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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.

The Theory of General Relativity (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 [

A first proposal to extend GR was attempted by A. Einstein [

In [

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 [

In [

A general principle emerges; namely, that

The consequences will be discussed in Section

An algebraic extension of GR consists in a mapping of the real coordinates to a different type, as, for example, complex or pseudocomplex (pc) variables

In [

In what follows, some properties of pseudocomplex variables are resumed, which is important to understand some of the consequences.

The variables can be expressed alternatively as

The

Due to the last property in (

In both zero-divisor components (

In pc-GR the metric is also pseudocomplex

For a consistent theory, both zero-divisor components have to be connected! One possibility is to define a modified variational principle, as done in [

The infinitesimal pc length element squared is given by (see also [

The connection between the two zero-divisor components is achieved, requiring that the infinitesimal length element squared in (

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 [

The variation of the action with respect to the metric

The

The reason why the dark energy outside a mass distribution has to be an anisotropic fluid is understood contemplating the

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 [

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

With the assumed density, the metric for the Kerr solution changes to [

When no event horizon is demanded, the parameter

In [

The main results are resumed in Figures

The orbital frequency of a particle in a circular orbit for the case GR (upper curve) and for

The position of the

The upper curve in Figure

As one can see, the difference between

In Figure

In order to connect to actual observations [

A thin, infinitely extended accretion disc. This is a simplifying assumption. A real accretion disc can be a torus. Nevertheless, the structure in the emission profile will be similar, as discussed here. These discs are easier to calculate.

An energy-momentum tensor is proposed which includes all main ingredients, as mass and electromagnetic contributions.

Conservation laws (energy, angular momentum, and mass) are imposed in order to obtain the flux function, the main result of [

The internal energy of the disc is liberated via shears of neighboring orbitals and distributed from orbitals of higher frequency to those of lower frequency.

How to deduce finally the flux is described in detail in [

In order to understand within pc-GR the structure of the emission profile in the accretion disc, we have to get back to the discussion in the last subsection. The local heating of the accretion disc is determined by the gradient of orbital frequency, when going further inward (or outward). At the maximum, neighboring orbitals have nearly the same orbital frequency; thus, friction is low. On the other hand, above and especially below the position of the maximum the change in orbital frequency is large and the disc gets heated. At the maximum the heating is minimal which will be noticeable by a dark ring. Further inside, the heating increases again and a bright ring is produced.

The above consideration is relevant for

Some simulations are presented in Figure

Infinite, counterclockwise rotating geometrically thin accretion disc around static and rotating compact objects viewed from an inclination of

As a global feature, the accretion disc in pc-GR appears brighter, which is due to the fact that the disc reaches further inside where the potential is deeper, thus releasing more gravitational energy, which is then distributed within the disc.

The reason for the dark fringe and bright ring was explained above due to the variability of the friction. The dark ring is the position of the maximum of the orbital frequency. An observed position of a dark ring can, thus, be used to determine

The differences in the structure of an accretion disc give us clear observational criteria to distinguish between GR and pc-GR. There are still others, maybe more realistic disc models, e.g., a thick disk as described in [

Finally, in Figure

Infinite, counterclockwise rotating geometrically thin accretion disc around static and rotating compact objects viewed from an inclination of

In [

Using GR and the mass-point approximation for the two black holes, before the merging, a relation is obtained between the observed frequency and its temporal change to the chirping mass

The main result is that the source in pc-GR corresponds to two black holes with several thousand solar masses. This may be related to the merger of two primordial galaxies whose central black hole subsequently merges. One way to distinguish the two predictions is to look for light events very far way. If, for observed gravitational wave events in future, there is a consistent appearance of light events much farther away as the distance deduced from GR, then this might be in favor for pc-GR. However, all the prediction depends on the assumption that the point mass approximation is still more or less valid when the two black holes are near together, which is not very good! In [

In [

Axial gravitational modes in pc-GR. The vertical axis gives the real part of

Another problem is to distinguish between GR and pc-GR. It depends very much on the observation of the ring-down frequency of the merger [

The pc-Robertson-Walker model is presented in detail in [

The line element in gaussian coordinates has the form

The corresponding Einstein equations were solved and an equation for the radius

Two particular solutions are shown in Figure

The left panel shows a case where the acceleration of the universe approaches a constant value and the right panel shows a case where the acceleration slowly approaches zero for

These results are not very predictive, because one can obtain several possible outcomes, depending on the values of

For the description of the interior of a star one needs the equation of state of matter and the coupling of the dark-energy with the matter. For the equation of state one can use the model presented in [

In [

A particular result is shown in Figure

The figure shows the dependence of the mass of the star as a function of its radius (figure taken from [

In [

Dark energy density as a function of

Finally, in Figure

The mass of a star as a function on its radius

This model also suffers from the approximations made and a complete description cannot be given. Nevertheless, now stars with up to 200 solar masses can be stabilized, which shows that the inclusion of dark-energy in massive stars may lead to stable stars of any mass! (Though, only within a phenomenological model.)

A report on the recent advances of the pseudocomplex General Relativity (pc-GR) was presented. The theory predicts a nonzero energy-momentum tensor on the right hand side of the Einstein equation. The new contribution is related to vacuum fluctuations, but due to a missing quantized theory of gravitation one recurs to a phenomenological ansatz. Calculations in one-loop order, with a constant back-ground metric, show that the dark energy density has to increase toward smaller

Consequences of the theory were presented: (i) the appearance of a dark ring followed by a bright one in accretion discs around black holes, (ii) a new interpretation of the source of the first gravitational event observed, (iii) possible outcomes of the future evolving universe, and (iv) attempts to stabilize stars with large masses.

The only robust prediction is the structure in the emission profile of an accretion disc.

The author declares that they have no conflicts of interest.

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