Rastall Black Holes in Modified Gravity

• Rastall black holes are a family of solutions in modified gravity where non-standard energy-momentum conservation is driven by the matter trace.
• The modified field equations produce diverse geometries including static, rotating, and charged spacetimes with Rastall-dependent corrections in horizon structure and thermodynamics.
• Observational features such as shadows, quasinormal modes, and phase transition criticality offer practical tests to distinguish Rastall gravity from General Relativity.

Rastall black holes are black-hole solutions studied in Rastall gravity, a non-Einstein framework in which the usual covariant conservation law of the matter energy-momentum tensor is replaced by $T^{\mu\nu}{}_{;\mu}=\lambda R^{,\nu}$. In this setting the gravitational field equations can be written either as $$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}$$ or, equivalently, as Einstein equations with an effective source containing the trace $T$. The resulting black-hole sector is not a single geometry but a family of static, rotating, charged, higher-dimensional, AdS, and matter-dressed spacetimes. A recurrent structural theme is that traceless sectors, especially Maxwell electrodynamics, often reproduce the corresponding general-relativistic geometries, whereas traceful fluids, nonlinear electrodynamics, noncommutative sources, quintessence-like media, and string clouds generate genuinely Rastall-dependent effects in the metric, horizon structure, thermodynamics, and optical observables (Heydarzade et al., 2016, Heydarzade et al., 2017, Nashed, 2022).

1. Rastall framework and the role of the matter trace

The defining modification is the replacement of the standard conservation law by

$T^{\mu\nu}{}_{;\mu}=\lambda R^{,\nu},$

together with field equations of the form

$$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}.$$

Several notational conventions occur in the literature, including $\mu=\kappa\lambda$, $\psi=\kappa\lambda$, and $\epsilon$, but the structural point is unchanged: Rastall gravity couples curvature to matter through the trace sector. In one common rewriting,

$$G_{\mu\nu}=\kappa\left(T_{\mu\nu}-\frac{\kappa\lambda}{4\kappa\lambda-1}g_{\mu\nu}T\right),$$

so the deformation disappears when $T=0$. The Einstein limit is recovered for $$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}$$0, while values such as $$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}$$1 are excluded because the effective couplings diverge (Heydarzade et al., 2016).

This trace dependence is decisive for black-hole physics. In the electromagnetic sector, where the Maxwell stress tensor is traceless in four dimensions, Rastall corrections can become dynamically irrelevant. Static charged solutions sourced only by Maxwell fields reduce to the usual Reissner–Nordström geometry, and one 2022 analysis makes the point explicitly: the Rastall parameter does not affect the linear charged solution because $$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}$$2, whereas a nonlinear electrodynamics source with $$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}$$3 reactivates the coupling and produces a Rastall-dependent effective cosmological term (Nashed, 2022).

A second major structural consequence is that perfect-fluid or Kiselev-like matter does not keep its Einsteinian radial dressing. In the static charged family surrounded by a perfect fluid, the surrounding-field contribution appears as

$$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}$$4

so Rastall gravity changes the radial exponent itself. This motivates the effective-fluid language used throughout the literature: a source with bare equation-of-state parameter $$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}$$5 in Rastall gravity can mimic a GR source with a different effective equation of state (Heydarzade et al., 2017).

2. Static spherically symmetric families

A foundational result is the 2016 derivation of the charged static solution in Rastall gravity coupled to electromagnetism and an additional matter sector. Starting from

$$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}$$6

with $$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}$$7, the field equations imply a consistency condition because two reduced equations have identical left-hand sides. Excluding the unphysical case $$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}$$8, consistency forces

$$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}$$9

The non-electromagnetic matter must therefore behave as vacuum energy, and the solution becomes

$T$0

with

$T$1

In this construction the cosmological constant is not added by hand but emerges from consistency of the non-vacuum Rastall equations. Schwarzschild, Reissner–Nordström, and Schwarzschild–(A)dS appear as special subclasses (Heydarzade et al., 2016).

A broader static class was obtained in 2017 for charged and uncharged black holes surrounded by perfect fluids in the Kiselev sense. The charged metric function is

$T$2

and the corresponding density is

$T$3

with

$T$4

This family contains dust, radiation, quintessence, cosmological constant, and phantom surroundings. Radiation and cosmological-constant cases are exceptional because the metric form collapses to the corresponding Einsteinian expression, whereas dust, quintessence, and phantom sectors acquire Rastall-dependent exponents and effective equations of state (Heydarzade et al., 2017).

