Manifolds, maps, and metrics

4.1 A manifold is a place where local coordinates work#

Imagine finding a place on Earth using latitude and longitude. Two numbers locate a point on the surface, even though we draw the globe in three-dimensional space. The point needs two independent coordinates because we can move along the surface in two independent directions.

The labels have limits. At the north pole, all longitudes meet. A small map centered on that pole can instead use two directions across the map. The place is ordinary; our first set of labels was unsuitable there.

A manifold is a space that can be described this way: each point has a surrounding region with an ordinary coordinate list. The number of entries is its dimension. A sphere’s surface is two-dimensional; spacetime is four-dimensional, with one time coordinate and three space coordinates. We use a collection of overlapping maps when one set of coordinates cannot cover everything.

A coordinate chart is the rule assigning a coordinate list to each point in one such region. It must work both ways: each point has one list, and each list in the chart’s range identifies one point. Nearby points must have nearby lists, and vice versa. Symbolically,

Here $M$ is the manifold, $U$ is the region covered by this chart, and $x(U)$ is its set of coordinate lists. The notation $\mathbb R^n$ means lists of $n$ real numbers. In this formula $x$ names the whole chart, rather than just one horizontal coordinate.

An atlas is a collection of charts covering the space. Where two charts overlap, we can translate between their labels. Starting with one list, use the first chart’s inverse to find the point, then the second chart to label it:

The symbol $\cap$ means the common region of the two patches; $\circ$ means composition, applying the right-hand map first. This translation is a transition map. On a smooth manifold, the transition maps and their inverses can be differentiated repeatedly, with continuous derivatives of every order. That is what lets us use calculus consistently on overlapping maps.

A complete two-chart example. Describe the unit sphere temporarily by $X^2+Y^2+Z^2=1$ in ordinary three-dimensional space. Capital letters are only a convenient construction aid. The surface itself has two independent coordinates. Let $N=(0,0,1)$ and $S=(0,0,-1)$ be its poles.

On the sphere with $N$ removed, use the chart

Every finite pair $(u,v)$ corresponds to exactly one point of that patch. To see this without trusting a picture, put $s=u^2+v^2$ and write the inverse:

The squared components sum to one, and substituting them back into the chart returns $u$ and $v$. The denominator $1+s$ never vanishes. The north pole is approached only as the coordinate radius becomes unbounded; the south pole is the perfectly ordinary coordinate pair $(0,0)$.

The second chart removes $S$ instead:

It covers the missing north pole, where $(p,q)=(0,0)$. On the overlap, substitute the first chart’s inverse into the second chart. Since $1+Z=2s/(1+s)$, the transition is

The excluded origin represents $S$, which is outside the second chart. The transition applies only where both charts cover the point. Applying the same formula to $(p,q)$ returns $(u,v)$, so the transition has a smooth inverse everywhere on the overlap. Its Jacobian has determinant

The minus sign reverses the orientation of these coordinate lists; the nonzero value says no infinitesimal direction has been collapsed. Two overlapping charts therefore cover the whole sphere, even though neither chart does so alone. These coordinate lists locate points. To calculate distances between them, we still need a measuring rule.

As a numerical check, the first coordinates $(u,v)=(2,1)$ locate $(X,Y,Z)=(2/3,1/3,2/3)$ and give second coordinates $(p,q)=(2/5,1/5)$. Those are two addresses for one point. Chapter 2’s Jacobian rule tells us how a tangent’s components change between them; §4.3 will make the tangent itself precise.

A small globe and a globe twice its size can use the same latitude and longitude labels. The distance between two given labels doubles. This separates the two ingredients we need: the manifold and its charts locate points; a metric supplies the measuring rule.

We used an outside picture to construct the sphere’s charts, but the final coordinate translations work without it. In the same way, describing four-dimensional spacetime does not require an additional physical space surrounding it.

