Schwarzschild Spacetime and Horizon-Penetrating Charts

Schwarzschild spacetime requires an atlas because no single chart simultaneously keeps the static exterior diagonal, crosses both horizon branches, and covers the maximal analytic extension. This record compares the commonly used charts for a four-dimensional vacuum black hole of mass $M\gt 0$, with $m=GM$, Schwarzschild radius $r_s=2m$, and areal radius $r\gt 0$.

Helpful background. Lorentzian Metrics and Causal Cones supplies causal classification and time orientation; Exact Solutions of Einstein’s Equations supplies the chart-domain and invariant-check method.

Static and horizon-penetrating forms

Set

$$f(r)=1-\frac{r_s}{r}, \qquad r_*=r+r_s\ln\left\lvert\frac{r}{r_s}-1\right\rvert.$$

The static line element is

$$ds^2=-f\,dt^2+f^{-1}dr^2+r^2d\Omega_2^2.$$

The exterior static chart has $r\gt r_s$. At $r=r_s$, its $g_{rr}$ component diverges although the geometry is regular. Advanced and retarded null times,

$$v=t+r_*, \qquad u=t-r_*,$$

give

$$ds^2=-f\,dv^2+2\,dv\,dr+r^2d\Omega_2^2$$

and

$$ds^2=-f\,du^2-2\,du\,dr+r^2d\Omega_2^2.$$

The first is regular across the future horizon into a black-hole interior; the second is regular across the past horizon into a white-hole interior. The radial metric block in either case has determinant $-1$, providing a direct nondegeneracy check at $r=r_s$.

The ingoing Painlevé–Gullstrand time and its line element are

$$dT=dt+\frac{\sqrt{r_s/r}}{f}\,dr, \qquad ds^2=-dT^2+\left(dr+\sqrt{\frac{r_s}{r}}\,dT\right)^2+r^2d\Omega_2^2.$$

Its radial block also has determinant $-1$. Introduce a comoving Lemaître coordinate by

$$dR=dt+\frac{dr}{f\sqrt{r_s/r}}, \qquad r=\left[\frac32\sqrt{r_s}(R-\tau)\right]^{2/3}, \qquad \tau=T.$$

Then

$$ds^2=-d\tau^2+\frac{r_s}{r}\,dR^2+r^2d\Omega_2^2,$$

with $R-\tau\gt 0$. The future black-hole horizon is the regular line $R-\tau=2r_s/3$, while $R-\tau=0$ is the future $r=0$ singularity in this ingoing patch.

For the exterior isotropic radius,

$$r=\rho\left(1+\frac{r_s}{4\rho}\right)^2,$$

and

$$ds^2 =-\left(\frac{1-r_s/(4\rho)}{1+r_s/(4\rho)}\right)^2dt^2 +\left(1+\frac{r_s}{4\rho}\right)^4 \left(d\rho^2+\rho^2d\Omega_2^2\right).$$

The branch $\rho\gt r_s/4$ covers one exterior. The inversion-related branch $0\lt\rho\lt r_s/4$ maps another exterior copy rather than the black-hole interior, and the transformation degenerates at $\rho=r_s/4$.

