21) IntroductionWe 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 curvatureof spacetime tells the matter how to move.Field equations of GR are called Einstein equations.
3Einstein equations: energy-momentum tensor (describes distribution of matter in the spacetime)Einstein tensor(describes curvatureof spacetime)
42) 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 asthe proper theory.
5Proposed quantum gravity theories: Loop quantum gravity (LQG)non-perturbative quantization (renormalization problem does not exist)unifies gravity and quantum mechanics onlyweakness: presently, it is not clear how to explain the existence of dark energy and dark matter within LQG
6M-theoryit is supposed to unify all known interactions (including gravity)it is defined in 11 dimensionsfundamental 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 natureeven more serious weakness: anuniquence in compactification.
73) 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):Known 4dspacetimeCalabi –Yau space, result of the compacification of 6 dimensionsThe fifth dimension
8Topology of the fifth dimension: ππUsually 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 ofthe Calabi-Yau space.We have obtained 5d model
9This is called brane-world scenario. This 5 dimensional model have topology:bulkπFour dimensional 3-branes (3 space + time) at the ends of the fifthdimension. Our universe is on the one of these two branes.This is called brane-world scenario.
10Assumptions: 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 )
11Possible size of the fifth dimension. There are two possibilities:Additional dimension is so small that we cannotobserve it in our experiments (compactification).maybe possible in futureWe assume very special properties of the model(space in the bulk, cosmological constant...).Then we can have even infinite size of the fifth dimensionwithout violating known formula of gravitional force.an example: Randall-Sundrum model
12Some interesting proposals Cyclic modelIt is a cosmological theory alternative to the standard cosmology.Basic features:the branes can move and collide with each otherthe brane collison from our 4d perspective looks likea big crunch/big bangevolution of the universe is a sequence of quantumand classical phasesdark energy is described by radion field
13Cyclic model is promising because: It solves dark energy problem (proper potential of radion field).It solves dark matter problem (matter on the second brane isa dark matter from our perspective).Problem: What happens when two branes collide?It is expected that Quantum Gravity will answer this question.
144) Mathematical formalism A) Modifications of General RelativityThe simplest way to obtain different (classical) theory of gravityfrom 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 ≥ They are proportional to second (and higher) power of the curvature scalar.
15On 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 hasa non-trivial dynamical content is for n=2.It is called the Gauss-Bonnet term:
16B) Lagrangian formulation of GR Einstein equations can be derived from the LagrangianMaking use of the variational principle gives:
17Einstein equations are very difficult to solve because C) Stacking solutionsEinstein equations are very difficult to solve becausethey are nonlinear.making assumptions about symmetries of the metric help tofind solutionsStacking solutions are examples of this idea.It is a procedure to build d+1 dimensional solutions of GRstarting from d dimensional ones by stacking d dimensionalmetric into the extra dimension.
185) Some results of my work I have considered one of the simplest modelsof 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.
19Stacking solutions procedure is used. Metric tensor has the form:Any 4d solution ofEinstein equationsWarp factorTwo cases are considered:1) General Relativity in five dimensions2) GR with Gauss-Bonnet term (Einstein-Gauss-Bonnetgravity) in five dimensions
20My goal was to obtain the following objects: Warp factor:Cosmological constants on the branes:andThese objects plus the 4d metric tensor contain allthe information about considered model.But, considered 4d metric can be any solutionof Einstein equations.We have obtained a large class of 5d solutions.
21Specific results (1) (a comfortable convention) For: and Ad.1) An ordinary General RelativityFor positive sign of R:Where y0 is any non-zero number.
22Specific results (2)For negative sign of R:We have also:
23Specific results (3) Ad.2) Einstein-Gauss-Bonnet gravity For positive sign of R:For negative sign of R:Where:
24Specific results (4) We have also: Where α multiplies Gauss-Bonnet term.
256) Conclusions Generalization of GR Resulting brane-world scenario can be usedto model the universeOur central aim is to describe the classical phaseof the cyclic model.We should:introduce Standard Model fields to the braneslet the branes to move and introduce additionalvolume fields