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Black holes, cosmological solutions, future singularities, and their thermodynamical properties in modified gravity theories

A. de la Cruz-Dombriz, D. Saez-Gomez

arXiv:1207.2663v2gr-qcastro-ph.COhep-th

TL;DR

The review addresses how modified gravity affects black-hole solutions, cosmological evolution, thermodynamics, entropy bounds, and future singularities. It synthesizes results on f(R) black holes and FLRW thermodynamics, including stability analyses and generalized Cardy–Verlinde relations. The review reports standard AdS thermodynamic behavior, modified Kerr–Newman configurations and phase structure, and entropy-bound behavior near cosmological singularities.

  • Problem

    The review investigates black-hole, cosmological, thermodynamic, and entropy-bound properties that arise when gravity is modified beyond General Relativity.

  • Method

    The paper reviews static, charged, spinning, and cosmological solutions in f(R) gravity, derives thermodynamic quantities, studies stability, and examines generalized entropy bounds near singularities.

  • Results

    The review finds standard Schwarzschild–AdS solutions through second-order perturbation theory, supports massive charged spinning black holes in f(R) gravity, and identifies model-dependent thermodynamic stability and entropy-bound behavior.

  • Takeaways & Limitations

    Modified gravity can reproduce relevant black-hole and cosmological configurations while yielding thermodynamic and entropy phenomena that differ across models and singularity regimes.

Abstract

from arXiv · show

Along this review, we focus on the study of several properties of modified gravity theories, in particular on black-hole solutions and its comparison with those solutions in General Relativity, and on Friedmann-Lemaitre-Robertson-Walker metrics. The thermodynamical properties of fourth order gravity theories are also a subject of this investigation with special attention on local and global stability of paradigmatic f(R) models. In addition, we revise some attempts to extend the Cardy-Verlinde formula, including modified gravity, where a relation between entropy bounds is obtained. Moreover, a deep study on cosmological singularities, which appear as a real possibility for some kind of modified gravity theories, is performed, and the validity of the entropy bounds is studied.

I. INTRODUCTION

The review motivates modified gravity as an alternative framework for explaining cosmic acceleration, inflation, and matter-density observations that standard GR with usual matter does not fully account for. It surveys black-hole solutions, thermodynamics, cosmological dynamics, and entropy bounds in these theories.

  • Motivation: GR with usual matter does not explain the observed accelerated expansion, inflation, or the matter density inferred from cosmological observations.The review presents these discrepancies as motivations for considering alternatives to GR.
  • Alternative theories: Modified gravity theories alter the Einstein–Hilbert action through alternatives including Lovelock, string-inspired, scalar-tensor, and f(R) models.f(R) theories can mimic cosmological evolution from inflation through the current accelerated-expansion era.
  • Black holes: Black-hole studies in modified gravity test whether astrophysical solutions and their features are specific to GR or shared by covariant gravitational theories.Prior work addressed stability, static and rotating configurations, and anomalies in f(R) black holes.
  • Thermodynamics and cosmology: Black-hole thermodynamics is connected to gravitational field equations, and apparent-horizon thermodynamics has been used to derive FLRW equations in GR and modified theories.The review also considers thermodynamic properties and stability of f(R) black holes.
  • Entropy bounds: Entropy bounds associated with Cardy–Verlinde-type relations can fail near future singularities and, in some scenarios, well before the singularity occurs.The review examines extensions involving modified gravity and the validity of these bounds.
  • Review scope: The review uses modified gravity and gravitational thermodynamics to investigate unresolved connections that may inform reconstruction of consistent gravitational theories.Its scope includes black holes, FLRW metrics, thermodynamic stability, entropy bounds, and cosmological singularities.

III. STATIC AND SPHERICALLY SYMMETRIC BLACK HOLES IN f(R) GRAVITIES

For static, spherically symmetric vacuum configurations with constant negative curvature, f(R) gravity yields the Hawking–Page black hole in AdS space. The most general constant-curvature metric contains an additional term that vacuum f(R) models cannot reproduce, although it can correspond to a charged solution in four-dimensional f(R)-Maxwell theory.

