arXiv subjects

David S. Pereira

Publications and source records attributed to David S. Pereira.

At least 19 recordsLinked to original sources

arXiv2026

Gravitational Waves and Matter Constraints in Gravity from Entropy

We show that Gravity from Entropy (GfE) exhibits a tensor-propagation pathology on generic FLRW backgrounds. In vacuum, in the symmetry-inheriting single-multiplet sector, and for the algebraic perfect-fluid constitutive embedding, the strict-principal TT system has a nonzero real characteristic only when $\dot H=K/a^2$. Away from this locus the temporal principal roots become complex, signaling a high-frequency hyperbolicity obstruction. Under the stated isotropy assumptions, a rank theorem further shows that one foundational multiplet cannot provide the isotropic bivector response needed to remove this problem.

gr-qc↗

arXiv2026

Gravitational Baryogenesis in Energy-Momentum Squared~Gravity

We demonstrate that the matter sector itself can drive baryogenesis in energy-momentum squared gravity, $f(R,\mathcal T^2)$ with $\mathcal T^2\equiv T_{μν}T^{μν}$. High-density matter corrections provide new time-dependent sources for the baryon asymmetry through derivative couplings to $\mathcal T^2$ and to the full combination $f(R,\mathcal T^2)$. Notably, the decoupling temperature is not treated as a free parameter; instead, it is fixed by the freeze-out of the Weinberg $B-L$-violating operator, directly linking the asymmetry to the light-neutrino mass scale and to the modified expansion history. Analyzing representative powers $n=1/4$, $n=1/2$, $n=5/8$, and $n=1$, we find that entropy evolution sharply distinguishes the models. The $n=1/4$ branch cannot serve as a self-contained radiation-era model, the $n=1/2$ branch is entropy-safe but too weak to reproduce the observed asymmetry, and the $n=1$ branch survives only after entropy dilution. Remarkably, the $n=5/8$ branch provides the cleanest viable realization, generating the observed baryon-to-entropy ratio with controlled effective-field-theory hierarchies.

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arXiv2026

Gravitational Waves as Thermodynamic Shear Excitations in Scalar-Tensor Gravity

Can a propagating spin--2 mode participate in a local thermodynamic process without assigning an autonomous entropy or a unique local energy to the wave itself? We show that this occurs in Jordan-frame scalar--tensor gravity, where a timelike scalar gradient defines an internal gravitational thermodynamic medium. Building on the first-order TT deformation--response relation, we identify the local shear power $\mathcal W_ϕ=π_{ab}^{(ϕ)}σ^{ab}$ from scalar-frame traction mechanics and show that its leading pure-TT contribution is second order, gauge invariant, and enters the exact scalar-sector energy balance. We further show that the same effective anisotropic stress controls the non-GR tensor damping, so the local thermodynamic exchange is directly encoded in gravitational-wave propagation. In geometric optics the corresponding ratio of shear work to GR-normalized tensor energy is controlled by the running effective Planck mass, linking the local process to the modified gravitational-wave luminosity distance. Finally, we extend this shear-excitation mechanism to nonminimally coupled scalars, metric $f(R)$ gravity, and viable luminal Horndeski theories, and identify the condition required for the same interpretation in more general tensor sectors.

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arXiv2026

A Buchdahl solution generator for Einstein-sigma gravity and multiscalar-tensor gravity

Buchdahl transformations provide a geometric route from vacuum spacetimes to scalar-field solutions. We formulate this mechanism as a target-covariant solution generator for Einstein gravity coupled to nonlinear sigma models in $D>3$ dimensions. On any connected region in which a Ricci-flat seed admits a non-null hypersurface-orthogonal cyclic coordinate with fixed-sign nonzero norm, the logarithm of that norm defines a Buchdahl potential $Σ$. A one-parameter redistribution of the cyclic norm then generates the exact Ricci source $R_{AB}=C_D(β)\partial_AΣ\partial_BΣ$, with $C_D(β)=\frac{D-2}{D-3}(1-β^2)$, while $Σ$ remains harmonic in the deformed metric. Composing $Σ$ with an affinely parametrized geodesic of a non-degenerate target metric whose squared speed is $C_D(β)$ therefore yields an exact Einstein--sigma-model solution. For positive-definite targets, the corresponding pullback condition also implies that the nontrivial scalar map has one-dimensional image; for indefinite targets, the geodesic ansatz remains a sufficient solution-generating sector but need not exhaust all possible realizations of the same rank-one spacetime source. Through the Einstein/Jordan-frame correspondence, the construction gives vacuum tensor--multiscalar solutions as well. We illustrate the unified generator with flat, spherical, hyperbolic and axion--dilaton targets and with Tangherlini, Weyl, Kasner, Rosen-wave, Gowdy, Kaluza--Klein bubble and C-metric seeds. The result separates cleanly the geometric Buchdahl source from its target-space realization and extends this rank-one cyclic sector across static, cosmological, wave and accelerating geometries.

