Transcript Fundamental Physical Constants, Problems of G and Many

Integrable Cosmological Models
in Diverse Dimensions
Vitaly N. Melnikov
Center for Gravitation and Fundamental Metrology, VNIIMS,
and Institute of Gravitation and Cosmology, PFUR,
Moscow, Russia
Center for Gravitation and
Fundamental Metrology, (1967) present*
• Gravitation, Cosmology, Theory of Gravitational
Experiments, Gravimetry and Gradiometry, Space
Experiments
• Fundamental Physical Constants: G, h, k, c, m, …
and Basic Standards
• Solid State Physics, Liquid Crystalls, Surface
Phenomena, Synchrotron Radiation
• Statistical Physics
• Elementary Particles
Staff: 50  15 during last 15 years
----------------------------------------------------------------------*Russian State Committee for Standards
Topics (Gravitation and
Cosmology Group CCFM)
4D:
1. Exact Solutions with Fields: φ, e-m, their
interactions, Λ in GR: Particle-Like and
Cosmological Solutions (1970-79)
2. Quantum Cosmology (φ, Λ, 71-72)
3. Quantum Effects in Cosmology (bounce
due to SSB, 1978; Vacuum Polarization).
4. FPC, their stability, G measurements,
theory of space experiments. SEE (from
93).
Topics (Gravitation and
Cosmology Group CCFM)
Contd. DD:
4. Multidimensional Cosmology
(Fields, Λ, PF, Forms)
(VNM et al., 1988 – present)
5. Multidimensional BH’s, Stability of BH’s
(VNM + Bronnikov)
6. Stochastic Behavior. Billiards for systems:
PF, 94; p-branes, 99; (φ + V(φ)), 2003.
(VNM + Ivashchuk)
7. Observational Windows of Extra Dimensions
Topics Contd.
1.
2.
3.
4.
5.
NLED in GR. Dilaton interaction with e-m.
First QC model with Λ [creation from nothing]
(1972); first QC model with minimal and conformal
scalar fields (1971)
First nonsingular cosmological model with SSB
(1978-79) of conformal nonlinear scalar field.
First non-singular field particle-like solution with
gravitational field (1979);
Only G may vary with respect to atomic time (78)
(System of Measurements).
Introduction
The second half of the 20th century in the field of gravitation was devoted
mainly to theoretical study and experimental verification of general relativity
and alternative theories of gravitation
with a strong stress on relations between macro and microworld fenomena
or, in other words,
between classical gravitation and quantum physics.
Very intensive investigations in these fields were done in Russia by
M.A.Markov, K.P.Staniukovich, Ya.B.Zeldovich, A.D.Sakharov and
their colleagues starting from mid 60’s.
As a motivation there were:
- singularities in cosmology and black hole physics,
- role of gravity at large and very small (planckian) scales,
- attempts to create a quantum theory of gravity as for other physical fields,
- problem of possible variations of fundamental physical constants etc.
A lot of work was done along such topics as :
- particle-like solutions with a gravitational field,
- quantum theory of fields in a classical gravitational
background,
- quantum cosmology with fields like a scalar one, Λ, …
- self-consistent treatment of quantum effects in cosmology,
- development of alternative theories of gravitation:
scalar-tensor, gauge, with torsion, bimetric etc.
As all attempts to quantize general relativity in a usual manner failed and it
was proved that it is not renormalizable, it became clear that the promising
trend is along the lines of unification of all physical interactions which started
in the 70’s. About this time the experimental investigation of gravity in strong
fields and gravitational waves started giving a powerful speed up in
theoretical studies of such objects as pulsars, black holes, QSO’s, AGN’s,
Early Universe etc., which continue now.
But nowadays, when we think about the most important lines
of future developments in physics, we may forsee that
gravity will be essential not only by itself, but as a missing
cardinal link of some theory, unifying all existing physical
interactions: week, strong and electromagnetic ones.
Even in experimental activities some crucial next generation
gravitational experiments verifiing predictions of unified
schemes will be important.
Epoch Now: Unified Models
SG, S, SS, M-theory…
E-M WEAK STRONG
E-W
Gravity via
.
UT!!
Gravit
GUT
BH’s
Early
Basic
GW
Other
Universe
Fundam.
Exp.
Appl.
Lab for HEP SFO
Terrestrial,
G,G-dot, ISL
EP, Clocks,
Tests of UT models via G
2nd order…
Cosmology as a lab for Super
HEP! New revolution > 1998,
acceleration, DM, DE. 95% -???
Space
Experiments.
- new generation.
We may predict as well that a thorough study of
gravity itself and within the unified models will give in the next century and
millennium even more applications for our everyday life as electromagnetic
theory gave us in the 20th century
after very abstract fundamental investigations of Faraday, Maxwell, Poincare,
Einstein and others, which never dreamed about such enormous applications
of their works.
Other very important feature, which may be envisaged, is an increasing role of
fundamental physics studies, gravitation, cosmology and astrophysics in
particular, in space experiments.
Unique microgravity environments and modern technology outbreak give
nearly ideal place for gravitational experiments which suffer a lot on Earth
