Living on a braneworld

1
(No Transcript)
2
Gauss-Bonnet gravity
The Gauss-Bonnet combination,

• is topological in D4, but dynamical in Dgt4
• gives rise to second order field equations in
  the metric
• appears in the slope expansion for the heterotic
  string action

3
Field equations and vacua
4
(No Transcript)
5
Schwarzschild solutions in GR
Consider a spherically symmetric source of
inertial mass, M. Since Birkhoffs theorem
applies outside the source, the exterior
solution must be given by
For Mgt0, the singularity at r0 is shielded by an
horizon, in accordance with cosmic censorship
Gravitational (ADM) mass, E, of this solution
exactly matches inertial mass of source, ie EM
6
Schwarzschild solutions in GB gravity
For the same source, a generalized form of
Birkhoffs theorem applies outside, so that we
admit two branches of Schwarzschild like
solutions, with potentials given by
Note that there are naked singularities on the GB
branch () even when Mgt0. Cosmic censorship fails!
7
Asymptotic behaviour and gravitational mass
This would mean that Gauss-Bonnet vacuum is not a
true local ground state of the theory, and is
therefore unstable Boulware Deser PRL (1985),
Wheeler Nuc. Phys. B (1986), Myers Simon PRD
(1988), Deser Yang CQG (1989)
8
A perturbative ghost
Negative energy on GB branch was attributed to
perturbations around the GB vacuum having the
wrong sign.
9
Ghosts carry negative energy, and can even lead
to gravitational repulsion, but do they really
alter the gravitational mass of the
Schwarzschild solutions, as originally claimed by
Deser, Wheeler and all the others?
Answer No!

• Reason
• A negative energy ghost can only lower the
  gravitational energy if it is freely
  propagating.
• Schwarzschild solutions only excite scalar modes
  (Birkhoffs thm) but these do not propagate
  (there are no scalar grav. waves).
• Therefore scalars can only carry the energy given
  to them by the source.
• The tensor mode of the ghost can in principle
  propagate, but is not excited by the these
  solutions.

The gravitational mass of the Schwarzschild-like
solutions must be the same as the inertial mass
of the source!
10
Gravitational mass in GB gravity
This is now consistent with Deser Tekins more
recent work (eg hep-th/0205318)) on energy in
quadratic gravity theories. By developing new
techniques, they corrected the old results, and
found that So gravitational massinertial
mass, on both branches! This also agrees with
my Hamiltonian calculation (gr-qc/0303082) We
now understand why this had to be its just
because there are no scalar gravitational waves
propagating on the vacuum, even in GB gravity!
And because Birkhoffs theorem still applies.
11
So is the GB vacuum stable?
Deser Tekin argued that it was, since there was
no funny business with Schwarzschild masses,
contrary to what we believed in the 80s. But now
that we understand the role of the ghost, we know
this cant be right The spherically symmetric
sources only excite non-propagating scalars, and
not the propagating tensor ghost. But the ghost
hasnt gone away. Other sources can and will
excite the ghost. eg a binary system of higher
dimensional black holes
12
Some conclusions

• Scalar modes transfer the inertial mass of the
  source to the gravitational mass of all
  spherically solutions in Gauss bonnet gravity.
• There are tensor ghosts on the GB branch, but
  these do not get excited by the above.
• Since the ghost couples to ordinary matter, it
  leads to classical instabilities because it is
  excited by any source capable of emitting
  gravitational waves
• The ghost can even lead to an instability of the
  true vacuum through divergent ghost-nonghost
  particle production.
• There are no ghosts on the Einstein branch, so
  there are none of these pathologies

13
The Chern-Simons limit--some caveats and other
results

• These results rely on linearised theory, and can
  only be trusted when the linearised approximation
  is valid.
• When we are close to the Chern-Simons limit,
  linearised theory is only valid on very large
  scales.
• Close to Chern-Simons limit, one can study
  instanton transitions between branches in the
  thin wall approximation. It turns out there is
  large mixing between branches, so neither branch
  can accurately describe the true vacuum in this
  limit.
• In the exact Chern-Simons limit not clear whether
  there is a ghost or not. Does it go away, or does
  it mix with another mode and survive, as in
  chiral limit of topologically massive gravity in
  AdS, or in the zero tension limit of the DGP
  model.