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Report prepared for the seminar in memory of Victor Ogievetsky Dubna, 22.07.2008

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].

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Report prepared for the seminar in memory of Victor Ogievetsky Dubna, 22.07.2008

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  1. 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

  2. N.B.: The slides are from the reports. [1] [2]

  3. 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

  4. 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.

  5. 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)).

  6. 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.

  7. 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…

  8. Vecton Dilaton Gravity

  9. 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.

  10. 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.

  11. 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:

  12. 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…

  13. 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.

  14. Remarks In 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.

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