Modified general relativity

TL;DR

This paper proposes a modified version of Einstein's general relativity by decomposing symmetric tensors and introducing a new energy-momentum tensor, addressing dark energy, energy localization, and explaining galactic rotation without dark matter.

Contribution

It introduces a new symmetric tensor for gravitational energy-momentum, modifies Einstein's equations, and offers explanations for dark energy and galactic dynamics without dark matter.

Findings

Addresses energy localization in gravity.

Provides a natural explanation for dark energy.

Derives the baryonic Tully-Fisher relation without dark matter.

Abstract

In a Lorentzian spacetime there exists a smooth regular line element field $(\bm{X},-\bm{X}) $ and a unit vector $ \bm{u} $ collinear with one of the pair of vectors in the line element field. An orthogonal decomposition of symmetric tensors can be constructed in terms of the Lie derivative along $ \bm{X} $ of the metric and a product of the unit vectors; and a linear sum of divergenceless symmetric tensors. A modified Einstein equation of general relativity is then obtained by using the principle of least action, the decomposition and a fundamental postulate of general relativity. The decomposition introduces a new symmetric tensor $ \varPhi_{\alpha\beta} $ which describes the energy-momentum of the gravitational field. It completes Einstein's equation and addresses the energy localization problem. Variation of the action with respect to $ X^{\mu} $ restricts $u_{\mu}$ to a particular value, which defines the possible Lorentzian metrics. $ \Phi $, the trace of $ \varPhi_{\alpha\beta} $, describes dark energy. The cosmological constant is dynamically replaced by $ \Phi $. A cyclic universe that developed after the Big Bang is described. The dark energy density provides a natural explanation of why the vacuum energy density is so small, and why it dominates the present epoch. Assuming dark matter does not exist, a solution to the modified Einstein equation introduces two additional terms into the Newtonian radial force equation, from which the baryonic Tully-Fisher relation is obtained.

Equations

$$w_{\alpha \beta }=v_{\alpha \beta }+{\Phi }_{\alpha \beta }$$

$$g_{\alpha \beta }=g_{\alpha \beta }^{+}-2u_{\alpha }{u}_{\beta }.$$

$$w_{\alpha \beta }=v_{\alpha \beta }+\frac{1}{2}{\pounds }_{\xi }{g}_{\alpha \beta }^{+}$$

$$\begin{array}{r}{\nabla }_{\alpha }{v}_{\beta }^{\alpha }-{\nabla }_{\alpha }^{+}{v}_{\beta }^{\alpha }=v^{\alpha \lambda }{\partial }_{\beta }(u_{\alpha }{u}_{\lambda }).\end{array}$$

$$v^{\alpha \lambda }{\partial }_{\beta }(u_{\alpha }{u}_{\lambda })=0$$

$$w_{\alpha \beta }=v_{\alpha \beta }+\frac{1}{2}{\pounds }_{\xi }{g}_{\alpha \beta }+{\pounds }_{\xi }{u}_{\alpha }{u}_{\beta }$$

$$w_{\alpha \beta }=v_{\alpha \beta }+{\Phi }_{\alpha \beta }$$

$${\Phi }_{\alpha \beta }:=\frac{1}{2}({\nabla }_{\alpha }{X}_{\beta }+{\nabla }_{\beta }{X}_{\alpha })+u^{\lambda }(u_{\alpha }{\nabla }_{\beta }{X}_{\lambda }+u_{\beta }{\nabla }_{\alpha }{X}_{\lambda }).$$

$$S=S^F+S^{EH}+S^G$$

$$\delta S^F=\int (\frac{\delta L^F}{\delta g^{\alpha \beta }}-\frac{1}{2}{L}^F{g}_{\alpha \beta })\delta g^{\alpha \beta }\sqrt{-g}{d}^{4}x$$

$${\tilde{T}}_{\alpha \beta }:=-2c(\frac{\delta L^F}{\delta g^{\alpha \beta }}-\frac{1}{2}{L}^F{g}_{\alpha \beta }).$$

$$C_{\alpha \beta }=\frac{a}{c}{\tilde{T}}_{\alpha \beta }+b{w}_{\alpha \beta }$$

$$w_{\alpha \beta }=v_{\alpha \beta }+{\Phi }_{\alpha \beta }$$

$$T_{\alpha \beta }={\tilde{T}}_{\alpha \beta }+\frac{bc}{a}{\Phi }_{\alpha \beta }$$

$$S^G:=-b\int g_{\alpha \beta }{\Phi }^{\alpha \beta }\sqrt{-g}{d}^{4}x.$$

$$C_{\alpha \beta }=\frac{a}{c}{T}_{\alpha \beta }+b{v}_{\alpha \beta }$$

$$-\frac{8\pi G}{c^4}{\tilde{T}}_{\alpha \beta }+R_{\alpha \beta }-\frac{1}{2}{g}_{\alpha \beta }R+\Lambda g_{\alpha \beta }+{\Phi }_{\alpha \beta }=0$$

$$S=\int L^F(A^{\beta },{\nabla }^{\alpha }{A}^{\beta },...,g^{\alpha \beta })\sqrt{-g}{d}^{4}x+b\int (R-2\Lambda )\sqrt{-g}{d}^{4}x-b\int {\Phi }_{\alpha \beta }{g}^{\alpha \beta }\sqrt{-g}{d}^{4}x.$$

$$\begin{gathered} \delta S=\int [-\frac{1}{2c}{\tilde{T}}_{\alpha \beta }+b(R_{\alpha \beta }-\frac{1}{2}{g}_{\alpha \beta }R)+b\Lambda g_{\alpha \beta }+b({\nabla }_{\alpha }{X}_{\beta }+2u^{\lambda }{u}_{\beta }{\nabla }_{\alpha }{X}_{\lambda } \\ +{\nabla }_{\mu }{X}_{\nu }(-u_{\alpha }{u}_{\beta }{g}^{\mu \nu }+u^{\mu }{u}^{\nu }{g}_{\alpha \beta }))]\delta g^{\alpha \beta }\sqrt{-g}{d}^{4}x+2\int {\nabla }_{\alpha }(u^{\alpha }{u}^{\beta })\delta X_{\beta }\sqrt{-g}{d}^{4}x \end{gathered}$$

$$-\frac{1}{2c}{\tilde{T}}_{\alpha \beta }+b(R_{\alpha \beta }-\frac{1}{2}{g}_{\alpha \beta }R)+b\Lambda g_{\alpha \beta }+b{\Phi }_{\alpha \beta }=0$$

$${\nabla }_{\alpha }(u^{\alpha }{u}^{\beta })=0.$$

$$u^{\alpha }{\nabla }_{\alpha }{X}_{\mu }+4u^{\lambda }{u}_{\mu }{u}^{\alpha }{\nabla }_{\alpha }{X}_{\lambda }-3u^{\lambda }{\nabla }_{\mu }{X}_{\lambda }=\Phi u_{\mu }$$

$$X^{\alpha }{X}_{\alpha }=-f^2,$$

$$u_{\mu }=\frac{3}{\Phi }{\partial }_{\mu }f.$$

$$\int g_{\alpha \beta }{\Phi }^{\alpha \beta }\sqrt{-g}{d}^{4}x=\int \Phi \sqrt{-g}{d}^{4}x=0$$

$$T^{\alpha \beta }={\tilde{T}}^{\alpha \beta }-\frac{c^4}{8\pi G}{\Phi }^{\alpha \beta }.$$

$$\int (-\frac{1}{2c}{\tilde{T}}^{\alpha \beta }+b{\Phi }^{\alpha \beta })\delta g_{\alpha \beta }\sqrt{-g}{d}^{4}x=0$$

$$\int {\nabla }_{\alpha }(-\frac{1}{2c}{\tilde{T}}^{\alpha \beta }+b{\Phi }^{\alpha \beta })Y_{\beta }\sqrt{-g}{d}^{4}x=0$$

$${\nabla }_{\alpha }{T}^{\alpha \beta }=0$$

$$G_{\alpha \beta }+\Lambda g_{\alpha \beta }=\frac{8\pi G}{c^4}{T}_{\alpha \beta }.$$

Full text

This is an update of an article published in General Relativity and Gravitation. The original authenticated version is available online at: https://doi.org/10.1007/s10714-019-2537-y.

