Schwarzschild and Reissner–Nordström

Schwarzschild and Reissner–Nordström are the reference spacetimes for learning how an exact solution becomes a geometric object rather than merely a formula. Under stated symmetry and field-equation hypotheses one first derives a local metric-field family. One must then determine where its coordinates work, which apparent boundaries are regular, how local patches join, which invariants diagnose singular behavior, and which limits are actually well-defined. The extremal Majumdar–Papapetrou family completes the chapter by showing that the same Einstein–Maxwell equations admit genuine multi-centre equilibrium.

Helpful background. The Einstein Equation supplies the gravitational field equation; Maxwell’s Equations in Covariant Form explains the conserved electric and magnetic fluxes; The Riemann Curvature Tensor supplies invariant diagnostics; Exact Solutions of Einstein’s Equations introduces the method for specifying a spacetime and its matter fields; Causality and Global Structure develops extension language; and Affine and Variational Geodesics supports horizon-crossing checks.

The metric families and the questions they leave open

All pages use four-dimensional Einstein or Einstein–Maxwell theory with $\Lambda=0$, signature $(-,+,+,+)$, and explicit $G$. For the spherical nonconstant-radius families, vacuum Birkhoff and electrovac Birkhoff supply the additional hypersurface-orthogonal Killing field. After its time parameter is normalized at asymptotic infinity, a diagonal areal-radius chart adapted to that Killing field has the familiar line element

$$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_2^2, \qquad r>0.$$

Here $r$ is the areal radius: every symmetry sphere has area $4\pi r^2$. Schwarzschild and dyonic Reissner–Nordström use

$$f_{\mathrm{Schw}}(r)=1-\frac{2GM}{r}, \qquad f_{\mathrm{RN}}(r)=1-\frac{2GM}{r} +\frac{G(Q_e^2+Q_m^2)}{r^2}.$$

The mass $M$ is physical mass, while $Q_e$ and $Q_m$ are Gaussian flux charges. It is often cleaner to define the geometric parameters

$$m=GM, \qquad q^2=G(Q_e^2+Q_m^2),$$

so that $f_{\mathrm{RN}}=1-2m/r+q^2/r^2$. The diagonal expression itself is valid only where $f\ne0$; it is static on components where $f>0$, while $t$ and $r$ exchange their timelike and spacelike roles where $f<0$. It must not be used to decide whether a root of $f$ is singular. The black-hole root classification in this overview assumes $M>0$. The derivation page keeps the zero- and negative-mass Schwarzschild branches logically separate.

Family	Exact data	Characteristic form	Main geometric question
Schwarzschild	ADM mass $M$ after asymptotic time normalization	$f=1-2m/r$	Is $r=2m$ a chart boundary or a curvature singularity?
Dyonic Reissner–Nordström	$M$, $Q_e$, and $Q_m$	$f=1-2m/r+q^2/r^2$	How do two roots, one double root, or no real roots change the extension?
Extremal Reissner–Nordström	$q^2=m^2$	$f=(1-m/r)^2$	Why does a degenerate horizon require its own extension and near-horizon scaling?
Majumdar–Papapetrou	positive $M_i$, positions $\mathbf{x}_i$, and one common charge sign	$U=1+\sum_i GM_i/\lvert\mathbf{x}-\mathbf{x}_i\rvert$	How can several positive poles become regular degenerate horizon ends without material struts or charged matter sources in the electrovac exterior?

For Majumdar–Papapetrou, $Q_i=\sigma\sqrt{G}\,M_i$ with one $\sigma=\pm1$, and $U$ must remain positive on the punctured Euclidean base. A negative residue or a mixture of charge signs is not another regular member of the same equilibrium class. This is the purely electric representative used here; one common electromagnetic duality rotation gives aligned dyonic representatives.

Other sources often write $G=1$ or absorb $G$ and $4\pi$ into charge. Comparing formulas before translating the action, flux definition, geometric charge, and time normalization is a reliable way to manufacture false disagreements. The Black-Hole Geometry mechanics pages use rationalized charge. Apply the complete Gaussian-to-rationalized Maxwell map before comparing their $Q$, $A$, $\Phi_H$, or geometric charge parameter with this chapter.