4.2 Where coordinates fail#

On an ordinary flat plane, polar coordinates describe a point by its distance $r$ from the origin and the angle $\theta$ from the positive $x$ axis. We measure the angle in radians: arc length divided by radius, so a full turn is $2\pi$. Before calculating, try changing the distance and angle separately. A change in distance takes you along a spoke; a change in angle takes you around a circle.

The horizontal and vertical sides of the triangle from the origin to P are $r\cos\theta$ and $r\sin\theta$. Thus the same point has Cartesian coordinates

At $r=0$, every value of $\theta$ labels the same point. Polar coordinates fail there. A Cartesian map still describes the origin normally.

Likewise, longitude fails to distinguish directions at a sphere’s poles. One can use another chart near a pole. The original labels were inadequate; the geometry is regular.

We will meet the same distinction at a black-hole horizon in Chapter 17. A coordinate expression can become infinite while a different chart describes the same location regularly. To establish a physical problem, we need more than the behavior of one set of labels.

A coordinate list is also different from a vector. Consider the simple relabeling $x'=x+10$. A point formerly labeled 2 is now labeled 12. But a displacement of 3 coordinate units is still a displacement of 3 units: the added 10 cancels when we subtract the endpoints. The point labels and displacement components follow different rules.

For nonlinear changes, even subtracting two widely separated coordinate lists does not generally produce a vector. Tangent vectors use the local Jacobian rule from Chapter 2. We now make their local meaning precise.

4.3 Tangent vectors live at events#

Take a curve through an event $p$, written $x^\mu(\lambda)$. Its tangent components are

We can identify this tangent by the rate of change it produces in a scalar field $f$ along the curve:

For example, take $f(x,y)=x+2y$ and a tangent with components $(3,4)$. Then $V[f]=3+2(4)=11$, the same vector–covector pairing we calculated in Chapter 2. Here we are viewing the vector as an instruction to differentiate a function in a specified direction. The brackets in $V[f]$ mean applying that instruction to $f$.

The operator satisfies linearity and the product rule,

All tangent vectors at $p$ form the tangent space $T_pM$. A coordinate basis is

Its dual basis is $dx^\mu|_p$, with $dx^\mu(e_\nu)=\delta^\mu{}_{\nu}$.

There is an additional question when we compare two different points. A vector at $p$ belongs to $T_pM$; a vector at a neighboring event $q$ belongs to $T_qM$. They are elements of different vector spaces. Subtracting them requires a rule for comparing those spaces.

On a flat Cartesian grid, an obvious translation rule is already operating in the background. On a general manifold there is no preferred rule supplied by the manifold alone. A connection will provide the missing comparison procedure. We will construct and use that rule in Chapters 6 and 7.

4.4 The metric is a local measuring operation#

In Chapter 3, the Minkowski metric turned a small displacement into an interval. Now allow the measuring rule to vary from place to place. We also give it two vector inputs, because this lets us describe their relation as well as each vector separately.

The ordinary dot product is a familiar example: $V\cdot V$ gives squared length and $V\cdot W=|V||W|\cos\theta$ relates two lengths to the angle between the arrows. If $W$ has unit length, the latter measures the component of $V$ along $W$. Expanding the square of $V+W$ gives

So the rule for squared lengths already determines a rule for pairs. The same identity works for a symmetric bilinear form even when its squared values can be negative. At every event, the metric is such a form on the tangent vectors:

It is symmetric,

and nondegenerate: no nonzero vector pairs to zero with every vector. Here orthogonal means $g(V,W)=0$, extending the zero-dot-product condition. In GR it has Lorentzian signature $(-,+,+,+)$. At a point, there is a basis in which its components are $\eta_{\mu\nu}$, with one negative and three positive eigenvalue directions. The number of positive and negative directions cannot be changed by a nonsingular real basis transformation.

Nondegenerate does not mean $g(V,V)$ is nonzero for every nonzero $V$. Null vectors have $g(V,V)=0$, but a nonzero null vector still has nonzero pairing with some other vectors. For example, with the two-dimensional matrix $\operatorname{diag}(-1,1)$, $V=(1,1)$ pairs to zero with itself but gives $g(V,W)=-2$ for $W=(1,-1)$. The matrix is invertible despite having nonzero vectors with zero self-pairing.

