Types
of Actions for General Relativity| Types
of Actions for General Relativity |
In General > s.a. gravitation; higher-order
gravity.
* Possible contributions
in four dimensions: [@ Zumino in(86)]
L0,4 = ea eb ec ed 
abcd :
Cosmological term ,
L1,2 = Rab ec ed abcd :
Einstein-Hilbert term ,
L2,0 =
Rab Rcd abcd
: Euler invariant .
Hilbert-Einstein Action > s.a. 2D
gravity.
* Expression: The variable
is the metric (or its inverse); With a cosmological constant
,
the volume term is
SHE[g]
= (16
G)–1 
M d4x |g|1/2
(R – 2)
,
which contains – linearly – second derivatives of g.
@ References: Hilbert KNGWG(15); Katanaev GRG(06)gq/05 [with
|g| as variable, polynomial].
Gibbons-Hawking-York (tr K) Action (with all boundary
terms) > s.a.
extrinsic curvature.
* Idea: Obtained from
the Hilbert-Einstein action by subtracting the boundary terms containing normal
derivatives of the metric; To ensure that the induced metric is fixed
on all components of 
M,
we must include all corner terms; For a four-dimensional M with initial
and final hypersurfaces 
1
and 2,
and timelike boundary 
which
meets i in Bi, i =
1,
2,
SGHY[g]
= (16G)–1 {M d4v (R –
2) + 2 [Sigma_2 (K–K0)
d3s – Sigma_1 (K–K0)
d3s
–
Tau (K–K0)
d3s – B_2 
d2
+
B_1 d2]
} ,
where K (K0) is the trace of the extrinsic curvature induced
by g (g0)
on M, and
:= sinh–1(u ·
n), with u the future-pointing
normal to , and n the
outward-pointing normal to .
@ References: Gibbons & Hawking PRD(77);
York FP(86);
Hayward PRD(93);
Hawking & Hunter
CQG(96)gq [boundaries];
Pons GRG(03)gq/01 [Lagrangian,
Noether charges].
3+1 Metric Form
* Expression: If 
=
extrinsic curvature of ; K extrinsic
curvature
of t;
= u · n;
aa = ub
b ua
= acceleration of ,
S[q, N, N] = (16G)–1 dt d3x N q1/2 (3R + Kab Kab – K2 – 2)
+ (8G)–1Tau d3x |
|1/2 ( +
K – na aa)
.
@ References: Hawking & Hunter CQG(96)gq.
Other Forms > s.a. action [similar
theories]; first-order actions.
* Baierlein-Sharp-Wheeler
form: The one obtained when the Lagrange
multiplier (the lapse function) is eliminated from the Lagrangian and one
is
left
with a product of square roots.
@ BSW form: Carlini & Greensite PRD(95)gq; Ó Murchadha
IJMPA(02)-in.
@ Regge calculus inspired: Ambjørn et al NPB(97)ht/96.
@ Curvature-saturated: 
cs = EH/
(1
+ l4R2)1/2, l a
length parameter,
Kleinert & Schmidt GRG(02)gq/00.
@ Eikonal limit: Arcioni et al JHEP(01)ht [boundary
action].
@ Self-dual: Nieto & Socorro PRD(99)ht/98 [and
Yang-Mills, MacDowell-Mansouri formalism];
Nieto MPLA(05)ht/04 [various
versions]; > s.a. connection
formulation.
@ Related topics: Robinson IJTP(98)
[chiral]; Mei a0711 [with
positive kinetic energy term].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
12 jun 2008