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arXiv2026

Differential Obstructions to Curvature-Dependent Conformal Transformations

Curvature-dependent conformal rules are local forward assignments on known metrics, but they are not generically local changes of metric variables. We study the nondegenerate class $\widetilde g_{μν}=F(R[g])g_{μν}$, with $F>0$ and $F_R\neq0$. Introducing an independent auxiliary scalar localizes the forward map, while recovering the original metric requires a differential constraint. The complete inverse metric tangent map contains the nonpolynomial projector $ξ^μξ^ν/ξ^2$; hence no differentiable finite-jet inverse, i.e., a formula involving only finitely many derivatives at the same point, exists on an open set of unrestricted metrics. Branchwise functional inverses may nevertheless exist after boundary or Cauchy data are specified. Metric $f(R)$ gravity gives an explicit realization: its local Einstein-frame scalar--tensor representation is a parent theory, whereas a metric-only Einstein-side description requires a differential section governed by a normal operator. We derive the pulled-back classical Hessian and its off-shell embedding term, and in the quadratic model show how constrained Gaussian elimination produces the scalaron nonlocal kernel, the corresponding normal determinant for the displayed measure, and the zero-mode compatibility condition. Exact parent and metric solutions remain equivalent; the obstruction concerns locality and the off-shell variational and fluctuation domains. These results provide a precise framework for assessing curvature-dependent frame transformations in modified gravity and clarify their implications for effective actions, semiclassical analyses, and quantum frame equivalence.

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arXiv2026

Jordan-like invariant representations of scalar-tensor gravity and the return of imperfect-fluid thermodynamics

It has been shown that the thermodynamics of frame-invariant scalar-tensor gravity admits an Einstein-frame-like invariant representation, in which the scalar sector is minimally coupled and behaves as a perfect fluid with vanishing temperature. We show that this is not the only invariant thermodynamic organization available. Using instead the invariant matter metric, we construct a Jordan-frame-like invariant representation in which matter is minimally coupled and the gravitational action takes the Brans-Dicke-Bergmann-Wagoner form with invariant fields $Ψ=\mathcal I_1^{-1}$, $\mathcal U(Ψ)$, and $ω_{\mathrm{inv}}(Ψ)$. In this representation the metric field equations retain the Hessian sector characteristic of Jordan-frame scalar-tensor gravity. A $1+3$ decomposition of the corresponding effective stress tensor yields nonvanishing heat flux and anisotropic stress in a generic congruence. In the $Ψ$-comoving frame, the usual first-order thermodynamic identifications are recovered in invariant form, including $K_{\mathrm J}T_{\mathrm J}=-\dotΨ/(κ^2Ψ)$ and $η_{\mathrm J}=\dotΨ/(2κ^2Ψ)$, with the bulk channel obtained in homogeneous and isotropic sectors. Thus, frame-invariant scalar-tensor thermodynamics is not intrinsically restricted to an Einstein-frame-like perfect-fluid description: it also admits a Jordan-frame-like invariant formulation in which the effective imperfect-fluid interpretation is manifest.

