Physics Letters B 757 (2016) 454–461
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Hamiltonian analysis of Einstein–Chern–Simons gravity
L. Avilés and P. Salgado
∗
Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile
a r t i c l e i n f o a b s t r a c t
Article history:
Received
29 October 2015
Received
in revised form 12 April 2016
Accepted
14 April 2016
Available
online 18 April 2016
Editor: M.
Cveti
ˇ
c
In this work we consider the construction of the Hamiltonian action for the transgressions field theory.
The subspace separation method for Chern–Simons Hamiltonian is built and used to find the Hamiltonian
for five-dimensional Einstein–Chern–Simons gravity. It is then shown that the Hamiltonian for Einstein
gravity arises in the limit where the scale parameter l approaches zero.
© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
In the context of the general relativity the spacetime is a dy-
namical
object which has independent degrees of freedom and is
governed by dynamical equations, namely the Einstein field equa-
tions.
This means that in general relativity the geometry is dynam-
ically
determined. Therefore, the construction of a gauge theory of
gravity requires an action that does not consider a fixed space-time
background. An action for gravity fulfilling these conditions, albeit
only in odd-dimensional spacetime, d = 2n + 1, was proposed long
ago by Chamseddine [1,2], which is given by a Chern–Simons form
for the anti-de Sitter (AdS) algebra. Chern–Simons gravities have
been extensively studied; see, for instance, Refs. [3–12].
If
Chern–Simons theories are to provide the appropriate gauge-
theory
framework for the gravitational interaction, then these the-
ories
must satisfy the correspondence principle, namely they must
be related to general relativity.
Studies
in this direction have been carried out in Refs. [13–16]
(see
also [17,18]). In these references it was found that standard,
five-dimensional GR (without a cosmological constant) emerges as
the → 0 limit of a CS theory for a certain Lie algebra B
5
. Here
is a length scale, a coupling constant that characterizes different
regimes within the theory. The B
5
algebra, on the other hand, is
constructed from the AdS algebra and a particular semigroup by
means of the S-expansion procedure introduced in Ref. [19].
Black
hole type solutions and the cosmological nature of the
corresponding fields equations satisfy the same property, namely,
that standard black-holes solutions and standard cosmological so-
lutions
emerge as the → 0 limit of the black-holes and cos-
mological solutions
of the Einstein–Chern–Simons field equations
[14–16].
*
Corresponding author.
E-mail
address: pasalgad@udec.cl (P. Salgado).
The Einstein–Chern–Simons action was constructed using trans-
gression
forms and a method, known as subspace separation pro-
cedure
[20]. This procedure is based on the iterative use of the
Extended Cartan Homotopy Formula, and allows one to (i) system-
atically
split the Lagrangian in order to appropriately reflect the
subspaces structure of the gauge algebra, and (ii) separate the La-
grangian in
bulk and boundary contributions.
However
the Hamiltonian analysis of Einstein Chern–Simons
gravity
action as well as transgression forms is as far as we know
an open problem.
In
Ref. [21] was studied the Hamiltonian formulation of the
Lanczos–Lovelock (LL) theory. The LL theory is the most general
theory of gravity in d dimensions which leads to second-order
field equations for the metric. The corresponding action, satisfying
the criteria of general covariance and second-order field equations
for d > 4is a polynomial of degree
[
d
/2
]
in the curvature, has
[
(d − 1)/2
]
free parameters, which are not fixed from first princi-
ples.
In
Ref. [6] was shown, using the first order formalism, that
requiring the theory to have the maximum possible number of
degrees of freedom, fixes these parameters in terms of the gravita-
tional
and the cosmological constants. In odd dimensions, the La-
grangian
is a Chern–Simons forms for the AdS group. The vielbein
and the spin connection can be viewed as different components
of an (A)dS or Poincare connection, so that its local symmetry is
enlarged from Lorentz to (A)dS (or Poincare when = 0).
The
principal motivation of this work is, using the first order
formalism, find the Hamiltonian formalism for a Chern–Simons
theory
leading to general relativity in a certain limit.
In
the first-order approach, the independent dynamical vari-
ables
are the vielbein (e
a
) and the spin connection (ω
ab
), which
obey first-order differential field equations. The standard second-
order
form can be obtained if the torsion equations are solved for
the connection and eliminated in favor of the vielbein—this step,
http://dx.doi.org/10.1016/j.physletb.2016.04.028
0370-2693/
© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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