  • Metric setup: The most general D-dimensional static, spherically symmetric metric is parameterized by radial functions whose scalar curvature depends only on radius.The analysis imposes staticity and spherical symmetry before examining constant-curvature solutions.
  • General constant-curvature metrics: For constant scalar curvature and constant Φ(r), the general metric function contains two integration constants and a curvature-dependent r^2 term.The D=4, R0=0 case becomes Reissner–Nordström after identifying the constants with mass and charge.
  • Vacuum f(R) solutions: Vacuum f(R) equations remove the r^(2−D) term present in the general constant-curvature metric.This missing term prevents the vacuum solution from representing the most general static, spherically symmetric constant-curvature geometry.
  • Field equations: The constant-curvature equations reduce to a f(R)-independent linear second-order differential equation, while R0 and the resulting metric retain dependence on the f(R) model.The reduction assumes constant curvature and excludes pathological parameter cases.
  • Vacuum f(R) solutions: For negative R0, the vacuum solution with the required constant fixing is the D-dimensional Hawking–Page black hole in AdS space.The AdS scale satisfies l^2 = −D(D−1)/R0.

C. f(R) Solutions Combined with Electromagnetism

The review combines f(R) gravity with electromagnetism and analyzes perturbations around Einstein–Hilbert black-hole solutions. It finds charge-dependent metric corrections in four dimensions and recovers Schwarzschild–(A)dS geometries through second order under analyticity assumptions.

  • f(R) Solutions Combined with Electromagnetism: In four dimensions, supplementing f(R) gravity with the Maxwell action produces constant-curvature charged black-hole solutions.The restriction to D=4 is required because the Maxwell action cannot generate the relevant r^(2−D) term otherwise.
  • f(R) Solutions Combined with Electromagnetism: The black-hole charge contribution to the metric is multiplied by (1+f′(R0))^−1, which must be positive to preserve the Reissner–Nordström sign.A negative factor would alter observable effects such as geodesic motion around a charged black hole.
  • Perturbations Around Schwarzschild–(anti)-de Sitter Solutions: The perturbative analysis assumes the modified Lagrangian and metric functions are analytic in a small parameter α around the Einstein–Hilbert solution.The procedure inserts the expansions into the modified field equations and solves order by order.
  • Perturbations Around Schwarzschild–(anti)-de Sitter Solutions: Through second order in α, the resulting metric has constant scalar curvature and is the standard Schwarzschild–(A)dS geometry after a time-coordinate reparameterization.This conclusion holds for analytic solutions and Lagrangians with g(R) representing a small Einstein–Hilbert deviation.
  • Perturbations Around Schwarzschild–(anti)-de Sitter Solutions: Non-constant-curvature static, spherically symmetric solutions remain possible when the f(R) function is not analytic in α.Thus the perturbative result does not exclude non-analytic branches.

IV. KERR–NEWMAN BLACK HOLES IN f(R) THEORIES

The review examines constant-curvature Kerr–Newman black holes in f(R) gravity, whose geometry resembles the Carter solution but whose charge contribution is modified by f′(R0). Their horizon structures depend on curvature, spin, charge, and the horizon parameter.

  • In four dimensions, the axisymmetric, stationary, constant-curvature solution describes a black hole with mass, electric charge, and angular momentum, analogous to Carter’s solution.
  • The metric’s charge contribution differs from General Relativity by a factor (1 + f′(R0))^-1/2, motivating the normalized charge parameter Q̄² = Q²/(1 + f′(R0)).
  • Event Horizons: Event horizons are roots of the quartic equation Δr = 0, whose real solutions include interior, exterior, and possible cosmological horizons.
  • Event Horizons: For sufficiently positive curvature, the horizon parameter vanishes at upper and lower spin bounds, corresponding respectively to extremal and marginal extremal configurations.
  • Event Horizons: For R0 < 0, h > 0 gives well-defined horizons, h = 0 gives an extremal black hole, and h < 0 gives a naked singularity.
  • Event Horizons: Figure 2 maps the allowed a/M ranges for black-hole existence at fixed R0M² and for Q̄/M equal to 0 or 0.75.