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arXiv2026

Vacuum Gravity from Entropy: Stability, Spectra, and Exact Waves

We analyze the vacuum dynamics of Gravity from Entropy, including its algebraically constrained $G$-field formulation. Evaluating the curvature traces over zero-, one-, and two-form sectors, we show that the complete Minkowski Hessian is exactly that of the quadratic-gravity action $A R+B R_{μν}R^{μν}$, with $A=3β/\ell_{\rm P}^{4}$ and $B=5β^{2}/(2\ell_{\rm P}^{4})$. For diagonalizable curvature blocks, the same action reduces to a sum over eigenvalue logarithms and reproduces these coefficients exactly. A strict diagonal-curvature restriction on the perturbations is instead only a reduced subsector and excludes non-diagonalizable type-N wave curvatures. Linearizing the $G$-field equations and subsequently imposing the algebraic vacuum constraint reproduces the same reduced metric equation and covariant Minkowski Hessian. The spectrum contains the massless graviton, a scalar with $m_{0}^{2}=3/(5β)$, and an opposite-residue spin-2 branch with $m_{2}^{2}=-6/(5β)=-2m_{0}^{2}$. For the foundational choice $β>0$, conventional Einstein normalization therefore implies a tachyonic spin-2 instability. We also show that every four-dimensional Ricci-flat metric solves the local bulk equations through quadratic curvature order, while square-zero Ricci-flat pp-waves are exact local vacuum solutions of the analytic metric-only logarithmic branch. On the isolated massless transverse-traceless eigenspace, the quadratic translation current has the standard general-relativistic normalization.

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arXiv2026

First-order thermodynamics of multi-scalar-tensor gravity

We formulate a first-order thermodynamic description of Jordan-frame tensor--multi-scalar gravity. We derive the exact covariant $1+3$ split of the geometric sector and recast it as an effective imperfect fluid. The interpretation is Eckart-like: effective temperature, conductivity, entropy current, and entropy production are meaningful only on branches where the geometric dissipative variables satisfy matching and integrability conditions. In a generic frame the heat flux is $q_a^{(g)}=-χ(a_a+W_a)$, with $χ=-\dot{\mathcal{F}}/(8π\mathcal{F})$ and $W_a$ the residual gradient sector. In the $\mathcal{F}$-comoving frame this defines the inertial variable $χ_{\mathcal{F}}\equiv K_{\mathcal{F}}T_{\mathcal{F}}$, while a nonzero spatial term $W_a^{(\mathcal{F})}$ remains, sourced by scalar directions not aligned with the coupling. Thus the multi-field thermal description is not generically reducible to a single $KT$-type quantity. We derive transport equations for $χ_{\mathcal{F}}$, for the field-space thermal vector $χ^A$ and covector $χ_A$, and for the residual gradient sector. We introduce the diagnostics $\mathfrak D_χ=χ_Aχ^A$ and $\mathfrak D_{\rm grad}=\mathcal B_{AB}\mathrm{D}_a^{(\mathcal{F})}ϕ^A\mathrm{D}^a_{(\mathcal{F})}ϕ^B$. Their meaning depends on the kinetic matrix $\mathcal B_{AB}$: they are canonical contractions when it is nondegenerate, nonnegative norm-like diagnostics only when positive definite, and require extra structure if degenerate. With this qualification, they show that freezing the effective coupling is generally weaker than full relaxation to the GR sector. We construct the entropy current and entropy production in the coupling frame, state the assumptions for nonnegative entropy production, and show that homogeneous cosmology suppresses the spatial sector while retaining nontrivial time-like multi-scalar thermal dynamics.

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arXiv2026

Big Bang Nucleosynthesis constraints on $f(T,L_m)$ gravity

We investigate Big Bang Nucleosynthesis (BBN) in the framework of $f(T,L_m)$ gravity, where the gravitational Lagrangian depends on the torsion scalar $T$ and the matter Lagrangian $L_m$. Working within a semi-analytical BBN strategy, we encode departures from GR through the expansion-rate ratio $Z\equiv H/H_{\rm GR}$ evaluated at a characteristic freeze-out temperature and combine this with the freeze-out condition and the observationally inferred abundances of deuterium and helium-4 to constrain the free parameters of three representative EFT-motivated $f(T,L_m)$ models. A distinctive aspect of $f(T,L_m)$ cosmology is that the explicit $L_m$ dependence can induce an effective energy exchange between the standard component and the modified-gravity sector; we therefore derive the corresponding interaction term $Q$ and restrict our analysis to the adiabatic regime $\varepsilon\equiv |Q_{\rm rad}/(4Hρ)|\ll 1$ throughout the BBN window, ensuring internal consistency of the temperature-based BBN mapping. Finally, to connect the radiation-era constraints with the late-time background, we present a two-fluid (dust+radiation) analysis showing how the $L_m$-dependent corrections decouple as $\ell\propto (1+z)^4\to 0$, yielding torsion-only ($f(T)$ or TEGR) cosmologies at late times on the GR-connected branch. Our results provide transparent first-pass BBN bounds on torsion--matter EFT corrections and identify viable parameter regions consistent with early-Universe data providing a controlled starting point for further early-Universe phenomenology in $f(T,L_m)$ gravity.

