Pseudo-complex General Relativity Research Articles

Cover Picture: Astron. Nachr. 01/2023

Synthetic image of the accretion disk around the black hole at the center of the galaxy M87. The bright emission corresponds to the part of the disk approaching us. The simulation is based on the so-called pseudo-complex General Relativity (pcGR), an extension of the standard theory of General Relativity (GR) constructed to avoid singularities. At a spatial resolution of 20 μ arcsec (left image), comparable to the resolution of the Event Horizon Telescope (EHT) of 24 μ arcsec, the predictions of GR and pcGR are indistinguishable. Only at even higher spatial resolution, better than 5 μ arcsec (right image), the pcGR prediction of a dark ring followed further in by a bright ring could be revealed. For further details see the article by Maghlaoui and Hess, page e220067 in this issue.

Comparison of the predictions of the pseudocomplex general relativity to the observations of the Event Horizon Telescope collaboration

AbstractThe observation predictions of the pseudocomplex General Relativity, related to the structure of an accretion disk, are compared to the reported observations of the Event Horizon Telescope.

Review on the Pseudocomplex General Relativity and Dark Energy

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.

Predictions of the pseudo-complex theory of Gravity for EHT observations – II: theory and predictions

Abstract We present a resumé on the modified theory of gravity, called pseudo-complex General Relativity (pc-GR). It is the second in a series of papers, where the first one (Boller et al. 2019; referred to as Paper I) discussed the observational consequences of pc-GR. In this paper, we concentrate on the underlying theory. PC-GR involves an algebraic extension of the standard theory of GR and it depends on two phenomenological parameters. An element included in pc-GR that is not present in standard GR is the energy–momentum tensor corresponding to an anisotropic ideal fluid, which we call dark energy. The two parameters are related to the coupling of mass to the dark energy and its fall-off as a function of r. The consequences and predictions of this theory will be discussed in the context of the observational results of the Even Horizon Telescope, expected soon. Our main result is that due to the accumulation of dark energy near a large mass, the modified theory predicts a dark ring followed by a bright ring in the emission profile of the accretion disc. We also discuss the light ring in the equatorial plane.

Predictions of the pseudo-complex theory of gravity for EHT observations – I. Observational tests

Abstract A modified theory of gravity, avoiding singularities in the standard theory of gravitation, has been developed by Hess & Greiner, known as the pseudo-complex theory of gravitation. The pseudo-complex General Relativity (pc-GR) theory shows remarkable observational differences with respect to standard GR. The intensity profiles are significantly different between both theories, which is a rare phenomenon in astrophysics. This will allow robust tests of both theories using Event Horizon Telescope observations of the Galactic Centre. We also predict the time evolution of orbiting matter. In this paper we summarize the observational tests we have developed to date. The theory is described in the second paper of this series (referred to as paper II hereafter).

Compact stars in the pseudo-complex general relativity

The field equations of the pseudo-complex General Relativity (pc-GR) have an extra term, of repulsive character, which may halt the gravitational attractive collapse of matter distributions in the evolution process of compact stars. This additional extra term simulates the presence of dark energy in the Universe. On the first part of this contribution we consider the effect of this additional term of pc-GR on the structure of a white dwarf. Observations of type Ia supernova (SNe Ia) admit white dwarfs with masses as high as 2.3M⊙ − 2.6M⊙, surpassing this way the mass limit established by Chandrasekhar (1.44M⊙). In a more conventional theoretical description of a white dwarf, the electron degeneracy pressure ultimately stabilizes the star against gravitational collapse and establishes this way the Chandrasekhar maximum limit of the stelar mass; this mechanism can not explain however values of masses more expressive like those mentioned above. We investigate here a possible mechanism for carrying out such a hypothesis: the combination of the electron degeneracy and dark energy internal pressures. In this study we use a very simple model for the white dwarf mater which consists of nucleons and degenerate electrons, held together by the presence of the gravitational interaction and superimposed to a repulsive background of dark energy. In the pc-GR formalism, the corresponding modified Tolman-Oppenheimer-Volkoff (TOV) equations are solved and the mass-radius relations as well as the maximum mass of the white dwarf star are determined for different parameter configurations. In the second part of this contribution we explore the presence of this additional term and study the role of dark energy in the structure of neutron stars composed by nucleons, hyperons, mesons, and weakly interacting massive fermion dark matter particles (WIMPs) held together by the presence of the nuclear force and the gravitational interaction, superimposed to the repulsive background of dark energy. To describe the hadron-lepton sector we consider three different effective nuclear models, Zimanyi-Moszkowski, Boguta-Bodmer, and the analytic parametrized coupling model, which we extend to consider, in the baryonic sector, the presence of the whole fundamental baryon octet. By solving the corresponding pc-GR Tolman-Oppenheimer-Volkoff (TOV) equations we estimate the maximum gravitational mass of neutron stars.