Chart	Definition and domain	What it displays; what it hides
Schwarzschild $(t,r)$	Static exterior $r\gt r_s$; the same component formula can be used inside only with the causal roles of $t$ and $r$ exchanged	Diagonal static metric; neither horizon branch is crossed
Ingoing Eddington–Finkelstein $(v,r)$	$v=t+r_*$; exterior plus black-hole interior	Future-horizon regular and stationary; past horizon not covered by this patch
Outgoing Eddington–Finkelstein $(u,r)$	$u=t-r_*$; exterior plus white-hole interior	Past-horizon regular and stationary; future horizon not covered by this patch
Ingoing Painlevé–Gullstrand $(T,r)$	$dT=dt+\sqrt{r_s/r}\,dr/f$	Future-horizon regular with intrinsically flat $T$ slices; adapted to infall from rest at infinity
Ingoing Lemaître $(\tau,R)$	$r=[3\sqrt{r_s}(R-\tau)/2]^{2/3}$	Comoving freely falling time; horizon regular but not manifestly stationary
Isotropic $(t,\rho)$	$r=\rho(1+r_s/4\rho)^2$, exterior $\rho\gt r_s/4$	Conformally flat spatial metric; approaches but does not cross the horizon
Kruskal–Szekeres $(U,V)$	Right-exterior normalization $-UV=(r/r_s-1)e^{r/r_s}$, analytically continued through four sign sectors with $r\gt 0$	Both horizon branches, bifurcation sphere, two exteriors, and both interiors; staticity is no longer component-diagonal

Reall compares the static, horizon-penetrating, and maximal-extension domains in Reall 2013, §§ 12.1–12.4, printed pp. 131–141, PDF; Bičák supplies an independent exact-solution comparison in Bičák 2000, § 2, printed pp. 19–30, PDF.

Kruskal sectors

In the right exterior choose

$$U=-e^{-u/(2r_s)}, \qquad V=e^{v/(2r_s)}.$$

Then

$$-UV=\left(\frac{r}{r_s}-1\right)e^{r/r_s},$$

and

$$ds^2 =-\frac{4r_s^3}{r}e^{-r/r_s}\,dU\,dV +r^2d\Omega_2^2.$$

The coefficient is finite and nonzero at $UV=0$, so $U=0$ and $V=0$ are regular null hypersurfaces. Region-dependent signs extend the same analytic expression into four sectors:

Sector	Signs	Region
I	$U\lt 0\lt V$	right asymptotically flat exterior
II	$U\gt 0$, $V\gt 0$	black-hole interior
III	$V\lt 0\lt U$	left asymptotically flat exterior
IV	$U\lt 0$, $V\lt 0$	white-hole interior

The normalized stationary Killing field becomes

$$\partial_t =\frac{1}{2r_s} \left(V\partial_V-U\partial_U\right).$$

It is timelike in Regions I and III, spacelike in II and IV, null on each horizon component, and zero at the bifurcation two-sphere $U=V=0$.

Choose the global future direction so that future radial causal curves have nondecreasing $U$ and $V$. With this choice $\partial_t$ is future-directed in Region I but past-directed in Region III; the future static time translation in the left exterior is $-\partial_t$. In Region II, increasing $U$ and $V$ leads toward $UV=1$ and the future singularity.

Invariant checks and limitations

Every chart describes the same local vacuum geometry on its overlap:

$$R_{\mu\nu}=0, \qquad R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} =\frac{48m^2}{r^6}.$$

The curvature invariant is finite at $r=r_s$ and diverges at $r=0$. Thus the horizon is a removable chart boundary of static coordinates, whereas $r=0$ is a curvature singularity. The determinant, inverse metric, or one invariant alone does not establish maximality; chart overlaps and extendible causal curves must also be checked.

The four Kruskal sectors describe the eternal maximal analytic vacuum solution. A collapse spacetime can share the late-time Schwarzschild exterior while lacking the white-hole region and second asymptotic exterior. For the complete exact-spacetime treatment, including the electrovac comparison, see Schwarzschild and Reissner–Nordström. See The Maximal Schwarzschild Extension for the collapse distinction and Maximally Extended Schwarzschild Diagram for its causal incidence.

References

• Bičák, Jiří. “The Role of Exact Solutions of Einstein’s Equations in the Developments of General Relativity and Astrophysics: Selected Themes.” In Einstein’s Field Equations and Their Physical Implications, edited by Bernd G. Schmidt, 1–126. Lecture Notes in Physics 540. Springer (2000), § 2, printed pp. 19–30. Open PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013), §§ 12.1–12.4, printed pp. 131–141. PDF.