astro-ph.CO↗

arXiv2026

Gravitational baryogenesis beyond the spectator approximation

The standard gravitational-baryogenesis operator $λ\,\nabla_μR\,J^μ$, with $λ\equiv ε/M_\ast^{2}$, is usually treated as a spectator interaction that generates an effective chemical potential in a prescribed background. When included in the gravitational action, however, it defines a genuine curvature--matter-coupling variational problem, relevant for the baryon, lepton, and $B\!-\!L$ currents, whether described microscopically by particle-physics operators or macroscopically by a fluid current $J^μ=n_Xu^μ$. Up to a boundary term the interaction is equivalent to $-λR\nabla_μJ^μ$, making its $f(R,{\rm Matter})$ character manifest, but the metric equations remain open unless the metric dependence of $J^μ$ is specified. For an arbitrary local realization $J^μ(Ψ,g)$ we derive the universal part of the field equations and isolate the realization-dependent tensor generated by $δ_g J^μ$. In the vector-density realization the explicit $J^α\nabla_αR$ term cancels, but an algebraic term $-λg_{μν}R\nabla_αJ^α$ survives, so the theory admits only a partial effective-Planck-mass interpretation, $M_{\rm eff}^2=M_{\rm Pl}^2-2λ\nabla_μJ^μ$, and a time-dependent effective gravitational coupling during baryogenesis. Specializing to flat Friedmann-Lemaître-Robertson-Walker (FLRW) with a homogeneous current $J^μ=n_Xu^μ$, we obtain the modified Friedmann and Raychaudhuri equations, the associated continuity relation, and dimensionless diagnostics that quantify when the spectator approximation is controlled. We also discuss the implications for gravitational-baryogenesis studies in modified theories of gravity, providing a consistent General Relativity (GR) baseline for implementations in both standard cosmology and modified-gravity frameworks.

gr-qc↗

arXiv2026

Eckart heat-flux applicability in $F(Φ,X)R$ theories and the existence of temperature gradients

We show that in single--scalar theories of the form $\mathcal{L}=F(Φ,X)R+G(Φ,X)$, a generic nonminimal coupling $F(Φ,X)$ induces, in the scalar--comoving frame, an additional transverse contribution to the effective heat flux, proportional to $(F_X/8πF)V_{\perp a}$, where $V_a \equiv h_a{}^c\nabla_c\nabla_d X\,u^d$ and $V_{\perp a}$ denotes the component orthogonal to the 4--acceleration $a_a$. This term cannot in general be written as a spatial temperature gradient, and therefore obstructs a standard Eckart interpretation of the scalar sector for arbitrary timelike scalar configurations. As a result, requiring an Eckart heat flux $q_a = -K\bigl(D_a T_g + T_g\, a_a\bigr)$ for all such configurations is possible if and only if $F_X(Φ,X)\equiv 0$, i.e.\ $F(Φ,X)=F(Φ)$, resulting in a theory that is a subclass of Horndeski. Thus, only Jordan--like theories of the type $F(Φ)R+G(Φ,X)$ admit a global Eckart fluid picture of the scalar sector, while models with $F_X\neq 0$ can recover an Eckart--like form only on highly symmetric backgrounds where the transverse contribution vanishes or collapses to a single gradient direction. We also make a brief comment on the existence of temperature gradients $D_aT_g$.

gr-qc↗

arXiv2026

Thermal channels of scalar and tensor waves in Jordan-frame scalar--tensor gravity

