Self-Similar Spacetimes in General Relativity

• Self-similar spacetimes are Lorentzian manifolds defined by a homothetic vector field that scales metric and matter fields, reducing Einstein’s equations.
• They underpin a range of exact solutions in cosmology and gravitational collapse, with classifications (first, zeroth, second, infinite) guiding their analysis.
• The alignment of the self-similar vector (tilted, parallel, orthogonal) with fluid flows constrains admissible metric forms and matter models, influencing solution properties.

Self-similar spacetimes are Lorentzian manifolds admitting vector fields that generate scaling symmetries, such that metric and matter fields transform homogeneously under dilations. In such settings, geometric and dynamical quantities inherit invariance properties that yield significant simplifications of the Einstein field equations and associated matter models. Self-similarity underlies a wide range of solutions across cosmology, gravitational collapse, and mathematical relativity, often providing powerful, sometimes unique, route to exact or critical solutions in general relativity and its extensions.

1. Mathematical Definition and Classification

A spacetime $(\mathcal{M},g_{ab})$ is called self-similar if it admits a vector field $\xi^a$ (the homothetic or self-similar vector field) obeying

$\mathcal{L}_\xi g_{ab} = 2\delta\, g_{ab}\,,$

with constant $\delta \neq 0$ for proper homothety and $\delta=0$ for Killing fields (isometries). When matter is present, kinematic self-similarity further requires

$\xi^a U_a = \alpha\,,$

where $U^a$ is the fluid four-velocity and $\alpha$ a constant. The ratio $\alpha/\delta$ (the similarity index) governs the classification:

Similarity Kind	$\alpha/\delta$	Example (KSS vector components)
First	$=1$	$\xi^0 = t$, $\xi^i \propto x^i$
Second	$\neq 0,1$	$\xi^0 = a t$
Zeroth	$=0$	$\xi^0 = 1$
Infinite	$\delta=0$ (homothety degenerates)	special case with only spatial components

This generalizes self-similarity beyond isometries to scaling symmetries relevant for dynamical and cosmological processes.

2. Metrics, Matter Models, and the Kinematic Self-Similar (KSS) Vector

Self-similar solutions impose strong constraints on admissible metric and matter field forms. In the context of locally rotationally symmetric (LRS) spacetimes, three families of metrics are commonly analyzed:

1. Type I (simplest LRS, e.g., Eq. (7)):

$$ds^2 = dt^2 - A^2(t)\,dx^2 - B^2(t)\,[dy^2 + F^2(y,k)\,dz^2]\,,$$

with $F(y,k)=y,\,\sin y,\,\sinh y$ for $k=0,1,-1$.

1. Type II, III (generalizations; additional spatial functions or cross-terms in $g_{ij}$).

The KSS vector for a typical LRS spacetime has the generic form

$$\xi = \xi^0(t, y)\,\partial_t + \xi^2(t, y)\,\partial_y$$

with component constraints dependent on whether $\xi^a$ aligns with (parallel), is orthogonal to, or is neither (tilted relative to) the fluid flow.

Matter content commonly includes perfect fluids or dust, with field equations

$$T_{ab} = [\rho(t, y) + p(t, y)]\,U_a U_b + p(t, y)\,g_{ab}\,.$$

3. Orientation of the Self-Similar Vector: Tilted, Parallel, and Orthogonal Cases

The relative orientation of $\xi^a$ and $U^a$ is decisive:

• Tilted: Both time and spatial components are nonzero. Most general case; encompasses the parallel case as a limit, especially for certain metrics. Solutions of first, zeroth, second, and infinite kind exist, e.g. for metric (7). Example first kind solution:

$$ds^2 = dt^2 - a^2 t^2 dx^2 - b^2 t^2 (dy^2 + y^2 dz^2)\,,$$

with a radiation-fluid equation of state $p = \rho / 3$.

