Predictions of the pseudo-complex theory of Gravity for EHT observations- II. Theory and predictions

TL;DR

This paper explores pseudo-complex General Relativity, a modified gravity theory with two parameters, predicting distinctive dark and bright rings in black hole accretion disk images, relevant for EHT observations.

Contribution

It introduces the theoretical framework of pseudo-complex GR with novel parameters and predicts observable features in black hole imaging.

Findings

Predicts a dark ring followed by a bright ring in accretion disk emission profiles.

Suggests dark energy accumulation affects light rings around black holes.

Provides theoretical predictions for EHT observations based on modified gravity.

Abstract

We present a resum\'e 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.

Figures

Equations

$$d{\omega }^2$$

$$g_{00}^{\mathrm{K}}$$

$$g_{11}^{\mathrm{K}}$$

$$g_{22}^{\mathrm{K}}$$

$$g_{33}^{\mathrm{K}}$$

$$g_{03}^{\mathrm{K}}$$

$$B_n$$

$$r_0=\frac{2(n-1)}{n}m,$$

$${\omega }_n$$

$$h_3(r)$$

$$r_{{\omega }_{\max }}$$

Full text

††pubyear: 2019††pagerange: Predictions of the pseudo-complex theory of Gravity for EHT observations- II. Theory and predictions–Predictions of the pseudo-complex theory of Gravity for EHT observations- II. Theory and predictions

1 Introduction

In our first contribution (Boller, 2019, hereafter Paper I) we presented observational predictions of a modified theory of Gravity. In the present contribution, the theory is explained in greater detail.

General Relativity (GR) has passed many observational tests Will (2006) in weak gravitational fields. The first observation of gravitational waves Abbott (2016) can be considered as a first test in the strong gravitational field. Recently bright spots near the black hole in $SrgA^{*}$ were observed GRAVITY (2018) and simulations, assuming that GR is still valid, indicate that they appear close to the Innermost Stable Circular Orbit (ISCO), at six times the gravitational radius. The Event Horizon Telescope EHT (2016) collaboration are expected to publish their results on the observation of SgrA∗ and the black hole at the center of M87 in 2019. In Giddings (2018) the oscillations of a black hole are investigated as a consequence of quantum fluctuations. Distortions of the black hole shadow and the shape of a light ring are discussed. They predict that a bright light ring should be present near the event horizon. In Schönenbach (2013, 2014), based on Hess (2009), it is assumed that any mass provokes vacuum fluctuations nearby, as explained in the book by Birrell (1982). In Schönenbach (2013, 2014) the vacuum fluctuations are treated in a phenomenological way, assuming a particular increase with the mass, such that it is finite at the Schwarzschild radius. The vacuum fluctuations are modeled by two parameters, one which describes the coupling of mass with the dark-energy and the second one on how the dark-energy density falls-off as a function in $r$. Up to now, we assumed the simplest fall-off behavior such that it does not yet contradict solar system experiment. Recently in Nielsen (2018) the fall-off behavior was restricted with the result that a steeper slope has to be used in order not to contradict gravitational wave observations.

The consequences for the light emission of accretions discs were computed in Schönenbach (2014). The main result is that a dark ring exists in addition to a bright inner one in the emission profile of the accretion disc. It is a consequence of the dependence of the angular velocity of point particles in a circular orbit on the distance and will be explained in the next section. This feature is a consequence of vacuum fluctuations, but not of oscillations of a black hole, as proposed in Giddings (2018). The theory in Schönenbach (2013, 2014) is static. Also the recent observation of bright spots (GRAVITY, 2018) can be reinterpreted such that the circular orbit is near $\frac{4}{3}$ of the gravitational radius ($m$), for the lowest fall-off and a bit further away for stronger fall-offs. The theory is phenomenological, it treats the quantum fluctuations as an asymmetric perfect fluid, and up to now involved only a parameter describing the coupling of matter to the vacuum fluctuations, also called dark-energy. In this contribution we will discuss also stronger fall-offs of the dark energy and show how the structure obtained will change. Though it is phenomenological, it may shed some light on the effects of quantization on the standard theory. At the end, we will discuss the existence of a light ring in the equatorial plane as a function in $n$, a parameter related to the fall-off of the dark energy and which will be introduced in the next section.

