\nu \rho }={\delta }^{\mu }{}_{\rho }$, raises indices, and $g_{\mu \nu }$ lowers them, so the metric is the isomorphism between vectors and one-forms:

The invariant scalar product of two vectors is $g_{\mu \nu }{U}^{\mu }{V}^{\nu }=U^{\mu }{V}_{\mu }$, generalizing the Minkowski dot product to a position-dependent $g_{\mu \nu }$. Lengths of curves, angles between vectors, volumes (through the factor $\sqrt{|\det g|}$), and the classification of separations into timelike, null, and spacelike all descend from the metric.

Worked example: the 2-sphere

The surface of a sphere of radius $a$ is the canonical curved two-dimensional manifold. In the usual polar and azimuthal angles $(\theta ,\varphi )$, the induced line element is

The metric is diagonal but not constant: the coefficient of $\mathbf{d}{\varphi }^2$ shrinks toward the poles, capturing that circles of constant latitude get smaller as $\theta \to 0$ or $\pi$. Two features are worth separating. First, the intrinsic curvature is real — no coordinate change can flatten the sphere, as later lessons confirm by computing a nonzero Riemann tensor. Second, the apparent breakdown of the coordinates at $\theta =0,\pi$, where $\det g=a^4{\sin }^2\theta$ vanishes, is a coordinate singularity: the poles are perfectly ordinary points, but the chart $(\theta ,\varphi )$ fails there because longitude is undefined at a pole. A second chart, rotated so its poles lie elsewhere, covers the region smoothly. Distinguishing a coordinate singularity from a genuine one is a recurring theme; the Schwarzschild horizon is a coordinate singularity, while its centre is a real one.

Worked example: the Rindler chart

The uniformly accelerated frame of the previous lesson is a flat manifold in curved-looking coordinates, the reverse situation from the sphere. Its metric,

has a position-dependent $g_{\tau \tau }$ and vanishing $\det g$ at $\rho =0$. That zero is a coordinate singularity — the Rindler horizon — not a curvature singularity, since the underlying spacetime remains Minkowski. Computing the curvature (done systematically two lessons on) returns zero everywhere, confirming that a nonconstant metric does not by itself imply curvature. The metric matrix carries two kinds of information tangled together: the true geometry, and the choice of coordinates. Separating them requires the covariant derivative and the curvature tensor, developed next.

The manifold supplies the arena and the metric supplies lengths and times, but comparing vectors at different points — the operation every derivative of a vector field requires — has no meaning yet, because the tangent spaces at different events are distinct vector spaces. Supplying that comparison is the work of the connection and the covariant derivative, the subject of the next lesson.