The Energy-Momentum Tensor of the Gravitational Field as a Correction to the Einstein Equation

Abstract

The energy-momentum-stress tensor of the Riemannian space is found and defined in the general theory of relativity as a function of the metric tensor. In inertial space, this tensor is equal to zero, but in the Newtonian limit, its energy density value is calculated. The general theory of relativity makes it possible to include the field energy tensor in the gravitational field equation. The new gravitational field equation is asymptotically equal to the Einstein equation when applied in weak fields. The region $0<r<r_g$ is occupied by a potential well, which has an extremely low potential, this fact suggests that neutron stars and heavier objects have the same structure. The introduction of the new equation of the gravitational field and the energy-momentum tensor of the gravitational field into the classical Einsteinian relativistic theory of gravitation led us to a consistent continuation of the general theory of relativity.

1. Introduction

The proposed model is not an alternative gravitational model [1]. Criticism and comparison with these models are beyond the scope of this work. The work provides a response to the critical remarks of V.A. Fock [2] on the inadmissibility of solutions to the Einstein equation with special (singular) points.

Of particular importance were the results of Einstein himself when creating the gravitational field equation [3] and the work that followed immediately after this event. Our model is built in strict accordance with the principles obtained by A. Einstein [4] in 1913.

Einstein introduced the Riemannian space into physics as a physical object. This made it possible to consistently describe gravity and the phenomena accompanying it, at least in the region of fields which were not too large. The current material being presented does not go beyond the principles of the general theory of relativity, on the contrary, the theory has managed to include a full-fledged principle of conservation of energy, which is absent in Einstein’s theory of gravity.

In this work, the energy-momentum tensor of the gravitational field and the equation of the gravitational field are successively introduced. The principles of Einstein’s general theory of relativity were established in 1913 [4]. One of the main principles: the law of conservation of energy-momentum, provided for the conservation of energy, not only the conservation of the energy-momentum of matter, but also the energy of complete systems of matter and their gravitational fields. In modern terms, the covariant equation of the gravitational field, which satisfies the law of the conservation of energy, should have looked like this:

$$G_{\nu }^{\mu }=\kappa \left(T_{\nu }^{\mu }+f_{\nu }^{\mu }\right),$$E1

where $f_{\mathit{\mu \nu }}$ is the energy-momentum tensor of the gravitational field. A conservation law could be derived from this field equation. However, Einstein failed to solve this problem. As a result, the current, simplified Einstein equation was obtained (December 1915 [5]):

$$R_{\mathit{\mu \nu }}=\frac{8\mathit{\pi G}}{c^4}\left(T_{\mathit{\mu \nu }}-\frac{1}{2}{g}_{\mathit{\mu \nu }}T\right),$$E2

the following shows the Ricci tensor

$$R_{\mathit{\mu \nu }}=\frac{\partial {\Gamma }_{\mathit{\mu \nu }}^{\alpha }}{\partial x^{\alpha }}-{\Gamma }_{\mathit{\mu \beta }}^{\alpha }{\Gamma }_{\mathit{\nu \alpha }}^{\beta }-\frac{\partial {\Gamma }_{\mathit{\mu \alpha }}^{\alpha }}{\partial x^{\nu }}+{\Gamma }_{\mathit{\mu \nu }}^{\alpha }{\Gamma }_{\mathit{\alpha \beta }}^{\beta }.$$

Einstein’s equation is now more commonly used in the form

$$R_{\mathit{\mu \nu }}-\frac{1}{2}{g}_{\mathit{\mu \nu }}R=\frac{8\mathit{\pi G}}{c^4}{T}_{\mathit{\mu \nu }}.$$E3