• Parallel: KSS vector aligned with $U^a$. Spatial components vanish, simplifying the KSS conditions. Constitutes a subset of tilted solutions. The metric functions exhibit power-law or exponential dependence on $t$. In several cases, reduces to standard Bianchi models.
• Orthogonal: KSS vector orthogonal to $U^a$; for the LRS class considered, this invariably leads to a contradiction for all three metric types—i.e., the field equations admit no orthogonal kinematic self-similar solution in either perfect fluid or dust case.

4. Solution Cataloging and Constraints from Metric Structure

A comprehensive enumeration for the LRS metrics of types I–III reveals:

Metric type	Matter	Vector orientation	Number of indep. solutions	Notable features
(7)	Perfect fluid	Tilted/parallel	6	Power-law/exp. scale factors, incl. vac.
(7)	Dust	Tilted/parallel	3	Two vacuum solutions in dust case
(8)	Perfect fluid	Parallel only	4	Only parallel yields solutions
(8)	Dust	Parallel	1	First kind only
(9)	Both cases	Any	0	Contradiction in all cases

Vacuum solutions—for instance, those corresponding to $c=0$ in the infinite kind for $k=0$—recover Minkowski or stiff-fluid spacetimes, revealing that certain self-similar ansätze generate trivial geometries as limits.

The breakdown of existence according to the alignment of $\xi^a$ and the chosen metric demonstrates the rigidity of the Einstein field equations under scaling symmetries. The impossibility of self-similar solutions for metric (9) exemplifies the obstructions introduced by additional degrees of anisotropy or cross terms.

5. Physical and Cosmological Significance

Self-similar solutions possess physical relevance in multiple domains:

• Cosmology: The models generalize standard FLRW or Bianchi universes, yielding solutions with scale-invariant dynamics, analytically tractable for examining early universe or asymptotic behaviors.
• Gravitational collapse: Self-similar spacetimes are central in studying critical phenomena, threshold dynamics, and possible formation of naked singularities or cosmic censorship [see related constructions in spherically symmetric scalar field collapse].
• Simplifying field equations: The imposition of self-similarity reduces the field equations to systems of ODEs for the metric functions, considerably lowering their complexity and often permitting a comprehensive classification.

The orientation and existence criteria derived in these studies clarify under which geometric and fluid-flow conditions physical (and physically interesting) self-similar dynamics can exist.

6. Implications, Limitations, and Further Directions

The classification in LRS geometries (Sharif et al., 2010) provides a template for broader self-similar studies:

• The emergence of self-similar solutions is highly sensitive to the chosen geometric ansatz and the alignment of the KSS vector; certain cross-term or anisotropic metrics are inherently incompatible with self-similarity.
• Orthogonal self-similar vectors, in the context of LRS metrics and simple fluid sources, are generically disallowed.
• The catalog of seventeen independent solutions, including two vacuum cases, offers a reference point for identifying physically realized models in gravitational and cosmological applications, as well as for validating numerical relativity codes.
• The reduction of the field equations for self-similar ansätze is instrumental for gaining analytical insight into possible behaviors near singularities and understanding the role of symmetry in the control of gravitational dynamics.

Developments in more general settings (non-LRS, non-perfect fluids, or higher curvature corrections) could reveal new classes of self-similar solutions or additional obstructions. Quantitative analysis of perturbations around these backgrounds provides information about their stability and their role as long-time or late-time attractors in more general evolutions.

7. Summary Table of Solution Types for LRS Metrics

Metric	Tilted (PF)	Parallel (PF)	Tilted (Dust)	Parallel (Dust)	Orthogonal (any)	Vacuum Cases
(7)	3	3	2	1	None	2
(8)	0	4	0	1	None	0
(9)	None	None	None	None	None	0

PF: Perfect fluid

This comprehensive structural overview and solution classification highlight the explicit mechanisms, algebraic and geometric, that enable or obstruct the existence of self-similar spacetimes in the context of Einstein's field equations with locally rotationally symmetric ansatz. Further extension to less symmetric families, alternative matter contents, and non-standard energy conditions remain avenues for ongoing and future research.