The linear-versus-nonlinear matter distinction was sharpened further in 2022. For linear Maxwell electrodynamics, the charged solution remains Reissner–Nordström. For a special nonlinear electrodynamics model, however, the exact solution becomes

$T$5

The resulting spacetime is an Anti-de-Sitter Reissner–Nordström geometry in which the Rastall parameter is absorbed into an effective cosmological constant. The mechanism is explicitly tied to the non-vanishing trace of the nonlinear source (Nashed, 2022).

3. Rotating, charged, and NUT-generalized geometries

The astrophysically relevant extension is the rotating sector. A 2017 construction used the Newman–Janis algorithm to generate a rotating counterpart of the static charged perfect-fluid Rastall black hole. The seed metric is

$T$6

The resulting spacetime is Kerr–Newman-like, with the surrounding-fluid term dressed by the Rastall-dependent exponent $T$7. The paper emphasizes that Rastall coupling changes the effective gravitational behavior of the ambient matter and thereby modifies the horizon equation, ergosphere, critical spin $T$8, temperature, and specific heat. It also notes an unusual feature: the horizon equation is generally $T$9-dependent (Kumar et al., 2017).

A closely related rotating solution surrounded by anisotropic fluid employs the Azreg-Aïnou modified Newman–Janis algorithm and takes

$T^{\mu\nu}{}_{;\mu}=\lambda R^{,\nu},$0

This geometry can admit up to three horizons,

$T^{\mu\nu}{}_{;\mu}=\lambda R^{,\nu},$1

with $T^{\mu\nu}{}_{;\mu}=\lambda R^{,\nu},$2 an additional cosmological-like horizon. The same paper develops an analytic shadow formalism and finds that increasing $T^{\mu\nu}{}_{;\mu}=\lambda R^{,\nu},$3 or $T^{\mu\nu}{}_{;\mu}=\lambda R^{,\nu},$4 decreases the shadow size and tends to suppress Kerr-like asymmetry (Kumar et al., 2017).

The most general rotating family in the supplied literature is the Kerr–Newman–NUT–Kiselev solution in Rastall gravity. Its Boyer–Lindquist-type form is governed by

$T^{\mu\nu}{}_{;\mu}=\lambda R^{,\nu},$5

together with

$T^{\mu\nu}{}_{;\mu}=\lambda R^{,\nu},$6

This introduces rotation $T^{\mu\nu}{}_{;\mu}=\lambda R^{,\nu},$7, NUT charge $T^{\mu\nu}{}_{;\mu}=\lambda R^{,\nu},$8, electric and magnetic charges, and a Rastall-deformed quintessence term. The 2019 analysis argues that this solution is not equivalent to an Einstein black hole with a redefined cosmological constant because the Ricci scalar is not constant and the radial deformation $T^{\mu\nu}{}_{;\mu}=\lambda R^{,\nu},$9 is not the same as the $$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}.$$0 modification of Kerr–NUT geometries (Sakti et al., 2019). A 2021 Kerr/CFT treatment of the same background shows that the Rastall coupling propagates into the near-horizon coefficient

$$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}.$$1

and hence into the CFT temperatures, central charges, absorption cross-sections, and echo time delays in exotic-compact-object extensions (Sakti et al., 2021).

These rotating constructions are not identical. Some use perfect fluids, some anisotropic fluids, and some Kiselev or quintessential sectors with NUT twist. This suggests that “rotating Rastall black hole” designates a class of Kerr-like nonvacuum solutions rather than a unique canonical metric.

4. Regular, noncommutative, conformally flat, and decoupled extensions

Regularity is not automatic in Rastall gravity. For noncommutative geometry inspired black holes with Gaussian-smeared matter, the Schwarzschild-like ansatz

$$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}.$$2

leads to

$$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}.$$3

Unlike the GR noncommutative Schwarzschild solution, this spacetime is not regular: the metric function stays finite at the origin, but the Ricci scalar diverges, the geometry has at most one horizon, and evaporation ends in a point-like massive zero-temperature remnant. A second, more general ansatz,

$$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}.$$4

combined with the equation of state $$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}.$$5, yields a regular Rastall black hole whose geometric structure and temperature are close to the standard GR noncommutative case (Ma et al., 2017).

