Observable consequences of pseudo-complex General Relativity
OObservable consequences of pseudo-complexGeneral Relativity
P. O. Hess , Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico,Circuito Exterior, C.U., A.P. 70-543, 04510 M´exico D.F., Mexico Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe Universit¨at,Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany
January 13, 2021
Abstract
A review of the pseudo-complex General Relativity (pc-GR) is pre-sented, with the emphasis on observational consequences. First it isargued why to use an algebraic extension and why the pseudo-complexis a viable one. Afterward, the pc-GR is formulated. Posterior, sev-eral observational consequences are discussed, as the perihelion shiftof Mercury, Quasi Periodic Objects, the emission profile of accretiondiscs, the pc-Robertson-Walker model of the universe, neutron starsand gravitational ring-down modes of a black hole.
The
General Relativity (GR) is up to now the most successful theoryof Gravitation. Many observations have been confirmed, especially inweek gravitational fields as in the solar system [1]. The existence ofgravitational waves were also confirmed [2, 3] and the shadow of ablack holes was observed in [4, 5, 6, 7, 8, 9]. Though, the last two ob-servations are related to very strong gravitational fields, we will showthat the pc-GR is also able to describe them. The reason lies in themethod to deduce masses and distances from the observations, whichis based on the assumption that only GR describes the observations. a r X i v : . [ g r- q c ] J a n ttempts to extend GR have a long history [10, 11, 12, 13, 14, 15],all with different motivations. For example, A. Einstein wanted tounify gravity with electrodynamics [10, 11]. On the other hand, M.Born approached a conceptional problem, namely that in QuantumMechanics coordinates and momenta are treated on an equal footing,while in GR coordinates are predominant. The main motivation forpc-GR [16, 17] was to extend GR via algebraic means and to searchfor the possibility to eliminate the event-horizon. Why this is impor-tant? This depends on the view of the beholder: While some arguethat the event horizon is just another coordinate singularity, otherscomplain that from a black hole, even nearby, no information can getout. The no-hair theorem implies that all information is lost, violatingfundamental concepts. Also the singularity in its center is bothersomeand theories have been proposed [18, 19] to remedy it.Though, the event horizon is only a coordinate singularity, its ori-gin is the strong gravitational field and there is no reason to believethat GR is still valid. Thus, a search for an extension of GR is justifiedwhen the limits of the theory are reached.In this contribution, we will elaborate on the motivations for propos-ing the pc-GR and explain its mathematical structure. Apart frompresenting a review on the pc-GR, the main body of the text con-centrates on observational consequences. Some of the examples werealready discussed in earlier publications, but some are new, as willbe noted in its course. We will see, that several predictions cannotbe distinguished from GR, due to either low resolution, or too smallcorrections, or alternative explanations within GR. In these cases, onehas to wait for further, improved observations.This contribution is organized as follows: In Section 2 algebraicextensions in general are briefly discussed. In section 3 the pc-GRis resumed. In section 4 the observational predictions and conse-quences are discussed, which includes: The Perihelion shift of Mer-cury , Quasi periodic objects , light emission structure of accretion discs ,the pc-Robertson-Walker cosmology , neutron stars and gravitationalring-down modes of a black hole. Some of the results were obtainedearlier and some are new, as resolving the structure of the pseudo-imaginary part of the coordinates and the perihelion shift of Mercury.In 5 conclusions will be drawn.The convention G = c = 1 is used throughout this contribution,as the metric signature ( − + + +) Algebraic extensions
A possible option to modify GR is to extend the space-time coordi-nates x µ ( µ = 0 , , ,
3) to a different kind, which is called an algebraicextension . The questions to address are: How many algebraic exten-sions exist? Which of those are consistent with basic principles? If analgebraic extension is consistent was addressed in [20] where all kindsof coordinates where investigated. A more detailed explanation canbe found in [21, 22].
