Coordinate Form versus Local Isometry

Two very different line elements represent the same local spacetime when a diffeomorphism between open regions pulls one metric back to the other. This is stronger than a visual resemblance and weaker than global equality: the coordinate map must be invertible on a stated overlap, while topology, identifications, and maximal domains may still differ.

Required background. The Riemann Curvature Tensor supplies coordinate-independent curvature tests; Killing Fields and Isometry Groups supplies the definition of an isometry and its symmetry algebra; The Einstein Equation supplies the field-equation data that an equivalence must preserve.

Helpful background. Frames, Coframes, and Anholonomy makes it possible to compare curvature components after local Lorentz freedom is fixed.

Local equivalence is a pullback statement

Let $(U,g)$ and $(\widetilde U,\widetilde g)$ be open Lorentzian regions. They are locally isometric if there is a diffeomorphism $\psi:\widetilde U\to U$ such that

$$\widetilde g=\psi^*g, \qquad \widetilde g_{ab}(y) =\frac{\partial x^\mu}{\partial y^a} \frac{\partial x^\nu}{\partial y^b}g_{\mu\nu}(x(y)).$$

If matter fields are part of the solution, the same map must carry them as well: $\widetilde\Phi=\psi^*\Phi$, up to declared internal gauge transformations. A matching line element with a mismatched Maxwell field is not equivalence of Einstein–Maxwell solutions.

A passive coordinate change uses this formula on one geometric region. An active isometry maps points within or between regions while preserving the metric. The component algebra is identical, but the interpretation differs. In either case, a vanishing Jacobian or a map that is not one-to-one marks the limit of the claimed overlap.

Rindler form is still flat spacetime

In the right Minkowski wedge $X>|T|$, define

$$T=\rho\sinh(a\tau), \qquad X=\rho\cosh(a\tau), \qquad \rho>0,\quad \tau\in\mathbb R,$$

with $a>0$. Differentiation gives

$$-dT^2+dX^2=-a^2\rho^2d\tau^2+d\rho^2,$$

so four-dimensional Minkowski spacetime becomes

$$ds^2=-a^2\rho^2d\tau^2+d\rho^2+dy^2+dz^2.$$

The factor $g_{\tau\tau}\to0$ at $\rho=0$ does not signal curvature. The inverse map,

$$\rho=\sqrt{X^2-T^2}, \qquad a\tau=\operatorname{artanh}(T/X),$$

exists only in that wedge; $\rho=0$ is its null boundary. Direct calculation gives $R^\rho{}_{\sigma\mu\nu}=0$, independently confirming local equivalence. The “Rindler horizon” is an observer- and chart-domain boundary inside globally regular Minkowski spacetime, not a curvature singularity.

The Jacobian supplies a sharper domain check. Its determinant in the $(\tau,\rho)$ plane is $a\rho$, so the transformation is a local diffeomorphism precisely for $\rho>0$ and fails at the limiting null surface. This failure does not contradict regularity of the target coordinates $(T,X)$; it says only that the accelerated chart cannot be extended there with the same coordinate labels. Thus a component zero, a singular inverse map, and a geometric singularity are three logically distinct statements.

Screening before solving for the map

The pullback equations are nonlinear partial differential equations, so a direct search for $\psi$ is usually inefficient. Start with necessary conditions:

• same dimension, signature, field content, and coupling constants;
• same Ricci and Weyl algebraic types;
• compatible scalar-invariant relations;
• isomorphic local Killing algebras with the same orbit dimensions;
• compatible normalized curvature derivatives.

Any mismatch proves inequivalence. Agreement does not generally prove equivalence, because distinct vanishing-scalar-invariant spacetimes can share every scalar polynomial invariant. Mergulhão and Batista make this one-way logic explicit and then replace it with invariant frame fixing Mergulhão and Batista 2021, §§I–II, printed pp. 1–5, PDF. In four-dimensional Lorentzian geometry, the degenerate Kundt class is the systematic exceptional family for scalar characterization Coley, Hervik, and Pelavas 2009, §§7–9, printed pp. 21–36, PDF.

Local sameness does not fix the global spacetime

Consider the flat metric $-dt^2+dx^2$ on $\mathbb R^2$. Identifying $x\sim x+L$ produces a flat cylinder. Every sufficiently small neighborhood is isometric to a Minkowski neighborhood, yet the cylinder has closed spacelike loops in a nontrivial homotopy class. A boost identification instead produces a different quotient and may introduce causal pathology. Local curvature data cannot distinguish these global choices.

Likewise, one coordinate chart may cover only part of a larger spacetime. Showing that a static chart is locally de Sitter does not show that its coordinate domain is the complete de Sitter hyperboloid. Conversely, two globally different quotients may be locally indistinguishable. Bičák’s discussion of flat toroidal identifications is a concise example of this local/global separation Bičák 2000, §1.3, printed pp. 8–10, Open PDF.

Transfer exercise

Replace $\rho$ above by $\rho=e^\xi/a$. What metric results, and has the geometry changed?

Solution

Since $d\rho=(e^\xi/a)d\xi$ and $a^2\rho^2=e^{2\xi}$,

$$ds^2=e^{2\xi}(-d\tau^2+a^{-2}d\xi^2)+dy^2+dz^2.$$

The components now look conformally curved in the $(\tau,\xi)$ plane, but this is only another invertible coordinate change for $-\infty<\xi<\infty$. Composing it with the Rindler-to-Minkowski map proves that the Riemann tensor still vanishes. The limit $\xi\to-\infty$ corresponds to $\rho\to0$ and remains a chart boundary rather than a point of the chart.

References

• Bičák, Jiří. “Selected Solutions of Einstein’s Field Equations: Their Role in General Relativity and Astrophysics.” Einstein’s Field Equations and Their Physical Implications, Lecture Notes in Physics 540 (2000), 1–126. arXiv:gr-qc/0004016, §§1.3–1.5, printed pp. 8–16.
• Coley, Alan, Sigbjørn Hervik, and Nicos Pelavas. “Spacetimes Characterized by Their Scalar Curvature Invariants.” Classical and Quantum Gravity 26 (2009), 025013. doi:10.1088/0264-9381/26/2/025013.
• Mergulhão, Thiago M., and Carlos Batista. “On the Equivalence of Spacetimes, the Cartan–Karlhede Algorithm.” General Relativity and Gravitation 53 (2021), 28. doi:10.1007/s10714-021-02820-4.