Minkowski Spacetime and Its Coordinate Atlas

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Minkowski spacetime has no geometric horizon or curvature singularity, yet spherical, Rindler, rotating, and compactified coordinates all develop apparent boundaries. Each becomes harmless once its domain and coordinate map are stated: spherical axes are basis degeneracies, Rindler horizons bound an accelerated chart, the rotating light cylinder limits stationary observers, and conformal infinity is a boundary of a rescaled metric rather than a point of the physical spacetime.

Required background. The Riemann Curvature Tensor supplies the flatness test; The Einstein Equation identifies Minkowski as the $\Lambda=0$ vacuum reference solution.

Helpful background. Killing Fields and Isometry Groups distinguishes chart time vectors from global inertial symmetries.

Inertial and null coordinates

In four dimensions, inertial Cartesian coordinates cover $\mathbb R^4$ :

ds^2=-dt^2+dx^2+dy^2+dz^2, \qquad (t,x,y,z)\in\mathbb R^4.

The Riemann tensor vanishes and the ten Killing generators form the Poincaré algebra. Spherical coordinates $r\ge0$ , $0\le\theta\le\pi$ , $\phi\sim\phi+2\pi$ give

ds^2=-dt^2+dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2).

The determinant vanishes at $r=0$ and on the polar coordinate axes, but the Cartesian map is regular there after the usual angular redundancy is removed. These are not missing spacetime points.

With $u=t-r$ and $v=t+r$ ,

ds^2=-du dv+\frac{(v-u)^2}{4}d\Omega_2^2, \qquad v\ge u.

The equality $u=v$ is the regular spherical origin, while constant $u$ and $v$ are radial null hypersurfaces. The factor convention in $-du dv$ means $g_{uv}=g_{vu}=-1/2$ .

Rindler wedges

Retarded and advanced null coordinates are not a single smooth global chart at the spherical center because the angular coordinates collapse there. They remain excellent coordinates on radial orbit space and make outgoing versus ingoing null propagation explicit. Their restriction illustrates a recurring atlas rule: regularity of the four-geometry need not imply regularity of one radial chart combined with one angular chart.

On the right wedge $X>|T|$ , set

T=\rho\sinh(a\tau), \qquad X=\rho\cosh(a\tau), \qquad \rho>0.

Then

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

Worldlines of fixed $\rho$ have proper acceleration $1/\rho$ . The null surfaces $X=\pm T$ occur at the limiting coordinate value $\rho=0$ and bound what these uniformly accelerated observers can exchange signals with; they do not bound Minkowski spacetime. The coordinate pullback and $R_{abcd}=0$ are independent checks.

Rindler time is generated by the Lorentz-boost Killing field $X\partial_T+T\partial_X$ , scaled by $a$ . That generator exists throughout Minkowski spacetime, but it is timelike only in the right and left wedges, null on their boundaries, and spacelike in the future and past wedges. The horizon is tied to a Killing flow and an accelerated observer family, not to a failure of the metric. A freely falling inertial worldline crosses the null boundary in finite proper time without encountering a distinguished curvature event.

The left wedge requires a separate choice of signs in the transformation. The same positive $\rho$ and real $\tau$ cannot cover both wedges one-to-one. Stating “Rindler coordinates” without specifying the wedge loses precisely the domain information needed to interpret the horizon.

Null Cartesian coordinates expose a further subtlety. In the right wedge,

T-X=-\rho e^{-a\tau}, \qquad T+X=\rho e^{a\tau}.

Sending $\rho$ to zero at fixed $\tau$ reaches only the bifurcation two-plane $T=X=0$ (with arbitrary transverse coordinates). A generic point on either Rindler horizon requires a correlated limit in which $\tau$ tends to positive or negative infinity. Therefore the single coordinate value $\rho=0$ does not label the whole horizon one-to-one. This is why merely “adding the endpoint” cannot repair the Rindler chart; overlapping inertial or null charts are required.

Rotating coordinates and the light cylinder

Let inertial cylindrical angle be $\Phi=\phi+\Omega t$ . The metric becomes

ds^2=-(1-\Omega^2r^2)dt^2 +2\Omega r^2dt d\phi+dr^2+r^2d\phi^2+dz^2.

