Gravitational Bubbles in Conformal Gravity

• Gravitational bubbles are nontrivial, compact spacetimes in conformal gravity defined by a vanishing Weyl tensor and constant positive curvature.
• They serve as crucial test cases for higher-derivative gravitational models, offering insight into universe nucleation via their instanton-like, zero-action properties.
• Their conformally flat, closed topology (with S³ spatial sections) sharply contrasts with pure vacuum solutions in standard General Relativity.

Gravitational bubbles refer, in their most precise usage, to spherically symmetric, compact, pure-vacuum solutions of Weyl conformal gravity—theories where the gravitational action is built from the square of the Weyl tensor rather than the Ricci scalar. These solutions, with vanishing Weyl tensor and no matter or cosmological constant, are spatially closed and conformally flat, forming a unique branch of classical spacetimes forbidden in standard General Relativity. Gravitational bubbles feature centrally in discussions of universe nucleation “from nothing,” due to their instanton-like nature and finite action. They also provide a fundamental test case for higher-derivative, generally covariant gravitational models (Berezin et al., 2014, Berezin et al., 2015).

1. Definition and Fundamental Properties

A gravitational bubble is defined as a nontrivial, compact, pure-vacuum solution to the conformal gravity field equations. The action in four dimensions is

$$S_{\rm W} = -\alpha_0 \int C_{\mu\nu\lambda\sigma} C^{\mu\nu\lambda\sigma} \sqrt{-g} d^4x\,,$$

where $C_{\mu\nu\lambda\sigma}$ is the Weyl tensor and $\alpha_0$ is a dimensionful coupling. The associated field equations, the Bach equations,

$$B^{\mu\nu} \equiv C^{\mu\sigma\nu\lambda}{}_{;\lambda;\sigma} + \tfrac{1}{2} C^{\mu\lambda\nu\sigma} R_{\lambda\sigma} = \tfrac{1}{8\alpha_0} T^{\mu\nu}\,,$$

are fourth-order in derivatives and only traceless matter (${T^\mu}_\mu=0$) can couple consistently.

A gravitational bubble possesses a spherically symmetric metric, with a $2+2$ decomposition: $$ds^2 = \gamma_{ij}(x^k) dx^i dx^j - r^2(x^k) (d\theta^2 + \sin^2\theta\, d\varphi^2)\,.$$ Conformal symmetry permits factoring out $r^2(x)$, yielding a truncated metric $\tilde{\gamma}_{ij}$ for the two-dimensional part. Crucially, the Weyl tensor squared reduces to

$$C_{\mu\nu\lambda\sigma} C^{\mu\nu\lambda\sigma} = \tfrac{1}{3} (\tilde{R} - 2)^2\,,$$

where $C_{\mu\nu\lambda\sigma}$0 is the scalar curvature of the truncated metric.

2. Bach Equations and Vacuum Solutions

In the absence of matter ($C_{\mu\nu\lambda\sigma}$1), the Bach equations for spherically symmetric metrics take the form

$C_{\mu\nu\lambda\sigma}$2

where “$C_{\mu\nu\lambda\sigma}$3” denotes the covariant derivative with respect to $C_{\mu\nu\lambda\sigma}$4. The vacuum conditions require

$C_{\mu\nu\lambda\sigma}$5

hence $C_{\mu\nu\lambda\sigma}$6 or $C_{\mu\nu\lambda\sigma}$7.

The solution with $C_{\mu\nu\lambda\sigma}$8 is conformally flat ($C_{\mu\nu\lambda\sigma}$9), forms a compact spacetime, and is the canonical “gravitational bubble.”

3. Explicit Metric Structure

The gravitational bubble admits an explicit metric in double-null coordinates $\alpha_0$0: $\alpha_0$1 For $\alpha_0$2, the full four-dimensional metric becomes

$\alpha_0$3

which is spatially compact and conformally flat. Upon coordinate transformation, it can be cast into closed Robertson–Walker form: $\alpha_0$4 with spatial sections topologically $\alpha_0$5.

4. Prohibition in Einstein Gravity

No closed, pure-vacuum solution of positive spatial curvature exists in standard Einstein gravity unless a positive cosmological constant is present. The Einstein equations $\alpha_0$6 enforce Minkowski or black-hole geometries only. By contrast, conformal gravity’s Bach tensor permits nonzero constant curvature even in strict vacuum: $\alpha_0$7 For zero matter, constant $\alpha_0$8 yields a consistent solution (Berezin et al., 2014, Berezin et al., 2015).

