Scalar-Supported Exact Spacetimes

Scalar fields support exact static geometries in ways that perfect fluids do not. A minimally coupled massless scalar can turn the would-be Schwarzschild horizon into a naked curvature singularity; a conformally coupled scalar can leave a familiar black-hole metric while the scalar itself diverges on a horizon; a tuned self-interaction can add cosmological horizons. The metric, scalar action, field profile, and domain must therefore be checked together.

Required background. Hilbert Stress–Energy from Metric Variation supplies stress tensors for metric-coupled fields.

Helpful background. The Einstein Equation fixes the gravitational equations; The Riemann Curvature Tensor supplies invariant singularity tests; Exact Solutions of Einstein’s Equations places these families; GR Formulations develops their initial-value setting; and Continuum Kinematics and Balance Laws provides a comparison with material sources.

This page uses geometrized units $G=c=1$, including in the displayed actions. Every mass symbol is a length, $M_{\mathrm{geom}}=G M_{\mathrm{phys}}$; thus a factor $1-2M/r$ would restore to $1-2G M_{\mathrm{phys}}/r$.

Minimally coupled massless scalar: the JNW family

Take

$$S=\int d^4x\sqrt{-g} \left[\frac{R}{16\pi}-\frac12(\nabla\phi)^2\right].$$

The equations are $R_{ab}=8\pi\nabla_a\phi\nabla_b\phi$ and $\Box\phi=0$. A static spherical solution is

$$\begin{aligned} ds^2={}&-f(r)^\nu dt^2+f(r)^{-\nu}dr^2 +r^2f(r)^{1-\nu}d\Omega^2,\\ \phi={}&\phi_0+\frac{\sqrt{1-\nu^2}}{\sqrt{16\pi}}\ln f(r), \qquad f(r)=1-\frac{b}{r}, \end{aligned}$$

with $b>0$, $0<\nu\le1$, and domain $r>b$. Its ADM mass is $M=\nu b/2$. The coordinate $r$ is not areal radius; the symmetry spheres have radius

$$R_{\mathrm{area}}=r f^{(1-\nu)/2}.$$

When $\nu=1$, the scalar is constant, $b=2M$, and the metric is Schwarzschild. For every $0<\nu<1$, $R_{\mathrm{area}}\to0$ and $\phi$ diverges as $r\to b^+$. Moreover

$$R=8\pi(\nabla\phi)^2 =\frac{(1-\nu^2)b^2}{2r^4}f^{\nu-2},$$

which diverges because $\nu-2<0$. Thus the zero of $g_{tt}$ at $r=b$ is not a regular horizon. It is a curvature singularity. Janis, Newman, and Winicour identified this scalar-field generalization and its Schwarzschild limit in Janis, Newman, and Winicour 1968, pp. 878–880.

The family is often written with a scalar-charge parameter instead of $\nu$. Such parameters are convention-dependent; the invariant information is the asymptotic scalar coefficient together with the ADM mass. State the scalar kinetic normalization before translating formulas.

Conformal coupling and a singular horizon field

Now consider the conformally coupled action

$$S=\int d^4x\sqrt{-g} \left[ \frac{R}{16\pi}-\frac12(\nabla\phi)^2 -\frac{1}{12}R\phi^2 \right].$$

The Bocharova–Bronnikov–Melnikov–Bekenstein solution has

$$ds^2=-\left(1-\frac{M}{r}\right)^2dt^2 +\left(1-\frac{M}{r}\right)^{-2}dr^2+r^2d\Omega^2,$$ $$\phi=\sqrt{\frac{3}{4\pi}}\frac{M}{r-M}.$$

The metric is exactly the extremal Reissner–Nordström geometry even though no Maxwell field is present, but the scalar diverges at $r=M$. Polynomial curvature invariants of the metric remain finite there. The example shows why “regular metric horizon” and “regular solution of the full matter–gravity theory” are different assertions. The effective gravitational coupling multiplying the Einstein tensor also becomes delicate when $1-(4\pi/3)\phi^2$ crosses zero. Bekenstein’s derivation and discussion of the conformal stress tensor appear in Bekenstein 1974, pp. 535–547.

A self-interacting de Sitter example

Add a cosmological constant and a quartic potential,

$$S=\int d^4x\sqrt{-g} \left[ \frac{R-2\Lambda}{16\pi}-\frac12(\nabla\phi)^2 -\frac{1}{12}R\phi^2-\alpha\phi^4 \right].$$

For the tuned coupling $\alpha=-2\pi\Lambda/9$, a neutral exact solution retains the same scalar profile but has

$$ds^2=-F(r)dt^2+\frac{dr^2}{F(r)}+r^2d\Omega^2, \qquad F(r)=\left(1-\frac{M}{r}\right)^2-\frac{\Lambda r^2}{3}.$$

Depending on $M\sqrt{\Lambda}$, $F$ can have black-hole and cosmological roots. The scalar pole remains at $r=M$, so a physically admissible static domain must be identified by comparing that pole with the ordered roots of $F$. The solution is not obtained by freely adding $-\Lambda r^2/3$ to the metric: the quartic coupling is fixed by the scalar equation and Einstein equation together Martínez, Troncoso, and Zanelli 2003, §§ II–III.

Checks and limits

For any scalar-supported claim, perform four separate checks: vary the stated action to fix the improved stress tensor; verify the scalar equation as well as Einstein’s equation; locate candidate horizons using the metric; and test the scalar field and effective coupling there. A Schwarzschild-looking limit is insufficient if the scalar or action becomes singular.

The JNW limit $\nu\to1$ removes scalar charge continuously. The conformal solution instead has a metric that already equals a familiar electrovac geometry, so its discriminating check is the nonminimal stress tensor rather than the metric alone. With self-interaction, $\Lambda\to0$ must be taken together with $\alpha\to0$ under the tuning above. These are inequivalent mechanisms, and they should not be grouped as a single notion of “scalar hair.”

Exercises

Show that the JNW singular surface shrinks to zero areal radius for nonzero scalar charge but becomes the Schwarzschild horizon in the scalar-free limit.

Solution

For $0<\nu<1$,

$$R_{\mathrm{area}}=r\left(1-\frac br\right)^{(1-\nu)/2} \longrightarrow0$$

as $r\to b^+$ because the exponent is positive. The Ricci scalar displayed above diverges, confirming a central singularity. At $\nu=1$, however, the exponent vanishes, $R_{\mathrm{area}}=r$, the scalar coefficient is zero, and $b=2M$; $r=b$ is then the regular Schwarzschild Killing horizon. The limits $r\to b$ and $\nu\to1$ are therefore not interchangeable.

References

• Jacob D. Bekenstein, “Exact Solutions of Einstein-Conformal Scalar Equations,” Annals of Physics 82, 535–547 (1974). DOI.
• Allen I. Janis, Ezra T. Newman, and Jeffrey Winicour, “Reality of the Schwarzschild Singularity,” Physical Review Letters 20, 878–880 (1968). DOI.
• Cristian Martínez, Ricardo Troncoso, and Jorge Zanelli, “de Sitter Black Hole with a Conformally Coupled Scalar Field in Four Dimensions,” Physical Review D 67, 024008 (2003). DOI.