Einstein–Scalar Collapse

A real, massless scalar field is the cleanest dynamical matter model for studying the boundary between dispersal and black-hole formation. It has no material surface, equation-of-state ambiguity, or shock, yet it can concentrate enough energy to create trapped spheres. A one-parameter family of smooth spherical data typically has a critical amplitude $p_*$: data with $p<p_*$ disperse, while data with $p>p_*$ form a black hole.

Required background. Use The Einstein Equation to relate scalar stress-energy to the evolving spherical geometry.

Helpful background. Black-Hole Geometry and Mechanics supplies trapped-surface diagnostics, Black-Hole Perturbation Theory supplies mode language near the critical solution, Formulations of General Relativity supplies constraint and gauge concepts, Relativistic Matter and Compact Objects contrasts scalar and fluid matter, and Computational Relativity develops production evolution methods.

The Einstein–scalar system

For a minimally coupled real field $\phi$, take

$$S=\int d^4x\,\sqrt{-g}\left[ \frac{R}{16\pi G}-\frac{1}{2}g^{ab}\nabla_a\phi\nabla_b\phi \right].$$

Variation gives

$$G_{ab}=8\pi G\left( \nabla_a\phi\nabla_b\phi- \frac{1}{2}g_{ab}\nabla_c\phi\nabla^c\phi \right), \qquad \nabla^a\nabla_a\phi=0.$$

In polar-areal coordinates before trapped-surface formation,

$$ds^2=-\alpha(t,r)^2dt^2+a(t,r)^2dr^2+r^2d\Omega^2.$$

Define $\Phi=\partial_r\phi$ and $\Pi=(a/\alpha)\partial_t\phi$. The wave equation becomes the first-order pair

$$\partial_t\Phi=\partial_r\left(\frac{\alpha}{a}\Pi\right), \qquad \partial_t\Pi=\frac{1}{r^2}\partial_r \left(r^2\frac{\alpha}{a}\Phi\right).$$

The two radial metric equations close this reduction:

$$\frac{\partial_r a}{a} =\frac{1-a^2}{2r}+2\pi G r(\Pi^2+\Phi^2), \qquad \frac{\partial_r\alpha}{\alpha} =\frac{a^2-1}{2r}+2\pi G r(\Pi^2+\Phi^2).$$

Thus, given the scalar data, one integrates for $a$ outward from the regular center and fixes the multiplicative lapse freedom by one clock condition. Here use $\alpha\to1$ at infinity, so $t$ measures asymptotic proper time. The central clock is instead $dt_c=\alpha(t,0)\,dt$. Choptuik’s original central-lapse normalization is another valid choice; both clocks cannot generally be set equal at once. At the center, regularity requires $a=1+O(r^2)$, $\Phi=O(r)$, and finite even $\Pi$. At large radius, asymptotically flat data approach $a\to1$ and have finite total mass. The equations are Gundlach and Martín-García 2007, §3.1, Eqs. (20)–(25), pp. 12–13, PDF, with $G$ restored for the scalar normalization used here.

The invariant spherical mass is defined by

$$1-\frac{2Gm(t,r)}{r}=g^{ab}\nabla_ar\nabla_br=\frac{1}{a^2}.$$

Differentiating the mass definition and using the radial constraint gives

$$\partial_r m=4\pi r^2\rho_{\rm E}, \qquad \rho_{\rm E}=\frac{\Pi^2+\Phi^2}{2a^2}\geq0,$$

where $\rho_{\rm E}$ is the energy density measured by the slice normal. This checks both the mass normalization and the sign of the gravitational source. Growth of $2Gm/r$ toward unity signals approach to a marginal sphere. Polar-areal coordinates do not smoothly cross a trapped region; continuation requires a suitable change of slicing or coordinates. Coordinate failure alone is not evidence of curvature singularity.

A characteristic description

Double-null coordinates follow the two radial light-ray families rather than the polar slices. Write

$$ds^2=-\Omega(u,v)^2\,du\,dv+r(u,v)^2d\Omega_2^2, \qquad g_{uv}=-\frac{\Omega^2}{2}.$$

Take increasing $u$ and $v$ future directed, with $r_u\lt0$ and $r_v\gt0$ in a regular untrapped region. The scalar equation and the two null constraints become

$$\phi_{uv}=-\frac{r_u\phi_v+r_v\phi_u}{r},$$ $$r_{uu}-2\frac{\Omega_u}{\Omega}r_u=-4\pi G r\phi_u^2, \qquad r_{vv}-2\frac{\Omega_v}{\Omega}r_v=-4\pi G r\phi_v^2.$$

The radial evolution equation is $r_{uv}=-r_ur_v/r-\Omega^2/(4r)$. These equations show focusing directly: scalar energy makes the rescaled derivatives $r_u/\Omega^2$ and $r_v/\Omega^2$ nonincreasing along their respective null directions. The same invariant mass now satisfies

$$1-\frac{2Gm}{r}=-\frac{4r_ur_v}{\Omega^2}.$$

An outgoing marginal sphere has $r_v=0$ with $r_u\lt0$; a trapped sphere has both derivatives negative. This is the same geometry that made $a$ diverge in polar-areal coordinates, expressed in variables that can remain regular at a horizon. Characteristic data are prescribed on intersecting null segments with their constraint and center compatibility conditions; changing coordinates does not remove the need to resolve the focusing region.