A different line of development appears in generalized Rastall theory. There the conformally flat condition gives

$$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}.$$6

In the original Rastall framework this metric is supported by an anisotropic source with

$$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}.$$7

while in the generalized subclass the same functional form can be interpreted as Einstein-like gravity with an induced cosmological-constant sector. The same 2019 study also constructs a non-singular generalized Rastall black hole from the Dymnikova-type density profile

$$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}.$$8

leading to

$$G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu\nu}.$$9

Regularity requires $\mu=\kappa\lambda$0 and $\mu=\kappa\lambda$1 (Moradpour et al., 2019).

A newer constructive approach uses gravitational decoupling via extended geometric deformation. Starting from Schwarzschild,

$\mu=\kappa\lambda$2

the metric potentials are deformed by

$\mu=\kappa\lambda$3

with a dual matter source. Three extended solutions are obtained from restrictions on the metric potentials and extra source. The abstract-level conclusion is severe: none of the obtained models satisfy the energy conditions, while only the model corresponding to the barotropic equation of state mimics an asymptotically flat spacetime (Sharif et al., 6 Apr 2025).

5. Geodesics, shadows, quasinormal modes, and observational diagnostics

The geodesic sector is unusually rich because the same Rastall parameter that changes the metric exponent also changes the radial polynomial governing test-particle motion. For the charged static black hole surrounded by a perfect fluid, the geodesic equation can be solved exactly in selected cases. Null trajectories reduce to Weierstrass elliptic functions, while timelike motions with quintic or sextic radial polynomials are solved with Kleinian sigma functions. The orbit taxonomy includes terminating orbits, escape orbits, bound orbits, two-world escape orbits, and many-world bound orbits; the Reissner–Nordström and Schwarzschild limits are recovered when $\mu=\kappa\lambda$4 and $\mu=\kappa\lambda$5, respectively (Soroushfar et al., 2018).

Shadow phenomenology is not universal across Rastall black-hole models. In the rotating anisotropic-fluid solution, increasing the Rastall coupling $\mu=\kappa\lambda$6 or the structure parameter $\mu=\kappa\lambda$7 decreases the shadow size and can make it more symmetric about the projected rotation axis (Kumar et al., 2017). By contrast, a 2026 study of a different rotating Rastall black hole parameterized by $\mu=\kappa\lambda$8, $\mu=\kappa\lambda$9, and $\psi=\kappa\lambda$0 finds that increasing $\psi=\kappa\lambda$1 enlarges the shadow radius and decreases the distortion parameter, while increasing $\psi=\kappa\lambda$2 enlarges the contour and shifts it toward positive $\psi=\kappa\lambda$3. In that model the predicted angular diameters remain compatible with the EHT confidence intervals for both M87$\psi=\kappa\lambda$4 and Sgr A$\psi=\kappa\lambda$5 over the parameter ranges studied (Sultan et al., 14 Jun 2026). The difference is model-dependent rather than contradictory: distinct surrounding matter prescriptions produce distinct optical trends.

Static AdS Rastall black holes with a cloud of strings and a quintessence-like field add another optical pattern. With

$\psi=\kappa\lambda$6

for $\psi=\kappa\lambda$7, the null effective potential decreases with increasing $\psi=\kappa\lambda$8 and $\psi=\kappa\lambda$9, the photon-sphere radius decreases slightly with increasing $\epsilon$0, but the shadow radius

$\epsilon$1

increases with both $\epsilon$2 and $\epsilon$3. The same model shifts the ISCO outward as either parameter increases (Moreira et al., 18 Sep 2025).

Quasinormal modes provide a complementary diagnostic. For charged black holes with non-linear-electrodynamic sources in Rastall gravity, the effective scalar potential and the QNM spectrum depend strongly on the surrounding field. The metric is $\epsilon$4-independent for cosmological-constant and radiation backgrounds, but dust, quintessence, and phantom cases retain explicit Rastall dependence. The non-linear electrodynamic black hole often has smaller real frequencies than the ordinary charged Rastall black hole, and the imaginary part is typically more sensitive to the structural parameter $\epsilon$5 than the real part (Gogoi et al., 2021).