Four distinctions that prevent common mistakes

A root is not a curvature diagnosis. At a positive root of $f$, the diagonal coefficient $g_{rr}=f^{-1}$ diverges. A horizon-regular null chart can nevertheless have a nondegenerate metric there, while scalar curvature invariants remain finite. The physical centre $r=0$ is a different question.

Crossing one horizon is not constructing the maximal extension. Relative to a chosen exterior, advanced coordinates cross the adjoining future outer horizon and retarded coordinates cross the past outer horizon. In charged spacetime, the role of an analogous chart at an inner Cauchy horizon must be read from its domain and analytic block. Kruskal-type charts organize more branches, but their construction and domain depend on whether a root is simple or degenerate.

An extremal parameter value is not a near-horizon limit. Setting $q^2=m^2$ produces the extremal Reissner–Nordström solution. Obtaining its local Bertotti–Robinson, or Poincaré-patch $\mathrm{AdS}_2\times S^2$, throat additionally requires a correlated radial zoom and inverse time rescaling; substituting $r=m$ into the old chart loses the geometry one is trying to resolve. The scaling is displayed explicitly in Bardeen and Horowitz 1999, §4, printed p. 12, Eqs. (4.1)–(4.2), Open PDF.

An eternal exact solution is not a formation or stability theorem. An analytic extension describes the ideal solution allowed by the equations. It does not by itself show that generic collapse forms that extension or that an inner horizon or multi-centre equilibrium survives perturbations.

Reall develops the Schwarzschild chart and extension sequence in Reall 2013, §12.1, printed pp. 131–133, and §§12.5–12.6, pp. 141–148, Open PDF, while Bičák compares the Schwarzschild, charged, extremal, and multi-centre families in Bičák 2000, §2, printed pp. 19–30; §§3.1–3.2, pp. 31–39; and §5, pp. 52–54, Open PDF.

Routes through the chapter

For a calculation-first route, begin with deriving the Schwarzschild metric. The reduced equations show exactly which integration constant is mass and which only rescales time. Continue to the Schwarzschild coordinate atlas to transform among static, null, infalling, isotropic, and Kruskal coordinates, then repeat the field-equation calculation with charge on the Reissner–Nordström solution.

For an extension-first route, start with the coordinate atlas and proceed to the maximal Schwarzschild extension. The contrast between a regular future-horizon chart and a four-region Kruskal chart prepares the more intricate Reissner–Nordström horizons and extensions, where two simple roots merge into a double root at extremality.

For an invariant-and-limits route, use invariants, singularities, and controlled limits to compare the Kretschmann, Ricci, Weyl, and electromagnetic scalars. Its extremal near-horizon scaling leads locally to a Poincaré patch of $\mathrm{AdS}_2\times S^2$. Finish with Majumdar–Papapetrou multi-centre equilibria, where a positive harmonic function replaces the radial metric function and every positive isolated pole becomes a degenerate horizon end.

What each calculation establishes

The seven pages form one chain rather than seven independent metric summaries:

1. Derivation: static spherical vacuum reduction makes the physical spherical mass function constant and equal to $M$; asymptotic normalization removes the remaining time-rescaling constant.
2. Atlas: a coordinate transformation is accepted only with its domain, selected horizon branch, and inverse-metric regularity.
3. Schwarzschild extension: the connected Kruskal extension contains a second exterior and black- and white-hole interiors, while $r=0$ remains singular.
4. Electrovac field: Maxwell flux gives the $q^2/r^2$ term and, for $q\ne0$, a nonzero Ricci tensor even though $R=0$.
5. Charged extension: the sign of $m^2-q^2$ distinguishes subextremal, extremal, and superextremal geometries.
6. Invariant comparison: a regular horizon-crossing chart together with finite scalars at each positive root separates those Killing horizons from the central curvature blowup of every nontrivial Schwarzschild or Reissner–Nordström member. The joint $M=q=0$ limit is Minkowski spacetime and fills in a regular centre.
7. Multi-centre balance: $U=1+\sum_iGM_i/\lvert\mathbf{x}-\mathbf{x}_i\rvert$ is harmonic away from its poles, and the nonlinear equations reduce consistently because every centre obeys the aligned extremal ratio $Q_i=\sigma\sqrt G M_i$ with one common sign $\sigma$.

