Continuous and Discrete Self-Similarity

Self-similarity is the geometric statement that a solution reproduces itself under changes of spacetime scale. Continuous self-similarity supplies a homothety for every scale change; discrete self-similarity supplies only an integer sequence of echoes. In critical collapse these symmetries describe the intermediate threshold solution in a shrinking local region, not the entire asymptotically flat spacetime.

Required background. The Einstein equation fixes how metric scaling couples to matter, while differential equations supplies flows, fixed points, and limit cycles.

Helpful background. Black-Hole Geometry and Mechanics supplies invariant curvature and horizon scales; Black-Hole Perturbation Theory develops modes on self-similar backgrounds; Formulations of General Relativity explains adapted coordinates and gauge; and Computational Relativity develops echo extraction.

Continuous homothety

A continuously self-similar spacetime admits a vector field $\xi^a$ satisfying

$$\mathcal L_\xi g_{ab}=2g_{ab}.$$

This is a homothety, not generally an isometry. In adapted dimensionless coordinates $(\tau,x^i)$ with $\xi=-\partial_\tau$, the metric can be written

$$g_{\mu\nu}(\tau,x)=e^{-2\tau}\widetilde g_{\mu\nu}(x).$$

The coordinate $\tau=-\ln(L/L_0)$ increases toward smaller physical scale $L$. Scale-invariant geometric variables are independent of $\tau$; a quantity $Q$ with length dimension $s$ has the form $Q=L_0^s e^{-s\tau}\widetilde Q(x)$. A curvature scalar therefore grows even though its dimensionless profile remains fixed. A matter field may transform under an additional internal symmetry: for a massless scalar, for example, $\phi$ may acquire an additive term proportional to $\tau$ while its stress tensor has the required scaling. Self-similarity of the geometry does not require every raw matter variable to be independent of scale.

Matter must be compatible with the homothety. A dimensionful mass, potential scale, or equation-of-state scale breaks exact continuous self-similarity, although it can become asymptotically irrelevant as $L$ shrinks. Boundary conditions at infinity also introduce a scale, so an exact asymptotically flat critical evolution is usually only locally self-similar near its accumulation event.

Discrete echoes

A discretely self-similar spacetime admits a diffeomorphism $\Phi$ and a positive echoing period $\Delta$ such that

$$\Phi^*g_{ab}=e^{-2\Delta}g_{ab}.$$

In adapted coordinates,

$$g_{\mu\nu}(\tau,x) =e^{-2\tau}\widetilde g_{\mu\nu}(\tau,x), \qquad \widetilde g_{\mu\nu}(\tau+\Delta,x) =\widetilde g_{\mu\nu}(\tau,x).$$

Thus a dimensionless profile repeats after the physical scale shrinks by $e^{-\Delta}$. The massless scalar critical solution is the standard DSS example. Gundlach and Martín-García define both symmetries and their adapted-coordinate forms in Gundlach and Martín-García 2007, §2.2, pp. 4–7.

Some matter fields return only up to an internal symmetry or sign, so the shortest metric period may be smaller than the full matter-field period. In the Choptuik solution, the usual $\Delta\simeq3.44$ denotes a full scalar-field echo. The scalar changes sign after $\Delta/2$, whereas the metric and quadratic stress tensor already repeat. Consequently metric observables have a half-echo period $\Delta/2\simeq1.72$ Gundlach 1997, §IV.B, Eqs. (71)–(81), pp. 12–13, PDF. Coordinate periodicity alone is insufficient: gauge conditions can generate artificial repetition.

Fixed points, limit cycles, and observations

After quotienting by overall scale, CSS appears as a fixed point of the evolution in $\tau$, while DSS appears as a limit cycle. Linear perturbations of CSS have factors $e^{\lambda_i\tau}$. For DSS, Floquet theory gives $e^{\lambda_i\tau}Z_i(\tau,x)$ with periodic $Z_i$. A shift of phase along the DSS cycle is a neutral direction, not a growing physical disturbance. After removing such symmetry directions, exactly one independent real growing amplitude gives codimension one; angular degeneracies or a complex conjugate pair must be counted by their real amplitudes.

To detect self-similarity, select geometrically defined dimensionless quantities—for example, curvature invariants multiplied by powers of an areal-radius scale—and compare profiles at successive central proper times approaching an accumulation time $t_*$. Define

$$\tau=-\ln\left(\frac{t_*-t}{L_0}\right).$$

Profiles should align under spatial rescaling and, for DSS, recur at a common $\Delta$. The inferred $t_*$, $\Delta$, and phase must converge with resolution and agree across several fields. Shifting $t_*$ can manufacture or destroy apparent periodicity, so it must be fitted jointly and tested on withheld echoes.

Limitations and checks

Exact self-similarity typically ends at an accumulation singularity. The critical solution need not be a globally regular, asymptotically flat spacetime by itself; it is approached in a local region by regular asymptotically flat data. Take the family parameter $p$ dimensionless, or replace its difference by $(p-p_*)/p_0$ for a fixed scale $p_0$. Fine tuning controls how many full echoes occur before departure:

$$N\simeq-\frac{\gamma}{\Delta}\ln\lvert p-p_*\rvert+\text{constant}.$$

Here $\gamma=1/\lambda_0$, with $\lambda_0$ the unique growing exponent in logarithmic time. The number is finite for every imperfect tuning. A coupling that becomes relevant at short distances can terminate echoes; an ordinary scalar mass instead becomes less important when the local length is much smaller than its Compton wavelength. Numerical truncation terminates the resolved echoes when gradients reach the grid scale.

A useful consistency triangle compares the echo period from local profiles, the periodic fine structure in the mass-scaling law, and the Floquet spectrum. Agreement supports a geometric DSS interpretation. Failure may indicate gauge contamination, threshold error, insufficient dynamic range, or a solution that is only approximately self-similar.

Check your understanding

A scalar DSS solution has full field period $\Delta=3.44$, with a sign reversal after half a period. By what factors does its length scale shrink between full field echoes and between metric half-echoes? How many additional echoes result from improving $\lvert p-p_*\rvert$ by a factor $10^6$ if $\gamma=0.374$?

Solution

The scale factor for a full field echo is $e^{-\Delta}\simeq0.0320$. Between metric half-echoes it is $e^{-\Delta/2}\simeq0.179$. The full-echo gain is

$$\Delta N=\frac{\gamma}{\Delta}\ln(10^6) \simeq\frac{0.374}{3.44}(13.816) \simeq1.50.$$

Thus six more decimal digits of tuning reveal only about one and a half additional full field echoes, or three metric half-echoes.

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.