V. BLACK HOLES THERMODYNAMICS IN f(R) THEORIES

The AdS thermodynamics of f(R) black holes is developed through Euclidean-action methods and temperature-based thermodynamic relations. Under R0 + f(R0) < 0, the resulting stability structure is analogous to the Hawking–Page behavior in Einstein gravity.

  • The Euclidean action provides the free energy and, together with temperature, enables derivation of energy, entropy, and heat capacity.
  • The alternative surface-gravity definition agrees with the Euclidean temperature, which depends only on the metric near the horizon.
  • The temperature diverges for rH → 0, grows linearly for rH → ∞, and has a minimum T0 separating distinct thermodynamic regimes.
  • For R0 + f(R0) < 0, large AdS black holes have C > 0 while small ones have C < 0, with divergent heat capacity near rH0.
  • Violating R0 + f(R0) < 0 makes the mass and entropy negative, so the corresponding AdS black-hole solutions are regarded as unphysical.
  • For T < T0 no stable black-hole solution exists; for T > T0 small black holes are unstable, while sufficiently large black holes become globally favored when T > T1.

B. BH Thermodynamics for KN Configuration

Kerr–Newman–AdS black-hole thermodynamics in f(R) gravity is obtained with Euclidean methods and exhibits fast and slow branches. Its qualitative stability behavior resembles General Relativity, while physical quantities acquire model-dependent corrections.

  • The temperature can be derived from the Euclidean action or surface gravity, and the two definitions coincide for Kerr–Newman configurations.
  • For zero spin and charge, the Kerr–Newman temperature reduces to the nonrotating uncharged expression.
  • The electric potential and physical charge definitions are f(R)-model independent, whereas extremal black holes make the Euclidean action singular because their temperature vanishes.
  • Positive mass and entropy require 1 + f′(R0) > 0; under this condition, the free energy changes from positive below a limiting horizon or mass to negative above it.
  • Fast black holes have C > 0 without phase transitions, whereas slow black holes exhibit two phase transitions at determined exterior-horizon radii.
  • Kerr–Newman–AdS black holes exist at any temperature, but configurations require M > Mmin and divide into fast and slow branches.
  • Slow black holes are unstable below Mlimit, while above Mlimit their negative free energy favors collapse of pure radiation into black-hole equilibrium.
  • Their thermodynamics is qualitatively analogous to General Relativity, but mass, angular momentum, and entropy scale by 1 + f′(R0), while free energy and heat capacity receive more complex charge-dependent deviations.

A. Thermodynamics in AdS

The review examines thermodynamic stability and black-hole existence across parameter regions of two f(R) models in AdS and Kerr–Newman settings. Stability depends on heat capacity, free energy, curvature, model parameters, and allowed spin ranges.

  • Model I: For Model I, physical parameter regions are restricted by the requirement 1 + f′(R0) > 0, yielding {α < 0, β > D/2} or {α > 0, β < 1}.The second region is narrowed from β < D/2 by the stronger constraint β < 1.
  • Thermodynamic stability: Heat capacity C and free energy F determine local and global stability across the parameter regions of the studied f(R) models.The parameter-space figures classify configurations by the signs of C and F.
  • Kerr–Newman black holes: In Model I Region 3, amax drops near α ≈0 and β ≈2, while away from the boundary black-hole existence is generally assured.Region 3 is defined by α < 0 and β > 2.
  • Kerr–Newman black holes: In Model I Region 4, amax approaches zero for 0 < β < 1 but gradually recovers higher values as β becomes more negative, reaching its usual value at β = −2.Region 4 is defined by α > 0 and β < 1.
  • Thermodynamic stability: Region 3 is most stable near α ≈0 and β ≈2, whereas Region 4 becomes more locally and globally stable as α increases.Region 3 transitions toward global and then local instability as |α| increases; Region 4 shows the opposite trend.
  • Model II: For Model II, valid black holes require κ > 1 and γ < 0, with an amax surface that declines as κ increases and more sharply as γ approaches zero.The allowed spin range is bounded above by amax; amin is discussed for the alternative region when present.

