Metrics, Causal Cones, and Volume Geometry

A Lorentzian metric supplies much more than a formula for an interval. Point by point it identifies timelike, null, and spacelike directions; together with an orientation it supplies volume and Hodge duality; on submanifolds it may induce either nondegenerate or degenerate geometry. This chapter develops those structures while deliberately stopping before global causal hierarchy and before any gravitational field equation.

Required background. Tensor Fields and Natural Operations develops tensors and pullbacks.

Helpful background. Calculus and Linear Algebra for Relativity reviews bilinear forms and determinants; Exterior Algebra and Differential Forms develops forms and orientations; Events, Intervals, and Minkowski Geometry supplies the flat causal model.

From a metric to geometric structure

A pseudo-Riemannian metric is a smooth, symmetric, nondegenerate bilinear form $g_p$ on every tangent space. Nondegeneracy makes the musical map $v\mapsto g(v,\cdot)$ invertible. Signature is invariant under changes of basis, and Lorentzian index one produces a double null cone rather than a positive norm. Choosing one timelike component continuously defines a time orientation.

Several constructions then depend on additional choices. Orientation and metric together determine the volume form. Metric, orientation, dimension, form degree, and index together determine the Hodge star. Pulling the metric onto a submanifold gives a genuine intrinsic metric only when the restriction is nondegenerate. A null hypersurface is the central exception: its normal direction is tangent and lies in the kernel of the pullback.

Tong treats the metric, Lorentzian signature, volume, and local causal classification in Tong 2019, §3.1, pp. 91–99, PDF. The chapter translates every statement to signature $(-,+,+,+)$ and positive orientation $\epsilon_{0123}=+\sqrt{-g}$.

Six pages in dependency order

The pages are best read in dependency order because each adds one structure without silently assuming the next.

• Pseudo-Riemannian Metrics and Signature separates nondegeneracy from positivity, proves signature invariance, and constructs index raising and lowering.
• Lorentzian Causal Cones and Time Orientation classifies vectors and covectors and proves convexity of a chosen future cone, while remaining pointwise.
• Induced Metrics and Degenerate Restrictions gives the criterion $TS\cap(TS)^\perp=\{0\}$ and calculates the null-hypersurface kernel.
• The Hodge Dual in Lorentzian Signature derives $\star^2=(-1)^{p(n-p)+s}$ with dimension, degree, index, and orientation visible.
• Metric Volume, Divergence, and Integration derives the invariant density, coordinate divergence formula, and boundary-flux identity.
• Conformal Metrics and Preserved Causal Structure identifies the cone and null-path data that survive $g\mapsto\Omega^2g$, and the scales that do not.

Readers interested primarily in connections should complete the first two pages and then continue to Connections, Covariant Differentiation, and Parallel Transport. Readers working with differential forms should add the Hodge and volume pages. Readers preparing for hypersurface geometry should add the induced-metric page before proceeding.

Recurring checks and distinctions

Three tests recur throughout the chapter.

First, an invariant statement must survive a coordinate change. The metric determinant itself is not a scalar, but $\sqrt{\lvert g\rvert}\,d^nx$ is an invariant density. Signature survives congruence even when diagonal entries change.

Second, null does not mean zero or degenerate. A nonzero vector may have zero Lorentzian norm while the ambient metric remains nondegenerate. Degeneracy of an induced metric is a different statement: it means a nonzero tangent vector pairs to zero with every tangent vector.

Third, conformal preservation must be stated object by object. Positive rescaling preserves causal classes and unparametrized null geodesics, but it changes proper lengths, volume by $\Omega^n$, the Hodge star on $p$-forms by $\Omega^{n-2p}$, and null affine parameters. Wendl’s systematic treatment of metrics, integration, and related constructions appears in Wendl 2021–2022, §§9–13 and §§22–24, pp. 69–118 and 184–211, PDF.

Standard diagnostic

Consider four-dimensional Minkowski space in coordinates $(u,v,x,y)$ with

$$ds^2=-2\,du\,dv+dx^2+dy^2$$

and positive orientation $du\wedge dv\wedge dx\wedge dy$. Classify $\partial_u$ and $\partial_u+\partial_v$, find the induced metric on $u=0$, and state what changes under $\widetilde g=\Omega^2g$ for $\Omega>0$.

Solution

The squared norm of $\partial_u$ is zero, so it is a nonzero null vector. Since $g(\partial_u+\partial_v,\partial_u+\partial_v)=-2$, the sum is timelike. On $u=0$, tangent vectors are spanned by $\partial_v,\partial_x,\partial_y$, and the pullback is

$$h=dx^2+dy^2.$$

Its kernel is $\operatorname{span}\{\partial_v\}$, even though the ambient metric has determinant $-1$ and is nondegenerate. Under $\widetilde g=\Omega^2g$, both causal classifications and this null kernel are unchanged. In four dimensions the volume form scales by $\Omega^4$; proper timelike and spacelike lengths scale locally by $\Omega$; null affine parameters generally change.

What this chapter enables

After an appropriate route, a reader can diagnose signature without relying on coordinate diagonals, classify causal directions, distinguish ambient from induced degeneracy, compute Lorentzian Hodge signs, construct invariant volume and divergence, and state precisely what conformal geometry preserves.

A productive way to review these outcomes is to use one object in several roles. For the null plane $u=0$ in Minkowski space, classify its normal covector, raise that covector, find the pullback kernel, dualize a basis two-form, and then repeat the classification after a nonconstant positive conformal rescaling. The answers should distinguish preserved lines from changed normalizations and preserved cone data from changed volume. If one calculation labels the ambient metric degenerate or produces an ordinary unit null normal, revisit the induced-metric page before moving on.

The chapter does not decide whether separated events are causally related, whether causal curves close, or whether a spacetime is globally hyperbolic. It also does not interpret curvature dynamically. Its next task is more basic: define how tensors at neighboring points can be differentiated and compared. That is the role of the connection chapter.

References

• Tong, David. General Relativity. University of Cambridge lecture notes, 2019, §3.1, pp. 91–99, and §4.4.1, pp. 168–170. Official course PDF.
• Wendl, Chris. Differential Geometry I and II. Humboldt-Universität zu Berlin lecture notes, 2021–2022, §§9–13, pp. 69–118, and §§22–24, pp. 184–211. Course PDF.