In addition, Einstein showed that choosing coordinates such that $\sqrt{-g}=1$ simplifies the equation. This brings the Einstein eq. (2) to the following form:

$$\left\{\begin{array}{c}\frac{\partial {\Gamma }_{\mathit{\mu \nu }}^{\alpha }}{\partial x_{\alpha }}-{\Gamma }_{\mathit{\mu \beta }}^{\alpha }{\Gamma }_{\mathit{\nu \alpha }}^{\beta }=\frac{8\mathit{\pi G}}{c^4}\left(T_{\mathit{\mu \nu }}-\frac{1}{2}{g}_{\mathit{\mu \nu }}T\right).\\ \sqrt{-g}=1.\end{array}\right.$$E4

This form of the equation was solved by Schwarzschild [6]. Due to the complexity of this solution, this version of Einstein’s equation has practically disappeared from the literature.

Landau and Lifshitz [7]¹ noticed, that there is no gravitational field energy density as a source on the right-hand side of Einstein’s field equations. In § 95, it is written, that.

Gravitational interaction plays a role only for bodies with a sufficiently large mass (due to the small gravitational constant),. . .

and in § 96 regarding the vanishing covariant derivative of the energy-momentum density tensor of matter, $T_{\mathit{ki};k}=0$, (that)

In this form, however, this (Einstein) equation does not generally express any conservation law whatever. This is related to the fact that in a gravitational field the four-momentum of the matter alone must not be conserved, but rather the four-momentum of matter plus gravitational field;

the latter is not included in the expression for $T_i^k$.

Indeed, Einstein initially introduced the energy-momentum density tensor of the gravitational field into his field eqs. [1]. However, it turned out, that this quantity, which is calculated from the conservation law of energy and momentum, is not a tensor. In this form, the expression for the gravitational field energy and momentum density was not covariant, which became a serious obstacle for creating the complete field equations. Within only two years, a way out of this dilemma was found, after Einstein simply had removed the gravitational field energy and momentum density from his theory. Then, the field equations became covariant and correctly described the motion of mercury.

2. Division of the ricci tensor

Following Einstein [6], we represent the Ricci tensor as the sum of two parts:

$$R_{\mathit{\mu \nu }}=A_{\mathit{\mu \nu }}+B_{\mathit{\mu \nu }};$$

$$A_{\mathit{\mu \nu }}=\frac{\partial {\Gamma }_{\mathit{\mu \nu }}^{\alpha }}{\partial x_{\alpha }}-{\Gamma }_{\mathit{\mu \beta }}^{\alpha }{\Gamma }_{\mathit{\nu \alpha }}^{\beta };$$

$$B_{\mathit{\mu \nu }}=-\frac{\partial {\Gamma }_{\mathit{\mu \alpha }}^{\alpha }}{\partial x^{\nu }}+{\Gamma }_{\mathit{\mu \nu }}^{\alpha }{\Gamma }_{\mathit{\alpha \beta }}^{\beta }.$$E5

However, we do not yet know whether parts of the Ricci tensor are tensor quantities. The following allows us thus to prove so.

Theorem 1. The magnitude of the simplified Christoffel symbol ${\mathrm{\Gamma }}_{\mathit{\mu \alpha }}^{\alpha }$ is a 4-vector, $A_{\mathit{\mu \nu }}$ and $B_{\mathit{\mu \nu }}$ are tensors.

This is Proof.

We used the formula for the coordinate transformation of Christoffel symbols (Eq. (85.15), [4]).

$${\Gamma }_{\mathit{\mu \alpha }}^{\iota }={\Gamma }_{\mathit{\nu \xi }}^{\prime \gamma }\frac{\partial x^{\iota }}{\partial x{\prime }^{\gamma }}\frac{\partial x^{\prime \nu }}{\partial x^{\mu }}\frac{\partial {x^{\prime }}^{\xi }}{\partial x^{\alpha }}+\frac{{\partial }^2{x}^{\prime \gamma }}{\partial x^{\alpha }\partial x^{\mu }}\frac{\partial x^{\iota }}{\partial x{\prime }^{\gamma }}$$

