The unified model of dark energy and dark matter proposed by

Albert Einstein (interpreted by A.T.Filippov)

Report prepared for the seminar in memory of

Victor Ogievetsky

Dubna, 22.07.2008

N.B.: The slides are from the reports.

[1]

[2]

This approach is applicable to any Lagrangian.

Thus, it can be shown that the connection obtained for

the special case is in fact general. The important thing

is that supposing the equation of motion to follow from

an action principle with the Lagrangian, which is any

function of the symmetric and anti-symmetric parts of

the curvature fixes the geometry (connection) and,

eventually, allows to fix some metric compatible with this

non-Riemannian connection. It does not coincide

with the Weyl connection. The affine spaces derived

by Weyl in the frame of his `unified’ theory of gravity and

electricity are usually called the Weyl spaces. In the

Eddington – Einstein approach the connection is a

bit different. With Einstein’s choice of the effective

Lagrangian, the original Lagrangian should be similar to

the effective one. Probably, this was not the best choice,

but this theory was left by its great author unfinished and

his choice of the Lagrangian should only be considered

as a reasonable guess compatible with the original

geometric idea. Let us formalize these statements.

i i j k i kjk j jk i

i ijk kj

General defines

parallel displacement of vecto

CONNECT

rs: δA = Γ A δx , δA = -

ION

Γ A δx .

Curvature tens

symmetric conne Γ = Γction

l l l r l r lijk ij,k ik, j ij rk ik rj

j r jij,k ij

j jik ijk r

j rik, j irk j ik k

or: R = - Γ + Γ - Γ Γ + Γ Γ

Ricci tensor: R = R = - + - + Γ R

Γ Γ Γ

General rep re

Γ Γ

sent

ijk

i il il1 1jk jk,l lj,k kl,j jkl ljk klj 2 2

iljk kj ijk ij;k lj

Γ :

,

where and (here = 1 , ; denotes

a "co

ation

vari

for

Γ = a ( - a + a + a ) - a ( - a + a + a )

a = a a a a a

jk

jk

jk ijk ij;k

ijk

ant derivative" with respect to the "metric" ) .

If then = 0

and term is the standard Christoffel symbol

g is the Riemann metric,

the first

a

a a g

.

ijk ij k jk i ki j

Let us introduce a special connection which we will call

th

e

)

e

D n

a = l a φ + m ( Weyl - Einstein connecti

a φ + a φon:

jkjk

i i i1k j j k2

ijk jk

ijk

oting , we find

For the Weyl connection l = m = 1 (so defined!)

For the Einstein connectio

g

a

= + [ (l-2m) φ - l δ φ - l δ φ ] . Γ g

1 13 3

k

n l = - , m =

(derived from the action principle, see below)

Both Weyl and Einstein relate to the electromagnetic potential. φ

Apparently, Einstein was disappointed in the theory with the

cosmological constant (Friedmann papers appeared in 1922-

1923), and also gradually realized that his interpretation of the

antisymmetric field as the electromagnetic field was not quite

satisfactory (indeed, even if the mass term is very, very small,

the theory is not the Maxwell theory). Anyway, he completely

abandoned this model. W.Pauli, in his addenda to English

translation (1956) of his famous book on general relativity, gave

a summary of main ideas of these papers but did not discuss the

concrete models, regarding them as irrelevant to physics of that

time. Somewhat earlier, to these ideas returned E. Schroedinger,

who thought it necessary to go to non-symmetric connections.

He devoted a few pages to this topic in the book `Space-Time

Structure’ (this book strongly influenced the attempt of B.A.Arbuzov and A.T.F. to

unify weak, electromagnetic, and gravitational interactions (1965-1967)).

Following the approach to DG developed in papers of V.De Alfaro and A.T.F. it is not difficult to derive these equations. Unfortunately, this dilaton gravity coupled to massive vector field is more complex than the well studied models of dilaton gravity coupled to scalar fields and thus it requires a separate study.

The first question is: are there exact analytical solutions like Schwarzschild or Reissner-Nordstroem black holes? If the vector field is constant we return to exactly soluble DG having explicit solutions with horizons. Otherwise, when the vector field is non-trivial, the answer is more difficult to find but it is worked out in some detail and briefly presented below. Thus, the simplest thing to do is to further reduce the theory to static or cosmological configurations. Consider first the static reduction.

