Theories of gravity in 5D brane-world scenarios

1) Introduction

• Geometry of spacetime is described by metric tensor g.

• Matter tells spacetime how to curve and curvature

of spacetime tells the matter how to move.

• Field equations of GR are called Einstein equations.

We know that the proper theory of gravity is General Relativity (GR).

Some basic features of GR:

Einstein equations:

Einstein tensor (describes curvature of spacetime)

energy-momentum tensor (describes distribution of matter in the spacetime)

2) Quantization of gravity

There is a serious problem with GR:

When we try to quantize GR in perturbation method (as one quantizes other fields in quantum field theory) we obtain a nonrenormalizable theory.

A nonrenormalizable theory can not be considered as the proper theory.

Proposed quantum gravity theories:

Loop quantum gravity (LQG)

• non-perturbative quantization (renormalization problem does not exist)

• unifies gravity and quantum mechanics only

• weakness: presently, it is not clear how to explain the existence of dark energy and dark matter within LQG

M-theory

it is supposed to unify all known interactions (including gravity)

• it is defined in 11 dimensions

• fundamental constituents of the universe are strings, membranes and hihger dimensional objects (p-branes)

• weakness: it makes sense only in the case when supersymmetry is realised in nature

• even more serious weakness: anuniquence in compactification.

3) Why 5 dimensions?

• 5 dimensional models are inspired by M-theory.

• In M-theory we can assume many different topology of additional dimensions.

• One of the most interesting (the simplest case):

Calabi –Yau space, result of the compacification of 6 dimensions

Known 4dspacetime The fifth dimension

Topology of the fifth dimension:

0 π

0 π

Usually it is assumed that size of the fifth dimension is much biggerthan the size of Calabi –Yau space.

In the 1st approximation we can ignore details of the Calabi-Yau space. We have obtained 5d model

This 5 dimensional model have topology:

0 π

Four dimensional 3-branes (3 space + time) at the ends of the fifth dimension. Our universe is on the one of these two branes.

This is called brane-world scenario.

bulk

Assumptions:

1. Only gravitational field and assumed „volume fields” (like radion field) can propagate in the fifth dimension.

2. Other fields and matter are confined to the branes.

3. In the simplest case branes cannot move (we want to omitte problems with colliding branes )

Possible size of the fifth dimension.

There are two possibilities:

1. Additional dimension is so small that we cannot observe it in our experiments (compactification). maybe possible in future

2. We assume very special properties of the model (space in the bulk, cosmological constant...). Then we can have even infinite size of the fifth dimension without violating known formula of gravitional force. an example: Randall-Sundrum model

Some interesting proposals

Cyclic model

It is a cosmological theory alternative to the standard cosmology.

Basic features:

• the branes can move and collide with each other• the brane collison from our 4d perspective looks like a big crunch/big bang• evolution of the universe is a sequence of quantum and classical phases• dark energy is described by radion field

Cyclic model is promising because:

It solves dark energy problem (proper potential of radion field).

It solves dark matter problem (matter on the second brane is a dark matter from our perspective).

Problem: What happens when two branes collide?

It is expected that Quantum Gravity will answer this question.

4) Mathematical formalism

The simplest way to obtain different (classical) theory of gravity from General Relativity is to add some additional terms to the Lagrangian of the Hilbert-Eistein action. Terms that we want to add should be important only in smallscales because in the big scale limit we want to obtain GR.

These terms are called Euler densities of rank n ≥ 2. They are proportional to second (and higher) power of the curvature scalar.

A) Modifications of General Relativity

On the other hand, Euler densities of rank higher than one enter, in some natural way (inspired M-theory), the brane-world scenario. In 5d spacetime the only Euler density that has a non-trivial dynamical content is for n=2.

It is called the Gauss-Bonnet term:

B) Lagrangian formulation of GR

Einstein equations can be derived from the Lagrangian

Making use of the variational principle gives:

C) Stacking solutions

Einstein equations are very difficult to solve because they are nonlinear.

making assumptions about symmetries of the metric help to find solutions

Stacking solutions are examples of this idea.

It is a procedure to build d+1 dimensional solutions of GR starting from d dimensional ones by stacking d dimensional metric into the extra dimension.

5) Some results of my work

I have considered one of the simplest models of the brane-world scenario.

Basic features:

There are two branes with no Standard Model fields on them.

The branes are assumed not to move with respect to each other.

Stacking solutions procedure is used.

Metric tensor has the form:

Warp factorAny 4d solution of Einstein equations

Two cases are considered:

1) General Relativity in five dimensions

2) GR with Gauss-Bonnet term (Einstein-Gauss-Bonnet gravity) in five dimensions

My goal was to obtain the following objects:

• Warp factor:

• Cosmological constants on the branes: and

These objects plus the 4d metric tensor contain all the information about considered model.

But, considered 4d metric can be any solution of Einstein equations.

We have obtained a large class of 5d solutions.

Specific results (1)

Ad.1) An ordinary General Relativity

For: and

For positive sign of R:

(a comfortableconvention)

Where y0 is any non-zero number.

Specific results (2)

We have also:

For negative sign of R:

Specific results (3)

Ad.2) Einstein-Gauss-Bonnet gravity

For positive sign of R:

For negative sign of R:

Where:

Specific results (4)

We have also:

Where α multiplies Gauss-Bonnet term.

6) Conclusions

Generalization of GR

Our central aim is to describe the classical phase of the cyclic model.We should:• introduce Standard Model fields to the branes• let the branes to move and introduce additional volume fields

Resulting brane-world scenario can be used to model the universe