Self-Similar Collapse with Elasticity

• Self-similar collapse with elasticity is defined as the class of scale-invariant solutions to the Einstein equations where elastic effects, such as anisotropic stresses and variable sound speeds, are incorporated.
• The methodology involves transforming the Einstein-matter system into a set of ODEs using self-similarity variables and solving them via a shooting method that ensures regularity at the sonic point.
• Elasticity introduces qualitative modifications—including negative radial pressures, multiple wave speeds, and stricter parameter bounds—that significantly diverge from the dynamics observed in perfect fluid collapse.

Self-similar collapse with elasticity refers to the existence and properties of continuously self-similar solutions to the Einstein and continuum matter equations when the matter content admits a (scale-invariant) elastic response. This theoretical scenario is motivated by the recognition that compact astrophysical objects—most notably neutron stars—depart from idealized perfect fluid behavior due to the presence of elastic (shear-supporting) crusts. Recent work (Rocha et al., 8 Sep 2025) has formulated and numerically constructed self-similar critical collapse solutions for fully nonlinear elastic matter models, revealing that elasticity can introduce qualitative modifications absent from the perfect fluid case.

1. Scale-Invariant Elastic Matter Models

The elastic matter model utilized in (Rocha et al., 8 Sep 2025) is constructed to preserve scale invariance and is formulated in terms of variables associated with the deformation operator. The key state variables are:

• $\delta$: normalized particle number density,
• $\eta$: averaged particle number.

The energy density is given by an explicit function: $$\hat{\rho} = \frac{n^2\,k}{n+1}\,\eta^{1+\frac{1}{n}}\Biggl[1-\frac{n+1}{n}\Bigl(1-3\frac{s}{n}\frac{1-\nu}{1+\nu}\Bigr)\Bigl(1-\frac{\delta}{\eta}\Bigr) -3\frac{s^2}{n^2}\frac{n+1}{s+1}\frac{1-\nu}{1+\nu}\Bigl(1-\Bigl(\frac{\delta}{\eta}\Bigr)^{1+\frac{1}{s}}\Bigr)\Biggr]$$ where $k$ is a positive constant, $n$ is the (generalized) polytropic index, $s$ is a “shear index” quantifying elastic effects, and $\nu$ is the (Lame) Poisson ratio.

This model reduces to a perfect fluid for $n = s$ and $\nu = 1/2$; deviations reflect nonzero elastic moduli (e.g., shear and Young’s moduli). Elasticity enters via additional terms that couple $\delta$ and $\eta$ and depend on $s$ and $\nu$, allowing for anisotropic stress states and the development of shear.

Importantly, the elastic longitudinal sound speed becomes position-dependent, and distinct transverse sound speeds appear. The model remains locally isotropic near the regular center, converging to a perfect fluid there, but exhibits genuine elastic behavior away from the center.

2. Self-Similar Solution Construction and Numerical Methods

The Einstein-matter system is rendered self-similar by a coordinate transformation to self-similarity variables: $x = \ln(-r/t)$ and a rescaled time $\tau$, leading to a system of ODEs for five dependent functions, including the normalized three-velocity $V(x)$ and metric functions $A(x), N(x)$.

The system is solved as a boundary value problem using a shooting method:

• The solution must be regular at $x \to -\infty$ (the center), where boundary conditions ensure analyticity and local isotropy.
• At $x = 0$, a sonic point is imposed by gauge, where regularity requires careful tuning of initial data.
• Beyond the sonic point, the solution is propagated numerically (e.g., via a fourth-order Runge–Kutta algorithm).

Solutions are discrete: a fundamental (nodeless) mode and countable overtones exist, classified by the number of sign changes (“nodes”) in the velocity profile $V(x)$. These numerically constructed solutions extend the known discrete self-similar solutions of perfect fluids to the elastic case.

3. The Sonic Point and Regularity Conditions

The sonic point plays a central role: it is the spatial location $x_\star$ where the principal characteristic speed (the local longitudinal wave speed $c_L$) matches the material velocity as measured in the self-similar frame: $v = \sqrt{1-\frac{(N^2-1)(1-V^2)}{(N+V)^2}}$ Regularity at $x = x_\star$ demands that a certain $2 \times 2$ coefficient matrix (relating derivatives of variables) has vanishing determinant: $$\det\begin{bmatrix} a & b \ c & d \end{bmatrix} = ad - bc = 0$$ This condition ensures that the profile passes through the sonic point smoothly, i.e., the system is non-singular there.