If we are to add the following to this expression, $\iota =\alpha$and $\gamma =\xi$, then the formula will be simplified as seen below:

$${\Gamma }_{\mathit{\mu \alpha }}^{\alpha }={\Gamma }_{\mathit{\nu \xi }}^{\prime \xi }\frac{\partial x^{\alpha }}{\partial {x^{\prime }}^{\xi }}\frac{\partial x^{\prime \nu }}{\partial x^{\mu }}\frac{\partial {x^{\prime }}^{\xi }}{\partial x^{\alpha }}+\frac{{\partial }^2{x}^{\prime \xi }}{\partial x^{\alpha }\partial x^{\mu }}\frac{\partial x^{\alpha }}{\partial {x^{\prime }}^{\xi }}={\Gamma }_{\mathit{\nu \xi }}^{\prime \xi }\frac{\partial x^{\prime \nu }}{\partial x^{\mu }}+\frac{{\partial }^2}{\partial x^{\mu }\partial {x^{\prime }}^{\xi }}{x}^{\prime \xi }$$

The last term is transformed according to the composite function differentiation theorem. According to the theorem on the equality of mixed derivatives, the order of differentiation can be changed. Then it turns out that the last term is equal to zero, and

$${\Gamma }_{\mathit{\mu \alpha }}^{\alpha }={\Gamma }_{\mathit{\nu \xi }}^{\prime \xi }\frac{\partial x^{\prime \nu }}{\partial x^{\kappa }}.$$

Thus, ${\mathrm{\Gamma }}_{\mathit{\mu \alpha }}^{\alpha }$ transforms as a 4-vector and is therefore a 4-vector.

The covariant differentiation ${\mathrm{\Gamma }}_{\mathit{\mu \alpha }}^{\alpha }$ generates the tensor

$$\frac{D{\Gamma }_{\mathit{\mu \alpha }}^{\alpha }}{\partial x_{\nu }}=\frac{\partial {\Gamma }_{\mathit{\mu \alpha }}^{\alpha }}{\partial x_{\nu }}-{\Gamma }_{\mathit{\mu \nu }}^{\alpha }{\Gamma }_{\mathit{\alpha \beta }}^{\beta },$$

the resulting tensor possesses and exactness up to the sign and is $B_{\mathit{\mu \nu }}$.

It follows from the proven theorem that since $R_{\mathit{\mu \nu }}$ is a tensor, it follows from $R_{\mathit{\mu \nu }}=A_{\mathit{\mu \nu }}+B_{\mathit{\mu \nu }}$ that $A_{\mathit{\mu \nu }}$ is also in fact a tensor quantity.

We’ll need a mixed tensor soon.

$$B_{\mu }^{\nu }=-\frac{D{\Gamma }_{\mathit{\mu \alpha }}^{\alpha }}{\partial x^{\nu }}=-\frac{\partial {\Gamma }_{\mathit{\mu \alpha }}^{\alpha }}{\partial x^{\nu }}-{\Gamma }_{\mathit{\mu \nu }}^{\alpha }{\Gamma }_{\mathit{\alpha \beta }}^{\beta }.$$E6

3. Connection of the part of the tensor $B_{\mathit{\mu \nu }}$ with the gravitational field

Consider the quantity $B_{\mathit{\mu \nu }}$ in the Newtonian approximation, which is described by the stationary metric:

$$d{s}^2=\left(1+\frac{2\phi }{c^2}\right)d{t}^2-d{x}^2-d{y}^2-d{z}^2,$$

here $\phi \left(x\right)$ is the Newtonian potential. Taking into account that the spatial components of the metric tensor and its determinant slightly differ from unity, we obtain:

$$B_0^0=-{\Gamma }_{00}^1{\Gamma }_{10}^0=-\frac{1}{c^2}\frac{\partial \phi }{\partial x}\bullet \frac{1}{c^2+2\phi }\frac{\partial \phi }{\partial x}\approx -\frac{1}{c^4}{\left(\frac{\partial \phi }{\partial x}\right)}^2.$$

