Conformal Vacuum Metric

• Conformal Vacuum Metric is a metric that solves the vacuum field equations in a conformally invariant gravity theory, leading to the Bach equations.
• Its analysis employs a 2+2 decomposition and spherically symmetric solutions, yielding features like gravitational bubbles and variable curvature classes.
• The framework illustrates how conformal invariance and a traceless energy–momentum tensor offer alternate insights into vacuum structure and cosmological constants.

A conformal vacuum metric is a metric that solves the vacuum field equations of a conformally invariant theory of gravity, typically in the absence of any traceful matter sources. In four-dimensional Weyl conformal gravity, for instance, the action is constructed from the square of the Weyl tensor, and vacuum solutions satisfy the Bach equations. Unlike general relativity, conformal gravity allows for a richer space of vacuum solutions, including compact curved spacetimes without cosmological constant (“gravitational bubbles”) and metrics conformally related to, but distinct from, the Einstein class. The interplay between conformal invariance, the structure of the Bach tensor, and the requirements on the energy-momentum tensor profoundly shapes both the mathematical structure and physical implications of conformal vacuum metrics.

1. Action, Field Equations, and Conformal Invariance

The central action in four-dimensional Weyl conformal gravity is

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

where $C_{\mu\nu\lambda\sigma}$ is the Weyl tensor. The conformal symmetry arises under the transformation $g_{\mu\nu} \to e^{2\omega(x)} g_{\mu\nu}$, with the Bach tensor and energy-momentum tensor transforming covariantly: $$B^\mu_{\ \nu} \to e^{-4\omega} B^\mu_{\ \nu}, \quad T^\mu_{\ \nu} \to e^{-4\omega} T^\mu_{\ \nu}.$$ The field equations are fourth order,

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

with the tracelessness condition $\mathrm{Tr}(T^{\mu\nu}) = 0$. In vacuum ($T^{\mu\nu}=0$), the field equations reduce to $B_{\mu\nu} = 0$.

2. Spherically Symmetric Metrics and the 2+2 Decomposition

For spherically symmetric spacetimes, the metric is commonly decomposed as

$$ds^2 = \gamma_{ik} dx^i dx^k - r^2(x) (d\theta^2 + \sin^2\theta\, d\phi^2),$$

or, after extracting a conformal factor,

$ds^2 = r^2(x) [d\hat{s}_2^2 - d\Omega^2],$

where $d\hat{s}_2^2$ is a two-dimensional “truncated” metric. This factorization leverages the invariance of the action under conformal transformations and permits the field equations (Bach equations) to be separated and simplified in terms of the lower-dimensional truncated metric.

The truncated metric, in double-null coordinates $(u,v)$, takes the form

$ds_2^2 = 2e^{2\omega} du\, dv.$

3. Classification of Pure Vacuum Solutions

The full set of spherically symmetric vacuum solutions in Weyl conformal gravity comprises two distinct classes:

(a) Constant Curvature Solutions:

Here, the two-dimensional truncated scalar curvature $R$ is constant, specifically $R = \pm 2$. The corresponding metric can be written, after suitable redefinitions, in a Robertson–Walker-like form: $$d\hat{s}_2^2 = A\, d\eta^2 - \frac{dR^2}{A},\qquad A=\frac{1}{6}(R^3 - 12R + C_0),$$ with $C_0$ an integration constant. When $R = +2$, the solution is both compact and has $C_{\mu\nu\lambda\sigma} = 0$, yielding a curved, empty “gravitational bubble” spacetime with no material sources or cosmological constant. Such configurations are not possible in general relativity and exemplify the “creation from nothing” feature enabled by conformal symmetry.

(b) Varying Curvature Solutions:

In these solutions, $R$ varies. Introducing $R$ as a coordinate and solving the reduced Bach equations leads to metrics of the form

$$ds^2 = F(r) dt^2 - \frac{dr^2}{F(r)} - r^2(d\theta^2 + \sin^2\theta\, d\phi^2),$$

with $F(r)$ a polynomial in $r$. This family conforms to or generalizes the Mannheim–Kazanas solution, which has three parameters and reproduces both Schwarzschild-like behavior at small $r$ and de Sitter asymptotics at large $r$. The one-parameter family obtained in this approach can be conformally covered by the Mannheim–Kazanas solution, with the conformal factor $r^2(x)$ embedding the full spherical geometry (Berezin et al., 2014, Berezin et al., 2015).