1 Introduction

It has been over a century since Einstein [1, 2, 3] formulated general relativity (GR) in 1915. He was aware that the gravitational field must interact with itself, but was unable to produce a symmetric tensor to properly describe the energy-momentum of the gravitational field. Instead, a non-covariant pseudo-tensor was introduced. However, the difficulties associated with this pseudo-tensor led to the problem of the localization of energy in GR. Over the decades, other pseudo-tensors were developed and different approaches to describe the energy-momentum of the gravitational field were investigated, [4, 5, 6](and references therein) but the energy localization problem still exists today. Despite this deficiency, general relativity is one of the two cornerstones of physics.

GR was developed by Einstein on a four-dimensional Riemannian manifold with the understanding that spacetime was locally Minkowskian under free fall. Today, we more properly describe spacetime on a time-oriented Lorentzian manifold with metric. The Lorentzian metric can be associated with a Riemannian metric by using the line element field, $(\bm{X},\bm{-X})$ that exists on a non-compact paracompact Hausdorff manifold. A classical result in Riemannian geometry, namely the Berger-Ebin theorem [7], can then be adapted to spacetime. This results in the Orthogonal Decomposition Theorem (ODT): an arbitrary second rank symmetric tensor on a time-oriented Lorentzian manifold with a torsionless and metric compatible connection can be orthogonally decomposed into a linear sum of divergenceless tensors and a new tensor, $\varPhi_{\alpha\beta}$. It is a symmetric tensor constructed from the Lie derivative along $X$ of both the metric and a product of unit line element covectors.

The left hand side of Einstein’s equation $G_{\alpha\beta}+\Lambda g_{\alpha\beta}=\frac{8\pi G}{c^{4}}T_{\alpha\beta}$ involves symmetric divergenceless tensors. The right hand side is defined by the variation of the action functional for all matter fields with respect to the metric. This generates a divergenceless symmetric tensor that must describe all interactions of the gravitational field with the matter fields, and with the energy-momentum of the gravitational field itself; otherwise, it would not be locally conserved. However, there is nothing in this definition that deals explicitly with the energy-momentum of the gravitational field. If we define $\tilde{T}_{\alpha\beta}$ as a symmetric energy-momentum tensor generated from the matter fields without the requirement that it completely describes the energy-momentum of the gravitational field as well, it cannot be locally conserved and would not be divergenceless. Consequently, this second rank symmetric tensor can be set proportional to an arbitrary symmetric tensor $w_{\alpha\beta}$, which is then orthogonally decomposed by the ODT into a linear sum of divergenceless tensors and $\varPhi_{\alpha\beta}$. Lovelock’s theorem [8] proves that in four dimensions, the divergenceless tensors composed from the metric and its first two derivatives can only consist of the metric and the tensor named after Einstein, $G_{\alpha\beta}$. Therefore, Einstein’s equation in a four-dimensional Lorentzian spacetime should be expressed more completely by including the $\varPhi_{\alpha\beta}$ term.

It will be proved that $G_{\alpha\beta}+\Lambda g_{\alpha\beta}+\varPhi_{\alpha\beta}=\frac{8\pi G}{c^{4}}\tilde{T}_{\alpha\beta}$ and that the tensor $T_{\alpha\beta}=\tilde{T}_{\alpha\beta}-\frac{c^{4}}{8\pi G}\varPhi_{\alpha\beta}$ is divergenceless which allows Einstein’s equation to be recovered. In that sense, $\varPhi_{\alpha\beta}$ is hidden in GR. Thus, general relativity is not complete; it is possible to construct a symmetric tensor from the metric and a regular vector field, that is independent of the energy-momentum tensor of the matter fields, and represents the energy-momentum of the gravitational field itself.

This differs with the presently and generally accepted belief that GR is complete. However, if that notion was true, GR should be able to describe particular features of dark matter. That unfortunately is not the case and is the reason why physicists invented the generally well accepted theory of Lambda cold dark matter ($\Lambda$CDM) to explain, in particular, the flat rotation curves of some galaxies, while leaving GR intact. Modified general relativity can describe those and other galactic rotation curves as discussed in section 7.

Since Lie derivatives have the same form when expressed with covariant or partial derivatives, $\varPhi_{\alpha\beta}$ does not vanish when the connection coefficients vanish. The metric can be locally Minkowskian, as in free fall, without forcing $\varPhi_{\alpha\beta}$ to vanish. This contrasts free fall in GR where the connection coefficients vanish locally and the gravitational field locally disappears; hence the well known conception that the energy-momentum of the gravitational field is not localizable [9]. $\varPhi_{\alpha\beta}$ has the structure to describe local gravitational energy-momentum. In free fall, the effective force of gravity disappears locally but the self-energy of the gravitational field is intact accordingly.

In section 2, the Orthogonal Decomposition Theorem is proved. In section 3, a modified equation of GR is derived by using the principle of least action, the ODT and a fundamental postulate of GR. $\varPhi_{\alpha\beta}$ appears naturally alongside the Einstein tensor and introduces $X^{\mu}$ from the line element field and its collinear unit vector $u^{\mu}$ as dynamical variables independent of the Riemannian metric. Variation of the action functional with respect to $X^{\mu}$ leads to the Lorentz invariant expression $u_{\mu}=\frac{3}{\Phi}\partial_{\mu}f$ where $\Phi$ is the trace of $\varPhi_{\alpha\beta}$ and $f\neq 0$ is the magnitude of $X^{\alpha}$. The myriad of possible line element covectors is restricted to those satisfying this condition.

Section 4 discusses the conservation equation for the divergenceless energy-momentum tensor $T_{\alpha\beta}=\tilde{T}_{\alpha\beta}-\frac{c^{4}}{8\pi G}\varPhi_{\alpha\beta}$ where $\tilde{T}_{\alpha\beta}$ is the total matter energy-momentum tensor describing all types of matter including baryonic and dark matter, massive neutrinos and any other possible particle; if dark matter particles exist.

In section 5, the cosmological constant is discussed. Using the global constraint $\int\Phi\sqrt{-g}d^{4}x=0$, it is shown that the cosmological constant $\Lambda$ is dynamically replaced with $\Phi$.

Section 6 is a discussion of the modified Einstein equation of GR in the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, and dark energy. A gravitationally repulsive condition is described by $\Phi>-2\Lambda_{d}$ where $\Lambda_{d}$ is the dark energy density. $\Phi$ defines dark energy. Dark energy describes the inflation of the universe immediately after the Big Bang when no matter of any type was present. The dark energy density then tends to the present value of the vacuum energy density. A cyclic universe is born with maximum and minimum values of the cosmological scale factor in the FLRW metric. Dark energy explains the small value of the vacuum energy density and why it now dominates the expansion and acceleration of the present universe.

