Rainbow Metrics in Quantum Gravity

• Rainbow metrics are energy-dependent generalizations of spacetime geometry that assign a distinct metric to particles based on their energy levels.
• They are derived from modified dispersion relations in Doubly Special Relativity, utilizing functions f and g to encode Planck-scale deformations and effectively alter kinematical properties.
• These models provide an effective parametrization for studying black-hole thermodynamics and cosmological singularities, though their predictive power depends critically on the operational definition of the probe energy E★.

Rainbow metrics, also called rainbow gravity, are an energy-dependent generalization of spacetime geometry motivated by modified dispersion relations and Doubly Special Relativity (DSR). Instead of a single geometry $g_{\mu\nu}(x)$, one works with a family of metrics $g_{\mu\nu}(x;E)$ or $g_{\mu\nu}(E_\star)$, so that different energy quanta “see” different effective geometries. In the literature summarized here, rainbow metrics are used to encode Planck-scale deformations of relativistic kinematics in black-hole physics, cosmology, and relative-locality constructions, and are often treated as an effective parametrization rather than as a fundamental theory (Boumali, 16 Feb 2026, Santos et al., 2015).

1. Kinematical origin in DSR and modified dispersion relations

DSR deforms special-relativistic kinematics while preserving the relativity principle by introducing a second invariant scale, typically the Planck energy $E_{\rm Pl}$, in addition to the speed of light. At the one-particle level, this is encoded in a modified dispersion relation (MDR) of the form

$\mathcal{C}(E,\mathbf{p};E_{\rm Pl})=m^2,$

and a widely used parametrization is

$$E^2 f^2\!\left(\frac{E}{E_{\rm Pl}}\right)-p^2 g^2\!\left(\frac{E}{E_{\rm Pl}}\right)=m^2,$$

with $f(0)=g(0)=1$ so that ordinary special relativity is recovered at low energy. For massless particles, the ratio $g/f$ controls the relation between energy and momentum (Boumali, 16 Feb 2026).

Two models recur in the literature. In the Amelino–Camelia-type MDR, the leading correction is proportional to $E p^2/E_{\rm Pl}$, and in the $f,g$ parametrization one has $g_{\mu\nu}(x;E)$0 and $g_{\mu\nu}(x;E)$1. In the Magueijo–Smolin invariant,

$g_{\mu\nu}(x;E)$2

one has $g_{\mu\nu}(x;E)$3, so $g_{\mu\nu}(x;E)$4 (Boumali, 16 Feb 2026).

The central curved-spacetime problem is then how to couple such MDRs to gravity without reinstating a preferred frame. The literature reviewed in (Boumali, 16 Feb 2026) distinguishes three common extensions beyond flat spacetime: MDRs in local orthonormal frames on fixed backgrounds, phase-space or Hamiltonian geometry with relative locality, and rainbow metrics.

2. Construction of energy-dependent geometry

The defining move in rainbow gravity is to replace a single metric by an energy-dependent family of effective metrics. Technically, one introduces energy-dependent orthonormal frames

$g_{\mu\nu}(x;E)$5

where $g_{\mu\nu}(x;E)$6 is an energy-independent orthonormal frame and $g_{\mu\nu}(x;E)$7 labels the probing scale. The metric seen by a probe of energy $g_{\mu\nu}(x;E)$8 is then

$g_{\mu\nu}(x;E)$9

The name “rainbow” reflects the idea that different energy quanta propagate on different “colored” metrics (Boumali, 16 Feb 2026).

For a static, spherically symmetric background

$g_{\mu\nu}(E_\star)$0

the corresponding rainbow-deformed metric is

$g_{\mu\nu}(E_\star)$1

with $g_{\mu\nu}(E_\star)$2 and $g_{\mu\nu}(E_\star)$3 treated as constants with respect to $g_{\mu\nu}(E_\star)$4 once $g_{\mu\nu}(E_\star)$5 is fixed (Boumali, 16 Feb 2026).

