Gravitational Field Equations and Theory of Dark Matter and Dark Energy

The main objective of this article is to derive a new set of gravitational field equations and to establish a new unified theory for dark energy and dark matter. The new gravitational field equations with scalar potential $\varphi$ are derived using the Einstein-Hilbert functional, and the scalar potential $\varphi$ is a natural outcome of the divergence-free constraint of the variational elements. Gravitation is now described by the Riemannian metric $g_{ij}$, the scalar potential $\varphi$ and their interactions, unified by the new gravitational field equations. Associated with the scalar potential $\varphi$ is the scalar potential energy density $\frac{c^4}{8\pi G} \Phi=\frac{c^4}{8\pi G} g^{ij}D_iD_j \varphi$, which 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 produces attraction, and the positive part produces repelling force. This potential energy density is conserved with mean zero: $\int_M \Phi dM=0$. The sum of this new potential energy density $\frac{c^4}{8\pi G} \Phi$ and the coupling energy between the energy-momentum tensor $T_{ij}$ and the scalar potential field $\varphi$ gives rise to a new unified theory for dark matter and dark energy: The negative part of this sum represents the dark matter, which produces attraction, and the positive part represents the dark energy, which drives the acceleration of expanding galaxies. In addition, the scalar curvature of space-time obeys $R=\frac{8\pi G}{c^4} T + \Phi$. Furthermore, the new field equations resolve a few difficulties encountered by the classical Einstein field equations.


Introduction and Summary
The main aim of this article is an attempt to derive a new theory for dark matter and dark energy, and to derive a new set of gravitational field equations.
The primary motivation of this study is the great mystery of the dark matter and dark energy. The natural starting point for this study is to fundamentally examine the Einstein field equations, given as follows: where R ij is the Ricci curvature tensor, R is the scalar curvature, g ij is the Riemannian metric of the space-time, and T ij is the energy-momentum tensor of matter; see among many others [1]. The Einstein equations can also be derived using the Principle of Lagrangian Dynamics to the Einstein-Hilbert functional: whose Euler-Lagrangian is exactly R ij − 1 2 g ij R, which is the left hand side of the Einstein field equations (1.1). It is postulated that this Euler-Lagrangian is balanced by the symmetric energy-momentum tensor of matter, T ij , leading to the Einstein field equations (1.1). The Bianchi identity implies that the left hand side of the Einstein equations is divergence-free, and it is then postulated and widely accepted that the energy-momentum tensor of matter T ij is divergence-free as well.
However, there are a number of difficulties for the Einstein field equations: First, the Einstein field equations failed to explain the dark matter and dark energy, and the equations are inconsistent with the accelerating expansion of the galaxies. In spite of many attempts to modify the Einstein gravitational field equation to derive a consistent theory for the dark energy, the mystery remains.
Second, we can prove that there is no solution for the Einstein field equations for the spherically symmetric case with cosmic microwave background (CMB). One needs clearly to resolve this inconsistency caused by the non-existence of solutions.
Third, from the Einstein equations (1.1), it is clear that where T = g ij T ij is the energy-momentum density. A direct consequence of this formula is that the discontinuities of T give rise to the same discontinuities of the curvature and the discontinuities of space-time. This is certainly an inconsistency which needs to be resolved. Fourth, it has been observed that the universe is highly non-homogeneous as indicated by e.g. the "Great Walls", filaments and voids. However, the Einstein equations do not appear to offer a good explanation of this inhomogeneity.
These observations strongly suggest that further fundamental level examinations of the Einstein equations are inevitably necessary. 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-dimensional 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 in this case is the Einstein-Hilbert functional as explained in many classical texts of general relativity.
The key observation for our study is a well-known fact that the Riemannian metric g ij is divergence-free. This suggests two important postulates for deriving a new set of gravitational field equations: • The energy-momentum tensor T ij 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, with divergence defined at the extremum Riemannian metric g: Here D i g is the contra-variant derivative with respect to the extremum point g, and X ij are the variational elements. Namely, for any X = {X ij } with D i g X ij = 0, lim λ→0 1 λ [F (g ij + λX ij ) − F (g ij )] = (δF (g ij ), X) = 0.
For this purpose, an important part of this article is to drive an orthogonal decomposition theorem of tensors on Riemannian manifolds, which we shall explain further in the last part of this Introduction.
Under these two postulates, using the orthogonal decomposition theorem of tensors, we derive the following new set of gravitational field equations with scalar potential: where the scalar function ϕ : M → R is called the scalar potential.
