Advances in High Energy Physics

Volume 2019, Article ID 1840360, 11 pages

https://doi.org/10.1155/2019/1840360

## Review on the Pseudocomplex General Relativity and Dark Energy

Correspondence should be addressed to Peter O. Hess; xm.manu.seraelcun@sseh

Received 26 March 2019; Accepted 30 May 2019; Published 20 June 2019

Guest Editor: Cesar A. Vasconcellos

Copyright © 2019 Peter O. Hess. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

A review will be presented on the algebraic extension of the standard Theory of Relativity (GR) to the pseudocomplex formulation (pc-GR). The pc-GR predicts the existence of a dark energy outside and inside the mass distribution, corresponding to a modification of the GR-metric. The structure of the emission profile of an accretion disc changes also inside a star. Discussed are the consequences of the dark energy for cosmological models, permitting different outcomes on the evolution of the universe.

#### 1. Introduction

The Theory of General Relativity (GR) [1] is one of the best tested known theories [2], mostly in solar system experiments. Also the loss of orbital energy in a binary system [3] was the first indirect proof for gravitational waves, which were finally detected in [4]. On April 10,* The Event Horizon Telescope* collaboration announced the first picture taken from the black hole in M87. This gives us an opportunity to compare results from the pc-GR to GR.

Nevertheless, the limits of GR may be reached when strong gravitational fields are present, which can lead to different interpretations of the sources of gravitational waves [5, 6].

A first proposal to extend GR was attempted by A. Einstein [7, 8] who introduced a complex valued metric , with . The real part corresponds to the standard metric, while the imaginary part defines the electromagnetic tensor. With this, A. Einstein intended to unify GR with Electrodynamics. Another motivation to extend GR is published in [9, 10], where M. Born investigated on how to recover the symmetry between coordinates and momenta, which are symmetric in Quantum Mechanics but not in GR. To achieve his goal, he introduced also a complex metric, where the imaginary part is momentum dependent. In [11] this was more elaborated, leading to the square of the length element ()which implies maximal acceleration (see also [12]). The interesting feature is that a minimal length “” is introduced as a* parameter* and Lorentz symmetry is, thus, automatically maintained; no deformation to small lengths is necessary!

In [13] the GR was algebraically extended to a series of variables and the solutions for the limit of weak gravitational fields were investigated. As a conclusion, only real and pseudocomplex coordinates (called in [13]* hyper-complex*) make sense, because all others show either tachyon or ghost solutions, or both. Thus, even the complex solutions do not make sense. This was the reason to concentrate on the pseudocomplex extension.

In pc-GR, all the extended theories, mentioned in the last paragraph, are contained and the Einstein equations require an energy-momentum tensor, related to vacuum fluctuations (dark energy), described by an asymmetric ideal fluid [12]. Due to the lack of a microscopic theory, this dark energy is treated phenomenologically. One possibility is to choose it such that no event horizon appears or barely still exists. The reason to do so is that, in our philosophical understanding,* no theory should have a singularity*, even a coordinate singularity of the type of an event horizon encountered in a black hole. Though it is only a coordinate singularity, the existence of an event horizon implies that even a black hole in a nearby corner cannot be accessed by an outside observer. Its event horizon is a consequence of a strong gravitational field. Because no quantized theory of gravitation exists yet, we are led to the construction of models for the distribution of the dark energy.

In [14] the pc-GR was compared to the observation of the amplitude for the inspiral process. As found, the fall-off in of the dark energy has to be stronger than suggested in earlier publications. We will discuss this and what will change, in the main body of the text. We also will compare EHT observations with pc-GR, taking into account the low resolution of , as obtained by the EHT. The main question is if one can discriminate between GR and pc-GR.

A general principle emerges; namely, that* mass not only curves the space* (which leads to the standard GR)* but also changes the space- (vacuum-) structure in its vicinity*, which in turn leads to an important deviation from the classical solution.

The consequences will be discussed in Section 3. There, also the cosmological effects are discussed, with different outcomes for the evolution of the dark energy as function of time/radius of the universe. Another application treats the interior of a stars, where first attempts will be reported on how to stabilize a large mass. In Section 4 conclusions will be drawn.

#### 2. Pseudocomplex General Relativity (pc-GR)

An algebraic extension of GR consists in a mapping of the real coordinates to a different type, as, for example, complex or pseudocomplex (pc) variableswith and where is the standard coordinate in space-time and is the complex component. When it denotes* complex* variables, while when it denotes pseudocomplex (pc) variables. This algebraic mapping is just* one possibility* to explore extensions of GR.

In [13] all possible extensions of real coordinates in GR were considered. It was found that only the extension to pseudocomplex coordinates (called in [13]* hyper-complex*) makes sense, because all others lead to tachyon and/or ghost solutions, in the limit of weak gravitational fields.

In what follows, some properties of pseudocomplex variables are resumed, which is important to understand some of the consequences.(i)The variables can be expressed alternatively as(ii)The satisfy the relations(iii)Due to the last property in (4), when multiplying one variable proportional to by another one proportional to , the result is zero; i.e., there is a* zero-divisor*. The variables, therefore, do not form a* field* but a* ring*.(iv)In both zero-divisor components () the analysis is very similar to the standard complex analysis.

