Unified Theory of Dark Energy and Dark Matter Tian Ma, 1 Shouhong Wang 2 1 Department of Mathematics, Sichuan University, Chengdu, P. R. China 2 Department of Mathematics, Indiana University, Bloomington, IN 47405 To whom correspondence should be addressed; E-mail: showang@indiana.edu. June 24, 2012 New field equations of gravitations with a scalar potential ϕ are discovered, and are uniquely derived from the Einstein-Hilbert functional using the principle of Lagrangian dynamics subject to the divergence-free constraint of the Riemannian metric of space-time. The scalar potential ϕ induces a scalar potential energy density Φ, unifying dark energy and dark matter. In fact, Φ represents a new type of energy caused by the non-uniform distribution of matter in the universe. The negative part of this potential energy density Φ represents the dark matter, which produces attraction, and the positive part represents the dark energy, which drives the acceleration of expanding galaxies. In addition, Φ is conserved with mean zero: ΦdM = 0. M This report is based on a new discovery on the theory of dark energy and dark matter, and on field equations, developed by the authors [1]. 1. Field equations with scalar potential. Motivated by the great mystery of the dark matter and dark energy, a careful fundamental level of examination of the Einstein equations enable us 1
to establish a new field equations of gravitation with scalar potential ϕ: R ij 1 2 g ijr = 8πG c 4 T ij D i D j ϕ, (1) where R is the scalar curvature, R ij is the Ricci curvature tensor, G is the gravitation constant, c is the speed of light, T ij is the energy momentum tensor of matter, and g ij is the Riemannian metric of space time. 2. New scalar potential energy. From the above new field equations, the conservations of matter, energy and momentum are given by div (D i D j ϕ + 8πG c 4 T ij) = 0, (2) and the momentum density T = g ij T ij and the scalar potential energy density Φ = g ij D i D j ϕ satisfy R = 8πG T + Φ, (3) c4 Φ gdx = 0. (4) M The scalar potential energy density Φ has a number of important physical properties: (1) This scalar potential energy density Φ represents a new type of energy caused by the nonuniform distribution of matter in the universe. This scalar potential energy varies as the galaxies move and matter of the universe redistributes. Like gravity, it affects every part of the universe as a field. (2) This scalar potential energy density Φ consists of both positive and negative energies. The negative part of this quantity Φ represents the dark matter, which produces attraction, and the positive part represents the dark energy, which drives the acceleration of expanding galaxies. 2
(3) The conservation law (4) amounts to saying that the the total scalar potential energy is conserved. (4) The universe with uniform distributed matter leads to identically zero scalar potential energy density, and is unstable. It is this instability that leads to the existence of the dark matter and dark energy, and consequently the high non-homogeneity of the universe. (5) The curvature R is always balanced by this scalar potential energy Φ in the entire space by (3), and the entire space is no longer flat. Namely the entire space is also curved and is filled with dark energy and dark matter. In particular, the discontinuities of R induced by the discontinuities of the momentum energy density T, dictated by the Einstein field equations, are no longer present thanks to the balance of the scalar potential energy density Φ. (6) This scalar potential energy density should be viewed as the main cause for the nonhomogeneous distribution of the matter/galaxies in the universe, as the dark matter (negative scalar potential energy) attracts and dark energy (positive scalar potential energy) repels different galaxies. 3. Interaction force formula. Consider a central matter field with the total mass M and radius r 0 and spherical symmetry. With the new field equations, the force excerted on an object with mass m is given by F = mmg [ 1 r 1 ( 2 + δ ) dϕ 2 δ r dr + Rr ], R = Φ for r > r 0. (5) δ where δ = 2GM/c 2, R is the scalar curvature, and ϕ is the scalar potential. The first term is the classical Newton gravitation, the second term is the coupling interaction between matter and the scalar potential ϕ, and the third term is the interaction generated by the scalar potential energy density Φ (R = Φ for r > r 0 ), as indicated in (3). In this formula, the negative and positive 3
values of each term represent respectively the attracting and repelling forces. It is then clear that the combined effect of the second and third terms in the above formula represent the force generated by dark matter, dark energy and their interaction with normal matter. Also, importantly, this formula is a direct representation of the Einstein s equivalence principle. Namely, the curvature of space-time induces interaction forces between matter. In addition, one can derive a more detailed version of the above formula: F = mmg [ 1r ( + 2 + δ ) εr 2 + Rr 2 r δ + 1 ( 2 + δ ) r 2 δ r ] r 2 Rdr, (6) where ε > 0. The conservation law (4) of Φ suggests that R behaviors as r 2 for r sufficiently large. Consequently the second term in the right hand side of (6) must dominate and be positive, indicating the existence of dark energy. To illustrate the main ideas, the above formula can be further simplified to derive the following approximate formula for r 0 < r < r 1 10 21 10 22 km: [ F = mmg 1 r k ] 0 2 r + k 1r, (7) k 0 = 4 10 18 km 1, k 1 = 10 57 km 3. (8) Again, in (7), the first term represents the Newton gravitation, the attracting second term stands for dark matter and the repelling third term is the dark energy. 4. Derivation of the field equations. It is clear that any modification of the Einstein field equations should obey three basic principles: the principle of equivalence, the principle of general relativity, and the principle of Lagrangian dynamics. The first two principles tell us that the spatial and temporal world is a 4-dimentional Riemannian manifold (M, g ij ), where the metric {g ij } represents gravitational potential, and the third principle determines that the Riemannian metric {g ij } is an extremum point of the Lagrangian action. There is no doubt that the most natural Lagrangian action is the Einstein-Hilbert functional. 4
The key point for our study is a well-known fact that the Riemannian metric g ij is divergencefree. This suggests two important postulates for deriving a new set of gravitational field equations: The momentum tensor of matter need not to be divergence-free due to the presence of dark energy and dark matter; and The field equations obey the Euler-Lagrange equation of the Einstein-Hilbert functional under the natural divergence-free constraint: F (g ij + λx ij ) F (g ij )] lim λ 0 λ = (δf (g ij ), X) = 0 X = {X ij } with div X = 0. (9) Here δf (g ij ) is the Euler-Lagrange variation of the Einstein-Hilbert functional. As the variational elements X are divergence-free, (9) do not imply δf (g ij ) = 0, which is the classical Einstein equations. In fact, (9) amounts to saying that δf (g ij ) is orthogonal to all divergencefree tensor fields X, and consequently it is balanced by scalar potential term D i D j ϕ for a scalar potential function ϕ. The existence and uniqueness of ϕ are based on the orthogonal decomposition of tensor fields into gradient and divergence parts proved in [1]. Namely, the new field equations (1) are then uniquely determined by the principle of Lagrangian dynamics applied to the Einstein-Hilbert functional with divergence-free constraint. 5. Summary. In a nutshell, our theory indicates that dark energy and dark matter are respectively the positive and negative parts of the scalar potential energy caused by the non-uniform distribution of the universe, which fills every part of the universe. The total scalar potential energy density is conserved with mean zero. In other works, the unified energy for dark energy and dark matter is conserved. The new field equations are the unique set of equations derivable from the Einstein-Hilbert functional following the principle of Lagrangian dynamics, and can be used to resolve a number of difficulties encountered by the classical Einstein equations. 5
References and Notes 1. Tian Ma and Shouhong Wang, Gravitational Field Equations and Theory of Dark Matter and Dark Energy, arxiv:1206.5078v1 [gr-qc], 2012. 6