Cyclic universes have been reported in the literature [10, 11, 12, 13, 14]. Dark energy has been described by various scalar theories such as the quintessential [15], k-essence [16] (and references therein) phantom or quintom theories [17] (and further references therein). Dark energy in this article is not a scalar theory; it is the energy generated from the interaction of the gravitational field with its energy-momentum tensor.

The existence of a tensor that describes the energy-momentum of the gravitational field brings into question the subject of dark matter. Since Einstein’s equation is incomplete without the tensor $\varPhi_{\alpha\beta}$, the plausibility of dark matter is questionable; its existence is based on the assumption that general relativity is a complete theory. Although the self-interactions in a weak gravitational field may be extremely small, in the gravitational field of a galaxy, they may be significant enough to explain dark matter.

In section 7, the modified equation of GR is calculated with a spheroidal metric in a region of spacetime outside of matter with the assumption that dark matter does not exist. Two additional terms appear in the modified Newtonian force equation that provides it the flexibility to describe various types of galaxies. By balancing the dark energy force with the Newtonian force, the Tully-Fisher relation is established and the acceleration parameter in MOND is expressed in terms of the dark energy radial force parameter.

2 Orthogonal Decomposition of Symmetric Tensors

The Lorentzian spacetime is described on a four-dimensional time-oriented non-compact paracompact Hausdorff manifold with metric, $(M,g_{\alpha\beta})$. The connection on the manifold is torsionless and metric compatible and the metric has a $+2$ signature. The manifold admits a smooth regular line element field $(\bm{X},-\bm{X})$ and a unit vector $\bm{u}$ collinear with one of the pair of regular vectors in the line element field [18, 19, 23]. The spacetime is assumed to admit a Cauchy surface and is therefore globally hyperbolic. This forbids the presence of closed causal curves [23].

The orthogonal decomposition of symmetric tensors on Riemannian manifolds has been documented in the literature [7, 20, 21, 22]. However, a decomposition of symmetric tensors on a time-oriented Lorentzian manifold is required.

Theorem 2.1**.**

An arbitrary (0,2) symmetric tensor $w_{\alpha\beta}$ in the symmetric cotangent bundle $S^{2}T^{\ast}M$ on an n-dimensional time-oriented Lorentzian manifold $(M,g_{\alpha\beta})$ with a torsionless and metric compatible connection can be orthogonally decomposed as

[TABLE]

where $v_{\alpha\beta}$ represents a linear sum of symmetric divergenceless (0,2) tensors and $\varPhi_{\alpha\beta}=\frac{1}{2}\pounds_{X}g_{\alpha\beta}+\pounds_{X}u_{\alpha}u_{\beta}$ where the unit vector $\bm{u}$ is collinear to one of the pair of regular line element vectors $(\bm{X},-\bm{X})$.

Proof.

Let the Lorentzian manifold $(M,g_{\alpha\beta})$ be non-compact paracompact Hausdorff and orientable. A smooth regular line element field $(\bm{X},\bm{-X)}$ exists as does a unit vector $\bm{u}$ collinear with one of the pair of line element vectors. Let M be endowed with a smooth Riemannian metric $g^{+}_{\alpha\beta}$. The smooth Lorentzian metric $g_{\alpha\beta}$ is constructed [19, 23] from $g^{+}_{\alpha\beta}$ and the unit covectors $u_{\alpha}$ and $u_{\beta}$ by setting

[TABLE]

Let $w_{\alpha\beta}$ and $v_{\alpha\beta}$ belong to $S^{2}T^{\ast}M$, the cotangent bundle of symmetric $(0,2)$ tensors on M. In the compact neighborhood of a point $p$ in an open subset of $S^{2}T^{\ast}M$ which contains $g^{+}_{\alpha\beta}$, an arbitrary $(0,2)$ symmetric tensor $w_{\alpha\beta}$ can be orthogonally and uniquely decomposed by the Berger-Ebin theorem [7] according to

[TABLE]

where $\bm{\xi}$ is an arbitrary vector and $v_{\alpha\beta}$ represents a linear sum of symmetric divergenceless (0,2) tensors: ${\nabla^{+}}^{\alpha}v_{\alpha\beta}=0$.

The divergence of $v_{\beta}^{\alpha}$ in the mixed tensor bundle can be written as $\nabla_{\alpha}v_{\beta}^{\alpha}=\partial_{\alpha}v_{\beta}^{\alpha}+\frac{v_{\beta}^{\lambda}}{2g}\partial_{\lambda}g-\frac{1}{2}v^{\alpha\lambda}\partial_{\beta}g_{\alpha\lambda}$. Since the determinant of $g_{\alpha\beta}$, $g$, is related to that of $g^{+}_{\alpha\beta}$ by $g=-g^{+}$

[TABLE]

The left hand side of (4) is a (0,1) tensor but the right hand side is not, which demands:

[TABLE]

where $\partial_{\beta}u_{\alpha}\neq 0$. This guarantees $\nabla^{\alpha}v_{\alpha\beta}=0$ because $\nabla^{+\alpha}v_{\alpha\beta}=0$. Hence,

[TABLE]

where $\nabla^{\alpha}v_{\alpha\beta}=0$. $\xi^{\lambda}$ is an arbitrary vector which can be chosen to be collinear to $u^{\lambda}$. Without loss of generality, $\xi^{\lambda}$ can then be replaced by $X^{\lambda}$. Using $X^{\lambda}=fu^{\lambda}$ where $f\neq 0$ is the magnitude of $X^{\lambda}$, the expression $X^{\lambda}\nabla_{\lambda}(u_{\alpha}u_{\beta})$ in the last term of (6) then vanishes in an affine parameterization and

[TABLE]

where

[TABLE]

The decomposition is orthogonal: $<v_{\alpha\beta},\varPhi_{\alpha\beta}>=0$.

∎

3 Derivation of the modified equation of general relativity

A modified equation of general relativity of the form $C_{\alpha\beta}=0$ is sought which contains a linear combination of symmetric tensors that define the Einstein equation, and a new tensor which can describe the energy-momentum of the gravitational field itself. This can be achieved by using the principle of least action, the Orthogonal Decomposition Theorem (1), and a fundamental postulate of GR.

First, the field equations contained in $C_{\alpha\beta}$, which are sought to describe general relativity and the energy-momentum of the gravitational field, must be derivable from the action functional

[TABLE]

where $S^{F}$ and $L^{F}$ refer to the action and Lagrangian, respectively, for all types of matter fields including those of dark matter if dark matter particles exist. $S^{EH}$ is the Einstein-Hilbert action for general relativity and $S^{G}$ is the action for the energy-momentum of the gravitational field with Lagrangian $L^{G}$. The variation of $S^{F}$ with respect to $g^{\alpha\beta}$

[TABLE]

generates the symmetric energy-momentum tensor $\tilde{T}_{\alpha\beta}$ which represents the interaction of all types of matter fields and associated radiation in a gravitational field, but does not specifically include the energy-momentum of the gravitational field:

[TABLE]

$C_{\alpha\beta}$ must then be expressed as

[TABLE]

where $w_{\alpha\beta}$ is an unknown symmetric tensor independent of $\tilde{T}_{\alpha\beta}$; and a and b are arbitrary constants.