Pseudo-Complex General Relativity

The present status of the pseudo-complex General Relativity is presented. The pcGR includes many known theories with a minimal length. Restricting to its simplest form, an energy-momentum tensor is added at the right hand side of the Einstein equations, representing a dark energy, related to vacuum fluctuations. We use a phenomenological ansatz for the density and discuss observable consequences: Quaisperiodic Oscillations (QPO), effects on accretion disks and gravitational waves.

Mimetic Dark Matter in Pseudo-Complex General Relativity

We investigate the dark matter problem in the context of Pseudo-complex General Relativity. A form of gravitational dark matter has recently been studied, the mimetic dark matter, which is a scalar tensor extension for gravity where the conformal degree of freedom is isolated in a covariant way. For such, we perform a combination of both approaches to reveal non trivial results even in the absence of matter. Solutions for different scenarios and possible interpretations are presented.

Neutron stars with dark energy

After a short review on the possible experimental observations to verify pseudocomplex General Relativity, neutron stars as a particular object of interest are investigated. Dark energy is added to the structure of a neutron star, while for the nuclear part the chiral SU(3) model is used. For the coupling of matter to dark energy a special assumption is made. The consequences are discussed. We show that neutron stars of up to six solar masses are obtained, which already behave similar to a black hole.

Neutron stars within pseudo-complex general relativity

The properties of neutron stars within the theory of pseudo-complex general relativity are studied. The pc-Tolman–Oppenheimer–Volkoff equations are numerically integrated in order to understand the structure of these objects. The interior Λ-term energy density has been linearly coupled to the respective baryonic quantity εm. Two models for the exterior region have been analyzed attending the continuity of the energy density at the boundary. Energy conditions have been studied for both regions. Solutions have been presented for different values of the coupling parameter. It is shown that accumulation of the Λ-term component allows the theoretical existence of larger and more massive neutron stars.

Ray-tracing in pseudo-complex General Relativity

Motivated by possible observations of the black hole candidate in the centre of our Galaxy and the galaxy M87, ray-tracing methods are applied to both standard General Relativity (GR) and a recently proposed extension, the pseudo-complex GR (pc-GR). The correction terms due to the investigated pc-GR model lead to slower orbital motions close to massive objects. Also the concept of an innermost stable circular orbit is modified for the pc-GR model, allowing particles to get closer to the central object for most values of the spin parameter a than in GR. Thus, the accretion disc, surrounding a massive object, is brighter in pc-GR than in GR. Iron Kα emission-line profiles are also calculated as those are good observables for regions of strong gravity. Differences between the two theories are pointed out.

Experimental tests of pseudo-complex General Relativity

Based on previous publications exploring pseudo-complex General Relativity (pc-GR) we present a selection of observable consequences of pc-GR and possible ways to experimentally access them. Whenever possible we compare the results to Einstein's GR and differences are worked out in detail. We propose experimental tests to check the predictions of pc-GR for the orbital frequency of test particles, the gravitational redshift effect and the last stable orbit. We will show that the orbital frequency of test particles at a given radius in pc-GR is in general lower compared to standard GR. Also the effect of frame dragging is modified (weakened) in pc-GR. Concerning the gravitational redshift of a radiation emitting object we find that it is also lower in pc-GR than in standard GR. Eventually the classical concept of a last stable orbit has to be modified in pc-GR.

PSEUDO-COMPLEX GENERAL RELATIVITY: SCHWARZSCHILD, REISSNER–NORDSTRÖM AND KERR SOLUTIONS

The pseudo-complex General Relativity (pc-GR) is further considered. A new projection method is proposed. It is shown that the pc-GR introduces automatically terms into the system which can be interpreted as dark energy. The modified pseudo-complex Schwarzschild solution is investigated. The dark energy part is treated as a liquid and possible solutions are discussed. As a consequence, the collapse of a large stellar mass into a singularity at r = 0 is avoided and no event-horizon is formed. Thus, black holes do not exist. The resulting object can be viewed as a gray star. It contains no singularity which emphasizes, again, that it is not a black hole. The corrections implied by a charged large mass object (Reissner–Nordström) and a rotating gray star (Kerr) are presented. For the latter, a special solution is presented. Finally, we will consider the orbital speed of a mass in a circular orbit and suggest a possible experimental verification.

THE ROBERTSON–WALKER METRIC IN A PSEUDO-COMPLEX GENERAL RELATIVITY

We investigate the consequences of the pseudo-complex General Relativity within a pseudo-complexified Robertson–Walker metric. A contribution to the energy–momentum tensor arises, which corresponds to a dark energy and may change with the radius of the universe, i.e., time. Only when the Hubble function H does not change in time, the solution is consistent with a constant Λ.