from its relatively strong gravitational field and gravitational fields of nearby
objects due to the fact that there is no ways of screening gravity.
In the developement of relativistic gravitation and dynamical cosmology after
A. Einstein and A. Friedmann, we may notice three distinct stages:
1. Investigation of models with matter sources in the form of a perfect fluid, as
was originally done by Einstein and Friedmann.
2. Studies of models with sources as diferent physical fields, starting from
electromagnetic and scalar ones, both in classical and quantum cases.
3. Which is really topical now, application of ideas and results of unified
models for treating fundamental problems of cosmology and black hole
physics, especially in high energy regimes, using ideas of extra dimensions
and p-branes as sources.
Multidimensional gravitational models play an essential role in the latter
approach.
The necessity of studying multidimensional models of gravitation and
cosmology is motivated by several reasons.
First, the main trend of modern physics is the unification of all known
fundamental physical interactions: electromagnetic, weak, strong and
gravitational ones. During the recent decades there has been a significant
progress in unifying weak and electromagnetic interactions, some more
modest achievements in GUT, supersymmetric and superstring theories.
Now, theories with membranes, p-branes and more vague M-theory are being
created and studied.
Having no definite successful theory of unification now, it is desirable to study
the common features of these theories and their applications to solving basic
problems of modern gravity and cosmology.
Moreover, if we really believe in unified theories, the early stages of the
Universe evolution and black hole physics, as unique superhigh energy
regions, are the most proper and natural arena for them.
Second, multidimensional gravitational models, as well as scalar-tensor
theories of gravity, are theoretical frameworks for describing possible
temporal and range variations of fundamental physical constants .
These ideas have originated from the earlier papers of E. Milne (1935) and P.
Dirac (1937) on relations between the phenomena of micro- and macroworlds, and up till now they are under thorough study both theoretically and
experimentally.
Lastly, applying multidimensional gravitational models to basic problems of
cosmology and black hole physics, we hope to find answers to such longstanding problems as
-singular or nonsingular initial states,
-creation of the Universe, creation of matter and its entropy,
-acceleration, cosmological constant, origin of inflation and specific scalar
fields which may be necessary for its realization,
-isotropization and graceful exit problems,
-stability and nature of fundamental constants ,
-possible number of extra dimensions, their stable compactification etc.
Bearing in mind that multidimensional gravitational models are certain
generalizations of general relativity which is tested reliably for weak fields up
to 0.0001 and partially in strong fields (binary pulsars), it is quite natural to
inquire about their possible observational or experimental windows. From
what we already know, among these windows are:
– possible deviations from the Newton and law, or new interactions,
– possible variations of the effective gravitational constant with a time rate
smaller than the Hubble one,
– possible existence of monopole modes in gravitational waves,
– different behaviour of SFO, such as multidimensional black holes,
wormholes, astrophysical sources, - standard cosmological tests etc.
- Possible non-conservation of energy in SFO and accelerators (e.g. LHC), if
BW-models ideas about gravity in the bulk turn out to be true…
------------------------------------------------------------------------------------------Since modern cosmology has already become a unique laboratory for testing
standard unified models of physical interactions at energies that are far beyond
the level of the existing and future man-made accelerators and other
installations on Earth, there exists a possibility of using cosmological and
astrophysical data for discriminating between future unified schemes.
As no accepted unified model exists, in our approach we adopted simple, but
general from the point of view of number of dimensions, models based on
multidimensional Einstein equations with or without sources of different nature:
– cosmological constant,
– perfect and viscous fluids,
– scalar and electromagnetic fields,
– their possible interactions,
– dilaton and moduli fields,
– fields of antisymmetric forms (related to p-branes) etc.
Our program’s main objective was and is to obtain
-exact self-consistent solutions (integrable models) for these
models and then
-- to analyze them in cosmological, spherically and axially
symmetric cases.
In our view this is a natural and most reliable way to study highly
nonlinear systems. It is done mainly within Riemannian
geometry.
Some simple models in integrable Weyl geometry and with
torsion were studied as well.
The Model of MGC
1. n Einstein spaces of constant curvature with
(m+1)-component perfect fluid, Λ (fields, formfields,…)
2. Metric (e.g. cosmological):
n