The line element

encodes the local geometry. For a timelike worldline, $d\tau=\sqrt{-ds^2}/c$. For a spacelike curve, its length is obtained from $\int\sqrt{ds^2}$. Defining a finite spatial distance between distant observers requires a specified spacelike slice or measurement protocol; the metric does not hand everyone the same universal “distance right now.”

The metric also defines the light cone by $g(V,V)=0$. The metric therefore tells us about clock readings and possible light signals as well as spatial lengths.

In four dimensions, a symmetric $4\times4$ metric has $4(4+1)/2=10$ independent components. Those are ten functions in a coordinate description, not ten independent propagating gravitational modes. Coordinate freedom and the structure of the field equations will substantially change that counting.

4.5 A complete metric calculation in flat polar coordinates#

Return to the angular-step view in the polar-coordinate experiment. Moving outward by a small amount changes your position by that same length. Turning through a small angle changes it by a length proportional to $r$: the little arc is longer on a larger circle. At the point where you stand, these radial and tangential directions are perpendicular.

This gives us a prediction. The two perpendicular legs of a very small displacement have lengths $dr$ and $r\,d\theta$, so Pythagoras should give $d\ell^2=dr^2+r^2d\theta^2$. The differentials describe the local limit; a finite straight chord need not equal a finite arc. We can now check the prediction by differentiating the coordinate formulas:

Square and add. The mixed terms cancel, and $\sin^2\theta+\cos^2\theta=1$ gives

Therefore the spatial metric is

The coefficient $r^2$ is necessary because one radian of angular change spans a longer arc farther from the origin. It is not evidence of curved space. We obtained the metric from the flat Euclidean plane by changing labels.

Units deserve attention. If $r$ is measured in meters and $\theta$ is dimensionless, $g_{rr}$ is dimensionless but $g_{\theta\theta}$ has units of square meters. It is the full line element that must have consistent units. The statement “the metric is dimensionless” is only true in suitable coordinate conventions.

Off-diagonal terms also have a simple meaning. On the same plane, choose oblique coordinates with position

Then

The cross term tells you the coordinate basis directions are not orthogonal. Its coefficient is $2g_{12}$ because the Einstein sum includes both $g_{12}dq^1dq^2$ and $g_{21}dq^2dq^1$.

4.6 Inverse metrics and raising or lowering indices#

The inverse metric is defined by

It is the matrix inverse, not component-by-component reciprocation except when the metric is diagonal. For our oblique metric,

Multiplying the two matrices verifies the identity directly.

The metric converts a vector into a covector:

The new covector asks another vector $W$ the question $V_\mu W^\mu=g(V,W)$. The inverse conversion is

For the oblique example, $V^i=(1,2)$ gives $V_i=(3,5)$, and

The corresponding ordinary Cartesian vector is $3\hat{\mathbf x}+2\hat{\mathbf y}$, whose Euclidean norm squared is indeed $9+4=13$. The components changed from $(1,2)$ to $(3,5)$ because the lowered object is a measuring rule. Applying it to the original vector gives the squared length in either description.

In inertial spacetime coordinates,

For example, $u_0=-\gamma c$ while $u^0=\gamma c$. Lowering the time index accounts for the minus sign in the energy contraction below.

The gradient distinction from Chapter 2 now resolves:

The differential was available before the metric. The gradient vector was not. In Lorentzian geometry it is also unsafe to carry over every Euclidean slogan about a gradient “pointing uphill most steeply,” because the unit-vector set and norm structure are different.