Second, $w_{\alpha\beta}$ can be orthogonally decomposed by the ODT into

[TABLE]

where $\varPhi_{\alpha\beta}$ is given by (8) and $\nabla^{\alpha}v_{\alpha\beta}=0$.

Third, Einstein concluded [1] that the metric should describe both the geometry of spacetime and the gravitational field. He postulated the totality of the matter energy-momentum tensor and the energy-momentum of the gravitational field, to be the source of the gravitational field. Adhering to this philosophy, the energy-momentum tensor $T_{\alpha\beta}$ describing the totality of all types of matter and the energy-momentum of the gravitational field, is postulated to be the source of the gravitational field.

$\varPhi_{\alpha\beta}$ is independent of $\tilde{T}_{\alpha\beta}$ and is not divergenceless. $\varPhi_{\alpha\beta}$ is therefore the sole candidate to describe the energy-momentum of the gravitational field. Thus,

[TABLE]

and the interaction of the gravitational field with its energy-momentum tensor can be defined with the action

[TABLE]

It was proved by Lovelock [8] that the only tensors in a four-dimensional spacetime which are symmetric, divergence free, and a concomitant of the metric tensor together with its first two derivatives are the metric and the Einstein tensor, $G_{\alpha\beta}=R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}R$. $v_{\alpha\beta}$ must therefore contain the Lovelock tensors.

$C_{\alpha\beta}$ is then formally decomposed as

[TABLE]

with $\nabla_{\alpha}v^{\alpha\beta}=0$ and $v_{\alpha\beta}:=G_{\alpha\beta}+\Lambda g_{\alpha\beta}$. $\Lambda$ is a global integration constant (in hindsight identified as the cosmological constant). With the collection of tensors $C_{\alpha\beta}$ defined to vanish, we obtain the modified Einstein equation of general relativity with cosmological constant $\Lambda$ and the gravitational energy-momentum term $\varPhi_{\alpha\beta}$

[TABLE]

by setting $a=-\frac{1}{2}$ and $b=\frac{c^{3}}{16\pi G}.$

Ma and Wang [22] obtained a similar result to (17) with $\Lambda=0$, but with an entirely different $\varPhi_{\alpha\beta}=\nabla_{\alpha}\partial_{\beta}\phi$ for some scalar $\phi$ by using a decomposition of symmetric tensors on a Riemannian manifold.

Equation (17) must be derived from the action functional (9). With (15):

[TABLE]

To calculate the variation of $S^{G}$ with respect to the inverse metric $g^{\alpha\beta}$, the following results are used: $g^{\alpha\beta}=g^{{+}{\alpha\beta}}-2u^{\alpha}u^{\beta}$ is the inverse of $g_{\alpha\beta}$; $g^{{+}{\alpha\beta}}\delta g_{\alpha\beta}=-g_{\alpha\beta}\delta g^{{+}{\alpha\beta}}$; $\delta g^{{+}{\rho\beta}}=-g^{{+}{\alpha\beta}}g^{\lambda\rho}\delta g^{+}_{\alpha\lambda}$; $g_{\alpha\beta}\delta(u^{\alpha}u^{\beta})=u_{\alpha}u_{\beta}\delta g^{\alpha\beta}$; and $\delta(u^{\alpha}u^{\beta})=u_{\lambda}u^{\beta}\delta g^{\lambda\alpha}$. The variation of S with respect to $g^{\alpha\beta}$ is then

[TABLE]

after calculating $\delta\Gamma^{\lambda}_{\alpha\beta}$ induced by the variations in the inverse metric, and integrating by parts several times. The last term in the variation with respect to $g^{\alpha\beta}$ vanishes which follows by writing the tensor in brackets, $-u_{\alpha}u_{\beta}g^{\mu\nu}+u^{\mu}u^{\nu}g_{\alpha\beta}$, as its equivalent, $\frac{1}{2}(g^{{+}{\mu\nu}}g_{\alpha\beta}-g^{+}_{\alpha\beta}g^{\mu\nu})$; and choosing an orthonormal basis $(e_{\alpha})$ at a point $p\in M$ for $g^{+}$ with $e_{0}=u$. Then, $u^{0}u_{0}=1$, $u^{i}u_{i}=0$, $g^{+}_{\alpha\beta}=\delta_{\alpha\beta}$, $g^{00}=-g^{{+}00}$ and $g_{00}=-g^{+}_{00}$, with all other components of the metric g equal to those of the metric $g^{+}$. Since $\delta{g^{\alpha\beta}}$ is symmetric, the second last term can be expressed as $b\varPhi_{\alpha\beta}$. With $\delta S=0$ and arbitrary variations for $\delta g^{\alpha\beta}$ and $\delta X_{\beta}$, we have

[TABLE]

and

[TABLE]

Setting $b=\frac{c^{3}}{16\pi G}$ yields the modified Einstein equation described in (17).

3.1 Some properties of $u_{\mu}$ and $\varPhi_{\alpha\beta}$

$X^{\beta}$ from the line element field and its collinear vector $u^{\beta}$ are dynamical variables independent of the Riemannian metric in (2). The dynamical properties of the line element fields are obtained by varying (18) with respect to $X^{\mu}$. This yields the equation

[TABLE]

using (17). With $X^{\alpha}=fu^{\alpha}$ where $f\neq 0$ is the magnitude of $X^{\alpha}$, the first two terms of (22) are then geodesics in an affine parameterization and can be set to zero. Since

[TABLE]

it follows that

[TABLE]

There is a myriad of regular vectors from the line element field for each Riemannian metric, and the associated Lorentzian metric is not unique. However, the variation of (18) with respect to $X^{\mu}$ restricts the line element fields to those given by (24) which in turn restricts the Lorentzian metric.

$\varPhi_{\alpha\beta}$ is expressed in terms of the Lie derivative of the metric and a product of unit line element covectors. Since Lie derivatives have the same form when expressed with covariant or partial derivatives, $\varPhi_{\alpha\beta}$ does not vanish when the connection coefficients vanish. The metric can be locally Minkowskian, as in free fall, without affecting $\varPhi_{\alpha\beta}$. It has the structure to describe local gravitational energy-momentum.

Using (21), it is straightforward to calculate the coupling of the gravitational field with its energy-momentum tensor:

[TABLE]

where $\Phi=\nabla_{\alpha}X_{\beta}(g^{\alpha\beta}+2u^{\alpha}u^{\beta})$. Equation (25) means the scalar $\Phi$ has local positive and negative values, all of which add to zero when integrated over the entire spacetime. $\Phi$ is globally conserved. Section 6 demonstrates that the positive values of $\Phi$ are attributed to the gravitationally repulsive properties of dark energy with the cosmological constant set to zero. The negative values represent the attractive part of the energy of the gravitational field interacting with its gravitational energy-momentum tensor. $\Phi$ is measurable; it can be expressed in terms of the density and pressure of total matter and the vacuum energy density as shown in section 6. The gravitational energy density is calculated from $\varPhi_{00}$ in section 7. The energy-momentum of the gravitational field is localizable and measureable.

