Higher-Order
Theories of Gravity| Higher-Order
Theories of Gravity |
In General > s.a. gravity; higher-order
quantum gravity; phenomenology of higher-order
gravity [including
solutions]; spin-2
field theories.
* 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.
* Metric vs affine or Palatini
forms: In general relativity, the purely affine picture and the two other
formulations are dynamically equivalent, at least classically; This
equivalence does not hold
for theories of gravity with a non-linear dependence on the curvature.
* 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)
[history and applications]; Sotiriou a0710-PhD;
Sotiriou & Faraoni a0805-RMP.
@ 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 a0712 [energy-momentum
complexes].
@ Noether symmetries: Sanyal
et al GRG(05)ap/03;
Modak et al GRG(05).
@ Metric vs Palatini (first- vs second-order):
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 a0706;
Iglesias et al PRD(07)-a0708 [how
(not) to Palatini]; Borunda et al a0804;
Faraoni PLB(08)-a0806 [Palatini
unphysical].
@ Cauchy / initial value problem:
Teyssandier & Tourrenc JMP(83)
[R + R2];
Jakubiec & Kijowski JMP(89)
[well-posedness]; Lanahan-Tremblay & Faraoni CQG(07)-a0709
[well-posed for metric version].
@ Related topics: Eliezer
NPB(89)
[Regge calculus]; Ghoroku pr(90) [new variables]; Bartoli et al CQG(99)gq/98 [gauge
fixing]; Woodard ap/06-in
[choice of theory]; Carloni & Dunsby JPA(07)gq/06-in
[dynamical systems approach]; Sotiriou gq/06-in
[matter couplings]; > s.a. hamilton-jacobi
theory, metric matching.
Examples of Theories > s.a. 2D
gravity; 3D gravity; actions
for gravity [curvature-saturated]; theories
of gravity; torsion in physics.
* And general relativity:
If
we
have
a
theory with Lagrangian
= |g|1/2 F(g,
Ric(g)) + m(g,
g, 
,
g)
,
a redefinition of the metric (which need no longer be Lorentzian) gives a scalar-tensor theory with action
= |h|1/2 R(h)
+ m + +(h,
h, ,h)
, where hab:=
|det(
/Rab)|–1/2 (/Rab)
.
* f(R)
theories: If
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.
* 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.
@ And general relativity, scalar-tensor: Goenner in(87); Magnano et
al GRG(87);
Jakubiec & Kijowski PRD(88),
JMP(89);
Magnano & Sokolowski PRD(94)gq/93,
Magnano gq/95-in;
Sokolowski gq/95-in;
Low GRG(95)
[singularities]; Núñez & Solganik ht/04;
Flanagan
CQG(04)gq [equivalence];
Sotiriou CQG(06)gq;
Bertolami & Páramos a0805 [and
non-trivial matter coupling].
@ R2 theories: Folomeshkin
CMP(71);
Simon PRD(90)
[non-locality]; Sanyal GRG(05)ht/04 [Hamiltonian];
Meng & Wang CQG(05)
[R2 + R–1].
@ R–1 theories: Chiba
PLB(03)ap [and
scalar-tensor]; Cline
ap/03-wd
[as extra term].
@ Lanczos/Lovelock gravity: Lanczos JMP(69); Madore CQG(86)
[action]; > s.a. thermodynamics.
@ Other theories: Havas GRG(77)
[linearized]; Caprasse et al IJMPD(93)
[cubic]; Hindawi et al PRD(96)
[vacua and excitations]; Hurth et al PRD(97)
[sugra]; Borowiec et al CQG(98)gq/96 [Rab Rab Lagrangians];
Meng & Wang PLB(04)ht/03 [ln
R, Palatini]; Schmidt gq/06-in
[4th-order]; Navarro & Van Acoleyen JCAP(06)gq/05 [MOND-like,
and cosmology]; Gruzinov & Kleban CQG(07)ht/06 [causality
constraints]; Baghram et al PRD(07)ap [f(R)
= (R2 – R02)1/2];
Sotiriou PLB(08)-a0805,
Sotiriou & Faraoni a0805 [with
matter coupling to R]; Faraoni PLB-a0806 [f(R)
and conformal rescaling].
@ Higher-dimensional: Ezawa et al CQG(99)
[semiclassical stability]; Collins & Holdom
JHEP(02)
[4+1 with cosmological constant].
@ Action/Lagrangian: Nojiri & Odintsov PRD(00)ht/99 [surface
terms].
@ Hamiltonian formulation: Szczyrba JMP(87);
Ezawa et al CQG(99)gq/98;
Querella
gq/99-PhD;
Ezawa et al CQG(06)gq/05.
@ From quantum gravity effective theory: Bojowald & Skirzewski IJGMP(07)ht/06-ln.
> Other: see brans-dicke; BRST; Conformal, Gauss-Bonnet, Lovelock
Gravity; Metric-Affine; unified
theories [fourth-order Weyl].
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20 jul 2008