Table 1: Algebras of various algebraic coordinate extensions.Algebra Generators a i algebraReal 1 -complex 1, i i = − I I = 1quaternion 1, a , a , a a i a j = − δ ij + ε ijk a k hyper-quaternions 1, a , a , a a = a = 1, a a = − a An algebraic extension is defined through the mapping of real co-ordinates to x µ → X µ = x µ + a i y µ (1)where a sum over i is implicit and the a i satisfy the algebra a i a j = C ijk a k , (2)with a sum of repeated indexes. One example is the complex extensionwhich contains, besides a = 1, a = i with i = −
1. Another one isthe pseudo-complex (pc) extension, with the generators a = 1 and a = I with I = 1. The possible algebras are listed in Table 1.Several modifications of GR, mentioned in the introduction, arerelated to algebraic extensions, though not obvious at the first sight.In [22] these attempts were resumed, as the one proposed in [14, 15]which implies a maximal acceleration, or the more obvious one in [23],were the complex extension is proposed. ost of the algebraic extensions run in a serious problems, whenthe square of at least one of the a i is -1. It is shown in [20, 22] thatin the limit of nearly flat space the propagator of a gravitational wavehas the wrong sign and corresponds to a ghost solution, excluded onphysical grounds. Therefore, the main observation of [20] is that apartfrom the real coordinates, corresponding to GR, the only ones whichdo not imply ghost and/or Tachyon solutions are the pseudo-complexcoordinates. This is a very important finding because it implies thatthe path through algebraic extensions is only possible through the useof pc-coordinates.This is the reason why we stick to this extension and in what fol-lows we will expose some consequences, theoretical and observationalones. The pc-GR was introduced in [16]. Since then, several reviews werepublished in [21, 22]. In [17] the theory, their mathematical structureand some observational facts can be retrieved and in [24, 25] someobservational predictions were published. One centerpiece in pc-GRis that around any mass vacuum fluctuations accumulate, whose pres-ence is a consequence of semi-classical Quantum Mechanical calcula-tions [26, 27]. These vacuum fluctuations have the nature of a darkenergy and semi-classical Quantum Mechanics calculations result in afall-off of the dark energy density as a function in distance to the mass.Up to now, in pc-GR a phenomenological ansatz is used, assuming afall-off of the dark energy density as ∼ B n /r n +2 , where B n describesthe coupling of the central mass to the dark energy and r is the radialdistance. The n is a parameter, which was assumed to be 3 in [24, 25],because it is the next leading order correction to the metric which stilldoes not contradict solar system observations [1]. However, in [28, 29]it is shown that n has to be >
3, using the first observed gravitationalwave event [2, 3]. For that reason, we assume n = 4 in what follows,keeping in mind that this is a phenomenological assumption .One important consequence of the accumulation of dark energynear a central mass is its recoupling to the metric, which is changedsuch that no event horizon appears. It can rephrased into the prin-ciple that a mass not only curves the space nearby but also changesthe vacuum properties , which may give clues on the quantization of ravity.The pc-Gr is formulated in analogy of the GR, using instead ofreal coordinates x µ new ones, namely X µ = x µ + Iy µ , I = 1 X µ = X µ + σ + + X µ − σ − σ ± = 12 (1 + I ) , σ ± = 1 , σ + σ − = 0 , (3)where two representations are shown, the one in terms of 1 and I andthe other one in terms of the so-called zero-divisor basis , given by σ ± .The σ ± behave as projectors, with ( σ ± ) = 1 and σ − σ + = 0. Thelast property is important because it allows to perform calculationsindependently in each zero-divisor component [17].The metric has also two components and is of the form g µν ( X ) = g + µν ( X + ) σ + + g − µν ( X − ) σ − . (4)The length element is defined in terms of X µ in the same manneras it is defined in terms of x µ in GR: dω = g µν dX µ dX ν = (cid:110) g Sµν [ dx µ dx ν + dy µ dy ν ] + g Aµν [ dx µ dy ν + dy µ dx ν ] (cid:111) + I (cid:110) g Aµν [ dx µ dx ν + dy µ dy ν ] + g Sµν [ dx µ dy ν + dy µ dx ν ] (cid:111) , (5)with g Sµν = 12 (cid:16) g + µν + g − µν (cid:17) , g Aµν = 12 (cid:16) g + µν − g − µν (cid:17) . (6)We have used the expression of X µ in terms of x µ and y µ for rewritingthe length element dω in (5).Requiring that particles only move along real distances, a con-straint of a real length element is imposed, demanding the factor of I in (5) to vanish, i.e., σ + − σ − ) (cid:110) g Aµν [ dx µ dx ν + dy µ dy ν ] + g Sµν [ dx µ dy ν + dy µ dx ν ] (cid:111) = 0 . (7)It is illustrative to consider the GR-limit, i.e., g Aµν = 0 and with g Sµν = g µν now real, the constriction (7) reduces to g µν dx µ dy ν = 0 . (8)This is nothing but the dispersion relation, whose solution is y ν ∼ u ν ,the four velocity. For dimensional reasons y ν = lu ν , with l a lengthparameter, related to the minimal length of the theory.Here, it is important to note that pc-GR contains a minimal length ,which is a mere real number and is not affected by a Lorentz trans-formation. This simplifies enormously the mathematical structure ofthe theory, because it does not require a deformation of the Lorentztransformation, needed in theories with a physical minimal length. This observation hints to the alternative to formulate theories whichcontain a minimal length and still use Lorentz symmetry .In general, the solution of (7) is quite complicated and we will dis-cuss two of those in an approximate manner: For the pc-Schwarzschildcase and for the pc-Robertson-Walker metric.The action within pc-GR is defined in analogy to GR, namely S = (cid:90) dX √− g ( R + 2 α ) , (9)where R is the pc-Riemann scalar, defined in the same way as in GR,and α is a pc-constant for the pc-Robertson-Walker model and maydepend on the radial distance for the Schwarzschild case. Note, thatalso the volume element is pseudo-complex.Varying this action in each zero-divisor sector, using the constraint(7), leads to the equations of motion R ± µν − g ± µν R ± = 8 πT Λ ± µν (10)in each zero-divisor basis. The Λ refers intentionally to its propertyas a dark energy. he T Λ ± µν is given by8 πT Λ ± µν = λu µ u ν + λ ( ˙ y µ ˙ y ν ± u µ ˙ y ν ± u ν ˙ y µ ) + αg ± µν , (11)which can be rewritten such that its property as a, in general, anisotropicfluid is obvious. For that, we refer to [21, 22].When T Λ ± µν is mapped to the real part, one obtains8 πT Λ µν = λu µ u ν + λ ˙ y µ ˙ y ν + αg ± µν . (12)As we will see further below, the y µ can be written as being pro-portional to the 4-velocity u µ . This allows us to write the (12) propor-tional to λ as λ (cid:0) l A ( r ) (cid:1) u µ u ν (here for a central problem), where l describes the correction due to the ˙ y µ ˙ y ν term. The A ( r ) can bevery large near the event-horizon, as will be seen later.For the T -component this implies pc-corrections to the density ofthe dark energy proportional to l . This effect was already studied in[30], where the effective potential deduced there produces a repulsivepotential near the event horizon.Though, we will discuss some solutions, at the end it will be moreeffective to assume a phenomenological approach , with a specific r -dependence of the dark energy density.For the dark energy density we prefer to use a phenomenologicalansatz, namely (cid:37) Λ ∼ r n +2 . (13)The value of n has to be larger than 3, as discussed earlier. Theproportionality factor parametrizes the coupling of the central massto the dark energy.After these general consideration, we resume the metric of a rotat-ing star (Kerr metric), which was derived in [24]: g = − r − m r + a cos ϑ + B n ( n − n − r n − r + a cos ϑ ,g = r + a cos ϑr − m r + a + B n ( n − n − r n − , = r + a cos ϑ ,g = ( r + a ) sin ϑ + a sin ϑ (cid:16) m r − B n ( n − n − r n − (cid:17) r + a cos ϑ ,g = − a sin ϑ m r + a B n ( n − n − r n − sin ϑr + a cos ϑ . (14)The B n and a are the above mentioned coupling parameter of the massto the dark energy and the Kerr rotational parameter, respectively. Inwhat follows n = 4 will be assumed and (14) reduces to the standardKerr solution of GR when B n = 0.For n = 4 and b n = there is still for a = 0 an event horizonat m . Thus, if we want to eliminate completely this horizon, thevalue of b n has to be at least infinitesimal larger than . For practicalreasons b n is assumed to acquire the value . In that case, a massfunction m ( r ) can be defined, namely m ( r ) = (cid:32) − (cid:18) m r (cid:19) (cid:33) . (15)For the pc-Schwarzschild case the g component acquires the form g = (cid:16) − m ( r ) r (cid:17) , which clearly reduces to the well known expressionin GR, when term proportional to 1 /r in m ( r ) is eliminated. A more complete explanation of the group theoretical structure of thepc-coordinates and their importance in field theory can be found in[17, 31, 32, 33, 34].The pc-Lorentz transformation is e − ω µν L µν = e − ω µν + L + µν σ + + e − ω µν L − µν σ − , (16)where L ± µν is the generator of the Lorentz group SO ± (3 , ± zero-divisor component. Its explicit form is given by (¯ h = 1) L µν = X µ P ν − X ν P µ (cid:16) X + µ P + ν − X + ν P + µ (cid:17) σ + + (cid:16) X − µ P − ν − X − ν P − µ (cid:17) σ − = L + µν σ + + L − µν σ − with P µ = 1 i ∂∂X µ . (17)With this and [ X µ , P ν ] = iδ µν , the commutation relations of thepc-Lorentz transformation generators are [17][ L µν , L λδ ] = i ( g λν L δµ + g δµ L λν + g λµ L νδ + g δν L µλ ) . (18)Thus, the group structure is SO pc (3 ,