The simplest way to derive the correspondent equations is to suppose that all the functions in the equations depend on r=u+v . These are not the most general reductions! There exist more general reductions that allow one to simultaneously treat black holes, cosmologies and some waves. These generalized reductions were earlier proposed in our papers devoted to dilaton gravity coupled to scalar fields and Abelian gauge fields; here we only discuss in some detail static and cosmological reductions. In both cases it can be seen that the perturbed theory (with non the vanishing mass term) is qualitatively different from the unperturbed one. Indeed, the unperturbed theory is just dilaton gravity coupled to the electromagnetism. This model is equivalent to pure dilaton gravity which is a topological theory. In particular, it automatically reduces to one dimensional static or cosmological models that can be analytically solved. Static states are the Reissner-Nordstroem black holes perturbed by the cosmological constant and having two horizons, while the space between horizons may be considered as an unrealistic cosmology. This object is known for long time; I think it was familiar to Einstein in 1923 but he did not discuss the static configuration and apparently did not consider black holes or horizons as having any relation to physics. In addition, at that time there was no clear understanding of the gauge transformation although the foundations of the general gauge principle were by this time formulated by Weyl in his `unified theory of electromagnetism and gravity’ we briefly mentioned above…

2(2) 2 2 2 2

The general Vecton Dilaton Gravity Lagrangian is :

= -g φR W(φ) ( φ) + V(φ) - φ F - φ m A , (Ι)

where we denote (one can add scalar fields but they do not appe

L g

2 ij kl 2 ijik jl i j

ar in WEE) :

F g g F F A = g A A V(φ) = 2(1 + Λ)

Here we do not bother on correct normalization and choice of units what is

important for applications to bla

(2)

ck holes and cosmologies. The general

Lagrangian can describe spherically or cylindrically symmetric time

dependent solutions or axial static ones. Further reductions to the dimension

1+ 0 or

L

0 +1 describe static or cosmological solutions which we considered

in some detail elsewhere. Here we concentrate on static spherically symmetric

states and on isotropic homogeneous cosmologies.

Vecton Dilaton Gravity

2(2) 2 2 2 2

We first discuss static spherical solutions. For the spherical case we have

= -g φR 2 2Λ + ( φ) /2φ - F - φ m A . (ΙΙ)

Using the Weyl rescaling, one may write the eff

iL

2(2) w -1/2 1/2 -3/2 2 2 2

w 1/2ij ij

ective Lagrangian

= -g φR 2φ 2Λφ - φ F - φ m A , (ΙΙΙ)

where the metric g is replaced by g φ g . By varying w.r.t. the

diagonal metric coefficients

WL

wii

2

2 2i i i

g we first write the energy and momentum

constraints in the light cone (LC) metric, ds 4f(u,v) du dv :

[f (φ / f)] + φ m A 0, φ φ , i = u, v .

These equations shouli i

-1/2 1/2 3/2 -2 2uv

d be derived by varying the Lagrangian in the general

metric. The other equations may be derived directly in LC metric :

φ + f 2φ 2Λφ - φ f F / 2 0, u v

uv v u u v

3/2 -1 2 j ij j

where F A A , and the vecton field satisfies the equations

(φ f F ) φ m A , i, j = u, v .

From the last equation immediately follows that

v u u v

μμ

(φ A ) + (φ A ) 0.

In the original 4-dimensional theory this is ( -g A ) = 0 condition

eliminating spin 0. Weyl, Eddington and Einstein called this the Lorentz

condition although w

e know that its origin and meaning is quite different

from the gauge fixing condition in Maxwell's theory first introduced by

L.Lorenz and later popularized by the H.A.Lorentz.

Now it is not difficult to write the . We first define

two additional functions (in fact related to the momenta in the

Hamiltonian description) by the equations:

φ = χ

static equations

-3/2 A = f φ B

2 2 212

-1/2 1/2 -3/2 2

Then the other equations are

χ = - f U , B = - φ m A , f = (f / χ) f U + φ m A ,

where U 2φ 2Λφ 2 φ B .

