Mass Scaling and Critical Exponents

The black-hole mass law translates one unstable perturbation of a scale-invariant threshold solution into an observable power law. Its exponent is an eigenvalue statement, not merely the slope of one numerical fit. Family-dependent amplitudes, threshold error, discrete fine structure, finite resolution, and additional physical scales must all be separated before a measured slope can be called a critical exponent.

Required background. The Einstein equation supplies the constrained evolution and dimensional scale, while differential equations supplies linear eigenmodes and logarithmic fits.

Helpful background. Black-Hole Geometry and Mechanics supplies quasi-local mass; Black-Hole Perturbation Theory develops eigenvalue problems; Formulations of General Relativity explains scale-adapted variables; and Computational Relativity develops threshold bracketing and error analysis.

From one eigenvalue to a mass law

Let a regular one-parameter family cross the black-hole threshold transversely at $p_*$. Take $p$ dimensionless, or divide its difference by a fixed parameter scale before taking a logarithm. In logarithmic scale time $\tau$, assume the critical solution has exactly one growing physical perturbation,

$$Z(\tau,x)\simeq Z_*(\tau,x) +\kappa(p-p_*)e^{\lambda_0\tau}Z_0(\tau,x), \qquad \lambda_0>0.$$

Choose a fixed small departure amplitude. The departure scale satisfies

$$e^{-\tau_*}\propto\lvert p-p_*\rvert^{1/\lambda_0}.$$

If the subsequent supercritical evolution introduces no independent length, the black-hole mass in four dimensions is

$$GM_{\rm BH}=C_{\mathcal F}(p-p_*)^\gamma, \qquad \gamma=\lambda_0^{-1}.$$

Here $M_{\rm BH}$ is physical mass and $GM_{\rm BH}$ has length dimension; a convention that writes the law directly for $M_{\rm BH}$ simply puts the factor $G^{-1}$ into $C_{\mathcal F}$. The subscript $\mathcal F$ emphasizes that the coefficient depends on the initial-data family and parameterization. Under a smooth invertible reparameterization $\bar p=f(p)$ with $f'(p_*)\ne0$, the coefficient changes but $\gamma$ does not. A family tangent to the threshold instead has $C_0(p)\propto(p-p_*)^k$, with $k\gt1$, giving an apparent exponent $k\gamma$. A singular coordinate change with vanishing derivative is not an admissible smooth invertible reparameterization at threshold. These are reasons to check transversality, not new critical exponents.

Gundlach and Martín-García give this derivation in Gundlach and Martín-García 2007, §2.3, pp. 7–8. Koike, Hara, and Adachi independently computed the growing mode of the Evans–Coleman perfect-fluid critical solution and related its eigenvalue to the exponent in Koike, Hara, and Adachi 1995, pp. 5170–5173.

Other observables and dimensions

An observable $Q$ with length dimension $s$ scales as

$$Q=C_Q\lvert p-p_*\rvert^{s\gamma}$$

when it is controlled only by the departure scale. Thus horizon area scales with $2\gamma$, the maximum magnitude of a nonzero curvature scalar of dimension $L^{-2}$ has subcritical exponent $-2\gamma$, and the maximum magnitude of the Kretschmann scalar has exponent $-4\gamma$ when it is nonzero in the critical region. Conserved charge or angular momentum can be governed by a decaying nonspherical mode and acquire a distinct exponent; dimensional analysis alone then does not supply the full answer.

For discrete self-similarity the straight line in logarithmic variables carries periodic fine structure:

$$\ln\left(\frac{GM_{\rm BH}}{L_0}\right) =\gamma\ln(p-p_*)+c_{\mathcal F} +f\!\left(\gamma\ln(p-p_*)+c_{\mathcal F}\right),$$

where $L_0$ is a fixed reference length and $f$ is periodic. If its period in the displayed argument is $P$, its period on the horizontal axis $\ln(p-p_*)$ is $P/\gamma$: increasing that axis by $P/\gamma$ advances the argument by $P$. For a generic DSS solution one can use the full echo period $P=\Delta$. In the Choptuik scalar solution the metric repeats after the scalar’s half-echo, so $P=\Delta/2$ and the mass residual has period

$$\frac{\Delta}{2\gamma}\simeq \frac{3.44}{2(0.374)}\simeq4.60$$

in $\ln(p-p_*)$, or about $2.00$ in $\log_{10}(p-p_*)$. This factor of two is physical, arising from the scalar sign reversal, rather than a choice of logarithm Gundlach 1997, §IV.B, Eqs. (71)–(81), pp. 12–13, PDF.

Fitting without manufacturing an exponent

First bracket $p_*$ with convergent supercritical and subcritical classifications. A bias $\delta p_*$ curves the smallest-$\lvert p-p_*\rvert$ data and can imitate a change of exponent. Fit several windows, include the periodic term when statistically resolved, and report covariance between $p_*$, $\gamma$, phase, and intercept. Avoid treating many time samples from one evolution as independent families.

Second, repeat with at least two distinct initial-data families. They should yield compatible $\gamma$ and echo period but different $p_*$ and intercept. Third, fit a subcritical curvature observable and compare its exponent with the mass result. Fourth, compare $\gamma^{-1}$ with a direct eigenvalue calculation when available.

The scaling window has two ends. Far from threshold, decaying modes and family details have not disappeared. Too close to threshold, truncation error, roundoff, unresolved gradients, or a physical ultraviolet cutoff intervenes. A dimensionful coupling need not end the cascade: an ordinary scalar mass becomes less important at shorter lengths, as explained in Universality from Perturbation Modes. A long straight segment is evidence only after these boundaries and residual structure are shown.

Check your understanding

A fit gives area law $A_H\propto(p-p_*)^{0.74}$ and subcritical maximum absolute Ricci scalar $\max|R|\propto\lvert p-p_*\rvert^{-0.75}$ in a model with nonzero Ricci curvature. Are these exponents mutually consistent with one type II scale?

Solution

Area has dimension $L^2$, so it predicts $2\gamma=0.74$ and $\gamma=0.37$. A Ricci scalar has dimension $L^{-2}$, so it predicts exponent $-2\gamma=-0.74$. The difference between $-0.75$ and $-0.74$ is $0.01$. Agreement is justified only if the combined fit and systematic uncertainty covers that difference; the numbers alone do not supply an error estimate. Both fits must share a converged threshold and scaling window. A Ricci scalar vanishing identically, as in vacuum, would be an unsuitable diagnostic even though its dimensions are correct.

References

• 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.
• Gundlach, Carsten. “Understanding Critical Collapse of a Scalar Field.” Physical Review D 55 (1997): 695–713. doi:10.1103/PhysRevD.55.695. Open PDF.
• Koike, Tatsuhiko, Takashi Hara, and Satoshi Adachi. “Critical Behavior in Gravitational Collapse of Radiation Fluid: A Renormalization Group Analysis.” Physical Review Letters 74 (1995): 5170–5173. doi:10.1103/PhysRevLett.74.5170.