Conformally Invariant Gravitating Mirages

• Conformally invariant gravitating mirages are effective energy-momentum components emerging from strict symmetry constraints, producing dark matter and radiation mimics via induced particle creation.
• They arise from a unified variational principle that couples fluid dynamics with conformal field theory, enforcing a traceless energy-momentum tensor and fixed Weyl weight source terms.
• Observable implications include gravitational lensing and modified cosmological dynamics, offering testable signatures to distinguish them from conventional dark matter models.

Conformally invariant gravitating mirages are emergent effective matter components arising within gravitational theories whose dynamics and sources are constrained by conformal symmetry. They manifest as energy-momentum contributions, mimicking the gravitational effects of dark matter and radiation, generated not through fundamental particles but as a consequence of strictly conformally invariant particle creation laws and geometry-matter coupling. These mirages are best elucidated in models where induced gravity arises from a conformally invariant fluid dynamics action, with the source term for particle creation possessing a uniquely fixed Weyl weight. Gravitating mirages represent a nontrivial intersection between conformal field theory, induced gravity, and effective cosmological fluid dynamics, often appearing in scenarios where the creation law for particle number is dictated by the requirement of overall conformal invariance in the action, leading to apparent dust- or radiation-like gravitational sources even in the absence of underlying microphysical degrees of freedom (Berezin et al., 2024). Their observable consequences include classical gravitational lensing indistinguishable from traditional cold dark matter and effective contributions to Friedmann equations that drive cosmological expansion.

1. Conformal Invariance in Gravitational Theories

Conformal invariance requires that all fundamental interactions and sources transform with definite Weyl weights under local rescalings of the metric, $g_{\mu\nu} \to \Omega^2(x) g_{\mu\nu}$, with corresponding transformation rules for matter fields and number densities. The most general local, quadratic curvature invariant in four dimensions is the square of the Weyl tensor, $$C^2 = C_{\mu\nu\lambda\sigma} C^{\mu\nu\lambda\sigma}$$, which retains conformal invariance. For matter creation, conformal invariance forces the particle-number source term to take the form: $$\Phi = \alpha C^2 + \beta \left( \varphi \Box \varphi - \tfrac{1}{6} \varphi^2 R + \Lambda \varphi^4 \right) + \gamma_1 \varphi n + \gamma_2 n^{4/3}$$ with each term possessing Weyl weight 4 (Berezin et al., 2024). Here $n$ is the particle number density, $\varphi$ is a conformal scalar field, and $\alpha, \beta, \gamma_1, \gamma_2, \Lambda$ are parameters fixed by the theory.

This symmetry constraint not only dictates the allowed forms of interactions but also enforces a traceless energy-momentum tensor, $g_{\mu\nu} T^{\mu\nu} = 0$, resulting in the trace condition: $$\varphi \tfrac{\partial\varepsilon}{\partial\varphi} + 3n \tfrac{\partial\varepsilon}{\partial n} = 4\varepsilon$$ which restricts the equation of state of the effective fluid (Berezin et al., 2024).

2. Variational Principle and Field Equations

The fundamental action for these models is a single variational principle, extending the ideal fluid action with dynamical particle number: $$S_{\mathrm{tot}} = \int d^4x \sqrt{-g} \left\{ -\varepsilon(X, \varphi, n) + \lambda_0(u_\mu u^\mu - 1) + \lambda_2 X_{,\mu} u^\mu + \lambda_1 [ (n u^\mu)_{;\mu} - \Phi ] \right\}$$ where $\lambda_0, \lambda_1, \lambda_2$ are Lagrange multipliers enforcing unit-timelike velocity $$C^2 = C_{\mu\nu\lambda\sigma} C^{\mu\nu\lambda\sigma}$$0, vorticity suppression, and the generalized particle-number creation law, respectively (Berezin et al., 2024).

The resulting field equations couple fluid dynamics, scalar field evolution, and spacetime geometry via:

• Generalized continuity and creation: $$C^2 = C_{\mu\nu\lambda\sigma} C^{\mu\nu\lambda\sigma}$$1
• Fluid Euler and normalization equations,
• Scalar field equations involving conformal couplings and source terms,
• Metric equations taking the form $$C^2 = C_{\mu\nu\lambda\sigma} C^{\mu\nu\lambda\sigma}$$2, all collectively subject to the conformal trace constraint.

3. Emergence of Gravitating Mirages

The strict form of the creation law $$C^2 = C_{\mu\nu\lambda\sigma} C^{\mu\nu\lambda\sigma}$$3 introduces effective energy-momentum components with the correct scaling to mimic cold dark matter ($$C^2 = C_{\mu\nu\lambda\sigma} C^{\mu\nu\lambda\sigma}$$4) and dark radiation ($$C^2 = C_{\mu\nu\lambda\sigma} C^{\mu\nu\lambda\sigma}$$5) in the Einstein-like equations, despite originating solely from the backreaction of quantum-induced particle creation (Berezin et al., 2024). Specifically, for FRW cosmologies with $$C^2 = C_{\mu\nu\lambda\sigma} C^{\mu\nu\lambda\sigma}$$6, $$C^2 = C_{\mu\nu\lambda\sigma} C^{\mu\nu\lambda\sigma}$$7, and constant scalar $$C^2 = C_{\mu\nu\lambda\sigma} C^{\mu\nu\lambda\sigma}$$8, the energy density reads: $$C^2 = C_{\mu\nu\lambda\sigma} C^{\mu\nu\lambda\sigma}$$9 Consequently, $$\Phi = \alpha C^2 + \beta \left( \varphi \Box \varphi - \tfrac{1}{6} \varphi^2 R + \Lambda \varphi^4 \right) + \gamma_1 \varphi n + \gamma_2 n^{4/3}$$0 parameterizes an emergent pressureless dust-like component (dark matter mirage), while $$\Phi = \alpha C^2 + \beta \left( \varphi \Box \varphi - \tfrac{1}{6} \varphi^2 R + \Lambda \varphi^4 \right) + \gamma_1 \varphi n + \gamma_2 n^{4/3}$$1 determines an effective radiation bath (dark radiation mirage).