Finally, we point out a different path to extend the standard theory, using canonical transformations Struckmeier (2017). The Einstein equation has a contribution which represents the known form but also has additional terms which might represent the contribution of the energy-momentum tensor in pc-GR.

The paper is organized as follows: In Section 2 the main ingredients of the extended theory are presented, As the point of partition, we present results for the fall-off of the dark energy as done in past publications and show how these structures change for varying the $r$-dependence of the dark energy. The implications for particles in a circular orbit are discussed as well as the accretion disc emission profiles. After that, a simulation of a gas clouds falling into the black hole is presented. For any emitting material a dark ring followed by a bright one will be one of the main observable results, which will be presented in section 3 for SgrA∗. The properties of the emission profile of the accretion disc as predicted by pc-GR may be visible in EHT observations. In this case, the pc-GR theory can be tested directly via the observational tests discussed in Paper I.

2 The theory, consequences and predictions

Historically, there have been several attempts to extend GR. To our knowledge, Einstein (1945, 1948) was the first to extend algebraically the real coordinates in the 4-dimensional space to complex coordinates, in order to unify Electromagnetism with GR. Born (1948, 1949) had different reasons: He could not accept that in GR the coordinates play a dominant role, while in Quantum Mechanics coordinates and momenta are treated on an equal footing. This lead to the proposal to add to the length element squared a term which depends on the momentum of a particle. More recently, the complex expanded GR was discussed by Mantz (2008, 2011). However, as shown in Kelly (1986), only algebraic extensions to the pseudo-complex coordinates (referred to in Kelly (1986) as hyper-complex) make sense, because all others have tachyon or ghost solutions. This was the reason why we developed pc-GR.

In Hess (2009) the pc-GR was introduced, which is briefly described in this following section, for more details please see for example Hess (2015): The theory extends the real space-time components to the so-called pseudo-complex ones, namely $x^{\mu}$ $\rightarrow$ $X^{\mu}=x^{\mu}+Iy^{\mu}$, with $I^{2}=1$. Any function (or set of coordinates) can be expressed in terms of the so-called zero-divisor basis $\sigma_{\pm}=\frac{1}{2}\left(1\pm I\right)$, i.e., $F(X)=F_{+}(X_{+})\sigma_{+}+F_{-}(X_{-})\sigma_{-}$ and $X^{\mu}=X_{+}\sigma_{+}+X_{-}\sigma_{-}$. Because $\sigma_{-}\sigma_{+}=0$, the two zero-divisor components commute. One obtains the Einstein equations for each component, i.e. $G_{\mu\nu}^{\pm}-\frac{1}{2}g_{\mu\nu}^{\pm}R^{\pm}$ = $8\pi T_{\mu\nu}^{\pm}$ ($c=G=1$). In each component the theory appears the standard structure of a theory of General Relativity. All principles and symmetries remain satisfied. The connection is obtained requiring that the orbit of a particle is real ($d\omega^{2}$ is real), which introduces a constraint whose solution (Hess, 2017) requires an energy-momentum tensor of an an-isotropic ideal fluid. In the original version of the pc-GR (Hess, 2009), a modified variation principle was proposed, which is not necessary when the constraint is taken into account. The theory contains effects of a minimal length scale (maximal acceleration). This can be observed when one writes the pc-length element squared, using the same functional form of the metric in each zero-divisor component ($g_{\mu\nu}(X_{+})$ in the $\sigma_{+}$-component and $g_{\mu\nu}(X_{-})$ in the $\sigma_{-}$-component), i.e.,

[TABLE]

where a symmetric metric was assumed. This length element has a strong resemblance to the one proposed in Caianiello (1981), when we set $y^{\nu}=ldu^{\nu}$ ($c=1$), with the 4-velocity $u^{\nu}$. When this expression for $y^{\nu}$ is substituted into the pseudo-imaginary component of (1) and $dx^{\mu}$ is substituted by $\frac{dx^{\mu}}{d\tau}=u^{\mu}$ ($\tau$ as the Eigen-time), then the pseudo imaginary part is just $2lg_{\mu\nu}u^{\mu}du^{\nu}$. Demanding that this is zero results in the dispersion relation. However, the identification of $y^{\nu}$ to the 4-velocity is strictly speaking only valid in a flat space. This is the reason why in pc-GR the coordinate $y^{\nu}$ is unaltered.