The chart remains locally nonsingular at $r=|\Omega|^{-1}$ : the $(t,\phi)$ block determinant is $-r^2$ . What changes sign is the norm of $\partial_t$ , the tangent to a fixed $(r,\phi,z)$ rotating worldline. Beyond the light cylinder that curve would be spacelike, so no material observer can remain at rest in the rotating grid. This is neither an event horizon nor a signature change of the spacetime.

The off-diagonal term is essential. Looking only at $g_{tt}$ could suggest that time and angle exchange roles at the light cylinder; the nonzero shift term shows that the full Lorentzian block remains regular. Inertial Cartesian coordinates cover both sides, and causal curves cross the cylinder freely. What fails is the attempted congruence of globally rigid rotation: tangential speed $|\Omega|r$ would exceed unity outside.

Conformal compactification

Choose an arbitrary length $L\gt0$ and set $U=\arctan(u/L)$ and $V=\arctan(v/L)$ . For $-\pi/2<U\le V<\pi/2$ ,

ds^2=\frac{L^2}{4\cos^2U\cos^2V} \left[-4dU dV+\sin^2(V-U)d\Omega_2^2\right].

The bracketed metric extends to finite coordinate boundaries after multiplying by the squared conformal factor $\Omega^2=4\cos^2U\cos^2V/L^2$ . Null infinity lies at $U=-\pi/2$ or $V=\pi/2$ in the rescaled geometry. The arbitrary $L$ makes the arguments of the arctangents dimensionless and changes no boundary type. The physical metric does not extend there with that same factor, and conformal compactification does not turn infinity into an ordinary finite-distance event. Bičák compares the distinct timelike, null, and spacelike infinities of Minkowski spacetime in Bičák 2000, §1.3, printed pp. 9–10, Open PDF.

Different limiting directions reach different pieces of the compactified boundary. Outgoing null curves keep $u$ finite while $v$ tends to positive infinity and approach future null infinity. Timelike geodesics with finite subluminal asymptotic velocity approach future timelike infinity, while spacelike directions at fixed time approach spatial infinity. A complete uniformly accelerated timelike curve can instead be asymptotic to open null infinity, so causal type at finite points does not determine the endpoint by itself. This separation is invariantly meaningful even though the finite labels belong to the auxiliary rescaled geometry.

Atlas discipline

Every chart in this page has the same vanishing curvature, but different coordinate ranges and observer congruences. To evaluate an apparent horizon, record the inverse map, its Jacobian, the region covered, the norm of the relevant Killing field, and causal curves crossing the boundary in another chart. Global quotients require an additional identification record: a flat cylinder or torus is locally Minkowski yet globally distinct.

No chart in this atlas changes the field equation. Nor does any chart manufacture acceleration of a geometric worldline; acceleration depends on the chosen congruence. Fixed Rindler position accelerates, fixed inertial Cartesian position does not, and fixed rotating position accelerates until that putative worldline becomes null at the light cylinder. This observer distinction is the physical content hidden behind superficially similar zeros of metric components.

The atlas also separates local and global flatness. Periodically identifying one inertial spatial coordinate produces a flat cylinder; identifying three produces a flat spatial torus. Every small curvature check still returns zero, but winding curves and the global isometry group change. A complete Minkowski claim therefore needs the underlying affine manifold and the absence or presence of identifications, not merely the constant component matrix.

One useful consistency test compares the chart boundary with geodesic behavior. An inertial geodesic crossing a Rindler horizon or the rotating light cylinder has smooth Cartesian coordinates and finite affine parameter, so neither surface signals spacetime incompleteness. By contrast, future null infinity is approached only at infinite affine parameter for the physical metric even though $U$ or $V$ reaches a finite endpoint after compactification. The affine parameter, not the visual position of a boundary on a coordinate diagram, distinguishes an artificial finite edge from an ideal conformal endpoint.

Check your understanding

At the rotating light cylinder $r=1/|\Omega|$ , show that the metric is nondegenerate and explain what becomes null.

Solution

The $(t,\phi)$ block is

\begin{pmatrix} -(1-\Omega^2r^2)&\Omega r^2\\ \Omega r^2&r^2 \end{pmatrix},

whose determinant is $-r^2$ for every $r>0$ . Thus the full metric is nondegenerate at $r=1/|\Omega|$ . The vector $\partial_t$ has norm $g_{tt}=-(1-\Omega^2r^2)$ , so it becomes null there. A different timelike direction still exists; only the rigidly rotating coordinate observers reach the speed of light.

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, printed pp. 8–10.