The corresponding conservations of mass, energy and momentum are then replaced by (1.5) div (D i D j ϕ + 8πG c 4 T ij ) = 0, and the energy-momentum density T = g ij T ij and the scalar potential energy density The scalar potential energy density c 4 8πG Φ has a number of important physical properties: 1. Gravitation is now described by the Riemannian metric g ij , the scalar potential ϕ and their interactions, unified by the new gravitational field equations (1.4). 2. This scalar potential energy density c 4 8πG Φ represents a new type of energy/force caused by the non-uniform distribution of matter in the universe. This scalar potential energy density varies as the galaxies move and matter of the universe redistributes. Like gravity, it affects every part of the universe as a field. 3. This scalar potential energy density c 4 8πG Φ consists of both positive and negative energies. The negative part of this potential energy density produces attraction, and the positive part produces repelling force. The conservation law (1.7) amounts to saying that the the total scalar potential energy density is conserved. 4. The sum of this new potential energy density c 4 8πG Φ and the coupling energy between the energy-momentum tensor T ij and the scalar potential field ϕ, as described e.g. by the second term in the right-hand side of (1.9), gives rise to a new unified theory for dark matter and dark energy: The negative part of ε represents the dark matter, which produces attraction, and the positive part represents the dark energy, which drives the acceleration of expanding galaxies. 5. The scalar curvature of space-time obeys (1.6). Consequently, when there is no normal matter present (with T = 0), the curvature R of space-time is balanced by R = Φ. Therefore, there is no real vacuum in the universe. 6. The universe with uniform distributed matter leads to identically zero scalar potential energy, 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.
Hereafter, we further explore a few direct consequences of the above new gravitational field equations.
First, the new field equations are consistent with the spherically symmetric case with cosmic microwave background (CMB). Namely, the existence of solutions in this case can be proved.
Second, our new theory suggests that the curvature R is always balanced by Φ in the entire space-time by (1.6), and the space-time is no longer flat. Namely the entire space-time is also curved and is filled with dark energy and dark matter. In particular, the discontinuities of R induced by the discontinuities of the energymomentum density T , dictated by the Einstein field equations, are no longer present thanks to the balance of Φ.
Third, this scalar potential energy density should be viewed as the main cause for the non-homogeneous 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; see (1.9) below.
Fourth, to further explain the dark matter and dark energy phenomena, we consider a central matter field with total mass M and radius r 0 and spherical symmetry. With spherical coordinates, the corresponding Riemannian metric must be of the following form: where u = u(r) and v = v(r) are functions of the radial distance. With the new field equations, the force exerted on an object with mass m is given by 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 c 4 8πG Φ as indicated in (1.6) (R = Φ for r > r 0 ). In this formula, the negative and positive 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 dark matter, dark energy and their interactions 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: where ε > 0. The conservation law (1.7) of Φ suggests that R behaviors as r −2 for r sufficiently large. Consequently the second term in the right hand side of (1.10) must dominate and be positive, indicating the existence of dark energy. In fact, the above formula can be further simplified to derive the following approximate formula: Again, in (1.11), the first term represents the Newton gravitation, the attracting second term stands for dark matter and the repelling third term is the dark energy.
The mathematical part of this article is devoted to a rigorous derivation of the new gravitational field equations.
First, as mentioned earlier, the field equations obey the Euler-Lagrange equation of the Einstein-Hilbert functional under the natural divergence-free constraint, with divergence defined at the extremum Riemannian metric g: As the variational elements X are divergence-free, (1.13) does not imply δF (g ij ) = 0, which is the classical Einstein equations. In fact, (1.13) amounts to saying that δF (g ij ) is orthogonal to all divergence-free tensor fields X.
Hence we need to decompose general tensor fields on Riemannian manifolds into divergence-free and gradient parts. For this purpose, an orthogonal decomposition theorem is derived in Theorem 3.1. In particular, given an (r, s) tensor field u ∈ L 2 (T r s M ), we have The gradient part is acting on an (r, s − 1) tensor field ψ. Second, restricting to a (0, 2) symmetric tensor field u, the gradient part in the above decomposition is given by Then using symmetry, we show in Theorem 3.2 that this (0, 1) tensor ψ can be uniquely determined, up to addition to constants, by the gradient of a scalar field ϕ: and consequently we obtain the following decomposition for general symmetric (0, 2) tensor fields: Finally, for the symmetric and divergence free (0, 2) field δF (g ij ), which is the Euler-Lagrangian of the Einstein-Hilbert functional in (1.13) and is orthogonal to all divergence-free fields, there is a scalar field ϕ ∈ H 2 (M ) such that which, by adding the energy-momentum tensor T ij , leads to the new gravitational field equations (1.4).
