1. University of Oslo & Caltech
Four-dimensional General Relativity from higher-dimensional Gauss-Bonnet gravity
Ingunn Kathrine Wehus
University of Oslo & Caltech
Cosmo08, Madison
August 26th, 2008
Based on Carroll and Wehus 2008, in preparation

2. Four-dimensional General Relativity from higher-dimensional Gauss-Bonnet gravity
Natural generalization of GR in higher dimensions
Special case of Lovelock gravity
Compactification of the extra dimensions
Give GR coupled to a scalar field in 4 dimensions
Scalar field may be stabilized
Gives pure GR, but with modified Planck mass
May be relevant to the hierarchy problem

3. Gauss-Bonnet
The Gauss-Bonnet theorem in differential geometry connects the geometry (curvature) of a surface to its topology (Euler characteristic)
In 2 dimensions the Ricci scalar R is a topological invariant.
The action gives trivial field equations.
However, in higher dimensions (D > 2) the same Lagrangian gives GR
Similarily, in 4 dimensions the Gauss-Bonnet scalar B is a topological invariant
The action gives trivial field equations for D=4.
For D > 4 it gives Gauss-Bonnet gravity.

4. Lovelock gravity
In D = 2n (+1 if odd) dimensions the most general action yielding second order field equations for a dynamical spacetime metric is given by the Lovelock Lagrangian
For example, the lowest order cases are

5. Gauss-Bonnet gravity
The combined GR and Gauss-Bonnet action in D dimensions,
gives the equations of motion
where
For a maximally symmetric spacetime this simplifies to
For D > 4, there are two vacuum solutions,
and

6. Compactification
We now Kaluza-Klein compactify the extra dimensions, to obtain an
effective theory for our ordinary four dimensional spacetime
We then find

7. Dimensional Reduction
Putting the previous expressions into the action
and integrating over the extra dimensions gives
where

8. From Jordan frame to Einstein frame
The Weyl transformation,
brings us to Einstein frame,
where

9. Stabilization by matter
Can various forms of matter help stabilize the scalar field?
Some available components:
Cosmological constant
Magnetic field wrapped around extra dimensions
Scalar fields
Casimir energy due to the compact space
Adding all these,
one obtains the combined potential

10. Pure Gauss-Bonnet
We first consider the special case of pure Gauss-Bonnet,
which, after dimensional reduction, gives the following effective action
In n = 2 extra dimensions, this action is already in Einstein frame

11. Scales and hierarchies
Adding matter gives the following potential for the scalar field
Choosing for instance
For b stabilized, the field equations reduce to
where

12. Summary
Gauss-Bonnet gravity is a natural extension of General Relativity when considering higher dimensions
Higher-dimensional Gauss-Bonnet gravity compactified to four dimensions reduces to General Relativity, coupled to a scalar field
The details of this scalar-tensor theory differ from the GR case
After stabilizing the scalar field, the four dimensional theory of gravity is pure General Relativity, but with a modified Planck mass