The minimal length parameter $l$ is not noticeable in present day observations and this is why its contributions, encoded in $y^{\mu}$, were neglected in subsequent treatments, which is equivalent to project the Einstein equations to their real part. The final equations have the same structure as the standard equations, save that on the right hand side there must appear an energy-momentum tensor, which describes an an-isotropic ideal fluid, as mentioned above. The pc-GR cannot determine how this fluid has to behave near a central mass and we are dependent on a phenomenological treatment.

The appearance of a non-zero energy-momentum tensor is a consequence of the pc-GR formulations and implies the conjecture that the mass not only curves space-time but also changes vacuum properties. Therefore, in pc-GR the effects of the vacuum fluctuations are associated with the the energy-momentum tensor $T_{\mu\nu}$. In fact, it is shown in semi-classical calculations (Birrell, 1982; Visser, 1996). that the vacuum fluctuations, with a static metric, behave proportional to $1/\left[r^{6}\left(1-\frac{2m}{r}\right)^{2}\right]$, which has an infinity at the Schwarzschild radius. The vacuum fluctuations increase with the strength of the gravitational field and problems arise when it is large, because vacuum fluctuations are represented by an energy density, which in turn changes the metric. This has to be taken into account. Because the calculations do not include such back-reaction effects of the vacuum fluctuations to the metric, the behavior near the horizon is not physical and requires a modification. Due to the lack of a functional quantized theory of gravity we are left to treat dark-energy in a phenomenological way, assuming for it a non-isotropic perfect and classical fluid (Schönenbach, 2014).

The theory contains two phenomenological parameters, $B_{n}$ and $n$: The coupling of the dark-energy to the mass ($B_{n}=bm^{n}$) (c.f. Eq. 3) and its fall-off toward a greater distance, i.e., $\sim 1/r^{n}$. This is the simplest Ansatz and one easily can add further complicated dependencies in $r$. Up to now, the dark energy density was modeled by a $\sim\frac{B}{r^{5}}$ dependence, which falls off strong enough in order not to be detected yet by solar system observations.

In Nielsen (2018), the pc-GR was tested on the results of the first observed gravitational wave event Abbott (2016). The corresponding data were not accessible before and are important to restrict the range of the parameters in pc-GR. Looking at the end of the amplitude evolution of the inspiral event, the pc-GR could not describe it well. Therefore, the used Ansatz for the $r$-dependence has to be modified. A direct new Ansatz is to increase the power of the fall-off, but even other dependencies may be used. Hence, pc-GR is not excluded! Therefore, we do study a number of different r-dependencies, as done in Nielsen (2018).

The evolution of certain structures, as a function of a parameter $n$, will be explained starting from a particular $n=3$, corresponding to the former ansatz. The main point is that the structures will be identical for all $n$, though numerical numbers will change. This is what we describe as robust. An important objective is to show, what will happen when dark-energy is accumulated around a large mass and indicate the signatures.

The parameter $B_{n}$ is chosen such that the metric component $g_{00}$ does not become zero, thus no event horizon appears in this theory. The reasoning for this choice is twofold: i) At the Schwarzschild radius the gravitational field is extremely strong, compared to the field in the solar system, and one has to count on the possibility that GR has to be extended in order to include further effects (like quantum effects). ii) The second reason is of philosophical nature and depends on the eye of the beholder, namely that the event horizon in GR excludes part of the space to be accessible to a nearby observer in the exterior and has several, for us, undesirable consequences like the information paradox. Both observations indicate that maybe GR reaches its limits or rather that something is missing. Therefore, our motivation to investigate what happens when there is no event-horizon for a so-called black hole.