g   exp 2 (t ) dt  dt   exp 2 x i (t ) g ( i )
i 1
3.
M  R  M1  ... M n
Manifold :
where (
Mi
of dimension
,
Ni
g
 
(i )
) is the Einstein space
:
Rmi ni g (i )  i gm(ii)ni , i  1,..., n ; n  2
4. EMT:
m
T  T
N
M
 0
M ( )
N

TNM ( )  diag   ( ) (t ), p1( ) (t ) km1 1 ,..., pn( ) (t ) kmn n

  0,..., m
 N TMN ( )  0
---------------------------------------------------------------------- ui( )  ( )
(

)
5.EOS :
  (t )
pi (t )  1 
ui( )
Ni 

 const , i  1,..., n ,   0,..., m
and/or EMT for fields (scalar, e-m, forms,…), Λ,…
6. Non-zero components of Ricci tensor:
n


R00   N i X i  X i  ( X i ) 2 ,
i 1
 i
Rmi ni  g mi ni   exp 2 X i  2

i  1,..., n

  i  i  n
 
i

 X  X   N i X    
 i 1
 


7. Conservation law constraint for

( )
 0,..., m
  N i X i  ( )  pi( )   0
n
i 1
Using EOS
( )
( )

( )
i
 (t )  A exp  2 Ni X (t )  u X (t )
( )
A  const
i
i

n
 0   Ni X i
8. Let
(harmonic time gauge),
i 1
1 M
M
2 M
R

then Einstein multidimensional Eqs. N 2  N R   TN
are equivalent to Lagrange Eqs. with Lagrangian
1
L  exp     0 ( x) Gij X i X j  exp    0 V  X 
2
minisuperspace metric
and potential n
Gij  N i ij  N i N j



m
1
i
i
V ( X )     N i exp  2 X  2 0 ( X )    2 A( ) exp ui( ) X i
2 i 1
 0
m

  A exp ui( ) X i
 0


m  m  n , A   2 A( ) ,   0,..., m
Exponential potential !!! Combined form for PF, Λ and λi
• Curvature terms:
Am  i  
1 i
 Ni
2
u
( m i )
j
 2 
i
j
 Nj

where i, j  1,..., n
----------------------------------------------------------------Cosmological constant term: A0   and
u
( 0)
j
 2 N j , j  1,..., n
So,
pi( 0)    ( 0) , i  1,..., n
And 0-component describes Λ
-----------------------------------------------------------------i
j
10. Diagonalization of metric G  Gij dx  dx
It has signature
,,..., so by linear
Transformation Z a  eia X i we diagonalize it
n 1
G  ab dz  dz  dz  dz   dz i  dz i
a
b
0
0
i 1
a, b  0,..., n  1
ab    ab   diag  1,1,...,1 and
• We come to:
 ij
n
1
G 

