Higher-Order Theories of Gravity: Types and Examples |
Variational Principles
> s.a. actions for gravity [curvature-saturated];
theories of gravity; torsion in physics.
@ General references:
Allemandi & Francaviglia IJGMP(07).
@ Action / Lagrangian: Nojiri & Odintsov PRD(00)ht/99,
Dyer & Hinterbichler PRD(09)-a0809 [surface terms];
Krasnov PRD(10)-a0911 [with two propagating degrees of freedom, effective Lagrangians];
Caravelli & Modesto PLB(11)-a1001 [from black-hole entropy];
Biswas et al PRL(12)-a1110
[the most general covariant ghost-free theories].
@ Perturbation theory: Havas GRG(77) [linearized];
Multamäki et al PRD(10)-a0910 [stability of solutions];
Bueno & Cano PRD(16)-a1607 [simpler calculations].
@ Boundary terms: Guarnizo et al GRG(10)-a1002;
Fatibene et al a1106;
Teimouri et al JHEP(16)-a1606;
Sáez-Chillón a2011 [Palatini f(R) gravity];
> s.a. Lanczos-Lovelock and scalar-tensor gravity below.
@ Hamiltonian formulation: Szczyrba JMP(87);
Ezawa et al CQG(99)gq/98;
Querella PhD(98)gq/99;
Ezawa et al CQG(06)gq/05;
Deruelle et al PRD(09)-a0906,
PTP(09)-a0908;
Fatibene et al CQG(10)-a1003 [self-dual connection formulation, constraints];
Sanyal et al CQG(12)-a1108;
Mandal & Sanyal PRD(17)-a1709 [inequivalent canonical structures];
Sanyal AP(19)-a1807 [ambiguity and the Dirac approach].
f(R) Theories > s.a. energy-momentum
pseudotensor; entropic gravity; unimodular gravity.
* Ostrogradski theorem:
The only potentially stable, local modification of general relativity is
to make the Lagrangian an arbitrary function of the Ricci scalar.
* Field equations:
(Extended gravity theories) If \(\cal L\) is a non-linear function of the
curvature scalar, then the field equations are of fourth order in the metric;
These equations were first discussed by H Weyl in 1918, as an alternative
to Einstein's theory; Palatini f(R) theories are
mathematically equivalent to Brans-Dicke theories, but not physically.
* R−1
theories: Motivated by the fact that they lead to cosmological acceleration;
A 1/R action by itself is ruled out by constraints on scalar-tensor gravity,
but could work as an extra contribution to the action.
@ Reviews: Perez Bergliaffa a1107-proc;
Capozziello & De Laurentis a1307-MG13;
Sporea a1403 [intro].
@ General references: Schmidt IJGMP(07)gq/06-ln [4th-order];
Faraoni PLB(08)-a0806 [f(R) and conformal rescaling];
Jaime et al PRD(11)-a1006 [with R as extra variable];
Yang EPL(11)-a1103 [conformal transformations];
Jorás IJMPA(11)-in [shortcomings];
Paliathanasis et al PRD(11)-a1110 [constraints and solutions];
Tamanini & Koivisto PRD(13)-a1308 [non-minimally coupled, consistency];
Guendelman et al a1312-proc;
de Haro EPL(14)-a1403 [holonomy corrections];
Carloni JCAP(15)-a1505 [as dynamical system];
Woodard a1506-en [Ostrogradski theorem];
Capozziello et al a1512-MG14 [frame equivalence];
Nayem & Sanyal IJMPD(18)-a1708 [different canonical forms];
Li et al PRD(18)-a1711 [causality and a-theorem constraints];
Chakraborty et al PRD(19)-a1812 [equivalence of Jordan and Einstein frames].
@ Solutions: Chakraborty & SenGupta EPJC(16)-a1604 [technique for solving field equations];
> s.a. phenomenology.
@ And scalar-tensor gravity:
Ezawa & Ohkuwa a1204-ch [and quantization];
Fatibene & Francaviglia IJGMP(14)-a1302;
Fatibene & Garruto IJGMP(14),
Morais Graça & Bezerra MPLA(15)-a1312 [and Brans-Dicke theories];
Langlois & Noui JCAP(16)-a1512 [Hamiltonian analysis];
Fatibene & Garruto IJGMP(16)-a1601-conf;
Nayem & Sanyal IJMPD(17)-a1609;
Ruf & Steinwachs PRD(18)-a1711 [quantum equivalence];
Castañeda & Velásquez a1808 [including boundary terms].