We study first-order scalar and tensor perturbations of Jordan-frame scalar--tensor gravity about a spatially flat FLRW background using the Einstein-like effective-fluid decomposition of the scalar sector. In the scalar-gradient frame, we derive the perturbed effective density, pressure, heat flux, and anisotropic stress, and show that they admit an exact Eckart-type constitutive identification at linear order. We then show that these same quantities appear explicitly and exhaustively in the linearized field equations: the scalar Hamiltonian, momentum, trace, and traceless Einstein-like equations are governed, respectively, by the effective density, heat-flux, pressure, and anisotropic-stress channels, while the tensor propagation equation is governed by the transverse-traceless anisotropic-stress channel. In particular, the Jordan-frame modification of gravitational-wave damping is identified with the effective transverse-traceless anisotropic stress of the scalar sector. We also derive the perturbed evolution equation for the invariant product $κT$, clarify its gauge behavior, and show that flux matching on FLRW fixes only the background value $\overline{κT}$, not its perturbation. These results leave open the possibility that gravitational waves in scalar--tensor gravity admit a deeper thermodynamic characterization, perhaps even an intrinsic one, although the present analysis establishes this only at the level of an effective constitutive description.

gr-qc↗

arXiv2026

Novel scalar degrees of freedom emerging from hybrid metric-Palatini gravity

Hybrid metric-Palatini gravity unifies the metric and Palatini formalisms while preserving a propagating scalar degree of freedom, offering a compelling route to modified gravity consistent with current observations. Motivated by this success, we consider an extended framework -- the hybrid metric-Palatini scalar-tensor (HMPST) theory -- in which an additional scalar field $ϕ$ modulates the curvature couplings, enriching the dynamics and enabling nontrivial self-interactions through scalar potentials. We focus on the analytically tractable linear-$f(\hat{R})$ subclass and study its cosmological, strong-field, and weak-field regimes. In homogeneous and isotropic settings, we identify de Sitter and matter-dominated cosmological solutions describing accelerated expansion and early-universe behavior. For static, spherically symmetric configurations, the field equations yield analytic solutions generalizing the Janis-Newman-Winicour and Buchdahl metrics, including the Schwarzschild-de Sitter limit. In the weak-field regime, linearized perturbations around Minkowski space lead to Yukawa-type corrections to the gravitational potential, with an effective Newton constant $G_{\rm eff}$ and post-Newtonian parameter $γ$ that recover General Relativity for heavy or weakly coupled scalars. These results show that the linear-$f(\hat{R})$ HMPST subclass provides a consistent and unified description of gravity across cosmological, astrophysical, and Solar System scales, offering a fertile framework for connecting modified gravity to observations and effective field-theoretic extensions.

gr-qc↗

arXiv2026

Scalar-tensor baryogenesis: a scalar-tensor completion of gravitational baryogenesis

We propose \emph{Scalar-Tensor Baryogenesis} (STB), in which the $C\!P$-violating bias needed for baryogenesis is sourced by the \emph{gravitational} scalars that appear in scalar-tensor representations of modified gravity. Derivative couplings $M_\ast^{-d}\nabla_μf(ϕ_i)\,J^μ_{B-L}$ act as an effective chemical potential $μ_{B-L}\propto\dot f$ in an FRW background, driving the plasma to a nonzero equilibrium $B\!-\!L$ density while $B\!-\!L$-violating reactions are active. The asymmetry freezes in at the dynamically determined decoupling temperature $T_D$ fixed by $Γ_{B-L}(T_D)=H(T_D)$, giving $n_b/s\propto[\dot f/(M_\ast^d T)]_{T_D}$ up to sphaleron conversion. A key structural result is an explicit \emph{on-shell/background} map -- through the Legendre relations defining the scalar potential -- between curvature-based geometric Gravitational baryogenesis operators and their scalar-tensor counterparts, together with a canonical Einstein-frame description closely paralleling spontaneous/quintessential baryogenesis, but with a gravitational (not ad hoc matter) biasing field. The map is not a mere change of variables: it imposes consistency conditions (existence of the scalar--tensor branch, local invertibility of the Legendre map, and validity of the spectator regime), thereby restricting the admissible operator space and tying $μ_{B-L}\propto\dot f$ to the modified-gravity dynamics once $F$ is specified. As an illustration, we implement STB in $F(R)=R^{1+\varepsilon}$ with $B\!-\!L$ violation from the dimension-five Weinberg operator, and reproduce the observed baryon asymmetry for $\varepsilon=\mathcal{O}(10^{-6})$ with $T_D\simeq 8.5\times10^{13}\,\mathrm{GeV}$ and negligible backreaction, while satisfying nucleosynthesis bounds and keeping the expansion arbitrarily close to the GR radiation solution.