A one-parameter threshold

A concrete center-regular initial-data family might set

$$\phi(0,r)=\frac{p}{\sqrt{4\pi G}}\left\{ \exp\!\left[-\frac{(r-r_0)^2}{\sigma^2}\right] +\exp\!\left[-\frac{(r+r_0)^2}{\sigma^2}\right] \right\},$$

with $\Pi(0,r)=0$ and fixed $r_0,\sigma\gt0$. The factor $1/\sqrt{4\pi G}$ makes $p$ dimensionless for the canonically normalized scalar. The reflected Gaussian makes $\phi$ even, so $\partial_r\phi=0$ at the center. Solve the metric constraints as part of preparing these data; adding a pulse to a flat metric is not a complete strong-field initial state. The amplitude $p$ is not an invariant universal number: rescaling the family changes $p_*$. Near a transverse crossing, tune it by bracketing a dispersing and a black-hole-forming solution and repeatedly bisecting the interval. A complicated family can cross the threshold more than once, so this is a local threshold construction rather than a global monotonicity theorem.

Well below threshold, the pulse passes through the center and radiates outward, $m(t,r)$ empties at fixed $r$, and no trapped sphere forms. Well above threshold, an outer marginal sphere appears and the final black-hole mass is finite. Near threshold on the supercritical side, Choptuik found

$$M_{\mathrm{BH}}=C_f(p-p_*)^\gamma$$

up to a small periodic fine structure in logarithmic variables. The family-dependent constant $C_f$ carries dimensions; for the spherical massless scalar system, the measured exponent is approximately $\gamma\simeq0.374$. The near-critical solution is discretely self-similar with full scalar-field echoing period approximately $\Delta\simeq3.44$ Choptuik 1993, pp. 9–12. These are numerical continuum claims supported across several data families, not closed-form exact values and not universal exponents for every matter model.

The dynamical explanation is developed in the critical-collapse chapter: a self-similar intermediate attractor has one growing perturbation. Tuning one parameter suppresses that mode for an increasingly long logarithmic time; its eventual departure sets the only small length scale and hence the black-hole mass Gundlach and Martín-García 2007, §§3.1–3.2, pp. 12–14.

Continuum observables and numerical validation

The continuum problem comprises the Einstein–scalar equations, a declared smooth data family, regular-center conditions, asymptotic falloff, and a gauge that remains usable in the region being measured. Primary invariant outputs are the Misner–Sharp mass, null expansions, black-hole mass, curvature scalars, and—after alignment in logarithmic scale—the echoing period. A coordinate profile at a fixed grid time is a diagnostic, not by itself a physical output.

A defensible resolution study uses at least three successively refined calculations or an equivalently discriminating adaptive-tolerance sequence. It monitors the Einstein constraints, compares independent mass measures, measures convergence of $p_*$ and $\gamma$, and quotes uncertainty from the fit window and truncation error separately. Since tuning increases the required scale range exponentially, adaptive mesh refinement or a scale-adapted characteristic method is not merely a convenience.

Useful benchmarks are deliberately simpler than the critical run. A weak pulse should converge to linear propagation on nearly Minkowski spacetime. The vacuum exterior of a compact pulse should approach Schwarzschild with constant mass. Independent polar-areal and characteristic implementations should agree on $\gamma$ and $\Delta$ within their stated errors. Reproduction of the published threshold values is a benchmark, not a substitute for constraint and convergence tests.

Failure modes and reproducibility boundary

Near-critical evolution amplifies small systematic errors. An outer boundary can reflect radiation back into the strong-field region. Insufficient refinement can cap the self-similar cascade and create a false minimum black-hole mass. A horizon finder can miss a small trapped sphere; polar slicing can stall before one is located. Roundoff and parameter-search error eventually dominate when $p-p_*$ is too small. Imposed spherical symmetry also removes every nonspherical instability by construction.

This page compares version-identified published calculations with the continuum equations; it does not report an independent rerun of the historical simulations. A reproducible rerun would record the code revision, formulation, gauge, grid and refinement criteria, outer boundary, floating-point precision, initial-data generator, search algorithm for $p_*$, fit interval, raw invariant diagnostics, and machine-readable convergence results. It would first reproduce weak-field and Schwarzschild benchmarks, then regenerate $\gamma$, $\Delta$, and their uncertainty estimates.

Check your understanding

Suppose two supercritical data sets in the same family satisfy $(p_2-p_*)=10^{-3}(p_1-p_*)$. Ignoring fine structure, estimate $M_2/M_1$ for $\gamma=0.374$. Why does this comparison not require knowing $C_f$?

Solution

The scaling law gives

$$\frac{M_2}{M_1} =\left(\frac{p_2-p_*}{p_1-p_*}\right)^\gamma =(10^{-3})^{0.374}\simeq7.55\times10^{-2}.$$

Both points belong to the same data family, so the family-dependent constant $C_f$ cancels in the ratio. Comparing different families would require aligning their parameter normalization and would not generally cancel distinct prefactors.

References

• Choptuik, Matthew W. “Universality and Scaling in Gravitational Collapse of a Massless Scalar Field.” Physical Review Letters 70 (1993): 9–12. doi:10.1103/PhysRevLett.70.9.
• Gundlach, Carsten, and José M. Martín-García. “Critical Phenomena in Gravitational Collapse.” Living Reviews in Relativity 10 (2007): 5. doi:10.12942/lrr-2007-5. Open PDF.