Pseudo-Riemannian Metrics and Signature

A pseudo-Riemannian metric is a smoothly varying, symmetric, nondegenerate bilinear form on each tangent space. Nondegeneracy—not positivity—is the indispensable hypothesis: it makes index raising and lowering reversible. In Lorentzian signature the metric still measures invariant quadratic data and supplies orthogonality, contractions, and a volume density, but it also admits nonzero vectors of zero norm. That single difference produces causal cones and invalidates several familiar Euclidean inequalities.

Required background. Tensor Fields and Natural Operations explains how a tensor is distinguished from its coordinate components and how tensors transform.

Helpful background. Calculus and Linear Algebra for Relativity supplies bilinear forms and diagonalization; Exterior Algebra and Differential Forms supplies forms and orientation; Events, Intervals, and Minkowski Geometry supplies the flat-spacetime causal model.

Nondegenerate metric fields

On a smooth $n$-manifold $M$, a pseudo-Riemannian metric is a section

$$g\in\Gamma(S^2T^*M)$$

such that $g_p:T_pM\times T_pM\to\mathbb R$ is nondegenerate at every $p$. Equivalently, if $g_p(v,w)=0$ for every $w$, then $v=0$. In coordinates,

$$g=g_{\mu\nu}\,dx^\mu\otimes dx^\nu, \qquad g_{\mu\nu}=g_{\nu\mu}, \qquad \det(g_{\mu\nu})\ne0.$$

The components change from chart to chart, whereas nondegeneracy and the numbers of positive and negative directions do not. Sylvester’s law of inertia gives a basis in which $g_p$ is diagonal with $r$ negative and $s$ positive entries, $r+s=n$. The pair $(r,s)$ is the signature; the number $r$ is the index in the convention used here. A Lorentzian metric in four dimensions has signature $(-,+,+,+)$ and index one. Tong develops this distinction between a metric field and its coordinate matrix in Tong 2019, §3.1, pp. 91–99, PDF.

Nondegeneracy defines the musical isomorphisms

$$\flat:T_pM\longrightarrow T_p^*M, \quad v^\flat=g(v,\cdot), \qquad \sharp=(\flat)^{-1}.$$

Thus $v_\mu=g_{\mu\nu}v^\nu$ and $\alpha^\mu=g^{\mu\nu}\alpha_\nu$, with $g^{\mu\rho}g_{\rho\nu}=\delta^\mu{}_\nu$. This construction survives every signature. It fails for a degenerate bilinear form because its matrix has no inverse and a nonzero kernel vector is sent to the zero covector.

What positivity adds

A Riemannian metric is positive definite: $g(v,v)>0$ for every nonzero $v$. It therefore defines the norm $\lVert v\rVert=\sqrt{g(v,v)}$, a positive distance after minimizing curve lengths, and the usual Cauchy–Schwarz inequality. A general pseudo-Riemannian metric does none of these by itself. For the Minkowski matrix

$$\eta=\operatorname{diag}(-1,1,1,1),$$

the nonzero vector $k=(1,1,0,0)$ satisfies $\eta(k,k)=0$. The vectors $u=(1,0,0,0)$ and $x=(0,1,0,0)$ have squared norms $-1$ and $+1$, respectively. No real-valued norm can be obtained by taking a square root of $g(v,v)$ on all tangent vectors.

What remains is substantial. The scalar $g(v,w)$ is coordinate independent; orthogonal complements, tensor contractions, traces, and the identification of vectors with covectors are well defined. With an orientation, the metric also determines a volume form. With Lorentzian index one, the sign of $g(v,v)$ separates timelike, null, and spacelike vectors. Positive definiteness is therefore an additional geometric property, not part of the definition of a metric used in relativity.

Signature and coordinate checks

Under a coordinate change with Jacobian $J^\mu{}_{\mu'}=\partial x^\mu/\partial x^{\mu'}$, the metric matrix transforms by congruence,

$$g'=J^{\mathsf T}gJ.$$

Consequently $\det g'=(\det J)^2\det g$. Its numerical determinant is not a scalar, but its nonvanishing and sign are chart independent. Congruence also preserves inertia, so no regular coordinate transformation can turn a Lorentzian metric into a positive-definite one.

As a discriminating example, consider on a two-dimensional tangent space

$$g=-2\,du\,dv.$$

Its matrix in the ordered coframe $(du,dv)$ is $\begin{pmatrix}0&-1\\-1&0\end{pmatrix}$. The eigenvalues are $+1$ and $-1$, so the form is nondegenerate and Lorentzian even though both coordinate basis vectors are null. This shows why inspecting only diagonal entries is unreliable. A change to $t=(u+v)/\sqrt2$ and $x=(u-v)/\sqrt2$ gives $g=-dt^2+dx^2$, displaying the invariant signature directly.

The corresponding general linear-algebra construction and the metric-induced maps are treated in Wendl 2021–2022, §18.5, pp. 154–159, PDF. Wendl orders the signature as positive directions followed by negative directions; here the index counts negative directions. Translating that ordering leaves nondegeneracy and the musical maps unchanged.

Local content and limitations

Signature is locally constant wherever the metric remains smooth and nondegenerate. On each connected component it is therefore constant. A family of matrices can change signature only by crossing $\det g=0$, where it ceases to be a pseudo-Riemannian metric. This is a useful failure test in coordinate ansätze.

The metric at one point does not decide whether two distant events can be joined by a causal curve, whether a time orientation exists globally, or whether geodesics are complete. Those are global questions. Nor does the metric alone specify gravitational dynamics: the Einstein equation is not a definition of Lorentzian geometry. The next page develops only the pointwise cone and time-orientation structures supplied by index one.

Exercises

For

$$G_a=\begin{pmatrix}a&1\\1&0\end{pmatrix},$$

determine for which real $a$ the form is nondegenerate and find its signature. Explain why the answer cannot be read from the sign of $a$.

Solution

The determinant is $-1$ for every $a$, so the form is always nondegenerate. The product of its two real eigenvalues equals $-1$, hence one eigenvalue is positive and the other negative. Its signature is therefore $(-,+)$ for all $a$. In particular, $a=0$ makes both coordinate basis vectors null while the form remains nondegenerate; diagonal entries depend on the chosen basis and do not determine inertia.

References

• Tong, David. General Relativity. University of Cambridge lecture notes, 2019, §3.1, pp. 91–99. Official course PDF.
• Wendl, Chris. Differential Geometry I and II. Humboldt-Universität zu Berlin lecture notes, 2021–2022, §18.5, pp. 154–159. Course PDF.