, D  1   Ni
Ni 2  D
i 1
ij
D is dimension of M.
11. Reduction to σ-model and Toda-like systems,
further to Liouville, Abel, Emden-Fowler Eqs. etc.
12. Behaviour of extra spaces:
- constant,
- dynamically compactified,
- torus,
- large, but with barriers, walls,…
Realized program in MGC (from 1988)
I.
Cosmology, exact solutions in DD for:
-
Λ, (Λ+φ):nonsingular, dynamically compactified, inflationary;
PF, (PF+φ) : nonsingular, inflationary;
Viscous fluid: nonsingular, generation of mass, entropy;
quintessence and coincidence in 2-component model;
Stochastic behaviour near the singularity, billiards in
Lobachevsky space, D=11 critical, φ destroys billiards (94);
Ricci-flat solutions above for any n, also with curvature in
one factor-space;
with curvatures in 2 factor-spaces only for N=10,11;
fields: scalar, dilatons, forms of arbitrary rank (JMP, 98) inflationary, Λ generation; billiads (99);
quantum variants (WDW-equation) for above cases;
dilatonic fields with potentials, billiads;
-
II. Solutions depending on r in DD:
- Generalized Schwarzchild, Tangerlini (BH’s), also with φ(no
BH’s),
- Generalized R-N (BH’s), plus φ (no BH’s);
- Multi-temporal;
- Dilaton-like interaction of φ and e.-m. fields (BH’s only for
special case);
- Stability studies (stable only BH’s case above);
- Same with dilaton-forms interaction, stability only in some
cases, e.g. for one form in particular;
-------------------------------------------------------------------------- Some simple axially-symmetric;
- with torsion;
- in integrable Weyl cosmology
Main Publications in 4D, DD
- VNM, Staniukovich. Hydrodynamics Fields and Constants in
Gravitation Theory, Moscow, Energoatomizdat, 1983 (in
Russian). English version of VNM part in CBPF-MO-02/02 (4D).
- VNM. Multidimensional Classical and Quantum Cosmology and
Gravitation: Exact Solutions and Variations of Constants.
CBPF-NF-051/93, also in “Cosmology and Gravitation”, Ed.
M.Novello, Frontieres, Singapore, 1994.
- VNM. Multidimensional Cosmology and Gravitation, CBPF-MO002/95, also in “Cosmology and Gravitation II”, Ed. M.Novello,
Frontieres, Singapore, 1996.
- VNM and Ivashchuk, in “Lecture Notes in Physics”, 2000, v.157,
p.214.
- VNM and Ivashchuk. Exact Solutions in Multidimensional
Gravity with Antisymmetric Forms. Class. Quant. Grav., 2001,
Topical Review, v.18, pp. R1-R66.
- VNM. Multidimensional Gravitation and Cosmology. II.
CBPF-MO-003/ 02, Rio de Janeiro. 2002.
--------------------------------------------------------------------------------------See also in gr-qc, hep-th for VNM.
---------------------------------------------------------------------------------------
Recent papers with scalar fields in 4D:
- VNM, Gavrilov, Dehnen. General Solutions for Flat Friedmann
Universe Filled by PF and Scalar Field with Exponential Potential.
Grav. & Cosm., 2003, v.9, p. 189 (acceleration, coincidence).
- VNM, Gavrilov, Abdyrakhmanov. Friedmann Universe with Dust
and Scalar Field with Multiple Exponential Potential. GRG,
2004, v.36, N 7, p. 1579 (acceleration, recollapse, coincidence).
In DD :
• VNM, Ivashchuk, Selivanov. Cosmological Solutions in
Multidimensional Model with SF, Multiple Exponential Potential.
JHEP, 0309 (2003) 059 (classical and quantum, acceleration,)
• VNM, Ivashchuk, S.-W. Kim. S-brane Solutions with
Acceleration in Models with Forms and Multiple Exponential
Potential. Crav. & Cosm., 2004, v. 10, N1-2, p. 141.
• VNM, Ivashchuk, Selivanov. Composite S-brane Solutions on
Product of Ricci-Flat Spaces. GRG, 2004, v. 36, N 7(accelerat.)
• VNM, Alimi,Gavrilov. Multicomponent Perfect Fluid with Variable
Parameters in n Ricci-Flat Spaces. JKPS, v.44, p. S148
(acceleration, isotropisation in 4D, compactification in DD)
• VNM, Alimi, Ivashchuk. Non-singular solutions in
multidimensional model with scalar fields and exponential
potential. Grav. & Cosm., 2005, N1-2.
G-dot in (4+N)-dimensional cosmology with multicomponent
anisotropic fluid (VNM+ Ivashchuk, 2003, JKPS)
We consider here a (4+N)-dimensional cosmology with an isotropic 3-space and an
arbitrary Ricci-°at internal space. The Einstein equations provide a relation between
and other cosmological parameters.
1 The model
Let us consider (4 + N)-dimensional theory described by the action
where
is the fundamental gravitational constant. Then the gravitational field
equations are
where
N + 3.
is a (4+N)-dimensional energy-momentum tensor,
, and M, P = 0, …,
For the (4 + N)-dimensional manifold we assume the structure
where
is a 3-dimensional space of constant curvature,
,
,
for k =
+1, 0, -1, respectively, and
is a N-dimensional compact Ricci-flat Riemann manifold.
The metric is taken in the form