This value of the energy density of the gravitational field, up to a constant factor, coincides with the known field density in the Newtonian approximation ([7] § 106, problem 1):

$$f=-\frac{{\left(\nabla \phi \right)}^2}{8\mathit{\pi G}}$$

or

$$B_0^0=\frac{8\mathit{\pi G}}{c^4}{f}_0^0=\kappa f_0^0.$$E7

On the other hand, the tensor $B_{\mathit{\mu \nu }}$ vanishes in the Minkowski space in which the fundamental tensor consists of ones with a positive or negative sign, or zeros:

$$B_{\mathit{\mu \nu }}=-\frac{\partial {\Gamma }_{\mathit{\mu \alpha }}^{\alpha }}{\partial x^{\nu }}+{\Gamma }_{\mathit{\mu \nu }}^{\alpha }{\Gamma }_{\mathit{\alpha \beta }}^{\beta }\to 0.$$

On the other hand, the tensor $B_{\mu }^{\nu }$ vanishes in Minkowski space, which does not contain gravitational fields. Relation (6) allows us to relate the tensor $B_{\mu }^{\nu }$ with the energy-momentum-stress tensor of the gravitational field $f_{\mu }^{\nu .}$

$$B_{\mu }^{\nu }=\kappa f_{\mu }^{\nu }.$$E8

4. Gravitational field equation

The introduction of Eq. (8) into the Einstein Eq. (1) leads to a new equation for the gravitational field:

$$\frac{\partial {\Gamma }_{\mathit{\mu \nu }}^{\alpha }}{\partial x_{\alpha }}-{\Gamma }_{\mathit{\mu \beta }}^{\alpha }{\Gamma }_{\mathit{\nu \alpha }}^{\beta }-\frac{1}{2}{g}_{\mathit{\mu \nu }}R=\frac{8\mathit{\pi G}}{c^4}{T}_{\mathit{\mu \nu }}.$$E9

In this case, additional unknowns do not appear, since the quantities entering Eq. (9) do not depend on the components of the tensor $f_{\mathit{\mu \nu }}$.

Theorem 2. As the determinant of the metric tensor g approaches the value of the Minkowski tensor, which already differs little from the value of the Minkowski tensor, the Eq. (9) comes to an indeterminate approach of Einstein’s Eq. (4).

Proof

As shown above the tensor $B_{\mathit{\mu \nu }}\to 0$ as the metric $g_{\mathit{\mu \nu }}$ approaches the Minkowski metric, we can write the asymptotic equality:

$$R_{\mathit{\mu \nu }}=\frac{\partial {\Gamma }_{\mathit{\mu \nu }}^{\alpha }}{\partial x_{\alpha }}-{\Gamma }_{\mathit{\mu \beta }}^{\alpha }{\Gamma }_{\mathit{\nu \alpha }}^{\beta }+B_{\mathit{\mu \nu }}\approx \frac{\partial {\Gamma }_{\mathit{\mu \nu }}^{\alpha }}{\partial x_{\alpha }}-{\Gamma }_{\mathit{\mu \beta }}^{\alpha }{\Gamma }_{\mathit{\nu \alpha }}^{\beta }.$$

Whence follows the asymptotic equivalence of Eq. (9) and the Einstein equation.

Theorem 3. From the equation of the gravitational field Eq. (9) follows the complete law of conservation of matter and the gravitational field.