In homogeneous and isotropic cosmology, the rainbow FLRW line element takes the form

$g_{\mu\nu}(E_\star)$6

so that the effective scale factor seen by an energy-$g_{\mu\nu}(E_\star)$7 probe is $g_{\mu\nu}(E_\star)$8. In cosmology the probe energy is itself time-dependent through redshift, so $g_{\mu\nu}(E_\star)$9 and $E_{\rm Pl}$0 introduce an additional energy-controlled time dependence into the metric (Santos et al., 2015).

3. Black-hole thermodynamics and the $E_{\rm Pl}$1 scaling

For static spherically symmetric black holes, the principal result of the recent comparative analysis is that two implementations of DSR/MDR effects are equivalent at the level of Hawking temperature if they are evaluated at the same energy scale $E_{\rm Pl}$2: a fixed background with local-frame MDR, and an energy-dependent rainbow metric. When the same prescription is used for the deformation energy scale, both yield

$E_{\rm Pl}$3

where $E_{\rm Pl}$4 is the classical surface gravity (Boumali, 16 Feb 2026).

In the rainbow-metric computation, the horizon radius remains defined by $E_{\rm Pl}$5, while the surface gravity rescales as $E_{\rm Pl}$6. The same temperature is recovered from the tunneling method in the fixed-background local-frame MDR picture. This establishes that, for static spherically symmetric black holes, rainbow metrics and local-frame MDRs are two equivalent parametrizations once the operational definition of $E_{\rm Pl}$7 is fixed (Boumali, 16 Feb 2026).

The model dependence is entirely carried by $E_{\rm Pl}$8. For the Amelino–Camelia-type MDR,

$E_{\rm Pl}$9

so the temperature is suppressed at fixed $\mathcal{C}(E,\mathbf{p};E_{\rm Pl})=m^2,$0 when $\mathcal{C}(E,\mathbf{p};E_{\rm Pl})=m^2,$1. For the Magueijo–Smolin invariant, $\mathcal{C}(E,\mathbf{p};E_{\rm Pl})=m^2,$2, hence $\mathcal{C}(E,\mathbf{p};E_{\rm Pl})=m^2,$3: the surface-gravity Hawking temperature is unchanged despite the underlying DSR structure (Boumali, 16 Feb 2026).

A central ambiguity is the meaning of $\mathcal{C}(E,\mathbf{p};E_{\rm Pl})=m^2,$4. The literature uses at least three prescriptions: the conserved Killing energy at infinity, a local energy in a regular near-horizon frame, or a self-consistent scale such as $\mathcal{C}(E,\mathbf{p};E_{\rm Pl})=m^2,$5. The recent review argues that many discrepancies in the literature arise mainly from different choices of $\mathcal{C}(E,\mathbf{p};E_{\rm Pl})=m^2,$6, not from a genuine conflict between fixed-background MDR and rainbow-metric implementations (Boumali, 16 Feb 2026).

The same review also emphasizes that evaporation is modified by more than the temperature shift alone. DSR effects can change mass thresholds, phase space and greybody factors, and composition laws. For macroscopic black holes, however, corrections are suppressed by ratios such as $\mathcal{C}(E,\mathbf{p};E_{\rm Pl})=m^2,$7 or $\mathcal{C}(E,\mathbf{p};E_{\rm Pl})=m^2,$8, and become significant only near the Planck regime (Boumali, 16 Feb 2026).

4. Cosmology, modified thermodynamics, and singularity analyses

In rainbow cosmology, the expansion dynamics can be written in terms of the effective expansion scalar

$\mathcal{C}(E,\mathbf{p};E_{\rm Pl})=m^2,$9

with modified Friedmann and continuity equations

$$E^2 f^2\!\left(\frac{E}{E_{\rm Pl}}\right)-p^2 g^2\!\left(\frac{E}{E_{\rm Pl}}\right)=m^2,$$0

The Raychaudhuri equation acquires both the usual focusing term and an additional term proportional to $$E^2 f^2\!\left(\frac{E}{E_{\rm Pl}}\right)-p^2 g^2\!\left(\frac{E}{E_{\rm Pl}}\right)=m^2,$$1, so the sign and magnitude of $$E^2 f^2\!\left(\frac{E}{E_{\rm Pl}}\right)-p^2 g^2\!\left(\frac{E}{E_{\rm Pl}}\right)=m^2,$$2 become dynamically relevant (Santos et al., 2015).