These "mirages" are not associated with microphysical entropy or fundamental particles; they instead represent the classical envelope of quantum particle-creation backreaction, entirely within the vacuum sector.

4. Connections to Optical Metrics, Lensing, and Null Geodesics

The concept of gravitating mirages aligns closely with the optical-mechanical analogy in general relativity. Light-ray geodesics in gravitational fields can be recast as geodesics of a conformally related optical metric $$\Phi = \alpha C^2 + \beta \left( \varphi \Box \varphi - \tfrac{1}{6} \varphi^2 R + \Lambda \varphi^4 \right) + \gamma_1 \varphi n + \gamma_2 n^{4/3}$$2, where $$\Phi = \alpha C^2 + \beta \left( \varphi \Box \varphi - \tfrac{1}{6} \varphi^2 R + \Lambda \varphi^4 \right) + \gamma_1 \varphi n + \gamma_2 n^{4/3}$$3 is an effective, position-dependent refractive index encapsulating the gravitational potential (2002.04390). This mapping preserves null geodesics under conformal transformations: $$\Phi = \alpha C^2 + \beta \left( \varphi \Box \varphi - \tfrac{1}{6} \varphi^2 R + \Lambda \varphi^4 \right) + \gamma_1 \varphi n + \gamma_2 n^{4/3}$$4 Therefore, gravitational lensing (“gravitational mirages”) can be fully described using gradient-index optics, with dark-matter-like effects arising from the structure of the effective refractive index—mathematically identical to the mirage density. Lensing observables, such as deflection angles and Einstein ring radii, emerge naturally in this formalism.

5. Conformal Invariance, Geometrized Shadows, and Observational Constraints

Null geodesic invariance under conformal transformations suggests that apparent lensing and shadow structures of black holes or compact objects would be invariant in form. However, observables measured in dimensionless units—especially shadow radii expressed as $$\Phi = \alpha C^2 + \beta \left( \varphi \Box \varphi - \tfrac{1}{6} \varphi^2 R + \Lambda \varphi^4 \right) + \gamma_1 \varphi n + \gamma_2 n^{4/3}$$5—depend on the definition of mass, which is not preserved under conformal rescaling. The ADM mass and active gravitational mass transform nontrivially: $$\Phi = \alpha C^2 + \beta \left( \varphi \Box \varphi - \tfrac{1}{6} \varphi^2 R + \Lambda \varphi^4 \right) + \gamma_1 \varphi n + \gamma_2 n^{4/3}$$6 where $$\Phi = \alpha C^2 + \beta \left( \varphi \Box \varphi - \tfrac{1}{6} \varphi^2 R + \Lambda \varphi^4 \right) + \gamma_1 \varphi n + \gamma_2 n^{4/3}$$7 is the conformal factor relating frames (Pal et al., 2021). Thus, the numerical values of shadow radii constrain conformally invariant gravity models, as shown by exclusion plots derived from Event Horizon Telescope measurements of M87*. A plausible implication is that strong-field lensing and shadow measurements can discriminate between conformally related gravitational theories even when the underlying null geodesics coincide.

6. Mirage Phenomenology in Gravitational Waves and Quantum Field Analogies

In cosmological and gravitational wave physics, conformal invariance of the perturbation sector guarantees that, for luminal propagation ($$\Phi = \alpha C^2 + \beta \left( \varphi \Box \varphi - \tfrac{1}{6} \varphi^2 R + \Lambda \varphi^4 \right) + \gamma_1 \varphi n + \gamma_2 n^{4/3}$$8), all wave dynamics and luminosity distances mirror GR—unless matter couples minimally to a frame other than the Einstein frame (Romano et al., 2023). Gravitating mirages thus represent an observational "mirage": in scenarios respecting strict conformal symmetry, all physical observables associated with propagation or quantum emission from horizons (e.g., in the moving mirror model) are invariant under conformal (or Möbius) transformations—yielding families of indistinguishable trajectories, fluxes, and spectra (Good et al., 2021).

This suggests that, at the level of energetics and entanglement, conformally invariant mirages are not distinguishable from real sources without direct access to the microscopics of the underlying sector.

7. Implications and Physical Interpretation

Conformally invariant gravitating mirages provide a framework in which effective cold dark matter and dark radiation components emerge from particle creation in strong fields, constrained entirely by symmetry. They serve as test beds for theories where induced gravity and conformal invariance determine the allowed form of energy-momentum sources, and provide a bridge connecting geometric, field-theoretic, and optical perspectives on gravitational lensing, horizon physics, and cosmological expansion. A plausible implication is that if the universe is described at a fundamental level by a conformally invariant gravity plus matter system, then some or all of the observed dark sectors may be gravitating mirages rather than real particle populations (Berezin et al., 2024). Observational strategies sensitive to mass definitions, strong-field lensing, and GW-EM luminosity distance discrepancies are critical in distinguishing these scenarios from non-conformal or microphysical dark matter models.