Proof

We know that the covariant derivative of the Einstein tensor is:

$$G_{\nu ;\mu }^{\mu }=0.$$

Hence from Eq. (1) follows the general conservation law:

$$T_{\nu ;\mu }^{\mu }+f_{\nu ;\mu }^{\mu }=0,$$

which includes not only matter but also the gravitational field. In fact, this is the conservation law of the $B_{\mathit{\mu \nu }}$ tensor, see Eq. (8). Thus, the equation of the gravitational field is obtained, which satisfies the postulates of Einstein + the law of conservation of energy-momentum.²

5. The Schwarzschild problem

Is solved in the same way as the Einstein equation. Looking for solutions in the form:

$$d{s}^2=s\left(r\right)d{t}^2-p\left(r\right)d{r}^2-r^{2}d{\theta }^2-r^2{\mathit{sin}}^2\mathit{\theta d}{\phi }^2.$$E10

The Christoffel symbols of the metric Eq. (10) have the meanings:

$${\Gamma }_{11}^1=\frac{p^{\prime }}{2p};{\Gamma }_{22}^1=\frac{-r}{p};{\Gamma }_{33}^1=-\frac{\mathit{\text{rsin}}{\left(\theta \right)}^2}{p};{\Gamma }_{10}^0={\Gamma }_{01}^0=\frac{s^{\prime }}{2s};{\Gamma }_{00}^1=\frac{s^{\prime }}{2p};$$

$${\Gamma }_{33}^2=-\mathit{sin}\left(\theta \right)\mathit{cos}\left(\theta \right);{\Gamma }_{12}^2={\Gamma }_{21}^2={\Gamma }_{13}^3={\Gamma }_{31}^3=\frac{1}{r};{\Gamma }_{23}^3={\Gamma }_{32}^3=\frac{\mathit{cos}\left(\theta \right)}{\mathit{sin}\left(\theta \right)}.$$

where the prime means the derivative with respect to r.

Equation for empty space:

$$A_{\mathit{\mu \nu }}=\frac{\partial {\Gamma }_{\mathit{\mu \nu }}^{\alpha }}{\partial x_{\alpha }}-{\Gamma }_{\mathit{\mu \beta }}^{\alpha }{\Gamma }_{\mathit{\nu \alpha }}^{\beta }=0.$$E11

From this equation, we find the components of the tensor Eq. (11), up to a nonzero factor, equal:

$$A_{00}\propto \frac{s^{\prime }}{p}\left(\frac{s^{\prime \prime }}{s^{\prime }}-\frac{p^{\prime }}{p}-\frac{s^{\prime }}{s}\right)=0,$$

$$A_{11}\propto {\left(\frac{p^{\prime }}{p}\right)}^{\prime }-\frac{4}{r^2}-\frac{s^{\prime }{s}^{\prime }}{2s^2}=0.$$

If we equate these components to zero, we get a system of ordinary differential equations. If eliminating from the above the value $\frac{p^{\prime }}{p}$ we thus obtain a third-order equation:

$${\left(\frac{s^{\prime \prime }}{s^{\prime }}\right)}^{\prime }-{\left(\frac{s^{\prime }}{s}\right)}^{\prime }-\frac{s^{\prime }{s}^{\prime }}{2s^2}=\frac{4}{r^2}.$$E12

We have not been able to find an analytical solution to this equation.

When solving the equation numerically, the asymptotic proximity with the Schwarzschild solution was used. This made it possible to use the values of the Schwarzschild solution $1-\frac{1}{r}$ as boundary conditions close to the expected solution. As expected, in the region of relatively small values, the solution follows the Schwarzschild solution. Then, similarly to the Schwarzschild solution, it rapidly decreases (Figure 1). But the difference from the Schwarzschild solution, the solution of Eq. (11) remains positive, although extremely minute. There is a potential well with an almost limiting value of the potential, which is equal to $\phi =-c^2$. The width of the potential well of the solution is noticeably larger than the Schwarzschild radius $r_g=1$ (Figure 1). No singularities or deviations from the monotonic dependence of the solution are observed in the surrounding neighborhood of the equation. Moreover, there is no indication that the expected singularity is present at the origin, similar to the singularity formed by a point source in known classical problems.

Figure 1.

Dependence of $g_{00}$ on the ratio $\frac{r}{r_g}$. The Schwarzschild solution of the Einstein equation (blue line) and the numerical solution of the exact equation of the gravitational field Eq. (9) (red line).