4.7 Measuring volume with a determinant#

On a plane, a narrow polar cell has physical area approximately $dr\times r\,d\theta$, so

For a small coordinate cell in more dimensions, the same idea applies. A matrix that converts coordinate components to components in a laboratory basis with metric matrix $\eta$ changes cell volumes by the absolute value of its determinant. The metric contains this conversion twice, once for each vector input, so its determinant contains the square of the volume factor. Let

With Lorentzian signature $(-,+,+,+)$, $g<0$ in every regular coordinate chart. Write $d^4x=dx^0dx^1dx^2dx^3$ for the product of the four small coordinate widths. This notation is not a fourth power of one coordinate. The invariant spacetime volume measure is

Why exactly this factor? Under $x\mapsto x'$, the covariant metric transforms as a matrix according to

where the first equation uses $g$ for the metric matrix and the second its determinant. Thus the square root picks up $|\det K|$. Meanwhile $d^4x'=|\det J|d^4x$, and $K=J^{-1}$. The factors cancel:

An oriented volume form additionally tracks the sign of orientation using a wedge product. The positive integration measure above uses absolute Jacobians. These are compatible viewpoints, but changing orientation is where their notation must be handled carefully. A formal treatment is available in Tong’s discussion of the metric volume form.

For a concrete four-dimensional example, keep $x^0=ct$ and describe space with spherical coordinates. The radius $r$ measures distance from the origin, $\theta$ measures angle down from the north axis, and $\phi$ measures angle around that axis:

Away from the origin and poles, the three coordinate directions are perpendicular. A radial step has length $dr$. A small change in $\theta$ has length $r\,d\theta$. A circle at fixed $r,\theta$ has radius $r\sin\theta$, so a small change in $\phi$ has length $r\sin\theta\,d\phi$. Squaring and adding these lengths, and including the time term, gives

so

The factor $r^2\sin\theta$ converts the two angular widths and the radial width into physical spatial volume. The factor $c\,dt$ supplies the fourth, time-directed width. The vanishing determinant at $r=0$ or at a polar axis signals failure of this chart there; it does not mean the regular Minkowski metric becomes physically degenerate.

4.8 Curvature, bending, and local frames#

A sheet can be rolled into a cylinder without stretching it. Distances and angles measured within a sufficiently small patch are unchanged, so its intrinsic curvature is zero. Its embedding in three-dimensional space is visibly bent. A flat sheet cannot smoothly cover a patch of a sphere while preserving all its lengths. The sphere’s intrinsic geometry differs from a plane’s.

Intrinsic curvature concerns measurements available to inhabitants of the geometry. Extrinsic curvature concerns how a chosen surface or other lower-dimensional space sits inside a larger geometry. Such an embedded space is called a submanifold. Both ideas occur in relativity: spacetime has intrinsic curvature, and a spatial slice can have extrinsic curvature within spacetime. They are different objects answering different questions.

Coordinate basis vectors need not be unit length or orthogonal. A physical laboratory instead likes an orthonormal frame $e_{\hat a}$ satisfying

Hats here label laboratory directions. For an observer of four-velocity $U$, choose $e_{\hat0}=U/c$. The other three frame vectors specify their instantaneous spatial axes.

On the polar plane, $g_{rr}=1$ and $g_{\theta\theta}=r^2$. Thus $\partial_r$ already has unit length, while $\partial_\theta$ has length $r$. Divide the latter by $r$ to obtain an orthonormal frame:

Thus $V^{\hat r}=V^r$ and $V^{\hat\theta}=rV^\theta$. The physical tangential component includes the conversion from angle to arc length.

There is a subtle distinction between making the metric’s components constant in a frame and finding coordinates in which they are constant throughout a neighborhood. The first is possible locally for a smooth Lorentzian metric. The second would make the neighborhood flat. A varying orthonormal frame generally cannot be the coordinate basis of one coordinate system; its axes can turn relative to one another from point to point. The connection keeps track of that turning.

At any ordinary event of a smooth spacetime, one can choose locally inertial coordinates with

This is stronger than merely choosing an orthonormal basis at the point, but it does not generally remove second-order departures from flatness. It is the mathematical doorway to the equivalence principle.