We remark here that the orthogonal decompositions (1.14) and (1.15) are reminiscent of the orthogonal decomposition of vectors fields into gradient and divergence parts, which are crucial for studying incompressible fluid flows; see among many others [7,8].
This article is divided into two parts. The physically inclined readers can go directly to the physics part after reading this Introduction. Part 1. Mathematics 2. Preliminaries 2.1. Sobolev spaces of tensor fields. Let (M, g ij ) be an n-dimensional Riemannian manifold with metric (g ij ), and E = T r s M be an (r, s)-tensor bundle on M . A mapping u : M → E is called a section of the tensor-bundle E or a tensor field. In a local coordinate system x, a tensor field u can be expressed component-wise as follows: where u j1···jr i1···is (x) are functions of x ∈ U . The section u is called C r -tensor field or C r -section if its components are C r -functions.
For any real number 1 ≤ p < ∞, let L p (E) be the space of all L p -integrable sections of E: For p = 2, L 2 (E) is a Hilbert space equipped with the inner product where (g ij ) is Riemannian metric, (g ij ) = (g ij ) −1 , g = det(g ij ), and √ −gdx is the volume element.
For any positive integer k and any real number 1 ≤ p < ∞, we can also define the Sobolev spaces W k,p (E) to be the subspace of L p (E) such that all covariant derivatives of u up to order k are in L p (E). The norm of W k,p (E) is always denoted by · W k,p .
As p = 2, the spaces W k,p (E) are Hilbert spaces, and are usually denoted by equipped with inner product (·, ·) H k and norm · H k .

2.2.
Gradient and divergent operators. Let u : M → E be an (r, s)-tensor field, with the local expression Then the gradient of u is defined as It is clear that the gradient ∇u defined by (2.4) is an (r, s + 1)-tensor field: ∇u : M → T r s+1 M. We define ∇ * u as For an (r + 1, s)-tensor field u = {u j1···l···jr i1···is }, the divergence of u is defined by Therefore, the divergence div u defined by (2.6) is an (r, s)-tensor field. Likewise, for an (r, s + 1)-tensor field u = {u j1···jr i1···l···is }, the following operator is also called the divergence of u, For the gradient operators (2.4)-(2.5) and the divergent operators (2.6)-(2.7), it is well known that the following integral formulas hold true; see among others [2].
where ∇ * u is as in (2.5) and div v is as in (2.6), the inner product (·, ·) is as defined by (2.1). If u is an (r, s − 1)-tensor and v is an (r, s) tensor, then where ∇u is as in (2.4) and divv is as in (2.7).
2.3. Acute-angle principle. Let H be a Hilbert space equipped with inner product (·, ·) and norm · , and G : H → H be a mapping. We say that G is weakly continuous if for any sequence {u n } ⊂ H weakly converging to u 0 , i.e.
If the operator G is linear and bounded, then G is weakly continuous. The following theorem is called the acute-angle principle [5].
Theorem 2.2. If a mapping G : H → H is weakly continuous, and satisfies for some constants α, β > 0, then for any f ∈ H there is a u 0 ∈ H such that

Orthogonal Decomposition for Tensor Fields
3.1. Main theorems. The aim of this section is to derive an orthogonal decomposition for (r, s)-tensor fields with r + s ≥ 1 into divergence-free and gradient parts. This decomposition plays a crucial role for the theory of gravitational field, dark matter and dark energy developed later in this article. Let M be a closed Riemannian manifold, and v ∈ L 2 (T r s M ) (r + s ≥ 1). We say that v is divergence-free, i.e., div v = 0, if for all ∇ψ ∈ L 2 (T r s M ), Theorem 3.1 (Orthogonal Decomposition Theorem). Let M be a closed Riemannian manifold, and u ∈ L 2 (T r s M ) with r + s ≥ 1. The following assertions hold true: (1) The tensor field u has the following orthogonal decomposition: where ϕ and v are as in (3.2), and h is a harmonic field, i.e.
In particular the subspace of all harmonic tensor fields in L 2 (T r s M ) is of finite dimensional: Here H is as in (3.5), and They are orthogonal to each other: Namely k is the integer that E can be decomposed into the Whitney sum of a k-dimensional trivial bundle E k = M × R k and a nontrivial bundle E 1 , i.e.