4 The conserved energy-momentum tensor $T_{\alpha\beta}$

The invariance of the action functional describing gravity, it’s self-energy-momentum and total matter fields under the symmetry of diffeomorphisms demands a symmetric divergenceless energy-momentum tensor

[TABLE]

This follows from an analysis of each term in the action functional $S$ defined in (9). The action $S^{EH}$ is independently invariant under a diffeomorphism. Variation of the action $S^{F}$ with respect to the metric contains only $\tilde{T}^{\alpha\beta}$ because the variations of $S^{F}$ with respect to each field and its derivatives vanish with the corresponding Euler-Lagrange equations. Variation of $S^{G}$ with respect to the metric yields $\varPhi^{\alpha\beta}$. Therefore, we can write

[TABLE]

where $b=\frac{c^{3}}{16\pi G}$. Under a diffeomorphism, the Lie derivative of the metric along a regular vector $Y^{\beta}$ generates the infinitesimal change in the metric $\delta g_{\alpha\beta}=\nabla_{\alpha}Y_{\beta}+\nabla_{\beta}Y_{\alpha}$. Integrating by parts then gives

[TABLE]

which requires

[TABLE]

for diffeomorphisms generated by $Y^{\beta}$.

Equation (29) is the local description of the conservation of energy and momentum in a modified theory of GR described by (17). The gravitational field has an intrinsic energy-momentum which is attributed to $\varPhi_{\alpha\beta}$. Being independent of $\tilde{T}_{\alpha\beta}$, $\frac{c^{4}}{8\pi G}\varPhi_{\alpha\beta}$ provides the additional self-energy-momentum of the gravitational field necessary to complete the source $T_{\alpha\beta}$ of the geometry of spacetime. $\varPhi_{\alpha\beta}$ completes the Einstein equation and leaves it intact in form:

[TABLE]

5 Cosmological Constant

The cosmological constant $\Lambda$ appears alongside the metric as the simplest and most basic Lovelock tensor. With a torsionless connection, the covariant derivative of the metric vanishes. Adding the metric to the Einstein equation seems trivial with the associated constant playing the role of a global integration constant. $\Lambda$ can then be interpreted as a constant global energy density. That seems very restrictive as energy densities are generally dynamic and not constant.

The regular vector fields that exist in a Lorentzian spacetime provide a dynamical background from which the energy-momentum of the gravitational field is constructed. It is not possible for a constant global energy density to represent the dynamic interaction of the metric with the energy-momentum tensor of the gravitational field. $\Lambda$ must therefore be dynamically replaced by a scalar.

Theorem 5.1**.**

The cosmological constant $\Lambda$ is dynamically replaced by the trace of $\varPhi_{\alpha\beta}$.

Proof.

Using (25), $S^{EH}$ with $\Lambda=0$ can be written as

[TABLE]

which generates the modified Einstein equation with no cosmological constant from (18). If $\Phi=2\Lambda$ locally, the Einstein equation with the cosmological constant is obtained accordingly. The trace of the tensor describing the energy-momentum of the gravitational field, dynamically replaces the cosmological constant but must obey the global equation (25).

∎

6 Energy-momentum of the gravitational field in the FLRW metric: Dark energy

Some properties of $\varPhi_{\alpha\beta}$ in the Friedmann-Lemaître-Robertson-Walker metric are now investigated. The FLRW metric is typically used to describe a spatially maximal symmetric universe according to the cosmological principle [24] whereby the universe is homogeneous and isotropic when measured on a large scale. This metric is given by

[TABLE]

where $a(t)$ is the cosmological scale factor which satisfies $a>0$ after the Big Bang at $t=0$. $\kappa$ is a constant used to describe a particular spatial geometry. The connection components of the FLRW metric are

[TABLE]

where $j=1,2,3$. The Ricci tensor components are

[TABLE]

and the Ricci scalar is

[TABLE]

It was proved in [24] that a maximally spatial form invariant symmetric second rank tensor $B_{\alpha\beta}$ has components in the form

[TABLE]

where $\varrho(t)$ and $p(t)$ are arbitrary functions of time. We therefore set,

[TABLE]

where $\varrho(t)$ and $p(t)$ are designated as the mass density and pressure functions, respectively, of total matter including dark matter; if dark matter particles exist. Similarly,

[TABLE]

where $\Lambda_{d}(t)$ and $P_{d}(t)$ refer to the energy density and pressure, respectively, of the tensor describing the energy-momentum of the gravitational field.

To obtain the Friedmann equations, we use the trace of the modified Einstein equation

[TABLE]

to rewrite the modified Einstein equation as

[TABLE]

from which we obtain

[TABLE]

from the $R_{00}$ component. The $R_{11}$ component gives

[TABLE]

and the conservation law for $T^{\alpha\beta}$ yields

[TABLE]

Inserting (41) into (42) produces a simpler equation

[TABLE]

Equations (41) and (44) are the Friedmann equations modified with $\varPhi_{\alpha\beta}$.

From (41), we immediately see that $\Phi+2\Lambda_{d}=\Lambda_{d}+\frac{3}{c^{2}}P_{d}>0$ tends to accelerate the universe; while all types of matter combined, with a positive mass density and pressure, tend to decelerate the universe. $\Phi>-2\Lambda_{d}$ is a gravitationally repulsive condition which relates dark energy to $\Lambda_{d}$. Hence, $\Lambda_{d}$ is called the dark energy density and $P_{d}$ the dark energy pressure. $\Phi$ tends to accelerate or decelerate the universe but has a net zero effect on it. $\varPhi_{\alpha\beta}$ and therefore $\Phi$, provide the flexibility to describe various eras in the evolution of the universe. The cosmological constant $\Lambda$, on the other hand, can be expressed as a fixed negative energy density which would have tended to accelerate the universe during all epochs.

One of the recent challenges in cosmology has been to find a natural mechanism that describes a small but positive vacuum energy density to explain the observed acceleration of the present universe. Dark energy provides a natural explanation of this challenge without the need of a cosmological constant.

After the discovery in 1929 by Hubble [25] that the universe was expanding, $\Lambda$ was not required to obtain a static solution to the Einstein equations with a positive mass density. Since the cosmological constant was vastly smaller than any value predicted by particle theory, most particle theorists simply assumed, that for some unknown reason, this quantity was zero [26]. This was widely believed to be true until the discovery of the presently accelerating universe in 1998-99 [27, 28]. $\Lambda$ was then considered to be associated with the dark energy conundrum. However, it is just a global integration constant in the modified Einstein equation and is replaced by $\Phi$ as proved in theorem 5.1. This is readily verified by restricting the dark energy variables to the constant values $\Lambda_{d}=-\Lambda$ and $P_{d}=c^{2}\Lambda$ in (41) and (44). The Friedmann equations with the cosmological constant $\Lambda$ are then recovered in accordance with theorem 5.1.

The Friedmann equations are now considered with $\kappa=1$ describing a closed universe:

[TABLE]

and

[TABLE]

To avoid confusion with $\Lambda$, we will denote the constant vacuum energy density as $\Lambda_{v}$ with the property $\Lambda_{v}>0$. In the present epoch, $\Lambda_{v}$ is measured to be $\approx 1.1\times 10^{-52}m^{-2}$. By defining

[TABLE]

and

[TABLE]

these equations can be simplified to

[TABLE]

and

[TABLE]

with the conservation equation

[TABLE]

Unless otherwise stated, $\varrho>0$ and $p>0$. Equation (49) requires $\tilde{\varrho}>0$.