5. Universe Creation “From Nothing” and Cosmological Implications

The gravitational bubble, as a compact Euclidean solution of the conformal gravity field equations, is naturally interpreted as an instanton for quantum cosmology. Its vanishing action $\alpha_0$9 renders nucleation exponentially favorable, circumventing the necessity for exotic matter or a cosmological term. The “bubble” provides the geometric seed for universe creation mechanisms as proposed by Vilenkin and others; its topology is that of a closed 3-sphere, with time oriented orthogonal to these spatial sections.

6. Relation to Other Solutions and Phenomenology

Gravitational bubbles contrast sharply with other vacuum solutions of the Bach equations, notably those with varying $$B^{\mu\nu} \equiv C^{\mu\sigma\nu\lambda}{}_{;\lambda;\sigma} + \tfrac{1}{2} C^{\mu\lambda\nu\sigma} R_{\lambda\sigma} = \tfrac{1}{8\alpha_0} T^{\mu\nu}\,,$$0 (not constant), which yield noncompact spacetimes conformally related to Mannheim–Kazanas metrics and possess nonvanishing Weyl tensor. In the presence of matter, the equations admit only traceless stress–energy, and the pure bubble solution remains unaffected.

In modified gravitational frameworks such as quadratic F(R) gravity, analogous spherically symmetric bubbles can manifest—e.g., pure double-layer bubbles at the matching surface of AdS–Schwarzschild patches with negative quadratic coefficient—however, the equivalence to the Weyl bubble occurs only in limit cases and with further restrictions (Eiroa et al., 2017).

7. Summary Table: Key Features of Gravitational Bubbles in Conformal Gravity

Feature	Value / Formula	Physical Interpretation
Scalar curvature	$$B^{\mu\nu} \equiv C^{\mu\sigma\nu\lambda}{}_{;\lambda;\sigma} + \tfrac{1}{2} C^{\mu\lambda\nu\sigma} R_{\lambda\sigma} = \tfrac{1}{8\alpha_0} T^{\mu\nu}\,,$$1	Constant, positive; selects compact branch
Weyl tensor	$$B^{\mu\nu} \equiv C^{\mu\sigma\nu\lambda}{}_{;\lambda;\sigma} + \tfrac{1}{2} C^{\mu\lambda\nu\sigma} R_{\lambda\sigma} = \tfrac{1}{8\alpha_0} T^{\mu\nu}\,,$$2	Conformally flat, no tidal forces
Matter content	$$B^{\mu\nu} \equiv C^{\mu\sigma\nu\lambda}{}_{;\lambda;\sigma} + \tfrac{1}{2} C^{\mu\lambda\nu\sigma} R_{\lambda\sigma} = \tfrac{1}{8\alpha_0} T^{\mu\nu}\,,$$3	Pure vacuum, traceless stress only allowed
Topology	$$B^{\mu\nu} \equiv C^{\mu\sigma\nu\lambda}{}_{;\lambda;\sigma} + \tfrac{1}{2} C^{\mu\lambda\nu\sigma} R_{\lambda\sigma} = \tfrac{1}{8\alpha_0} T^{\mu\nu}\,,$$4 (spatial sections)	Compact, closed universe
Metric (double null)	$$B^{\mu\nu} \equiv C^{\mu\sigma\nu\lambda}{}_{;\lambda;\sigma} + \tfrac{1}{2} C^{\mu\lambda\nu\sigma} R_{\lambda\sigma} = \tfrac{1}{8\alpha_0} T^{\mu\nu}\,,$$5	Standard bubble form
Possibility in GR	Not permitted	Needs $$B^{\mu\nu} \equiv C^{\mu\sigma\nu\lambda}{}_{;\lambda;\sigma} + \tfrac{1}{2} C^{\mu\lambda\nu\sigma} R_{\lambda\sigma} = \tfrac{1}{8\alpha_0} T^{\mu\nu}\,,$$6 or matter in Einstein gravity
Quantum cosmology	Instanton for universe nucleation	“Creation from nothing” scenario

Gravitational bubbles in Weyl conformal gravity thus uniquely supply compact, source-free universes of positive curvature. Their existence points towards fundamentally different mechanisms for cosmic birth and structure in gravitational theories extending beyond the Einstein–Hilbert paradigm (Berezin et al., 2014, Berezin et al., 2015).