Proof of Theorem 3.1. We proceed in several steps as follows.
Step 1 Proof of Assertion (1). Let u ∈ L 2 (E), E = T r s M (r + s ≥ 1). Consider the equation where ∆ is the Laplace operator defined by Without loss of generality, we only consider the case where div u ∈Ẽ = T r−1 s M . It is clear that if the equation (3.6) has a solution ϕ ∈ H 1 (Ẽ), then by (3.7), the following vector field must be divergence-free Moreover, by (3.1) we have Namely v and ∇ϕ are orthogonal. Therefore, the orthogonal decomposition u = v + ∇ϕ follows from (3.8) and (3.9). It suffices then to prove that (3.6) has a weak solution ϕ ∈ H 1 (Ẽ): To this end, let Then we define a linear operator G : H → H by It is clear that the linear operator G : H → H is bounded, weakly continuous, and Based on Theorem 2.2, for any f ∈ H, the equation has a weak solution ϕ ∈ H. Hence for f = divu the equation (3.6) has a solution, and Assertion (1) is proved. In fact the solution of (3.6) is unique. We remark that by the Poincaré inequality, for the space H = H 1 (Ẽ) \ {ψ|∇ψ = 0}, (3.12) is an equivalent norm of H. In addition, by Theorem 2.1, the weak formulation (3.10) for (3.6) is well-defined.
Step 2 Proof of Assertion (2). Based on Assertion (1), we have Define an operator∆ : is the canonical orthogonal projection. We know that the Laplace operator ∆ can be expressed as where B is the lower order derivative linear operator. Since M is compact, the Sobolev embeddings H 2 (E) ֒→ H 1 (E) ֒→ L 2 (E) are compact, which implies that the lower order derivative operator where the integer N is the dimension of the tensor bundle E. According to the elliptic operator theory, the elliptic operator in (3.14) is a linear homeomorphism. Therefore the operator in (3.14) is a linear completely continuous field ∆ : H 2 (E) → L 2 (E), which implies that the operator of (3.13) is also a linear completely continuous field: . By the the spectral theorem of completely continuous fields [6,8], the spacẽ It follows that u ∈H ⇔ ∇u = 0 ⇒ H =H, where H is the harmonic space as in (3.5). Thus we have . The proof of Theorem 3.1 is complete.

3.2.
Uniqueness of the orthogonal decomposition. In Theorem 3.1, a tensor field u ∈ L 2 (T r s M ) with r + s ≥ 1 can be orthogonally decomposed into Now we address the uniqueness problem of the decomposition (3.15). In fact, if u is a vector field or a co-vector field, i.e.
then the decomposition of (3.15) is unique.
We can see that if u ∈ L 2 (T r s M ) with r + s ≥ 2, then there are different types of the decompositions of (3.15). For example, for u ∈ L 2 (T 0 2 M ), the local expression of u is given by In this case, u has two types of decompositions: It is easy to see that if u ij = u ji then both (3.16) and (3.17) are two different decompositions of u ij . Namely If u ij = u ji is symmetric, u can be orthogonally decomposed into the following two forms: and ϕ and ψ satisfy (3.18) and (3.19) are the same, and ϕ = ψ. Therefore, the symmetric tensors u ij can be written as  (1) u has a unique orthogonal decomposition if and only if there is a scalar function ϕ ∈ H 2 (M ) such that u can be expressed as (2) u can be orthogonally decomposed in the form of (3.22) if and only if u ij satisfy Proof. We only need to prove Assertions (2) and (3).
We first prove Assertion (2). It follows from (3.20) that where ∆ = D k D k . By the Weitzenböck formula [4], and (δd + dδ) is the Laplace-Beltrami operator. We know that for ω = ϕ i dx i ,
Due to (4.10), we can define the derivative operators of the functional F , which are also called the Euler-Lagrange operators of F , as follows where W −m,2 (E) is the dual space of W m,2 (E), and δ * F, δ * F are given by For any given metric g ij ∈ W m,2 (M, g), the value of δ * F and δ * F at g ij are second-order contra-variant and covariant tensor fields respectively, i.e.
where (δ * F ) kl and (δ * F ) kl are the components of δ * F and δ * F respectively.
Proof. We only need to verify Assertion (3). Noting that g ik g kj = δ j i , we have the variational relation δ(g ik g kj ) = g ik δg kj + g kj δg ik = 0.