@ And other theories:
Granda a0812-in [from holographic principle];
Saltas & Hindmarsh CQG(11)-a1002
[equivalence of f(R) and Gauss-Bonnet theories];
Fabbri & Vignolo AdP(12)-a1012 [with torsion and ELKO matter];
Benachour a1202 [theories equivalent to general relativity];
Santos & de Souza Santos AIP(12)-a1212 [and its bimetric structure].
@ Hamiltonian formulation:
Olmo & Sanchis-Alepuz PRD(11) [à la Brans-Dicke theory];
Ohkuwa & Ezawa EPJP(15)-a1412 [using Lie derivatives];
Bombacigno et al a1911 [connection variables].
@ R2 theories:
Folomeshkin CMP(71);
Simon PRD(90) [non-locality];
Meng & Wang CQG(05)
[R2 + R−1];
Zhogin a0812-GRF [problems];
Edery & Nakayama PRD(19)-a1902 [Palatini formulation = Einstein gravity, no massless scalar].
@ R2 theories, Hamiltonian:
Sanyal GRG(05)ht/04;
Klusoň et al PRD(14)-a1311;
Debnath et al PRD(14)-a1408 [in higher dimensions].
@ R−1 theories:
Chiba PLB(03)ap [and scalar-tensor];
Cline ap/03-wd [as extra term].
@ Other forms: Meng & Wang PLB(04)ht/03 [ln R, Palatini];
Baghram et al PRD(07)ap [\(f(R) = (R^2 − R_0^2)^{1/2}\)];
Kruglov IJTP(13)-a1202 [\(f(R) = \{1 - (1-2\lambda R)^{1/2}\}/\lambda\), Born-Infeld-like];
Kruglov IJMPA(13)-a1204 [f(R)
= R eαR];
Kruglov ASS(15)-a1502 [\(f(R) = (1/\beta) \arcsin(\beta R)\), and cosmology];
Kumar a1611 [cosmologically viable].
> Related topics: see Birkhoff's Theorem;
geodesics [geodesic deviation]; metric matching;
singularities [quantum fields as probes].
Other Theories
> s.a. higher-order gravity [metric vs Palatini formulations]; scalar-tensor
theories [Jordan vs Einstein frames]; spin-2 field theories.
* Quadratic theories: Adding terms
quadratic in the curvature to the action renders gravity renormalizable; The price
to pay is the presence of a massive ghost.
* Lanczos-Lovelock gravity: The most
general theories of gravity in D dimensions which satisfy (a) the principle
of equivalence, (b) the principle of general covariance, and (c) have field equations
involving derivatives of the metric tensor only up to second order; The m-th
order Lanczos-Lovelock Lagrangian is a polynomial of degree m in the curvature
tensor; The field equations resulting from it become trivial in the critical dimension
\(D = 2m\) and the action itself can be written as the integral of an exterior derivative
of an expression involving the vierbeins.
@ Matter couplings:
Sotiriou in(08)gq/06;
Sotiriou PLB(08)-a0805,
Sotiriou & Faraoni CQG(08)-a0805 [with matter coupling to R];
Harko PLB(08)-a0810,
PRD(10) [f(R) theories];
Fabbri & Vignolo CQG(11)-a1012
[f(R) theories with torsion and Dirac fields];
Haghani et al IJMPD(14)-a1405-GRF [with
RabTab term].
@ Lanczos-Lovelock gravity: Lanczos JMP(69);
Madore CQG(86) [action];
Yale & Padmanabhan GRG(11)-a1008;
Deser & Franklin CQG(12)-a1110 [canonical analysis];
Kunstatter et al CQG(12) [Hamiltonian for spherically symmetric Lovelock gravity];
Padmanabhan & Kothawala PRP(13)-a1302 [rev];
Chakraborty & Padmanabhan PRD(14)-a1408 [evolution as departure from holographic equipartition];
Chakraborty et al GRG(17) [boundary terms in the action];
> s.a. black-hole geometry; lovelock gravity;
quasilocal energy; thermodynamics.