It is interesting to explore how the energy-momentum of the gravitational field can describe critical features of a Big Bang universe. Immediately after the event of the Big Bang, the universe violently accelerates and $\dot{a}>0$. For a very short time, there is no matter; $\varrho=0$ and $p=0$. In this very early stage of the evolution of the universe, it is possible that the constant vacuum energy density developed. If we set $\varrho=0$ in (45), the inequality

[TABLE]

must hold. From (43) and (46) with $\dot{a}\neq 0$,

[TABLE]

If $\Lambda_{d}\longrightarrow-\Lambda_{v}$ and $\frac{P_{d}}{c^{2}}\longrightarrow\Lambda_{v}$ just after the Big Bang, (46) requires $\frac{\ddot{a}}{a}$ to be constant. With those assumptions, equation (53) has the solution

[TABLE]

where $c_{1}$ is an arbitrary constant. Setting $c_{1}=-3$ and $\Lambda_{v}=\frac{3\ddot{a}}{ac^{2}}$,

[TABLE]

which satisfies (52) and tends to $-\Lambda_{v}$ as the universe expands. Dark energy can generate $\Lambda_{v}$ during this epoch of the universe. The expansion of the universe is then described by

[TABLE]

The pressure density of dark energy is $P_{d}=\frac{c^{2}}{a^{2}}+\Lambda_{v}c^{2}$ and the acceleration of the universe is

[TABLE]

The scalar $\Phi=\frac{6}{a^{2}}+4\Lambda_{v}$ is positive. $\Phi>0$ is the condition to be satisfied for an expanding and accelerating universe when no matter is present. Because this result depends entirely on dark energy, $\Phi>0$ defines dark energy.

With all types of matter appearing after the initial inflation, $\Lambda_{d}$ must obey the constraint

[TABLE]

With constant total matter, the equation

[TABLE]

is obtained from (43) and (46) with $\dot{a}\neq 0$. Since $\frac{\ddot{a}}{a}=\frac{d}{dt}(\frac{\dot{a}}{a})+(\frac{\dot{a}}{a})^{2}$, a slowly varying non-zero Hubble parameter $\frac{\dot{a}}{a}$ requires $\frac{\ddot{a}}{a}$ to be approximately constant. With that assumption, equation (59) has the solution

[TABLE]

with

[TABLE]

The dark energy pressure is

[TABLE]

A pure dark energy effect returns (56) and (57) as the expansion and acceleration, respectively. In a universe with essentially constant matter, which is assumed to be the case of the present era, this demonstrates why $\Lambda_{v}$ is important. As expected, $\Phi=\frac{6}{a^{2}}+4\Lambda_{v}+\frac{8\pi G}{c^{2}}(-\varrho+\frac{3p}{c^{2}})$ is positive or negative.

Riess et al. [29] used the Hubble telescope “to provide the first conclusive evidence for cosmic deceleration that preceded the current epoch of cosmic acceleration”. Given the violent acceleration after the Big Bang, this observation evidences the cyclic nature of the universe to this point in time. The cosmological scale factor must have had maximum and minimum values in the past because of the observed changes in sign of its second derivative; there were extremums at $\dot{a}=0$. In general, this requires $\Lambda_{d}=-\frac{3}{a^{2}}+\frac{8\pi G\varrho}{c^{2}}$ from equation (45). The Hubble parameter vanishes and (60) must change because (61) is not constant at the extremum. Dark energy in the amount of $\Lambda_{v}$ must be transferred to $\Lambda_{d}$ from the dark energy pressure; $\frac{P_{d}}{c^{2}}$ in (62) decreases by $-\Lambda_{v}$ with an offsetting change by that amount to $\Lambda_{d}$ in (60). This allows an extremum to occur while keeping $\Phi$ unchanged. Then, cosmic acceleration can change to a decelerating epoch, and conversely with the opposite exchange of dark energy.

The maxima or minima of the cosmological scale factor follows directly from equations (49) and (50). The second derivative of $a$ must satisfy

[TABLE]

when $\dot{a}=0$. The value of $\tilde{p}$ in equation (63) governs the condition for a maximum or minimum of $a$. With $-\Lambda_{d}$ having a small fixed value of $\Lambda_{v}$ determined early in the evolution of the universe, the variation in $\Phi$ is determined by $P_{d}$. The constraint (25) on $\Phi$ can force $P_{d}$ to change, which can change the sign of $\tilde{p}$. Near the end of an acceleration phase, if the dark energy pressure decreases so that $P_{d}\leq\frac{8\pi Gp}{c^{2}}$, $\tilde{p}$ changes from positive to zero or negative, and the scale factor has a maximum value at $a_{max}$; $\tilde{p}\leq 0$ is satisfied in (63). The acceleration phase ends and the universe undergoes a deceleration. The scale factor then decreases toward a minimum value $a_{min}$ at which the dark energy pressure increases enough to satisfy $\tilde{p}>\frac{c^{2}}{2a^{2}}$. The deceleration phase changes to that of an acceleration and the cyclic process continues indefinitely. $\Phi$, governed by (25), smoothly controls the maximum and minimum values that the cosmological scale factor can have. The global constraint on $\Phi$ keeps the universe gravitationally in balance. This model of the universe starts with the Big Bang and then cycles to eternity. It does not suffer the catastrophes of the Big Crunch or the Big Rip.

Although recent data and analysis [30] suggests the observable universe is flat, the data likely represents a small fraction of the presently unknown entire universe. If the entire universe has a positive curvature, a measurement of it will appear to be nearly flat if data from large enough distances is not available. Therefore, at this time, the conjecture of a flat universe which expands forever based on observational evidence is less likely than the cyclic universe described and observed after the Big Bang and into this epoch.

Dark energy thus provides a natural explanation of why the vacuum energy density is minute, and why it dominates the present epoch of the universe.

7 Energy-momentum of the gravitational field: Dark matter

The $\Lambda$CDM model describes the formation of galaxies after the Big Bang from cooled baryonic matter gravitationally attracted into a dark matter skeleton. Dark matter in the $\Lambda$CDM model also provides the additional mass required to describe the flat rotation curves observed in many galaxies. However, no dark matter particles have been detected and there have been several attempts to explain the flat rotational curves without dark matter.

The leading candidate is a phenomenological model of Modified Newtonian dynamics (MOND) introduced by Milgrom [31]. The Newtonian force F is modified according to

[TABLE]

where $A_{0}$ is a fundamental acceleration $\approx 1.2\times 10^{-10}m/s^{2}$. $\mu$ is a function of the ratio of the acceleration relative to $A_{0}$ which tends to one for $a\gg A_{0}$ and tends to $\frac{a}{A_{0}}$ for $a\ll A_{0}$. MOND successfully explains many, but not all, mass discrepancies observed in galactic data. However, it has no covariant roots in Einstein’s equation or cosmological theory. MOND and $\Lambda$CDM were thoroughly discussed by McGaugh in [32].

Other alternatives to dark matter were reviewed by Mannheim in [33] with references therein. In particular, Moffat [34] used a nonsymmetric gravitational theory without dark matter to obtain the flat rotation curves of some galaxies. The bimetric theory of Milgrom [35] involved two metrics as independent degrees of freedom to obtain a relativistic formulation of MOND.