where i, j, k = 1, 2, 3; m, n, p = 4, …, N + 3;
,
the metrics and scale factors for
and
. For
multi-component (anisotropic) fluid form
,
and
are, respectively,
we adopt the expression of the
Under these assumptions the Einstein equations take the form
The 4-dimensional density is
where we have normalized the factor b(t) by putting
On the other hand, to get the 4-dimensional gravity equations one should put
. Consequently, the e®ective 4-dimensional gravitational "
constant" G(t) is defined by
whence its time variation is expressed as
2 Cosmological parameters
Some inferences concerning the observational cosmological parameters can be extracted
just from the equations without solving them [3]. Indeed, let us de¯ne the Hubble
parameter H, the density parameters
and the "deceleration" parameter q referring to a
¯xed instant
in the usual way
Besides, instead of G let us introduce the dimensionless parameter
Then, excluding b from (1.6) and (1.8), we get
with
where
When
is small we get from (1.15)
Note that (1.18) for N = 6, m = 1,
(so that
) coincides with the
corresponding relation of Wu and Wang [1] obtained for large times in case k = -1 (see
also [2]).
If k = 0, then in addition to (1.18), one can obtain a separate relation between
and
,
namely,
(this follows from the Einstein equation
combination of (1.6)-(1.8).
The present observational upper bound on
if we take in accord with [4, 5]
and
, which is certainly a linear
is
3 Two-component example: dust + (N -1)-brane
Let us consider two component case: m = 2. Let the first component (called "matter") be a
dust, i.e.
and the second one (called "quintessence") be a (N - 1)-brane, i.e.
We remind that as it was mentioned in [6] the multidimensional cosmological model on
product manifold
with fields of forms (for review see [8]) may be
decribed in terms of multicomponent "perfect" fluid [7] with the following equations of state
for
component:
if p-brane worldvolume contains
and
in opposite case. Thus, the field
of form matter leads us either to
, or to stiff matter equations of state in internal
In this case we get from (1.18) for small a
spaces.
and for k = 0 and small g we obtain from (1.19)
Now we illustrate the formulas by the following example when N = 6 (
Yau manifold) and
We get from (1.24)
may be a Calabi-
in agreement with (1.20).
In this case the second fluid component corresponds to magnetic (Euclidean) NS5-brane (in
D = 10 type I, Het or II A string models). Here we consider for simplicity the case of constant
dilaton field.
SCALAR-TENSOR COSMOLOGY AND VARIATIONS OF G
(VNM+Bronnikov, Novello, G&C, 2002)
The purpose of this note is to estimate the order of magnitude of variations of the
gravitationalconstant G due to cosmological expansion in the framework of scalar-tensor
theories (STT) ofgravity.
Consider the general (Bermann-Wagoner-Nordtvedt) class of STT where gravity is characterized by the metric
and the scalar field
; the action is
Here
is the scalar curvature,
, and U are certain functions of
varying from theory to theory,
is the matter Lagrangian.
,
This formulation of the theory corresponds to the Jordan conformal frame, in which
matter particles move along geodesics and hence the weak equivalence principle is valid,
and non-gravitational fundamental constants do not change. In other words, this is the
frame well describing the existing laboratory, geophysical and cosmological observations.
Among the three functions of
entering into (1) only two are independent since there is
a freedom of transformations
. We use this arbitrariness, choosing
as is done, e.g., in Ref. [1]. Another standard parametrization is to put
,
and
( the Brans-Dicke parametrization of the general theory (1)). In our parametrization
, the Brans-Dicke parameter , the Brans-Dicke parameter
is
; here and henceforth, the subscript denotes a derivative with respect to
The Brans-Dicke STT is the particular case
, so that in (1)
The field equations that follow from (1) read
where
is the D'Alembert operator, and the last term in (4) is matter energy-momentum tensor
odf matter.
Consider now isotropic cosmological models with the standard FRW metric
where
is the scale factor of the Universe, and k = −1, 0, 1 for closed, spatially
flat and hyperbolic models, respectively. Accordingly, we assume
and the
energy-momentum tensor of matter in the perfect fluid form
( is the density and is the presuure).
.
The field equations in this case can be written as follows:
To connect these equations with observations, let us fix the time t at the present epoch (
(i.e., consider the instantaneous values of all quantities) and introduce the standard