Hence (δ * F ) ij = −g ki g lj (δ * F ) kl . Thus Assertion (3) follows and the proof is complete. 4.2. Scalar potential theorem for constraint variations. We know that the critical points of the functional F in (4.6) are the solution (4.18) δF (g ij ) = 0, in the following sense where E = T M ⊗ T M . Hence, the critical points of functionals of Riemannian metrics are not solutions of (4.18) in the usual sense.
It is easy to see that L 2 (T M ⊗ T M ) can be orthogonally decomposed into the direct sum of the symmetric and contra-symmetric spaces, i.e. Thus, we can say that the extremum points of functionals of the Riemannian metrics are solutions of (4.18) in the usual sense of (4.21), or are zero points of the variational operators δF : W m,2 (M, g) → W −m,2 (E). Now we consider the variations of F under the divergence-free constraint. In this case, the Euler-Lagrangian equations with symmetric divergence-free constraints are equivalent to the Euler-Lagrangian equations with general divergence-free constraints. Hence we have the following definition.
. It is clear that an extremum point satisfying (4.22) is not a solution of (4.18). Instead, we have the scalar potential theorem for the extremum points of divergence free constraint (4.22), which is based on the orthogonal decomposition theorems. This result is also crucial for the gravitational field equations and the theory of dark matter and dark energy developed later. (δF (g ij )) kl = D k D l ϕ.
By Theorem 4.1, δF is symmetric. Hence we have It follows from (3.28) that (4.29) ∂ψ l ∂x k = ∂ψ k ∂x l . By assumption, the first Betti number of M is zero, i.e. the 1-dimensional homology of M is zero: H 1 (M ) = 0. It follows from the de Rham theorem that if d(ψ k dx k ) = ∂ψ k ∂x l − ∂ψ l ∂x k dx l ∧ dx k = 0, then there exists a scalar function ϕ such that dϕ = ∂ϕ ∂x k dx k = ψ k dx k . Thus, we infer from (4.29) that ψ l = ∂ϕ ∂x l for some ϕ ∈ H 2 (M ).
If the first Betti number β 1 (M ) = 0, then there are N = β 1 (M ) number of 1-forms: which constitute a basis of the 1-dimensional de Rham homology H 1 d (M ). We know that the components of ω j are co-vector fields: (4.31) ψ j = (ψ j 1 , · · · , ψ j n ) ∈ H 1 (T * M ) for 1 ≤ j ≤ N, which possess the following properties: or equivalently, Namely, ∇ψ j ∈ L 2 (T * M ⊗ T * M ) are symmetric second-order contra-variant tensors. Hence Theorem 4.2 can be extended to the non-vanishing first Betti number case as follows.
The proof of Theorem 4.3 is similar to Theorem 4.2, and is omitted here.
Remark 4.1. By the Hodge decomposition theory, the 1-forms ω j in (4.30) are harmonic: dω j = 0, δω j = 0 for 1 ≤ j ≤ N, which implies that the tensors ψ j in (4.32) satisfy According to the Weitzenböck formula (3.25), we obtain from (4.33) that for ψ j = (ψ j 1 , · · · , ψ j n ) in (4.32). The general theory of relativity is based on 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-dimensional 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, which is the Einstein-Hilbert functional. Let (M, g ij ) be an n-dimensional Riemannian manifold. The Einstein-Hilbert functional 1 where W 2,2 (M, g) is defined by (4.5), R = g kl R kl and R kl are the scalar and the Ricci curvatures, S ij is the stress tensor, G is the gravitational constant, and c is the speed of light. The Euler-Lagrangian of the Einstein-Hilbert functional F is given by where T ij is the energy-momentum tensor given by and the Ricci curvature tensor R ij is given by By (5.3)-(5.5), the Euler-Lagrangian δF (g ij ) of the Einstein-Hilbert functional is a second order differential operator on {g ij }, and δF (g ij ) is symmetric.

Einstein field equations. The General Theory of Relativity consists of two main conclusions:
1) The space-time of our world is a 4-dimensional Riemannian manifold (M 4 , g ij ), and the metric {g ij } represents gravitational potential. 2) The metric {g ij } is the extremum point of the Einstein-Hilbert functional (5.2). In other words, gravitational field theory obeys the principle of Lagrange dynamics.
The principle of Lagrange dynamics is a universal principle, stated as: Principle of Lagrange Dynamics. For any physical system, there are a set of state functions u = (u 1 , · · · , u N ), which describe the state of this system, and there exists a functional L of u, called the Lagrange action: such that the state u is an extremum point of L. Usually the function L in (5.6) is called the Lagrangian density.