These equations are apparently not integrable and thus we

should

study the asymptotic and other properties of their solutions. Lacking

time for this task, we consider the solutions of the equations near a

possible horizon following the approach of our work wi

0

th D.Maison.

The are defined by possible zeroes of the metric: f 0 for

finite values of φ φ . It is not diffi

H

c

ORIZONS

ult to understand

horizons that we also

should require that A is finite near horizons. To study the behavior of

the solutions near horizons it is convenient to consider the solutions as

functions of the dilaton φ. Further analysis shows that these functions

0

2 2

can be expanded in the power series of φ = φ - φ and that the

functions F = f χ and = A χ should be finite. Thus we have:

F = φ m F , (1) χ = - F U ,

2 -3/21

2

(2)

B = - φ m , (3) χ = F [ φ B + U ], (4)

where now the prime denotes differentiations in the new variable φ .

It is not d

0 0 0 0

-3/20 0 0 0 0 0 0

ifficult to show that φ , , B , F can be taken arbitrary

up to one relation that should be satisfied due to equation (4):

U + φ B = 0 , where U U(φ , B )

0

. (5)

This equation can be solved with respect to any of the parameters.

It is interesting that it may have two solutions for φ that means there

may exist two horizons as distinct from

0

the Schwarzschild black hole.

Note that the solutions with different F are equivalent because

the equations are invariant under the scale transformation,

F C F , χ C χ .

Now, following the method of our paper with D. Maison, one can find

several terms o

f the expansions of all the functions defined by the

equations (1) - (4) (it is not easy to construct the complete expansions

due to the nonlinearity of these equations). The corresponding static

solution in general has two horizons, like the Reissner-Norstroem BH

or like BH in higher dimensional supergravities.

Let us turn to cosm

COSMOL

olog

OGY

ical reductions. The simplest cosmology can

be obtained from the same reduction as above. Indeed it is well known

that the internal part of the Schwarzschild BH is a cosmology.

However, this cosmology does not coinside with the homogeneous isotropic Friedman type cosmology. In addition, it can be shown that cosmologies derived by such a naïve reduction are closed. If we wish to to get Friedman type cosmologies from the vecton dilaton gravity, corresponding to the spherically symmetric world, we must employ a more complex procedure of dimensional reduction to 1+0 dimension that we have described some time ago in some detail. In short, this procedure is the following. We do not use the LC variables and write equation (I) in the original coordinates r and t . Then one can see that using a generalization of the method of separation of variables we can obtain many more 1+0 and 0+1 dimensional reductions describing static states and cosmologies that actually depend on two variables. The important property of these reductions is that the dependence on two variables is important only when we wish to find the higher dimensional interpretation. Otherwise we can treat the equations of motion as effectively one-dimensional.

α+γ 2 3α-γ 3α+γ α-γ 2 2 α+γ 2

-

The effective Lagrangian for the can be

written as follows:

=

Fr

6ke 6α e 2Λe e A m e A ,

where the effective 1+1 dimensional metric

iedman type cosmo

is

o

-e

gy

[

l

ceffL

2γ -2α

0 1

0

1

, e ], and α, γ, A

depend on t (note that when A , A are independent of r, we have

A 0 as follows from the 1+1 dimensional equations of motion;

we thus denote A A).

The equation

s of this cosmological model are much simpler than the

static ones. In particular, we immediately see that γ is the Lagrangian

multiplier, the variation in which gives the Hamiltonian that must

be α

2 2 2 2 2

zero (we take e f and choose the gauge γ = 0):

= f [-6f A - 6k

Another useful gauge (related

+ 2Λf + m A ] = 0

to LC gauge) is of course α = γ .

ceffH

2α 2 2α 4α 2 2 2α 2

2 2 2 4 2 2 2

Then the are:

= 6ke 6α e Λe A m e A ,

= [- 6f A - 6k f + 2Λf + m f A ] = 0

Both form

effective action and

s o

the Hamiltonian constrai

f

ntceff

ceff

L

H

the constraint tell us that for Λ > 0 the static solution is

possible only if k>0 when the universe is closed. When k 0 the

universe must be expanding or shrinking. Using the Lagrangian in the

2 2 2 22 13 6

LC gauge it is not difficult to derive the equations of motion. In analogy

to the static case we write them in the first order form:

f = f F , F + F + k = Λf + m A , A = B , B = 2 2

2 2 2 2 4 2 2 2

- m f A ,

= [ - 6f F - 6k f B + 2Λf + m f A ] = 0 .