@ Quadratic theories: Borowiec et al CQG(98)gq/96 [\(R_{ab}^{~} R^{ab}\) Lagrangians];
Olmo et al PRD(09) [\(f(R, R^{mn}R_{mn}^{~})\)];
Biswas et al CQG(14)-a1308 [quadratic, non-local but ghost-free and UV asymptotically free theories];
Álvarez-Gaumé et al FdP(16)-a1505 [rev];
Álvarez et al a1710-conf [and UV completion],
EPJC(18)-a1802 [physical content];
Salvio FrPh(18)-a1807 [rev];
Morales & Santillán JCAP(19)-a1811 [Cauchy problem].
@ Cubic theories: Caprasse et al IJMPD(93);
Bueno & Cano PRD(16)-a1607 [Einsteinian cubic gravity];
> s.a. lensing models.
@ With curvature derivatives: Oliva & Ray PRD(10)-a1004 [six-derivative Lagrangians, classification];
Naruko et al CQG(16)-a1512
["scalar-tensor" theories, with 2 scalar and 2 tensor degrees of freedom].
@ Fourth-order gravity: Carloni et al GRG(09) [dynamical-systems approach];
Avalos et al a2102,
a2102 [energy, positive-energy theorem].
@ Other theories:
Hindawi et al PRD(96) [vacua and excitations];
Borowiec et al gq/00-conf [general Ricci-type gravitational Lagrangians];
Navarro & Van Acoleyen JCAP(06)gq/05 [MOND-like, and cosmology];
Gruzinov & Kleban CQG(07)ht/06 [causality constraints];
Ishak & Moldenhauer JCAP(09)-a0808 [minimal set of invariants];
Lü & Pope PRL(11)-a1101 ["critical gravity"];
Atazadeh & Darabi GRG(14)-a1302 [f(R,G), energy conditions and viability];
Colléaux & Zerbini Ent(15)-a1508 [with second-order equations of motion];
Hao & Zhao PRD(17)-a1512 [Ricci polynomial];
Karasu et al PRD(16)-a1602 [minimal extension of Einstein's gravity, quartic theory];
Stachowiak PRD(17)-a1703
[f(Rab), and cosmology];
Bueno et al JHEP(19)-a1906 [generalized quasi-topological gravities];
Amendola et al PLB(20)-a2006 [inverse of the Ricci tensor].
@ Lower-dimensional: Bergshoeff et al AP(10) [3D, and higher-spin gauge theories];
Güllü et al PRD(10)-a1002 [canonical structure];
Gürses et al PRD(12)-a1112 [3D f(Rab)
theories, constant-scalar-curvature Type-N and Type-D solutions];
Ohta CQG(12) [3D, classification of unitary and stable theories];
> s.a. 2D gravity; 3D gravity.
@ Higher-dimensional: Ezawa et al CQG(99) [semiclassical stability];
Collins & Holdom JHEP(02) [4+1 with cosmological constant];
Huang et al PRD(10) [and cosmological acceleration];
> s.a. kaluza-klein theory.
Quantum-Gravity Motivated
> s.a. higher-order theories of quantum gravity.
@ General references: Bonanno et al CQG(11)-a1006
[renormalization-group improved, inflationary solutions];
Pereira-Dias et al PRD(11)-a1009
[effect of Lorentz-symmetry violating Chern-Simons and Ricci-Cotton terms in the action];
Dzhunushaliev IJMPD(12)-a1201
[f(R) modified gravities from non-perturbative quantum effects].
@ Infinite-derivative, string-inspired: Biswas & Talaganis MPLA(15)-a1412 [rev];
Talaganis & Teimouri a1701 [Hamiltonian];
Talaganis a1704 [towards UV finiteness];
Gording & Schmidt-May JHEP(18)-a1807 [ghost-free];
Edholm a1904-PhD [phenomenological predictions];
Dimitrijevic et al proc(18)-a1902 [derivation of the equations of motion];
> s.a. gravitating bodies;
singularities [avoidance].
@ From loop-quantum-gravity effective theory:
Bojowald & Skirzewski IJGMP(07)ht/06-ln;
Olmo & Singh JCAP(09)-a0806 [lqc].
> Other: see brans-dicke theory;
BRST symmetries; Conformal Gravity;
Gauss-Bonnet Gravity [including f(G) theories];
hořava-lifshitz gravity; lovelock gravity;
Metric-Affine Theories; supergravity;
unified theories [fourth-order Weyl].
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