Thus, Einstein’s space finally received the status of an independent physical object with parameters specified by the energy-momentum-stress tensor

$$f_{\mathit{\mu \nu }}=-\frac{8\mathit{\pi G}}{c^4}\frac{D{\Gamma }_{\mathit{\mu \alpha }}^{\alpha }}{\partial x_{\nu }}.$$

Einstein’s equation is a good approximation but only for gravitational fields that are not too large, therefore it must be replaced by the exact equation of the gravitational field:

$$R_{\mathit{\mu \nu }}-\frac{1}{2}{g}_{\mathit{\mu \nu }}R=\kappa T_{\mathit{\mu \nu }}⟹\frac{\partial {\Gamma }_{\mathit{\mu \nu }}^{\alpha }}{\partial x_{\alpha }}-{\Gamma }_{\mathit{\mu \beta }}^{\alpha }{\Gamma }_{\mathit{\nu \alpha }}^{\beta }-\frac{1}{2}{g}_{\mathit{\mu \nu }}R=\kappa T_{\mathit{\mu \nu }}.$$

Formally, the elimination of these “small” errors leads to an equation, the solution of which must be a smooth metric tensor. Moreover, the law of conservation of energy-momentum follows directly the equation of the gravitational field.

6. Black holes and neutron stars

There is a potential well with a limiting value of the potential, which is equal to $\phi =-c^2$. The width of the potential well of the solution is noticeably larger than the radius Schwarzschild. But the difference from the solution Schwarzschild solution of Eq. (11) remains positive, although extremely small. The width of the potential well of the solution is noticeably larger than the radius Schwarzschild $r_g=1$ (Figure 2). But in the vicinity, there are features or deviations from the monotonic dependence of the solution. Moreover, there is no indication that there is a singularity at the origin, similar to the singularity formed by a point source in known classical fields (Figure 3).

Figure 2.

Stretched 5000 times Figure 1. The horizontal section of the solution of Eq. (9) has a positive value, but it is so small that it graphically merges with the coordinate axis.

Figure 3.

Dependence of $ln{g}_{00}$ on the ratio$\frac{r}{r_g}$. The numerical solution of the exact equation of the gravitational field Eq. (9).

For applications, the most interesting solutions are for large fields. There was confidence that the fundamental tensor entirely belongs to the original Riemannian space and does not contain invalid values.

The presented results solve the problem of describing super heavy compact objects - black holes and neutron stars. It can be expected that the gravitational field of such objects is a kind of potential wells with extreme but finite depth.

The ultimate black hole/potential hole test.

Unlike a black hole, thermal radiation can leave the potential well. However, due to the gravitational redshift, the emission maximum should shift toward longer wavelengths. For this reason, it is possible to observe such objects in the radio range.

The filling of the potential well with a substance depends on the equation of the state of the substance and its temperature. This should raise the pit fill level above the extreme value.

Figures 1 and 2 show that the edge of the potential well is slightly blurred. However, this does not prevent us from estimating the radius of the potential well. In Figure 2, we see that the radius of the well is slightly larger than $4r_g$. This value is close to the estimated radius of neutron stars, so we can make the assumption that the gravitational component of a neutron star is completely determined by Eq. (8).

Qualitatively, the characteristics of the near gravitational field are similar to those of the field of the Schwarzschild solution. Thus, it will be difficult for the observer to distinguish the details of the field by observing refraction in the electromagnetic bands. However, the observation of gravitational waves makes it possible to observe orbital characteristics up to the moment of collision, which will make it possible to differentiate the various laws of attraction. Although, we repeat, the observation of the low-frequency component of the radiation of heavy objects, in our opinion, remains the most promising way to prove the non-Schwarzschild nature of “black holes.”

7. Conclusion

This work was preceded by a series of preprints, starting with preprint [8] and ending with preprints [9] and [10]. The work may claim to change our understanding of both dense compact objects (black holes and neutron stars) and space-time as a physical object.