A central result of the singularity analysis is that consistent thermodynamics is indispensable. For a gas of particles obeying an MDR, the density of states is modified to

$$E^2 f^2\!\left(\frac{E}{E_{\rm Pl}}\right)-p^2 g^2\!\left(\frac{E}{E_{\rm Pl}}\right)=m^2,$$3

so the equation-of-state parameter $$E^2 f^2\!\left(\frac{E}{E_{\rm Pl}}\right)-p^2 g^2\!\left(\frac{E}{E_{\rm Pl}}\right)=m^2,$$4 becomes a nontrivial function of temperature. Earlier claims of nonsingular behaviour that used classical equipartition or fixed $$E^2 f^2\!\left(\frac{E}{E_{\rm Pl}}\right)-p^2 g^2\!\left(\frac{E}{E_{\rm Pl}}\right)=m^2,$$5 are therefore not consistent with the MDR framework (Santos et al., 2015).

Two explicit examples illustrate the range of behaviour. In an exponential MDR with $$E^2 f^2\!\left(\frac{E}{E_{\rm Pl}}\right)-p^2 g^2\!\left(\frac{E}{E_{\rm Pl}}\right)=m^2,$$6 and $$E^2 f^2\!\left(\frac{E}{E_{\rm Pl}}\right)-p^2 g^2\!\left(\frac{E}{E_{\rm Pl}}\right)=m^2,$$7, the high-temperature limit drives $$E^2 f^2\!\left(\frac{E}{E_{\rm Pl}}\right)-p^2 g^2\!\left(\frac{E}{E_{\rm Pl}}\right)=m^2,$$8, so radiation behaves effectively like dust in the ultraviolet and the approach to the singularity is slowed down, but not avoided. In a power-law MDR with $$E^2 f^2\!\left(\frac{E}{E_{\rm Pl}}\right)-p^2 g^2\!\left(\frac{E}{E_{\rm Pl}}\right)=m^2,$$9 and $f(0)=g(0)=1$0, one finds $f(0)=g(0)=1$1 and $f(0)=g(0)=1$2, the scaling expected for a thermal system in two spacetime dimensions; here the ultraviolet fluid behaves as a stiff fluid and the focusing condition is strengthened. In both cases the extrapolated rainbow-metric evolution remains singular (Santos et al., 2015).

A separate thermodynamic analysis of flat rainbow FRW cosmology studies whether Jacobson’s derivation of Friedmann equations survives when $f(0)=g(0)=1$3 and $f(0)=g(0)=1$4 are genuinely time-dependent through $f(0)=g(0)=1$5. The conclusion is negative in general: the first law of thermodynamics together with the classical area law, even supplemented by a logarithmic entropy correction,

$f(0)=g(0)=1$6

is not sufficient to recover the general modified Friedmann equations unless the rainbow functions satisfy an additional constraint. In the large-horizon limit this reduces to

$f(0)=g(0)=1$7

so $f(0)=g(0)=1$8 in the classical regime (Ashour et al., 2019).

Taken together, these results imply that rainbow cosmology substantially modifies early-universe kinematics and thermodynamics, but does not by itself establish singularity resolution. The effective description becomes unreliable precisely in the regime $f(0)=g(0)=1$9, where the probe energy grows without bound and full quantum-gravitational effects are expected to dominate (Santos et al., 2015, Ashour et al., 2019).

