Higher-Order Theories of Gravity  

In General > s.a. gravity theories; phenomenology of higher-order gravity [and solutions]; types of theories [and Hamiltonian formulation].
* Idea: Theories of gravitation, first discussed by H Weyl in 1918 as alternatives to Einstein's theory, that include quadratic or higher-order curvature terms in the action, e.g., R2, Rab Rab and Rabcd Rabcd (only two of these are independent – the integral of a combination of them gives the Euler number (M)).
* Motivation: One can alleviate the divergences in the quantum theory, and modify the predicted expansion history of the universe in cosmology.
* Action / Lagrangian: In 4D, the most general one with up to quadratic terms is

S = – ( Rab Rab – R2 + R) dv ;

any other term can be absorbed into these using the Gauss-Bonnet theorem [true only in positive-definite case?]; The only one which gives second-order equations in d > 4 is the Lanczos Lagrangian; But S could have a different curvature dependence.
* Ostrogradski theorem: The only potentially stable, local modification of general relativity is to make the Lagrangian an arbitrary function of the Ricci scalar.
* Bicknell theorem: A conformal relationship between fourth-order gravity and scalar-tensor theory.
* Field equations: [@ Stelle PRD(77)]

Hab:= (–2) R;ab– Rab;c;c – (–2) gab R + 2 Rmn Rambn +

– 2 R Rab – gab ( Rmn Rmn– R2) + Rab –   gab R = – Tab ,

where Hab satisfies Hab;b = 0 identically, as a consequence of the field equations.
@ Reviews: Cotsakis gq/97-MG8 [status]; Nojiri & Odintsov ht/06-ln [and dark energy]; Farhoudi GRG(06) [and trace anomaly relation]; Capozziello & Francaviglia a0706 [and phenomenology]; Schmidt IJGMP(07)gq/06-in [4th-order, history and applications]; Sotiriou a0710-PhD; Sotiriou & Faraoni RMP-a0805; Faraoni a0810-in [successes and challenges]; Capozziello et al a0909.
@ General references: Stelle GRG(78); Brans CQG(88) [and metric redefinitions]; Essén IJTP(90); Allemandi & Francaviglia IJGMP(07) [variational].
@ Energy: Boulware et al PRL(83); Deser & Yang CQG(89); Borowiec et al GRG(94); Deser & Tekin PRL(02)ht, PRD(03)ht/02, PRD(07)gq; Fatibene et al IJGMP(06) [same solution in different theories]; Multamäki et al CQG(08)-a0712 [energy-momentum complexes]; > s.a. gravitational energy.
@ Noether symmetries: Sanyal et al GRG(05)ap/03; Modak et al GRG(05).
@ Cauchy / initial-value problem: Teyssandier & Tourrenc JMP(83) [R + R2]; Jakubiec & Kijowski JMP(89) [well-posedness]; Lanahan-Tremblay & Faraoni CQG(07)-a0709, comment Capozziello & Vignolo CQG(09)-a0903, reply a0906 [well-posed for metric version]; Capozziello & Vignolo IJGMP(09)-a0901, CQG(09) [with perflect fluid].
@ Related topics: Eliezer NPB(89) [Regge calculus]; Ghoroku pr(90) [new variables]; Bartoli et al CQG(99)gq/98 [gauge fixing]; Woodard LNP(07)ap/06 [choice of theory]; Carloni & Dunsby JPA(07)gq/06-in [dynamical systems approach]; > s.a. hamilton-jacobi theory; metric matching.

Metric vs Palatini Formulation
* Idea: In general relativity, the metric (second-order) and the Palatini (first-order) formulations are dynamically equivalent, at least classically; This equivalence does not hold for theories of gravity with a non-linear dependence on the curvature, and the set of solutions of
the Palatini equations is a non-trivial subset of the solutions of the metric equations; One set of exception are the Lovelock gravity theories.
@ References: Shahid-Saless PRD(87) [R + R2 theory]; Olmo & Komp gq/04; Ezawa et al gq/03, NCB(04)gq [f(R) theories]; Deser CQG(06)gq, comment Kiriushcheva & Kuzmin CQG(07)gq/06 [obstacles with odd-derivative terms]; Sotiriou PLB(07) [instability in metric formulation]; Exirifard & Sheikh-Jabbari PLB(08)-a0705 [equivalence of formulations only for Lovelock gravity]; Poplawski IJMPA(08)-a0706; Iglesias et al PRD(07)-a0708 [how (not) to Palatini]; Borunda et al JCAP(08)-a0804; Faraoni PLB(08)-a0806 [Palatini unphysical]; Bastero-Gil et al a0901-in.

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send feedback and suggestions to bombelli at olemiss.edu – modified 11 oct 2009