gr-qc↗

arXiv2026

Gravitational baryogenesis in $f(T,L_m)$ gravity

The observed matter-antimatter asymmetry of the Universe remains a fundamental challenge in modern physics. In this work, we explore gravitational baryogenesis within the framework of $f(T,L_m)$ gravity, where the gravitational Lagrangian depends on both the torsion scalar $T$ and the matter Lagrangian $L_m$. We consider three representative models and examine their ability to generate the observed baryon-to-entropy ratio. Our analysis shows that couplings involving both torsion and the matter Lagrangian, $\partial_μ(-T-\frac{L_m}{L_0})$, can successfully account for the baryon asymmetry for decoupling temperatures in the range $10^{12}$-$10^{14}\,\text{GeV}$, while remaining consistent with small deviations from General Relativity. These results highlight the capacity of $f(T,L_m)$ gravity to provide novel mechanisms for baryogenesis, demonstrating that the interplay between torsion and matter-sector contributions can naturally generate the observed asymmetry. The framework also remains compatible with late-time cosmological evolution, offering a unified setting for early- and late-time dynamics.

gr-qc↗

arXiv2025

Baryon asymmetry from higher-order matter contributions in gravity

We investigate the observed asymmetry between matter and antimatter by incorporating higher-order matter contributions in gravity, specifically analyzing gravitational baryogenesis within the framework of $ f(R,\mathcal{T}^2) $ gravity, where $ R $ is the Ricci scalar and $ \mathcal{T}^2 \equiv T_{μν}T^{μν} $. We further explore the impact of high-order matter contributions by considering an interaction term analogous to those in gravitational and spontaneous baryogenesis, constructed using $ \mathcal{T}^2 $. The cutoff energy scale of the new interaction term is presented and its implications to Big Bang Nucleosynthesis (BBN) are discussed. The properties and implications of this term are analyzed within the frameworks of General Relativity and $f(R, \mathcal{T}^2)$ gravity. Furthermore, a connection to Big Bang Nucleosynthesis (BBN) is established, providing an observational constraint on the functional form of $f(R, \mathcal{T}^2)$. By introducing $\mathcal{T}^2$ into the gravitational action, we propose that these modifications could significantly influence the early Universe's dynamics, thereby altering the conditions necessary for baryogenesis to occur.

gr-qc↗

arXiv2024

Thermodynamics of the Primordial Universe

This review delves into the pivotal primordial stage of the universe, a period that holds the key to understanding its current state. To fully grasp this epoch, it is essential to consider three fundamental domains of physics: gravity, particle physics, and thermodynamics. The thermal history of the universe recreates the extreme high-energy conditions that are critical for exploring the unification of the fundamental forces, making it a natural laboratory for high-energy physics. This thermal history also offers valuable insights into how the laws of thermodynamics have governed the evolution of the universe's constituents, shaping them into the forms we observe today. Focusing on the Standard Cosmological Model (SCM) and the Standard Model of Particles (SM), this paper provides an in-depth analysis of thermodynamics in the primordial universe. The structure of the study includes an introduction to the SCM and its strong ties to thermodynamic principles. It then explores equilibrium thermodynamics in the context of the expanding universe, followed by a detailed analysis of out-of-equilibrium phenomena that were pivotal in shaping key events during the early stages of the universe's evolution.

gr-qc↗

arXiv2024

Extension of Buchdahl's Theorem on Reciprocal Solutions

Since the development of Brans-Dicke gravity, it has become well-known that a conformal transformation of the metric can reformulate this theory, transferring the coupling of the scalar field from the Ricci scalar to the matter sector. Specifically, in this new frame, known as the Einstein frame, Brans-Dicke gravity is reformulated as General Relativity supplemented by an additional scalar field. In 1959, Hans Adolf Buchdahl utilized an elegant technique to derive a set of solutions for the vacuum field equations within this gravitational framework. In this paper, we extend Buchdahl's method to incorporate the cosmological constant and to the scalar-tensor cases beyond the Brans-Dicke archetypal theory, thereby, with a conformal transformation of the metric, obtaining solutions for a version of Brans-Dicke theory that includes a quadratic potential. More specifically, we obtain synchronous solutions in the following contexts: in scalar-tensor gravity with massless scalar fields, Brans-Dicke theory with a quadratic potential, where we obtain specific synchronous metrics to the Schwarzschild-de Sitter metric, the Nariai solution, and a hyperbolically foliated solution.

gr-qc↗