Theories of gravity in 5D braneworld scenarios

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Theories of gravity in 5D brane-world scenarios
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1) Introduction
We know that the proper theory of gravity is
General Relativity (GR). Some basic features of
GR

• 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.

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Einstein equations
energy-momentum tensor (describes distribution
of matter in the spacetime)
Einstein tensor (describes curvature of
spacetime)
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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.
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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

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• 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.

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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 4d spacetime
The fifth dimension
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Topology of the fifth dimension
0
p
0
p
Usually it is assumed that size of the fifth
dimension is much bigger than the size of Calabi
Yau space.
In the 1st approximation we can ignore details of
the Calabi-Yau space. We have obtained 5d model
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This 5 dimensional model have topology
bulk
p
0
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.
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• Assumptions
• Only gravitational field and assumed volume
  fields
• (like radion field) can propagate in the fifth
  dimension.
• 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
  )

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Possible size of the fifth dimension.

• There are two possibilities
• Additional dimension is so small that we cannot
• observe it in our experiments (compactification).
• maybe possible in future
• 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

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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

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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.
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4) Mathematical formalism
A) Modifications of General Relativity
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 small scales
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.
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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 n2.
It is called the Gauss-Bonnet term
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B) Lagrangian formulation of GR
Einstein equations can be derived from the
Lagrangian
Making use of the variational principle gives
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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 d1 dimensional
solutions of GR starting from d dimensional ones
by stacking d dimensional metric into the extra
dimension.
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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.
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Stacking solutions procedure is used. Metric
tensor has the form
Any 4d solution of Einstein equations
Warp factor

• Two cases are considered
• 1) General Relativity in five dimensions
• 2) GR with Gauss-Bonnet term (Einstein-Gauss-Bonn
  et
• gravity) in five dimensions

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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.
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Specific results (1)
(a comfortable convention)
For
and
Ad.1) An ordinary General Relativity
For positive sign of R
Where y0 is any non-zero number.
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Specific results (2)
For negative sign of R
We have also
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Specific results (3)
Ad.2) Einstein-Gauss-Bonnet gravity
For positive sign of R
For negative sign of R
Where
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Specific results (4)
We have also
Where a multiplies Gauss-Bonnet term.
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6) Conclusions
Generalization of GR
Resulting brane-world scenario can be used to
model the universe

• 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