Recent presentations of pc-GR are given in Schönenbach (2014); Hess (2017) who apply pc-GR to thin, optically thick accretion discs. Though a thin optical disc is not always realized (except perhaps around SgrA∗), the effects discussed in this contribution apply equally to other disc models, which is also important for the black hole in the center of M87. The modified Kerr metric is given by Schönenbach (2014), Schönenbach (2014c).

[TABLE]

where the index K refers to Kerr (in Schönenbach (2012); Hess (2015) the $B_{n}$ is divided by an extra factor 2, which should not be there). For $B=0$ the Kerr solution of GR is obtained. The $n$ is the above mentioned parameter, which is related to the one in Nielsen (2018) $n_{\rm N}=n-1$. There is no event horizon for

[TABLE]

which for $n=3$ is $\frac{64}{27}m^{3}$ and for $n=4$, the next possible value, it is $\frac{81}{8}m^{4}$. For the equal sign in (3) there would be a horizon at the position

[TABLE]

which for $n=3$ acquires the value $r_{0}=\frac{4}{3}$ m and for $n=4$ it is $r_{0}=\frac{3}{2}$  m. The rotational parameter $a$ is in units of $m$ and ranges as usual from [math] to $1$ m. Often the rotational parameter in observations is redefined without units.

The orbital frequency for a particle in a circular orbit and for $n=3$ was given in Schönenbach (2013); Hess (2015). For $n$ arbitrary and for prograde orbitals, this is

[TABLE]

For $B_{n}$ given by the right-hand expression in (3), the $h_{n}(r)$ acquire the form

[TABLE]

In Fig. 1 the orbital frequency for $n=3$ and $n=4$, for $a$=0.995, is compared to GR.

One distinct feature of the modified theory is that the orbital frequency is always lower than in GR, though noticeable differences only appear near the Schwarzschild radius (Schönenbach (2014). Furthermore, the orbital frequency shows for $B_{\rm max}$ a maximum at

[TABLE]

which is independent of the value of $a$, after which it falls off toward the center and reaches zero at $r_{0}$ (Eq. (4)). For $n=3$ the value is approximately $r$=1.72  m and for $n=4$ it is $r$=1.89 m.

Taking these numbers one notes that the zero of the orbital frequency and the position of its maximum move further out with increasing $n$. The fall off of the orbital frequency towards smaller $r$ is explained by the increase of the dark energy, which effectively reduces the gravitational constant (it is a function in $r$, see also Hess (2015)). In Schönenbach (2014) it is shown that for $\rm a\lesssim 0.4~{}m$ ($n=3$) the last stable orbit is at a lower value of $r$ but follows approximately the behavior of GR. Because particles can now reach further inside, more gravitational energy is released, and as a consequence the disc appears to be brighter. Above $a=0.4~{}m$ stable orbits are possible up to the surface. Circular stable orbits are possible up to the maximum of the orbital frequency. Because at the maximum the gradient in the orbital frequency of neighboring orbits is small, less friction is present, resulting in less heating. Thus, a dark ring appears near the position $r_{\omega_{\rm max}}$, which is 1.72 for $n=3$. Further inside, the gradient becomes significantly stronger and a bright ring occurs (see Paper I).

What will change as a function in $n$ is the position of the dark and brighter inner ring. For small $a$ the pc-GR curve approaches the ISCO of GR and the point from which on there are stable orbits in pc-GR up to the surface approaches larger values. The main point is that the signatures of the dark and bright ring will not change. Note, that these features are independent on the detailed structures of the disc model. If a dark ring is observed, its position can be used to determine the optimal value of $n$ using Eq. 7. We use the model proposed by Page (1974) for the accretion disc. It assumes a thin disc which is optically thick at the equator and the loss of energy is only through the emission of light. However, we acknowledge that other accretion disc models may also be valid. The model exploits the laws of conservation of mass, energy, and angular momentum. Properties such as the viscosity must be implicitly included in the parameters of the model (for example the accretion rate). This has to be extracted from somewhere else.