Based on this principle, the gravitational field equations are the Euler-Lagrange equations of the Einstein-Hilbert functional: which are the classical Einstein field equations: By the Bianchi identities, the left hand side of (5.8) is divergence-free, i.e.
Therefore it is required in the general theory of relativity that the energy-momentum tensor {T ij } in (5.8) satisfies the following energy-momentum conservation law:

5.3.
New gravitational field equations. Motivated by the mystery of dark energy and dark matter and the other difficulties encountered by the Einstein field equations as mentioned in Introduction, we introduce in this section a new set of field equations, still obeying the three basic principles of the General Theory of Relativity.
Our key observation is a well-known fact that the Riemannian metric g ij is divergence-free. This suggests us two important postulates for deriving the new gravitational field equations: • The energy-momentum tensor of matter need not to be divergence-free due to the presence of dark energy and dark matter; and • The field equation obeys the Euler-Lagrange equation of the Einstein-Hilbert functional under the natural divergence-free constraint. Under these two postulates, by the Scalar Potential Theorem, Theorem 4.2, if the Riemannian metric {g ij } is an extremum point of the Einstein-Hilbert functional (5.2) with the divergence-free constraint (4.22), then the gravitational field equations are taken in the following form: is called the scalar potential. We infer from (5.9) that the conservation laws for (5.11) are as follows (5.12) div (D i D j ϕ + 8πG c 4 T ij ) = 0.
Using the contraction with g ij in (5.11), we have represent respectively the energy-momentum density and the scalar potential density. Physically this scalar potential density Φ represents potential energy caused by the non-uniform distribution of matter in the universe. One important property of this scalar potential is which is due to the integration by parts formula in Theorem 2.1. This formula demonstrates clearly that 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. We shall address this important issue in the next section.
5.4. Field equations for closed universe. The topological structure of closed universe is given by where S 1 is the time circle and S 3 is the 3-dimensional sphere representing the space. We note that the radius R of S 3 depends on time t ∈ S 1 , R = R(t), t ∈ S 1 , and the minimum time t 0 , is the initial time of the Big Bang. For a closed universe as (5.15), by Theorem 4.2, the gravitational field equations are in the form where ∆ = D k D k , ϕ the scalar potential, ψ = (ψ 0 , ψ 1 , ψ 2 , ψ 3 ) the vector potential, and α is a constant. The conservation laws of (5.16) are as follows

Interaction in A Central Gravitational Field
6.1. Schwarzschild solution. We know that the metric of a central gravitational field is in a diagonal form [1]: (6.1) ds 2 = g 00 c 2 dt 2 + g 11 dr 2 + r 2 (dθ 2 + sin 2 θdϕ 2 ), and physically g 00 is given by where ψ is the Newton gravitational potential; see among others [1]. If the central matter field has total mass M and radius r 0 , then for r > r 0 , the metric (6.1) is the well known Schwarzschild solution for the Einstein field equations (5.8), and is given by We derive from (6.2) and (6.3) the classical Newton gravitational potential 6.2. New gravitational interaction model. We now consider the metric determined by the new field equations (5.11), from which we derive a gravitational potential formula replacing (6.4). Equations (5.11) can be equivalently expressed as For the central matter field with total mass M and radius r 0 , by the Schwarzschild assumption, for r > r 0 , there exists no matter, i.e. (6.6) T ij = 0.
For the metric (6.7), the non-zero components of the Levi-Civita connection are as follows (6.11) Hence the Ricci tensor are given by (6.12) Furthermore, we infer from (6.7), (6.9) and (6.11) that (6.13) Thus, by (6.12) and (6.13), the equations (6.5) are as follows . (6.16) 6.3. Consistency. We need to consider the existence and uniqueness of solutions of the equations (6.14)-(6.16). First, in the vacuum case, the classical Einstein equations are in the form (6.17) R kk = 0 for k = 1, 2, 3, two of which are independent. The system contains two unknown functions, and therefore for a given initial value (as u ′ is basic in (6.17)): the problem (6.17) with (6.18) has a unique solution, which is the Schwarzschild solution Now if we consider the influence of the cosmic microwave background (CMB) for the central fields, then we should add a constant energy density in equations (6.17): Namely, where ρ is the density of the microwave background, whose value is ρ = 4 × 10 −31 kg/m 3 . Then it is readily to see that the problem (6.20) with (6.18) has no solution. In fact, the divergence-free equation which implies that u ′ = 0. Hence R 00 = 0, a contradiction to (6.20). Furthermore, if we regard ρ as an unknown function, then the equations (6.20) still have no solutions.