6. Thermodynamics, criticality, and interpretive debates

Many Rastall black-hole families preserve the area law for entropy while modifying temperature, pressure, and stability through altered horizon equations. For the rotating charged perfect-fluid black hole, the outer-horizon quantities are taken as

$\epsilon$6

with a generalized first law

$\epsilon$7

The Hawking temperature acquires direct $\epsilon$8 corrections, and the specific heat contains explicit dependence on the surrounding field and Rastall coupling (Kumar et al., 2017).

Higher-dimensional solutions broaden this picture. In $\epsilon$9-dimensional topological black holes sourced by a power-Maxwell field, quintessence, and a cosmological constant, Rastall gravity modifies the coefficient of the $$G_{\mu\nu}=\kappa\left(T_{\mu\nu}-\frac{\kappa\lambda}{4\kappa\lambda-1}g_{\mu\nu}T\right),$$0 term, the quintessence exponent, and the temperature and heat capacity, while the entropy is still taken to obey the area law. The solutions split into a generic branch $$G_{\mu\nu}=\kappa\left(T_{\mu\nu}-\frac{\kappa\lambda}{4\kappa\lambda-1}g_{\mu\nu}T\right),$$1 and a BTZ-like branch $$G_{\mu\nu}=\kappa\left(T_{\mu\nu}-\frac{\kappa\lambda}{4\kappa\lambda-1}g_{\mu\nu}T\right),$$2, and include spherical, planar, and hyperbolic horizons (Lin et al., 2018).

The AdS sector reveals full extended-phase-space criticality. For $$G_{\mu\nu}=\kappa\left(T_{\mu\nu}-\frac{\kappa\lambda}{4\kappa\lambda-1}g_{\mu\nu}T\right),$$3-dimensional AdS black holes surrounded by a perfect fluid, the metric function is

$$G_{\mu\nu}=\kappa\left(T_{\mu\nu}-\frac{\kappa\lambda}{4\kappa\lambda-1}g_{\mu\nu}T\right),$$4

The pressure is Rastall-dependent,

$$G_{\mu\nu}=\kappa\left(T_{\mu\nu}-\frac{\kappa\lambda}{4\kappa\lambda-1}g_{\mu\nu}T\right),$$5

and the equation of state displays van der Waals-like criticality. The critical point satisfies the Ehrenfest equations, and the Prigogine–Defay ratio is

$$G_{\mu\nu}=\kappa\left(T_{\mu\nu}-\frac{\kappa\lambda}{4\kappa\lambda-1}g_{\mu\nu}T\right),$$6

so the critical point is interpreted as a second-order equilibrium phase transition (Ali, 2019). A related 2025 AdS model with strings and quintessence-like matter shows both a heat-capacity divergence and a swallowtail Gibbs structure, indicating second-order criticality and a first-order small/large black-hole transition in the same extended thermodynamic setting (Moreira et al., 18 Sep 2025).

Thermodynamic behavior also depends on the matter sector. In the 2022 nonlinear charged solution,

$$G_{\mu\nu}=\kappa\left(T_{\mu\nu}-\frac{\kappa\lambda}{4\kappa\lambda-1}g_{\mu\nu}T\right),$$7

and the physical branch is thermodynamically stable, unlike the linear charged case with a second-order phase transition (Nashed, 2022). In generalized Rastall theory, the situation is subtler: the Euler relation can be made to hold, but the pressure and temperature inferred from thermodynamics depend on whether one uses the density-integral mass or the generalized Misner–Sharp mass. The thermodynamic pressure that satisfies Euler need not coincide with the pressure component appearing in the field equations, and equality of thermodynamic and Hawking temperatures occurs only under additional restrictions (Moradpour et al., 2019).

The conceptual debate over equivalence with GR is therefore sector-dependent. In traceless Maxwell backgrounds, multiple papers recover exactly Einsteinian geometries (Heydarzade et al., 2016, Nashed, 2022). In contrast, rotating nonvacuum solutions with quintessence or NUT charge have non-constant Ricci scalar, Rastall-dependent horizon structure, and modified Kerr/CFT data, and are presented as evidence that Rastall gravity is not merely Einstein gravity in disguise (Sakti et al., 2019, Sakti et al., 2021). A plausible synthesis is that Rastall black-hole physics is most distinctive precisely where the matter trace, ambient fluid profile, or effective non-conservation law cannot be removed by a formal redefinition of source terms.