Though the model was applied exclusively to GR up to now, it is easy to extend it to other metrics, such as the one used in pc-GR. This is because the procedure discussed in Page (1974) is formulated in such a way that it is independent of the kind of metric that is used. For the simulation we use the GYOTO routine Vincent (2011), which applies the ray-tracing method and allows the use of several models, including that used by Page and Thorne, permitting an arbitrary metric. In Vincent (2011) the theoretical background of the ray tracing technique is clearly described.

The basic concepts are that light-rays are incident on the detector. which is divided into pixels. The light rays are then followed back to the accretion disc, such that only rays coming from there are taken into account. In order to construct the paths of the photons a Hamilton-Jacobi method is applied and the so-called Carter constant is constructed, which is a conserved quantity, which has to be checked at various points along the path in order to minimize numerical errors. These equations also involve the metric and, thus, the details of the simulations will depend on the metric. The path of a photon depends on the metric and the intensity profile also depends on the radius of the last stable orbit, which also depends indirectly on the metric. The smaller the radius of the last stable orbit, the more energy can be released.

It is also possible to change the subroutines in GYOTO to be consistent with the model of Page (1974). Such an extension for pc-GR is available in Schönenbach (2014b). When the model of Page (1974) is not applicable and other models have to be used, the differences to GR, nevertheless, will be similar in structure because the effects only depend on the orbital frequency as a function in $r$.

Under the hypothesis that, for $n=3$ the, $B=\frac{64}{27}m^{3}$, simulations of accretion discs have been performed in Schönenbach (2014); Hess (2015). However, in these works, the total flux was calculated in bolometric units, the mass was normalized to the mass of the black hole in the Galactic Center, and the accretion rate was set to an arbitrary value. The EHT on the other hand observes well defined objects at a fixed wavelength of approximately 1.2 mm which corresponds to a frequency of 250 GHz EHT (2016). Also the field of view was chosen arbitrarily and the temperature of the disc near the inner edge has not been evaluated.

The results of the simulation depends on several well determined parameters, such as the accretion rate, the rotational parameter $a$, the field of view, and the mass of the black hole. The specific uncertainties for these parameters are calculated in relative units. Nevertheless, the important part is the relative comparison to GR. In pc-GR the intensities are much larger than in GR (see Paper I). We have performed simulations from $10^{o}$ to $80^{o}$ in steps of ten degrees. One such simulation is presented in Paper I, Figure 1. For the remainder of this section, we discuss the position of the light ring, in the equatorial plane) as a function of $n$. The position of the light ring is obtained demanding that the path of a photon (infinitesimal length element squared put to zero) is also geodesic Schönenbach (2012); Nielsen (2018). This corresponds to demand that two frequencies, one for the circular orbital geodesic orbit and for the path of the photon in a circular orbit, have to be set equal. We used the Eqs. (7) and (14) in Schönenbach (2013), with now $n$-dependent metric coefficients, and Eqs. (7) and (9) in Nielsen (2018) and compared the results for consistency. For $B_{n}$ we used the value given in (3). The properties we found are as follows: i) From a minimal value, $a_{\rm min}>0.8~{}m$ on until $a=1~{}m$, the condition for the existence of the light ring provides no real positive solution for $r$, i.e, no light ring is possible ii) For $n=3$ there is a light ring up to approximately $a=0.95$ m, starting at approximately $r=$ 2.93 m at $a=0$ and tending to lower $r$ values for increasing $a$. For larger $a$ the point up to which a light ring exists is lowered systematically, reaching for $n=8$ the value approximately 0.77 m. Whether this is relevant for SgrA∗ cannot be concluded because the lower limit for $a$ in SgrA∗ at 0.5 m Genzel (2003).