Similarly to our consideration of the static equation it is convenient to

change the independent variable: t ln(f)

ceffH

= . Denoting by the

prime we have the following simpler equations:

2 2 2α 2 24 13 3

2 2α

(F ) + 2F + 2k = Λe + m A

FA = B , FB = - m A

(we need not rewrite the constraint). It is instructive to rewrite the

equati

e

122 2α 2α1 1

3 3

2 2 2 2α-2φ13

ons by introducing the new functions,

G F + k - Λe , φ ln F ln (G - k + Λe ) :

G + 2G = m A , A + φ A + m A = 0

By looking at these no

e

α0 0 0 1

nlinear but reasonably simple equations with due

attention one can find that in the asymptotic region α - :

G exp[ 2(α + α )] + ..., φ (α + α ) + ..., A A + A e + ...

wh

0 0 1ere α , A , A are constants. One can construct an asymptotic

iterations scheme using linearity of the equations in G and A. The

real problem is to find the asymtotic solution for α + tha

0 0 1

t depend

on the parameters α , A , A , but this is rather a difficult problem.

-α -2 2α 2 4α 1/20 0 1

Nevertheless, let us write the a few terms of the asymptotic expansion

in the region α - . First, it is obvious that

F = C e {1 - k C e + C e + ...} ,

where C

2 2 21 10 1 03 6

5α

is an arbitrary constant, C = + m A .

With these terms in the expansion of F(α) one can find the terms

up to e in the expansion of A , and then further terms in F , etc.

Now, recalling that F = α , we find the expansion of the equation

for α(t) that we may try to solve.

Note, however, that the equations for A and f are nonlinear and

nonintegrable. In fa

ct, they look similarly to the well known

the solution of which demonstrate an

extremely complex behaviour including

To study such effects one should

Henon - Heiles equati

deve

o s

l

n

o

dynamical chaos. p a qualitative theory of our

problem including the region between the asymptotic ones.

2 2 2α 2α 2 2

2 23

In the asymptotics for α is easier

to derive. The equations of motion and the constraint are:

A = 6α e + 6k 2Λe m A ,

α + 2α =

γ =

Λ -

0 standard gauge

-2α 2 2 -2α 216

2 2 2 2 2α 2α

k e - m A e , A + αA + m A = 0 .

Denoting α = β and A = B , we pass to the first order equations

B + m A = 6 β e + 6k 2Λe ,

22 -2α 2 2 -2α

2

[c]

1 β 2 1 + 2 β = Λ - k e - m A e , [b]

2 α 3 6A B

β = B , β + βB + m A = 0 . α α

Denoting the differentiation in α by the prime

d

dd d

d d

2 2 2

, we rewrite the last

two equations as one second - order equation that is linear in A :

β A + ( β + β β ) A + m A = 0 . [a]

2 -2α13

2 -2α 2

Introducing β β + - ke we rewrite [b] as

1 β + 4β = m e A . [b]

3

One can find that the asymptotic behaviour of β and A for α

is

2 -2α13

the following. The function β(α) is decreasing so that

β - ke + ...

Then the asymptotic behavior of A is defined by the equation

A + (

+

-1 -2α 2 -1 -1 -2α

λα 2 -1 1/212

λ α -2α0 1

1 + 3 ke ) A + 3m (1 + 3 ke ) A = 0 .

Thus the main term in A is e , where λ = - [1 (1 - 12m ) ],

and the expansion is A = e (A + A e ...) plus the similar

e

-λ α

2

xpression with e . Using these expansions we can find the next

terms in the expansion of β etc. The problem of connecting the

expansions in the left and the right regions is not addressed here.

Discussion and perspectives

Let us first summarize the results and thoughts of Weyl, Eddington, and Einstein.

1. WEYL had a very clear and original geometric ideas, but: a) his physics was rightly criticized by Einstein, Pauli, and other physicists, b) he considered the theory as a unified theory of gravity and electromagnetism but his vector field was also not electromagnetic, c) his discussion of dynamics was incomplete and he himself regarded it as preliminary. Nevertheless, it is possible that not all the potential of the Weyl ideas is understood and used.