The chapter stops at exact geometry. Horizon mechanics belongs to Black-Hole Geometry; perturbations, mass inflation, and collapse from initial data belong to Black-Hole Dynamics and Mathematical Relativity; observational use requires Black-Hole Astrophysics.

Review the chapter

Coordinate diagnosis. A student points to the divergence of $g_{rr}=f^{-1}$ at a simple root and calls it a curvature singularity. Give two independent corrections.

Solution

First transform to advanced time $v=t+r_*$: the radial block becomes $-f\,dv^2+2\,dv\,dr$ with determinant $-1$, so the metric is regular. At the root, $g^{rr}=f=0$, so its normal $dr$ is null, while the Killing generator $\partial_v$ also becomes null. This identifies a smooth local Killing horizon. Second compute a scalar such as the Kretschmann invariant; it is finite at every positive root. These checks defeat the coordinate-component inference, but calling that Killing horizon an event horizon additionally requires the global causal structure.

Parameter calculation. Assume $M>0$ in Reissner–Nordström spacetime. Determine the root regimes and recover Schwarzschild.

Solution

Writing $Q^2=Q_e^2+Q_m^2$, the numerator equation $r^2-2GMr+GQ^2=0$ gives $r_\pm=GM\pm\sqrt{G^2M^2-GQ^2}$. There are two simple positive roots for $0<Q^2<GM^2$, one double root for $Q^2=GM^2$, and no real roots for $Q^2>GM^2$. At $Q=0$, only $r_+=2GM$ is a horizon. The formal value $r_-=0$ comes from multiplying the lapse equation by $r^2$ and lies at the excluded Schwarzschild singular endpoint. At fixed $r>0$ in a common chart, the metric converges to Schwarzschild as $Q\to0$.

Limit calculation. Why does setting $r=m$ in extremal Reissner–Nordström fail to produce the throat? Exhibit the required metric scaling.

Solution

For $q^2=m^2$, the lapse is $f=(1-m/r)^2$. Let $\lambda>0$ be dimensionless and introduce

$$r=m(1+\lambda y), \qquad t=\frac{m\tau}{\lambda},$$

holding the dimensionless coordinates $(\tau,y)$ fixed with $y>0$ as $\lambda\to0^+$. Then

$$f=\frac{\lambda^2y^2}{(1+\lambda y)^2}, \qquad dr=m\lambda\,dy,$$

so the metric tends to

$$ds^2\longrightarrow m^2\left( -y^2d\tau^2+\frac{dy^2}{y^2}+d\Omega_2^2 \right).$$

This is the exterior Poincaré patch of the locally $\mathrm{AdS}_2\times S^2$ Bertotti–Robinson throat, with equal curvature radii $m$. Directly imposing $r=m$ instead sets $dr=0$ in the old coordinates and produces no nondegenerate two-dimensional limit. When $Q_e\ne0$, a regular electric potential requires its accompanying gauge shift, developed on the controlled-limits page.

Transfer task. Why is the statement “several extremal centres remain at rest because forces cancel” incomplete?

Solution

The Newtonian force comparison checks only a weak-field limit. The exact result requires one positive harmonic function to solve both Maxwell’s equation and the nonlinear Einstein constraints, positive isolated residues, and extremal flux-to-mass ratios with one common sign. Away from the horizon ends the solution contains no supporting material source; in a collinear axisymmetric subclass, elementary flatness also rules out a conical strut. This proves a static electrovac geometry, not its formation or stability.

References

• Bardeen, James M., and Gary T. Horowitz. “The Extreme Kerr Throat Geometry: A Vacuum Analog of $\mathrm{AdS}_2\times S^2$.” Physical Review D 60, no. 10 (1999): 104030. DOI. Open PDF.
• Bičák, Jiří. “Selected Solutions of Einstein’s Field Equations: Their Role in General Relativity and Astrophysics.” In Einstein’s Field Equations and Their Physical Implications, edited by Bernd G. Schmidt, Lecture Notes in Physics 540, 1–126. Springer, 2000. DOI. Open PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes, 2013. Open PDF.