3 SgrA∗

The main parameters for Sgr A∗ are its mass and the accretion rate. The mass of the black hole is known to be about 4.3 million solar masses (Gillessen, 2009) and an upper limit of the accretion rate is estimated in Quataert (1999) with the value of ${\rm 8}\times 10^{-5}$ solar masses per year. Under the supposition that an accretion disc forms around SgrA∗, the same value for the accretion rate will be used. For SgrA∗ there is no proof for the existence of an accretion disc. In the case where no accretion disc exists, when a cloud approaches the black hole it will pass through regions where an accretion disc would show different intensities. We expect that the changes in intensity will be qualitatively reflected when the cloud approaches the black hole. We assume a spatial resolution of 0.1 $\mu{\rm as}$ over a field of 1000 pixels, resulting into a field of view of 100 $\mu{\rm as}$. The simulations provide figures for the specific intensity at the fixed frequency mentioned above. The unit for the specific intensity is ${\rm erg}~{}{\rm cm}^{-2}{\rm Hz}^{-1}{\rm sr}^{-1}$ (GYOTO manual 2015). The version of the ray tracing routine we used, provided by Vincent (2011), developed numerical instabilities for very large distances. Because the specific intensity is independent of the distance, the simulations were performed at a distance of 1000$m$. The field of view was correspondingly adjusted, which for SgrA$~{}*$ at a distance of 1000 $m$ is about 0.022 rad. Intensive ray-tracing simulations for the black hole in the Galactic Center have been obtained by Schönenbach (2014) for different spin and inclination values. Note, that the intensity in pc-GR is significantly larger than in GR. Also, at a distance (about $1.72~{}m$ for $n=3$, for larger values this position changes according to the discussion in the text) a dark ring occurs, which is a consequence of the maximum of the orbital frequency of a particle in a circular orbit Schönenbach (2013). Near this distance, friction is low (and therefore viscosity) and the emission of light reaches a minimum. This is also noted in Figure 2, where the intensity is plotted versus the pixel number. The distance between two pixels corresponds to approximately 0.022m. The intensity reaches a minimum at a distance of $1.72~{}m$ and for lower radial distances the intensity increases again, which corresponds to the bright ring shown in Fig. 1 in paper I Boller (2019). In GR this behavior is not seen.

4 Conclusions

We summarized the basics of a modified theory of General Relativity, which depends on two phenomenological parameters, namely the coupling of the dark energy to the mass ($B_{n}$) and the power of the fall off of the dark energy as a function of the radial distance. One of the main predictions is that the accretion disc emission will show both an outer dark ring and an inner bright ring. Though a thin accretion disc was assumed in this work, the results we obtained are robust and can also be applied for other types of emission. This is because these features depends only on the $r$-dependence of the orbital frequency of particles in a circular orbit. Another main difference is that in pc-GR the specific intensity is larger than in GR (see also the related discussion in Paper I). The reason for this is that stable orbits exist at smaller radii, and therefore, more energy is released. A further prediction is that the surface is at approximately $r_{0}=\frac{2(n-1)}{n}~{}m$, where $n$ is a parameter of the theory which describes the rise of the dark-energy density toward the center. This parameter can be restricted by gravitational wave analysis Nielsen (2018) and by observations of the Event Horizon Telescope EHT (2016). i.e. in the Schwarzschild case ($a=0$) the size of the black hole is smaller than in the standard theory and in the Kerr case with $a=1m$ it is larger. In addition, for the spin parameter $a$, below $a=1$ there is a range of values where there is no light ring possible. This range is for $n=3$ between 0.95 and 1 and increases its size for larger $n$.

These predictions and others (see also Paper I) should be verifiable by the Event Horizon Telescope and they are the same for the black hole in the center of M87.

Acknowledgements

TB and POH are grateful for the longstanding support and collaboration with the Frankfurt Institute of Advanced Studies. The authors thank Damien Coffey for a critical reading of the manuscript. POH acknowledges financial support from Programa de Apoyo a Proyectos de Investigación e Innovación Tecnolǵica (PAPIIT) (IN100418). H.St. acknowledges the Judah M. Eisenberg Professor Laureatus at the Fachbereich Physik endowed by the Walter Greiner-Gesellschaft zur Förderung der physikalischen Grundlagenforschung e.V. Frankfurt am Main. The authors would like to thank the referee whose comments greatly improved the content of this paper.