4. Gravitational Bubbles and Spontaneous Symmetry Breaking

The gravitational bubble solution, corresponding to $R = +2$, is a compact, vacuum, curved spacetime with $C_{\mu\nu\lambda\sigma} = 0$. In double-null coordinates, the truncated metric is

$ds_2^2 = 2 e^{2\omega} du\, dv,$

with $\omega$ satisfying a Liouville equation. After suitable coordinate transformations, these solutions reduce to familiar Robertson–Walker metrics. The existence of such solutions, both curved and compact without sources, is a unique feature of conformal gravity not paralleled in general relativity, which requires a cosmological constant for any curved vacuum solution. The simultaneous existence of two constant curvature vacua (gravitational bubble and $R = -2$ “anisotropic” vacua) illustrates spontaneous symmetry breaking in the vacuum structure of the theory—another phenomenon absent in Einstein gravity (Berezin et al., 2014, Berezin et al., 2015).

5. Energy–Momentum Tensor Structure and Transformation Properties

Conformal gravity requires the energy–momentum tensor to be traceless: $\mathrm{Tr}(T_{\mu\nu}) = 0,$ ensuring compatibility with local Weyl symmetry. The transformation law under conformal mappings $$g_{\alpha\beta} = e^{2\omega} \tilde{g}_{\alpha\beta}$$ is

$$T_{\alpha\beta} = e^{-2\omega} \tilde{T}_{\alpha\beta}.$$

In the vacuum sector, $T^{\mu\nu} = 0$, but the Bach equations generally yield nontrivial curved solutions because the geometric (gravitational) sector alone can sustain non-flat configurations. The role of the energy–momentum tensor, including matter couplings, is tightly restricted by this tracelessness, with mass generation and massive particle physics only entering through symmetry breaking or spontaneous mechanisms.

6. Conformal Equations, Invariants, and Observational Consequences

The explicit vacuum Bach equations,

$$C^{\mu\sigma\nu\lambda}_{\ ;\lambda;\sigma} + \frac{1}{2} C^{\mu\lambda\nu\sigma} R_{\lambda\sigma} = 0,$$

together with their 2+2 reductions, lead to solvable vectorial field equations in the spherically symmetric case: $$\left[2\tilde{\Delta} + \frac{1}{3}(\tilde{R}^3 - 12\tilde{R})\right]_{,i} = 0,$$ with $\tilde{\Delta}$ a quadratic form in the derivatives of the curvature. Solutions can be reconstructed and characterized by their Weyl invariants and symmetry properties. For the subclass with variable $R$, the resulting metrics directly connect to astrophysical applications, such as alternative explanations for galactic rotation curves, and allow for a wide variety of gravitational potentials unconstrained by the Einstein equations.

A summary of the key structures is provided below:

Type	Metric Form	Features
Gravitational Bubble	Compact form with $R=+2$, $C_{\mu\nu\lambda\sigma}=0$	Curved, vacuum, creation from nothing
Mannheim–Kazanas	$ds^2 = F(r) dt^2 - dr^2/F(r) - r^2 d\Omega^2$, with $F(r)$ cubic	Schwarzschild–de Sitter interpolation, 3-params
Varying Curvature	As above, with $R = -C_1/r + C_2$	Conformally covers Mannheim–Kazanas; generic case

7. Implications for Fundamental Theory and Cosmology

The existence of conformal vacuum metrics not possible in general relativity has significant implications for cosmological models and the theoretical foundations of gravity. Gravitational bubbles enable the construction of curved universes “from nothing,” changing the paradigm for the initial conditions of cosmological evolution. The spontaneous symmetry breaking between discrete vacuum states introduces new possibilities for cosmic phase transitions and the origin of cosmological structure. Moreover, the presence of integration constants in the vacuum solutions provides effective cosmological "constants" emergent from the integration structure rather than as explicit Lagrangian parameters, offering alternative perspectives on the cosmological constant problem.

These results highlight fundamental differences between Einstein gravity and conformal gravity regarding vacuum structure and the role of conformal invariance, as well as illustrating technically how complex, curved, vacuum spacetimes can emerge in the absence of conventional sources (Berezin et al., 2014, Berezin et al., 2015).