5. Relative locality and emergence from quantum spacetime

One route to rainbow metrics derives them from the Relative Locality framework rather than postulating them. In the $g/f$0-dimensional $g/f$1-Poincaré/$g/f$2-Minkowski setting, a momentum-dependent spacetime metric $g/f$3 can be constructed from the deformed symplectic structure through a generalized Pythagorean relation. In this formalism, rainbow metrics are equivalent to the Hamiltonian formulation of Relative Locality in flat spacetime, and the null condition reproduces the same momentum-dependent massless worldline,

$g/f$4

found in the Hamiltonian approach (Loret et al., 2015).

This construction also sharpens a conceptual distinction within the rainbow-gravity literature. For the first-order $g/f$5-Poincaré choice $g/f$6, $g/f$7, the associated Magueijo–Smolin line element

$g/f$8

is not invariant under the deformed boost that leaves the $g/f$9-Poincaré Casimir invariant. By contrast, the Relative Locality-based momentum-dependent metric is explicitly tied to the deformed symplectic sector and is constructed to yield an invariant line element under the deformed symmetries (Loret et al., 2015).

A second route derives rainbow metrics from quantum cosmology. In quantum field theory on a quantum FLRW spacetime, under a Born–Oppenheimer test-field approximation, each Fourier mode of a scalar field propagates as if it saw an effective classical FLRW metric whose lapse and scale factor depend on the mode momentum $E p^2/E_{\rm Pl}$0. Matching the matter Hamiltonian on the quantum background to the matter Hamiltonian on a classical FLRW geometry yields an effective metric $E p^2/E_{\rm Pl}$1, so that different modes propagate on different metrics (Assanioussi et al., 2014).

In the low-momentum regime, this gives a modified dispersion relation

$E p^2/E_{\rm Pl}$2

where the state-dependent parameter

$E p^2/E_{\rm Pl}$3

summarizes the relevant quantum-geometric expectation values. For massless quanta, the effective relation becomes $E p^2/E_{\rm Pl}$4, so all modes see the same high-energy metric but, in general, a different one from the low-energy metric. This derivation is theory-agnostic in the sense that it assumes only a quantum-gravitational Hilbert space $E p^2/E_{\rm Pl}$5 and suitable operators, not a specific ultraviolet completion (Assanioussi et al., 2014).

6. Conceptual status, ambiguities, and scope

Across these literatures, rainbow metrics occupy an intermediate conceptual position. They are directly built from DSR-inspired MDRs and are easy to apply in semiclassical settings, but they do not automatically encode the full momentum-space and interaction structure of DSR. More rigorous phase-space approaches, including Hamilton geometry and relative locality, suggest that the correct implementation of deformed relativistic kinematics in gravity may not literally be an energy-dependent metric (Boumali, 16 Feb 2026, Loret et al., 2015).

A recurrent source of ambiguity is the operational meaning of the energy argument. In black-hole problems, using the conserved energy at infinity, a static local energy, or a freely falling local energy gives physically different prescriptions. In cosmology, the energy entering $E p^2/E_{\rm Pl}$6 and $E p^2/E_{\rm Pl}$7 redshifts and can become arbitrarily large as $E p^2/E_{\rm Pl}$8. These are not mere calculational conventions; the recent literature treats the identification of the deformation scale as part of the physical content of the model (Boumali, 16 Feb 2026, Santos et al., 2015).

The status of rainbow metrics is therefore predominantly effective. In black-hole thermodynamics, the recent synthesis concludes that for macroscopic black holes the corrections are tiny and become relevant only near the Planck regime, where semiclassical approximations fail. In rainbow cosmology, the effective description becomes unreliable when probe energies approach the regime in which one would expect genuinely nonperturbative quantum gravity. In this sense, rainbow metrics provide a compact parametrization of MDR effects on propagation, horizons, and thermodynamic quantities, but they do not remove the need for a full quantum-gravitational description in the ultraviolet (Boumali, 16 Feb 2026, Santos et al., 2015, Ashour et al., 2019).