On the other hand, the new gravitational field equations (6.14)-(6.16) are solvable for the microwave background as the number of unknowns are the same as the number of independent equations. Equations (6.14)-(6.16) have the following equivalent form: equipped with the following initial values: It is classical that (6.21) with (6.22) possesses a unique local solution. In fact, we can prove that the solution exists for all r > r 0 .
As the interaction force F is given by it follows from (6.23) and (6.25) that Of course, the following energy balance and conservation law hold true as well: where R is the scalar curvature and Φ = g kl D k D l ϕ. Equation (6.27) is derived by solving ϕ ′′ using (6.28). Namely, for r > r 0 where T = 0, 6.5. Simplified formulas. We now consider the region: r 0 < r < r 1 . Physically, we have (6.29) |ϕ ′ |, |ϕ ′′ | << 1.
Hence u and v in (6.26) can be replaced by the Schwarzschild solution: As δ/r is small for r large, by (6.29), the formula (6.26) can be expressed as This is the interactive force in a central symmetric field. The first term in the parenthesis is the Newton gravity term, and the added second term −rϕ ′′ is the scalar potential energy density, representing the dark matter and dark energy. In addition, replacing u and v in (6.27) by the Schwarzschild solution (6.30), we derive the following approximate formula: Consequently we infer from (6.31) that 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. In this formula, the negative and positive values of each term represent respectively the attracting and repelling forces.
Integrating (6.32) yields (omitting e −δ/r ) where ε 2 is a free parameter. Hence the interaction force F is approximated by where ε = ε 2 δ −1 , R = Φ for r > r 0 , and δ = 2M G/c 2 . We note that based on (6.28), for r > r 0 , R is balanced by Φ, and the conservation of Φ suggests that R behaviors like r −2 as r sufficiently large. Hence for r large, the second term in the right hand side of (6.35) must be dominate and positive, indicating the existence of dark energy. We note that the scalar curvature is infinite at r = 0: R(0) = ∞. Also R contains two free parameters determined by u ′ and v respectively. Hence if we take a first order approximation as (6.36) R = −ε 1 + ε 0 r for r 0 < r < r 1 = 10 21 km, where ε 1 and ε 0 are free yet to be determined parameters. Then we deduce from (6.34) and (6.36) that Therefore, Physically it is natural to choose Also, ε 1 and 2ε 2 δ −1 r 2 are much smaller than (εδ + ε 1 δ −1 )r for r ≤ r 1 . Hence where k 0 and k 1 can be estimated using the Rubin law of rotating galaxy and the acceleration of the expanding galaxies: We emphasize here that the formula (6.38) is only a simple approximation for illustrating some features of both dark matter and dark energy.

Theory of Dark Matter and Dark Energy
7.1. Dark matter and dark energy. Dark matter and dark energy are two of most remarkable discoveries in astronomy in recent years, and they are introduced to explain the acceleration of the expanding galaxies. In spite of many attempts and theories, the mystery remains. As mentioned earlier, this article is an attempt to develop a unified theory for the dark matter and dark energy. A strong support to the existence of dark matter is the Rubin law for galactic rotational velocity, which amounts to saying that most stars in spiral galaxies orbit at roughly the same speed. Namely, the orbital velocity v(r) of the stars located at radius r from the center of galaxies is almost a constant: v(r) = constant for a given galaxy.
Typical galactic rotation curves [3] are illustrated by Figure 7.1(a), where the vertical axis represents the velocity (km/s), and the horizontal axis is the distance from the galaxy center (extending to the galaxy radius). However, observational evidence shows discrepancies between the mass of large astronomical objects determined from their gravitational effects, and the mass calculated from the visible matter they contain, and Figure 7.1 (b) gives a calculated curve. The missing mass suggests the presence of dark matter in the universe.
In astronomy and cosmology, dark energy is a hypothetical form of energy, which spherically symmetrically permeates all of space and tends to accelerate the expansion of the galaxies.