VII. COSMOLOGICAL SOLUTIONS IN MODIFIED GRAVITY

This section reviews how f(R) gravity can reconstruct cosmological histories, including ΛCDM-like expansion and phantom behavior without introducing corresponding extra fluids. It also presents reconstruction methods based on differential equations for the Hubble parameter and action.

  • ΛCDM reconstruction: f(R) gravity can reproduce the ΛCDM model without a cosmological constant, although the resulting hypergeometric action is more complicated than GR with a cosmological constant.The reconstruction uses the e-folding variable N and produces a hypergeometric equation for F(R).
  • Phantom cosmology: Phantom expansion ending in a Big Rip can be obtained in F(R) gravity without introducing a phantom fluid.The same scale-factor behavior a(t) = a0(ts −t)^−H0 is recovered in the modified-gravity framework.
  • Reconstruction method: The reconstruction approach selects a function P(φ), solves a first-order equation for g(φ) after neglecting matter, and obtains Q(φ) from the resulting relations.The scalar field is chosen as the time coordinate, with H = g(φ).
  • Scope: The review presents F(R) gravity as a candidate for explaining late-time cosmic acceleration and then examines its FLRW thermodynamics and entropy relations.The cosmological reconstruction is followed by analyses of horizon thermodynamics and the Cardy–Verlinde formula.

VIII. FIRST LAW OF THERMODYNAMICS AND FLRW EQUATIONS

The section derives FLRW equations from thermodynamic relations at the apparent horizon. The same strategy extends to f(R) gravity, where a modified horizon entropy recovers the corresponding gravitational equations.

  • Apparent-horizon framework: The apparent horizon is selected because its area-based entropy and energy flux support a unified first-law description.The internal-energy variation is expressed using the work density, energy flux, area, and volume.
  • General Relativity: Using apparent-horizon entropy, temperature, and the first law dE = −TdS yields the second FLRW equation.The horizon treatment identifies the surface gravity divided by 2π as the temperature and entropy proportional to A/4G.
  • General Relativity: Combining the derived second equation with the continuity equation also yields the first FLRW equation.Thus both FLRW equations follow from the thermodynamic construction.
  • Modified gravity: For f(R) gravity, the corresponding entropy relation leads through the same steps to the modified FLRW equations.The construction establishes a relation between horizon thermodynamics and modified gravitational field equations.

IX. GENERALIZATION OF CARDY–VERLINDE FORMULA

The review develops the Cardy–Verlinde relation from FLRW thermodynamics and examines when entropy formulas and cosmological bounds retain that correspondence. For radiation-like fluids, the relation connects the FLRW equation with the Cardy formula when the Casimir bound is saturated.

  • The Cardy–Verlinde formula relates FLRW equations for a conformal fluid to the Cardy formula of a two-dimensional conformal field theory.
  • For a single fluid, the total energy is decomposed into extensive and Casimir components with distinct rescaling behaviors under entropy and volume transformations.
  • The resulting entropy is expressed through the total energy and Casimir energy, providing the generalized Cardy–Verlinde construction.
  • When the Casimir bound EC ≤ EBH is saturated, the entropy equation coincides with the first FLRW equation.

A. Multicomponent Universe

For multicomponent fluids and modified gravity, the Cardy–Verlinde correspondence generally fails because the total energy depends on multiple scale-factor powers. It can be recovered only for special equations of state or selected de Sitter and effective-fluid configurations.

  • A. Multicomponent Universe: With multiple fluids, total entropy is the sum of component entropies and generally cannot be reduced to a function of total energy alone.
  • A. Multicomponent Universe: For two fluids, different equations of state produce total energies with two scale-factor powers, preventing a single-entropy representation.
  • A. Multicomponent Universe: If w1 = w2, the entropy formula reduces to the single-fluid result and coincides with the Cardy–Verlinde formula when weff = 1/n.
  • A. Multicomponent Universe: For inhomogeneous equations of state, the Cardy–Verlinde formula is reconstructed only for special choices such as w(a) = −1 and g(a) = −am.
  • A. Multicomponent Universe: In f(R) gravity, an effective-fluid representation can recover the entropy formula in selected cases, but the modified FLRW equation changes the associated cosmic thermodynamic quantities.
  • A. Multicomponent Universe: For de Sitter solutions in F(R) gravity, the entropy formula and universal bound can hold, and dynamical entropy bounds are not violated.