In pc-GR the metric is also pseudocomplexBecause in each zero-divisor component one can construct independently a GR theory.

For a consistent theory, both zero-divisor components have to be connected! One possibility is to define a modified variational principle, as done in [15]. Alternatively, one can implement a constraint; namely, a particle should always move along a real path; i.e., the pseudocomplex length element should be real.

The infinitesimal pc length element squared is given by (see also [16])as written in the zero-divisor components. In terms of the pseudoreal and pseudoimaginary components, we havewith and . The upper indices and refer to a* symmetric* and* antisymmetric* combination of the metrics. The case when , i.e., , leads to Identifying , where is an infinitesimal length and is the 4-velocity, one obtains the length element defined in [11]. It also contains the line element as proposed in [9, 10], where the is proportional to the momentum component of a particle. However, this identification of is only valid in a flat space, where the second term in (8) is just the scalar product of the 4-velocity () to the 4-acceleration ().

The connection between the two zero-divisor components is achieved, requiring that the infinitesimal length element squared in (7)* is real*; i.e., in terms of the components it is

Using the standard variational principle with a Lagrange multiplier, to account for the constraint, leads to an additional contribution in the Einstein equations, interpreted as an energy-momentum tensor.

The action of the pc-GR is given by [16]where is the Riemann scalar. The last term in the action integral allows to introduce the cosmological constant in cosmological models, where has to be constant in order not to violate the Lorentz symmetry. This changes when a system with a uniquely defined center is considered, which has spherical (Schwarzschild) or axial (Kerr) symmetry. In these cases, the is allowed to be a function in , for the Schwarzschild solution, and a function in and , for the Kerr solution.

The variation of the action with respect to the metric leads to the equations of motion in the zero-divisor component, denoted by the independent unit-elements . These equations still contain the effects of a minimal length parameter , as shown in [11]. Because the effects of a minimal length scale are difficult to measure, maybe not possible at all, we neglect them, which corresponds to mapping the above equations to their real part, giving

The , is real and is now given by [16]where and radial are the tangential and pressure, respectively. For an isotropic fluid we have = =. The are the components of the 4-velocity of the elements of the fluid and is a space-like vector () in the radial direction. It satisfies the relation . The fluid is anisotropic due to the presence of . The and are related to the pressures as [16]

The reason why the dark energy outside a mass distribution has to be an anisotropic fluid is understood contemplating the* Tolman-Oppenheimer-Volkov* (TOV) equations [17] for an isotropic fluid: The TOV equations relate the derivative of the dark-energy pressure with respect to (for an isotropic fluid, the tangential pressure has to be the same as the radial pressure, i.e., ) to the dark energy density . Assuming the isotropic fluid and that the equation of state for the dark energy is , the factor in the TOV equation for is zero; i.e., the pressure derivative is zero. As a result the pressure is constant and with the equation of state also the density is constant, which leads to a contradiction. Thus, the fluid has to be anisotropic, due to an additional term, allowing the pressure to fall off as a function on increasing distance. The additional term in the radial pressure , added to the TOV equation, is given by [18].

For the density one has to apply a phenomenological model, due to the lack of a quantized theory of gravity. What helps is to recall one-loop calculations in gravity [19], where vacuum fluctuations result due to the non-zero back ground curvature (Casimir effect). Results are presented in [20], where at large distances the density falls off approximately as . The semiclassical Quantum Mechanics [19] was applied, which assumes a* fixed* back-ground metric and is thus only valid for weak gravitational fields (weak compared to the solar system). Near the Schwarzschild radius the field is very strong which is exhibited by a singularity in the energy density, which is proportional to , with a constant mass parameter [20].

Because we treat the vacuum fluctuations as a classical ideal anisotropic fluid, we are free to propose a different fall-off of the negative energy density, which is finite at the Schwarzschild radius. In earlier publications the density did fall-off proportional to . However, in [14] it is shown that this fall-off has to be stronger. Thus, in this contribution we will also discuss a variety of fall-offs as a function of a parameter , i.e., proportional to , where describes the coupling of the dark energy to the mass.

With the assumed density, the metric for the Kerr solution changes to [21, 22]where is the spin parameter of the Kerr solution and = 3, 4, For the old ansatz is achieved. The Schwarzschild solution is obtained, setting . The parameter measures the coupling of the dark energy to the central mass. The definition of here is related to the in [14] by .

When no event horizon is demanded, the parameter has a lower limit given byFor the equal sign, an event horizon is located ate.g., for and for .

#### 3. Applications

##### 3.1. Motion of a Particle in a Circular Orbit

In [23] the motion of a particle in a circular orbit was investigated. This section was first discussed in [24, 25].

The main results are resumed in Figures 1 and 2. In Figure 1 the orbital frequency, in units of , is depicted versus the radial distance , in units of , for a rotational parameter of . The function for the orbital frequency, in prograde orbits, is given by