The High-Z Supernova Search Team in 1998 and the Supernova Cosmology Project in 1998 published their observations which reveal that the expansion of the galaxies is accelerating. In 2011, a survey of more than 2 × 10 5 galaxies from Austrian astronomers confirmed the fact. Thus, the existence of dark energy is accepted by most astrophysicist. 7.2. Nature of dark matter and dark energy. With the new gravitational field equation with the scalar potential energy, and we are now in position to derive the nature of the dark matter and dark energy. More precisely, using the revised Newton formula derived from the new field equations: we determine an approximation of the constants k 0 , k 1 , based on the Rubin law and the acceleration of expanding galaxies.
First, let M r be the total mass in the ball with radius r of the galaxy, and V be the constant galactic rotation velocity. By the force equilibrium, we infer from (7.2) that which implies that This matches the observed mass distribution formula of the galaxy, which can explain the Rubin law (7.3).
Second, if we use the classical Newton formula F = − mM G r 2 , to calculate the galactic rotational velocity v r , then we have which is consistent with the theoretic rotational curve as illustrated by Figure  7.1(b). It implies that the distribution formula (7.4) can be used as a test for the revised gravitational field equations.
Third, we now determine the constants k 0 and k 1 in (7.2). According to astronomic data, the average mass M r1 and radius r 1 of galaxies is about Based on physical considerations, (7.9) k 0 ≫ k 1 r 1 (r 1 as in (7.7)) By (7.7)-(7.9), we deduce from (7.4) that (7.10) Now we consider the constant k 1 . Due to the accelerating expansion of galaxies, the interaction force between two clusters of galaxies is repelling, i.e. for (7.2), F ≥ 0, r ≥r, wherer is the average distance between two galactic clusters. It is estimated that r = 10 8 pc ≃ 10 20 ∼ 10 21 km.
In summary, for the formula (7.2) with (7.12), if the matter distribution M r is in the form (7.13) M r = V 2 G r 1 + k 0 r , then the Rubin law holds true. In particular, the massM generated by the revised gravitation isM , r 1 as in (7.7), where M T = V 2 r 1 /G is the theoretic value of total mass. Hencẽ Namely, the revised gravitational massM is four times of the visible matter M r1 = M T −M . Thus, it gives an alternative explanation for the dark matter. In addition, the formula (7.2) with (7.12) also shows that for a central field with mass M , an object at r >r (r as in (7.11)) will be exerted a repelling force, resulting the acceleration of expanding galaxies at r >r.
Thus the new gravitational formula (7.2) provides a unified explanation of dark matter and dark energy. 7.3. Effects of non-homogeneity. In this section, we prove that if the matter is homogeneously distributed in the universe, then the scalar potential ϕ is a constant, and consequently the scalar potential energy density is identically zero: Φ ≡ 0.
The conclusion (7.23) indicates that if the universe is in the homogeneous state, then the scalar potential energy density c 4 8πG Φ is identically zero: Φ ≡ 0. This fact again demonstrates that ϕ characterizes the non-uniform distribution of matter in the universe.

Conclusions
We have discovered new gravitational field equations (1.4) with scalar potential under the postulate that the energy momentum tensor T ij needs not to be divergence-free due to the presence of dark energy and dark matter: With the new field equations, we have obtained the following physical conclusions: First, gravitation is now described by the Riemannian metric g ij , the scalar potential ϕ and their interactions, unified by the new gravitational field equations (1.4).
Second, associated with the scalar potential ϕ is the scalar potential energy density c 4 8πG Φ, which represents a new type of energy/force caused by the nonuniform distribution of matter in the universe. This scalar potential energy density varies as the galaxies move and matter of the universe redistributes. Like gravity, it affects every part of the universe as a field.
This scalar potential energy density c 4 8πG Φ consists of both positive and negative energies. The negative part of this potential energy density produces attraction, and the positive part produces repelling force. Also, this scalar energy density is conserved with mean zero: M ΦdM = 0.
Third, using the new field equations, for a spherically symmetric central field with mass M and radius r 0 , the force exerted on an object of mass m at distance r is given by (see (6.33)): where δ = 2M G/c 2 .
Fourth, the sum ε = ε 1 + ε 2 of this new potential energy density ε 1 = c 4 8πG Φ and the coupling energy between the energy-momentum tensor T ij and the scalar potential field ϕ gives rise to a new unified theory for dark matter and dark energy: The negative part of ε represents the dark matter, which produces attraction, and the positive part represents the dark energy, which drives the acceleration of expanding galaxies.
Fifth, the scalar curvature R of space-time obeys: R = 8πG c 3 T + Φ. Consequently, when there is no normal matter present (with T = 0), the curvature R of space-time is balanced by R = Φ. Therefore, there is no real vacuum in the universe.