2. EDDINGTON proposed to use, instead of the Weyl’s non-Riemannian `metric’ spaces, the most general spaces with symmetric affine connection (i.e. without torsion). He discussed possible invariants that can be used in physics, in particular, the square root of the determinant of Rij = gij + f ij . He proposed to consider the symmetric part of this curvature matrix, g, as the metric in the general space and the antisymmetric one, f, as the electromagnetic field tensor. In later works he discussed a possibility to use this as a Lagrangian (long before the proposal of Born and Infeld). However, he did not find a consistent approach to dynamics.

3. EINSTEIN started with formulating dynamics by use of the Hamilton principle similar to that of proposed by Palatini in general relativity. The new (and crucial) idea was that he propose not to introduce any metric at the beginning and not to fix any special form of the affine connection (apart of the symmetry condition). He soon realized (in paper II, that he does not need to use a concrete form of the Lagrangian that can be just any function (in fact, a tensor density) of g and f matrices. For any such Lagrangian he proved that the affine connection allows to introduce a symmetric metric and find the expression for connection. Both his expression and Weyl’s one are special cases of the general formula for The symmetric connection. The general expression is the following:

i i ir1jk jk jkr kjr rjk2

iij jk

ij

Γ Γ a (a a a )

where a is a symmetric tensor that is interpreted as a metric, and Γ is

the Christoffel symbol constructed of a and of its inverse,

irijk

iij jk

ij

ijk ij k

a , a is the

covariant derivative of a with the Chtistoffel connection Γ that itself

depends on a . We propose to to consider the class of connections for

which a = λ a α +

ik j jk kμ (a α + a α ) . If λ = -μ =1 2 we have

the Weyl connection; for λ = -μ = -1 3 we have the Einstein one.

Both expressions look very similar but do not coincide. In the paper I he derived from [2] the effective Lagrangian that was the sum of the square root of the symmetric metric, the standard scalar curvature, and of some additional term bilinear in the antisymmetric field. When the antisymmetric field is zero this gives the standard GR with the cosmological term. The effective theory [3], [4] was not derived from basic principles but incorporates naturally properties of the theory with the Lagrangian [2]. In this sense, it is connected to the original geometric considerations but it is less natural than the theory [2].

Weyl did not like the Einstein approach because he thought that geometry is a much more fundamental thing than the Hamilton principle. Eddington considered the Einstein contribution very important and the model attractive. However, in the end, all the characters of this drama shared the opinion of Pauli that this beautiful theory has no connection to Reality…

Today, Reality is not in obvious contradiction with the main points of the WEE ideas and the presented above Einstein model looks a reasonable candidate for explaining the origin of Dark Energy and Dark Matter (two in one!). The most important thing is that the Einstein model in fact predicted the existence of both long time ago. The question is – can this model or some similar model to explain these phenomena in more detail? Actually, Einstein made no attempt to study other models or find detailed properties of the theory [3], [4]. Also, I think that he did not realize that this theory has nothing to do with electromagnetism, but he certainly felt that something is wrong… Who knows? In any case, this direction is worth of pursuing.

RemarksIn conclusion, let us briefly summarize the results of our study of the simplest solutions of the WEE model. We only studied the spherically symmetric solutions, and in cosmology – only the homogeneous, isotropic model. As we noted elsewhere, even small deviations from the spherical symmetry may result in a qualitatively different theory. In particular, if we consider axially symmetric configurations infinitesimally deviating from the spherically symmetric ones, we will find additional scalar fields in the vecton gravity that may be very important in cosmological considerations. Also note that our consideration of the Friedman type model is incomplete. In fact, we have studied only asymptotic behavior of the solutions of the equations for the metric and vecton. As we mentioned above the complete solutions should reveal some sort of chaotic behavior. To study these phenomena we first need to carefully discuss the physical parameters of the theory. In the original formulation we have gravitational constant, cosmological constant and the vecton mass. In addition, the asymptotic boundary conditions possibly introduce some other parameters (for example, if we try to connect the left and the right asymptotic expansions we will find that this is strongly dependent on the parameters that characterize the influence of the nonlinear terms in the equations, up to producing chaotic effects). This requires a qualitative and numerical study of the equations.