observables:
where
is the critical density, or, in our model, the r.h.s. of Eq. (7) in case k = 0:
.
This is slightly different from the usual definition
where G is the Newtonian
gravitational constant. The point is that the locally measured Newtonian constant in STT
differs from
; provided the derivatives
and
are sufficiently small, one has [1]
Since, according to the solar-system experiments,
, for our order-of-magnitude
reasoning we can safely put
, and, in particular, our definition of
now
coincides with the standard one.
The time variation of G, to a good approxiamtion, is
where, for convenience, we have introduced the coefficient
of the Hubble parameter H.
expressing
in terms
Eqs. (6)(8) contain too many arbitrary parameters for making a good estimate of . Let
us now introduce some restrictions according to the current state of observational cosmology:
(i) k = 0 (a spatially flat cosmological model, so that the total density of matter equals
);
(ii) p = 0 (the pressure of ordinary matter is negligible compared to the energy density);
(iii)
(the ordinary matter, including its dark component, contributes to only 0.3 of
the critical density; unusual matter, which is here represented by the scalar field,
comprises the remaining 70 per cent).
Then Eqs. (7) and (8) can be rewritten in the form
Subtracting (8) from (7), we exclude the cosmological constant U , which can be quite large
but whose precise value is hard to estimate. We obtain
The first term in Eq. (13) can be represented in the form
and
can be replaced with
. The term can be neglected for our estimation
purposes. To see this, let us use as an example the Brans-Dicke theory, in which
. We then have
here the first term is the same as the first term in Eq. (13), times the small parameter
.
Assuming that
is of the same order of magnitude as
(or slightly greater), we see that,
generically,
. Note that our consideration is not restricted to the Brans-Dicke theory and
concerns the model (1) with an arbitrary function
and an arbitrary potential
.
Neglecting , we see that (13), divided by
, leads to an algebraic equation with respect
to :
where
.
According to modern observations, the Universe is expanding with an acceleration, so that
the parameter q is, roughly, -0.5 ± 0.2, hence we can take
.
In case
we simply obtain
. Assuming
and
, we come to the estimate
Where
is, by modern views, close to 0.7.
For nonzero values of
we arrive at the estimate
have instead of (15)
, solving the quadratic equation (14) and assuming
, so that, taking
and again
,
, we
We conclude that, in the framework of the general STT, modern cosmological observations,
taking into account the solar-system data, restrict the possible variation of G to values within
. This estimate may be considerably tightened if the matter density parameter
and the (negative) deceleration parameter
will be determined more precisely.
Thanks !
Merci !
Danke !
Спасибо !
Special Thanks to Local
Organizers
Multidimensional models
In the 80’s the supergravitational theories were “replaced” by superstring models.
Now it is heated by expectations connected with the overall M-theory. In all these
theories, 4-dimensional gravitational models with extra fields were obtained from
some multidimensional model by dimensional reduction based on the
decomposition of the manifold M = M4×Mint,
where M4 is a 4-dimensional manifold and Mint is some internal manifold (mostly
considered to be compact).
The earlier papers on multidimensional gravity and cosmology dealt with
multidimensional Einstein equations and with a block-diagonal cosmological or
spherically symmetric metric defined on the manifold M = R ×M0 × . . . ×Mn of the
form
where (Mr, gr) are Einstein spaces, r = 0, . . . , n. In some of them a
cosmological constant and simple scalar fields were also used. Such models
are usually reduced to pseudo-Euclidean Toda-like systems with the
Lagrangian
It should be noted that pseudo-Euclidean Toda-like systems are not yet wellstudied. There exists a special class of equations of state that gives rise to
Euclidean Toda models [9].
At present there exists a special interest to the so-called M- and F-theories etc.
These theories are “supermembrane” analogues of the superstring models in D
= 11, 12 etc. The low-energy limit of these theories leads to models governed
by the Lagrangian
where g is the metric, Fa = dAa are forms of rank na, and φa are scalar fields.
It was shown that, after dimensional reduction the manifold may be
M0 ×M1 × . . . ×Mn
and when the composite p-brane ansatz is considered, the problem is reduced
to the gravitating self-interacting σ-model with certain constraints.