A distinctive effect of elasticity is that $c_L$ is not constant. For sufficiently strong elasticity (especially for small $\nu$ or large $s$), $c_L(x)$ can vary so much that a second sonic point occurs for $x > 0$. Numerical integration fails to maintain regularity at this additional sonic point: too many constraints overdetermine the system, and singularities develop.

Thus, the requirement for a single, regular sonic point imposes strong bounds on the elastic parameters; for example, with $n = s = 3$, only $0.415 < \nu \leq 1/2$ allows regular solutions. This is unlike the perfect fluid case, where no such secondary singular behavior emerges.

4. Elasticity-Induced Modifications and Phenomenology

Elasticity alters the collapse landscape in several distinct ways:

• Negative radial pressures: For some parameter regimes (notably lower $\nu$), the radial pressure becomes negative near the sonic point. This is a phenomenon that cannot occur in perfect fluids with $p \propto \rho$.
• Multiple wave speeds: The elastic theory supports a position-dependent longitudinal speed

$c_L^2 = \frac{\partial p}{\partial \hat{\rho}}$

and two transverse speeds,

$$c_T^2 = \frac{p - \hat{p}_t}{\hat{\rho}+p}, \;\;\; \widetilde{c}_T^2 = \text{(different transverse mode)}$$

The departure of $c_T$ and $\widetilde{c}_T$ with increasing elasticity highlights the richer propagation structure (e.g., mode splitting).

• Increased compressibility: Increasing $s$ or decreasing $\nu$ enhances compressibility.
• Discreteness and bounds: The set of regular self-similar solutions is discrete and only exists within bounds set by the elastic parameters.

These findings highlight the intricate coupling between elasticity and spacetime geometry, fundamentally modifying collapse scenarios compared to perfect fluid models.

5. Relationship to the Broader Self-Similar Collapse Program

Self-similar collapse has been widely studied in the context of perfect fluids, scalar fields, and certain other matter models. The inclusion of elasticity addresses longstanding questions regarding the fate of more realistic astrophysical bodies, such as neutron stars with elastic crusts.

The discrete structure of solutions, the centrality of the sonic point, and the occurrence of critical phenomena (such as universality classes and scaling exponents) are familiar from the perfect fluid case, but the present results show elasticity can drive substantial deviations. For instance, the possible violation of regularity via a second sonic point has no analogue in perfect fluids (Rocha et al., 8 Sep 2025).

A plausible implication is that elasticity may modify the universality seen in gravitational critical phenomena, affecting the set of initial data that leads to black hole formation and altering critical exponents.

6. Connections and Outlook

The construction of elastic self-similar solutions is distinct from, yet related to:

• Nonlocal and fractal elasticity theories: Where self-similar, nonlocal Laplacians produce singular Green’s functions and scale-invariant kernels (Michelitsch et al., 2011).
• Buckling and pattern formation: In thin elastic sheets, self-similar, energy-minimizing patterns arise from geometric and elastic competition (Gemmer et al., 2016).
• Astrophysical and cosmological implications: Self-similar effective equations of state can be generalized to include elastic terms, which would alter collapse thresholds in protostellar and supernova scenarios (Lee, 18 Jan 2024).
• Criticality in modified gravity and with different matter: E.g., the role of matter fields with nontrivial “elastic” scaling response in Einstein–Maxwell–dilaton theory (Rocha et al., 2018).

The new findings on collapse with elasticity clarify parameter regimes leading to regularity versus breakdown; further work is needed on linear perturbations and the computation of scaling exponents to fully classify critical behavior and universality in gravitational collapse with elastic materials.

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In summary, self-similar collapse with elasticity introduces profound phenomenological and mathematical features—variable sound speeds, negative radial pressures, multiple wave speeds, and bounds on admissible elastic parameters—all absent from perfect fluid models. These results set the stage for systematic exploration of critical gravitational collapse in matter with realistic microphysics (Rocha et al., 8 Sep 2025).