Different approaches to the missing matter problem include dipolar dark matter, which was introduced by Bernard, Blanchet and Heisenberg in [36] to solve the problems of cold dark matter at galactic scales and reproduce the phenomenology of MOND. The theory involves two different species of dark matter particles which are separately coupled to the two metrics of bigravity and are linked together by an internal vector field. In [37], a theory of emergent gravity (EG) which claims a possible breakdown in general relativity, was introduced by Verlinde that provided an explanation for Milgrom’s phenomenological fitting formula in reproducing the flattening of rotation curves. Campigotto, Diaferio and Fatibenec [38] showed conformal gravity cannot describe galactic rotation curves without the aid of dark matter. On the other hand, a logical analysis based on observational data was presented by Kroupa in [39] to support the conjecture that dark matter does not exist.

The existence of dark matter is based on the assumption that general relativity is correct. However, Einstein’s equation is incomplete without the tensor $\varPhi_{\alpha\beta}$ describing the energy-momentum of the gravitational field. The validity of modified general relativity is now tested with the attempt to describe the additional gravitational attraction in various galaxies without dark matter.

7.1 Modified GR in a spheroidal spacetime

It is assumed dark matter does not exist and that baryonic matter and other possible sources of matter such as neutrinos, produce the gravitational field. In a region of spacetime where there is no matter, $\tilde{T}_{\alpha\beta}=0$ and the field equations must satisfy

[TABLE]

Spheroidal solutions to these nonlinear equations are now investigated. The spheroidal behaviour of the metric is to be determined from a particular solution to (65) in a spacetime described by a metric of the form

[TABLE]

where $\nu$ and $\lambda$ are functions of t, r and $\theta$. The non-zero connection coefficients (Christoffel symbols) are:

[TABLE]

The unit vectors $u^{\alpha}$ satisfy

[TABLE]

As a first step to understand this highly nonlinear set of equations given by (65) with the property (67) in this metric, $u_{3}$ is chosen to vanish. This requires

[TABLE]

because $u_{\alpha}$ is collinear with $X_{\alpha}$. All other components of $X_{\alpha}$ are non-zero.

Static solutions to (65) are sought which require the components of the line element field to satisfy

[TABLE]

and from the metric,

[TABLE]

The components of $\varPhi_{\alpha\beta}$ to be considered are then:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

the Ricci scalar, which from (65) equals $\Phi$, is

[TABLE]

and the corresponding components of the Einstein tensor are:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where the prime denotes $\partial_{1}$.

These equations are greatly simplified by setting

[TABLE]

which provides a potential correspondence to a Schwarzschild-like solution. Thus, a class of static spheroidal solutions to (65) are sought with the restrictions (68),(69),(70) and (80).

Since $e^{\lambda-\nu}(\varPhi_{00}+G_{00})+\varPhi_{11}+G_{11}=0$ from (65),

[TABLE]

From (65) $G_{22}+\varPhi_{22}=0$ gives

[TABLE]

and $G_{33}+\varPhi_{33}=0$ in the interval $0<\theta<\pi$ yields

[TABLE]

Subtracting (82) from (83) requires

[TABLE]

A solution to these equations, which depends on both $r$ and $\theta$ would be useful in the study of angular-dependent aspects of cosmology. However, before tackling that problem, the spherically symmetric solution to (83) with $\partial_{2}\lambda=0$ must be obtained. That is accomplished by expressing the bracketed term in (83), $re^{-\lambda}X_{1}+X_{2}\cot\theta$, as a power series in $r$. To determine a meaningful expression for the power series, there are two physical requirements that can be invoked. Firstly, the Tully-Fisher relation should be obtainable to describe the flat rotation curves of some galaxies within a universe filled with dark energy. This condition requires a term of $\ln r$ in $\lambda$ which will yield a $\frac{1}{r}$ term in $\lambda^{\prime}$. Secondly, the Newtonian gravitational energy density has a $\frac{1}{r^{4}}$ radial dependence. These conditions will be the guides to follow from solutions to (81) and (84). With the benefit of some hindsight, solutions of the form

[TABLE]

are sought where all of the parameters are arbitrary constants.

From (24), the static condition requires $\partial_{0}f=0$ so $u_{0}=0$ and $u_{1}u^{1}+u_{2}u^{2}=-1$. Equations (81) and (84) can then be combined as

[TABLE]

This is the equation from which power series expressions for $X_{1}$ and $X_{2}$ are sought which can then be used in (83) to generate $\lambda$ of the form given by (85).

Firstly, an expression for $X_{1}$ of the form $X_{1}=e^{\lambda}P$ is pursued where $P$ is a polynomial in $r$. By setting

[TABLE]

,

[TABLE]

where $c_{3}$ is an arbitrary constant. By demanding $P\lambda^{\prime}=\frac{P^{\prime}}{P}$, we get the Ricatti equation

[TABLE]

which leads to the expression

[TABLE]

where $c_{4}$ is an arbitrary constant. It is chosen to be an upper bound on $\lambda$: $\lambda\ll c_{4}$. Then,

[TABLE]

By setting $r=\frac{1}{x+1}$ with $\mid x\mid<1$, the $\ln r$ term in $\lambda$ of (85), and then $\lambda$, can be expanded to $O(r^{-2})$ which results in the power series for $X_{1}$:

[TABLE]

where all of the parameters have been absorbed into the arbitrary constants $a_{0},a_{1}$ and $a_{2}$ accordingly.

Equation (86), with $X_{1}$ given by (92), becomes

[TABLE]

which has a solution involving the hypergeometric function. However, a manageable approximate expression for $X_{2}$ can be obtained by observing that $\frac{2P^{\prime}}{\lambda^{\prime}P}\simeq 0$ for large $r$. Then

[TABLE]

where $N:=\frac{2rP^{\prime}}{\lambda^{\prime}}\simeq 2a_{1}+\frac{4a_{2}}{r}$ and $c_{5}$ is an arbitrary constant. In the equatorial plane, we can do the perturbation $\frac{\pi}{2}+2\theta$ around $\frac{\pi}{2}$ where $\theta<<1$ so $\ln\tan(\frac{\pi}{4}+\theta)\simeq 2\tan\theta$. $X_{2}$ then has the structure

[TABLE]

where $c_{5}=0$ and all parameters are absorbed into the arbitrary constants $b_{0}$ and $b_{1}$ accordingly. Equation $(\ref{phiG33})$ can then be written as

[TABLE]

By demanding

[TABLE]

the undesirable term with $\frac{\ln r}{r}$ in the solution for $\lambda$ can be avoided. Equation (96) then simplifies to

[TABLE]

where

[TABLE]

This has the exact solution in the equitorial plane as desired:

[TABLE]

where $c_{1}$ and $c_{2}$ are arbitrary constants. Equation (100) represents the extended Schwarzschild solution.

7.1.1 The energy density of the gravitational field

The total energy-momentum tensor given by (26) requires the energy density of the gravitational field to be described by

[TABLE]

where

[TABLE]

From (92) and (100),

[TABLE]

after calculating $\lambda^{\prime}$ from (100). By setting each of the coefficients of the $\frac{1}{r},\frac{1}{r^{2}},\frac{1}{r^{3}}$ terms to zero, we obtain $2b=a_{1}$, $a_{0}(a_{2}+c_{1})=2ba_{1}$, $2ba_{2}=c_{1}a_{1}$ respectively. Then it follows that

[TABLE]

and

[TABLE]

and

[TABLE]

where $c_{1}$ is chosen to be the parameter

[TABLE]

from the Schwarzschild solution. From (101), it follows that the energy density of the static gravitational field is

[TABLE]

We see that the radial term is twice the Newtonian gravitational energy density. This is not surprising considering the fact that in general relativity, it is calculated from the linearized field equations [41]. The perturbation expansion of the field equations in terms of $h_{\alpha\beta}$, a very small change in the metric relative to the flat spacetime Minkowski metric, contains an infinite number of very small terms involving $h^{2},h^{3},...h\partial h,...h\partial\partial h...$ to all possible powers of $h$ and its first two derivatives. These truncated self interactions of the perturbed field and the linear term, apparently contribute the same amounts of gravitational energy to $W$. But there is no way of knowing that in GR because it has no tensor that explicitly represents the energy-momentum of the gravitational field.