X. ON THE COSMOLOGICAL BOUNDS NEAR FUTURE SINGULARITIES

The review examines Verlinde-type entropy bounds near future cosmological singularities, where unusual effective equations of state can threaten the fundamental Casimir bound. It then considers quantum corrections as a mechanism that can avoid or postpone some violations.

  • X. ON THE COSMOLOGICAL BOUNDS NEAR FUTURE SINGULARITIES: Verlinde’s cosmological bound restricts the Casimir energy relative to the Bekenstein–Hawking energy and yields entropy bounds in strong- and weak-self-gravity regimes.
  • X. ON THE COSMOLOGICAL BOUNDS NEAR FUTURE SINGULARITIES: At saturation in the strong-self-gravity regime, S = SH and the FLRW equation coincides with the Cardy–Verlinde formula.
  • X. ON THE COSMOLOGICAL BOUNDS NEAR FUTURE SINGULARITIES: The review studies whether the fundamental bound remains valid near Big Rip, sudden, Type III, and Type IV singularities in a closed universe.
  • X. ON THE COSMOLOGICAL BOUNDS NEAR FUTURE SINGULARITIES: The analysis uses explicit singularity-ending FLRW solutions and incorporates conformal-anomaly corrections through an effective-action treatment of quantum effects.
  • X. ON THE COSMOLOGICAL BOUNDS NEAR FUTURE SINGULARITIES: For some singularity classes, quantum effects can avoid violation of the cosmic bound.

A. Big Rip Singularity

The Big Rip drives the scale factor and relevant energies toward divergent behavior, causing the Casimir bound to fail before the singularity in the classical treatment. Including conformal-anomaly corrections moderates the singularity and can avoid or postpone that violation.

  • A. Big Rip Singularity: The phantom regime w < −1 can produce a Big Rip singularity, with the scale factor diverging at finite time.
  • A. Big Rip Singularity: As the Big Rip is approached, EC ∝ an|w| grows faster than EBH ∝ an−2, so the bound must be violated before the singularity.
  • A. Big Rip Singularity: Conformal-anomaly corrections introduce a phase transition in the Hubble evolution near the Big Rip.
  • A. Big Rip Singularity: Because the corrected divergent energy density would become negative, it cannot diverge physically; numerical analyses therefore find that the anomaly moderates the singularity and can avoid or postpone bound violation.
  • A. Big Rip Singularity: For a modified-gravity singularity with higher-derivative divergences removed, EC may still outgrow EBH, requiring a critical universe size to maintain the bound.
  • A. Big Rip Singularity: In another analyzed regime, the Casimir energy can dominate the Bekenstein–Hawking energy near the singularity, although specific coefficients may preserve the bound.

C. Type III Singularity

Type III singularities can preserve a finite scale factor while producing divergent Casimir energy, causing the dynamical entropy bound to fail before the singularity. Across the reviewed singularity analysis, the bound is generally violated near future singularities, with special exceptions for some Type II and Type IV cases.

  • C. Type III Singularity: For −1 < m < 0, the scale factor remains finite at the singularity, making the solution similar to a Big Rip.As t approaches ts, a(t) approaches the finite value as.
  • C. Type III Singularity: The Casimir energy diverges at the singularity while EBH scales as a_n−2.
  • C. Type III Singularity: The entropy bound is violated long before the singularity because the relevant energy contribution grows while the scale factor remains finite.Maintaining the bound would require assuming that GR fails near or at the bound; quantum effects do not restore it for this singularity type.
  • C. Type III Singularity: Except for special Type II and Type IV cases, the dynamical entropy bound is generally violated near future singularities and is not universal.Including quantum effects of conformally invariant matter does not generally improve the situation.
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