The Tully-Fisher relation is derived in the next subsection.

7.1.2 The radial force and galactic rotation curves

The radial force on an object of mass m can now be calculated from (100) in the equitorial plane. Using the conventional relationship of the Newtonian potential $\phi$ to $g_{00}$,

[TABLE]

the radial force $F_{r}$ is

[TABLE]

From (107) where M represents the total mass of the galaxy composed of mainly baryonic matter and no dark matter, we arrive at the modified Newtonian force

[TABLE]

The correction terms to the Newtonian force come from the non-zero components of the line element field in the energy-momentum tensor $\varPhi_{\alpha\beta}$. The components of the line element field can change their sign, which means $a_{0}$ can change to $-a_{0}$. The second term is gravitationally attractive and represents the “dark matter” correction if $b>0$. It is the term that gives rise to the flat rotation curves. The third term is positive and repulsive if $a_{0}>0$. This describes the repulsive dark energy force in the present epoch. However, during a part of the previous decelerating epoch observed by Riess et al. [29], $a_{0}<0$. They used the Hubble telescope to provide the first conclusive evidence for cosmic deceleration that preceded the current epoch of cosmic acceleration.

Assuming a circular orbit about a point mass, it follows that the orbital velocity of a star rotating in the galaxy satisfies

[TABLE]

where $v_{N}^{2}$ is the Newtonian term

[TABLE]

Equation (112) demands an upper limit to r describing a large but finite galaxy.

Because $a_{0}\neq 0$, it is possible for the Newtonian force to balance the dark energy force. This requires

[TABLE]

and

[TABLE]

with $b>0$ describes a specific class of galaxies with a flat orbital rotation curve. From (113) and (114), we obtain the Tully-Fisher relation

[TABLE]

This result holds for any finite r in contrast to EG which holds only for large r as determined by Lelli, McGaugh and Schombert [41]. With $\frac{{c^{2}a_{0}}}{2}:=A_{0}$, the Tully-Fisher relation in MOND is evident.

The importance of the radial acceleration relative to the rotation curves of galaxies was discussed by Lelli, McGaugh, Schombert, and Pawlowski in [42] where it was determined that late time galaxies (spirals and irregulars), early time galaxies (ellipticals and lenticulars), and the most luminous dwarf spheroidals follow a tight radial acceleration relation which correlates well with that due to the distribution of baryons.

Equation (111), which does not include dark matter in this analysis, is general enough to describe the rotation curves of many types of galaxies. For example, galaxy NGC4261 has a relatively flat rotation curve but starts to rise at larger radii, reaching velocities of 700 km $s^{-1}$ at 100 kpc [42]. That requires $a_{0}$ in (112) to be negative which was interpreted above. As another example, both $a_{0}c^{2}r$ and $bc^{2}$ could be small enough relative to $\frac{GM}{r}$, or those terms could cancel one another, so that the Newtonian term is dominant. Galaxies with no flat rotation curves have recently been observed by van Dokkum et.al [43].

It should be remembered that the general equation (65) provides additional variables that may explain even more aspects of cosmology now attributed to dark matter. However, it is still possible that dark matter particles may exist. As a part of (17) in the total matter energy-momentum tensor $\tilde{T}_{\alpha\beta}$, they would contribute to the gravitational field outside of its source along with baryonic matter in equation (65) and therefore in (111). But any dark matter contribution to the gravitational field would play a much lesser role because of the existence of $\varPhi_{\alpha\beta}$.

From the expression for $\lambda$ given by (100), it is clear that in the absence of the line element field in GR, the Einstein tensors alone cannot represent the additional gravitational attraction attributed to dark matter. This explicitly shows why GR is incomplete and why $\Lambda$CDM was invented to describe dark matter.

8 Conclusion

The results in this article stem from the existence of the line element field in a Lorentzian spacetime. It is a fundamental part of the Lorentzian metric and provides the extra freedom to construct the symmetric tensor $\varPhi_{\alpha\beta}$ from the Lie derivative along the line element vector of both the metric and the unit line element vectors. That tensor, which is absent in GR, solves the problem of the localization of gravitational energy-momentum. $T_{\alpha\beta}$, the sum of the matter energy-momentum and $\varPhi_{\alpha\beta}$ is divergenceless. $\varPhi_{\alpha\beta}$ completes Einstein’s equation and leaves it intact in form.

The line element field is a dynamical variable independent of the Riemannian metric. Variation of the action functional with respect to $X^{\mu}$ restricts the covectors that can be a part of the Lorentzian metric to be those satisfying the Lorentz invariant expression $u_{\mu}=\frac{3}{\Phi}\partial_{\mu}f$.

The gravitational energy density is calculated from $\varPhi_{00}$. It is shown that the radial contribution is twice the Newtonian gravitational energy density calculated from the linearized field equations in GR. The energy-momentum of the gravitational field is localizable and measureable.

$\Phi$ has the global property $\int g^{\alpha\beta}\varPhi_{\alpha\beta}\sqrt{-g}d^{4}x=0$. The cosmological constant $\Lambda$ is dynamically replaced by $\Phi$.

Important features attributed to dark energy result from the investigation of the modified Einstein equation in the FLRW metric. $\Phi>0$ defines dark energy in the present epoch. The dark energy pressure explains the observed cyclic nature of the universe after the Big Bang. The dark energy density explains the initial inflation of the universe and provides a natural explanation of why the vacuum energy density is so small and why it now dominates the expansion and acceleration of the present universe.

The energy-momentum of the gravitational field is important in the description of dark matter. A static solution is obtained from the modified Einstein equation in a spheroidal metric describing the gravitational field outside of its source, which does not contain dark matter. The modified Newtonian force contains two additional terms: one represents the dark energy force which depends on the parameter $a_{0}$; and the other represents the $\frac{1}{r}$ “dark matter” force which depends on the parameter $b$. The baryonic Tully-Fisher relation is obtained by balancing the dark energy force with the Newtonian force. This condition describes the class of galaxies associated with MOND. The rotation curves for galaxies with no flat orbital curves, and those with rising rotation curves for large radii describe examples of the flexibility of the orbital rotation curve equation. The results obtained from the complete Einstein equation thus far are able to substantially describe the missing mass problem attributed to dark matter. Further mathematical and detailed numerical analyses to explore the ability of the energy-momentum tensor of the gravitational field to replace dark matter in cosmology, are fully warranted. This rigorous analysis with comparison to astronomical data may still point to the existence of dark matter to some extent. But even if that is the case, the gravitational role of dark matter is substantially reduced by the impact of the energy-momentum tensor of the gravitational field.

Thus, $\varPhi_{\alpha\beta}$ represents the energy-momentum of the gravitational field itself and explains particular features of dark energy and dark matter. It is the symmetric tensor that Einstein sought many years ago.

Acknowledgements

I would like to thank the anonymous referee for his/her constructive comments.