1) = SO + (3 , ⊗ SO − (3 , ⊃ SO (3 , , (19)i.e., the direct product of two Lorentz groups. This direct productreduces to the standard Lorentz group, when the real part of the pc-generators is projected.In conclusion, the mathematical structure is very similar to thestandard formulation, it only involves twice as much generators andparameters for the transformation, a reflection of the 8-dimensionalcoordinate space. In this section, some older, with additions, and some new predictionsof the pc-GR are presented. First of all, the general properties of pc-GR related to the y µ -components are discussed. As mentions above,the y µ variables can be identified, for the case of a flat space, by lu µ ,where the l is a length parameter introduced for dimensional reasons.An allowed ansatz for a general case is y µ = A ( x, y ) lu µ , (20)where A ( x, y ) is a function in x ν and y ν , yet to be determined, where y µ has the same tensorial properties as u µ and A ( x, y ) is a scalar.That (20) is correct will be seen in two examples to be discussed inthis paper, namely for the pc-Schwarzschild solution and for the pc-Robertson-Walker metric. .1 A general equation for y µ In what follows, the general structure is discussed and an equation forthe determination of (20) is presented. At the end, the y µ is obtainedwithin an approximation for the Schwarzschild case, the first example.A curve in the pseudo-complex manifold is described by the pc-4-vector V µ = dX µ ds = dX µ + ds σ + + dX µ − ds σ − , (21)with ”s” as the curve parameter, which is the same in both zero-divisorcomponents.In addition to the pc-coordinates X µ the local coordinates (or co-moving frame) (cid:101) X µ at a given point p are defined in analogy as X µ ± = x µ ± y µ , (cid:101) X µ ± = (cid:101) x µ ± (cid:101) y µ . (22)Transforming (21) into the co-moving frame in both components,we obtain dX µ ds = e µ + i d ˜ X i + ds σ + + e µ − i d ˜ X i − ds σ − , (23)where e i ± µ ( i =0,2,3,4) are the inverse matrix elements of e µ ± i . In thiscontribution only diagonal metrics are discussed, therefore the e i ± µ are the square roots of the diagonal metric elements.In a co-moving frame the space is locally flat and, as we saw earlier,the ˜ y µ assume the relation l d ˜ x µ ds , i.e., it is proportional to the 4-velocity.This results into a relation of the zero divisor components of X µ inthe local frame to x µ and y µ : d ˜ X i + ds = e i + µ dX µ + ds = d ˜ x i ds + d ˜ y i ds = (cid:18) l (cid:19) ˜ y i + d ˜ y i dsd ˜ X i − ds = e i − µ dX µ − ds = d ˜ x i ds − d ˜ y i ds = (cid:18) l (cid:19) ˜ y i − d ˜ y i ds . (24)Next, we solve for (cid:101) y µ and its derivative with respect to s , giving y i = l (cid:34) e i + µ dX µ + ds + 12 e i − µ dX µ − ds (cid:35) d ˜ y i ds = 12 e i + µ dX µ + ds − e i − µ dX µ − ds . (25)In a subsequent step, the first equation in (25) is derived further withrespect to the line element s and the result is set equal to the secondequation in (25). Then, all expression of the σ + and σ − componentsare collected on one side of the equation, resulting into l de i − µ ds dX µ − ds + e i − µ (cid:20) l d X µ − ds + dX µ − ds (cid:21) − e i + µ (cid:20) dX µ + ds − l d X µ + ds (cid:21) + l de i + µ ds dX µ + ds = 0 . (26)Let us consider, as a special case, a central and static problem: Set dX ± = 0 and dX ϑ ± = 0, dX ϕ ± = 0. As an approximation, all termsproportional to the minimal length are neglected compared to theothers, because we will not work in the regime of maximal acceleration.We also rescale, for this particular example, the mass parameter to m = 1.The 4-bein tensors e i ± µ are used, restricting to the pure r part: e i ± µ → e r ± r = 1 − m ± ( X r ± ) X r ± , m ± = 1 − X r ± . (27)The differential equation (26) then reduces to dy r dr = (cid:104) r − y r ) − r − y r ) (cid:105) − (cid:104) r + y r ) − r + y r ) (cid:105) (cid:104) r − y r ) − r − y r ) (cid:105) + (cid:104) r + y r ) − r + y r ) (cid:105) . (28)Furthermore, assuming a small y r , compared to r , we arrive at dy r dr ≈ ( r − r )1 + r − r y r . (29) hen the differentials dy r and dr are moved to opposite sides of theequation and we divide it by dτ , where τ is the time parameter, onenotes that the time derivative of y r is proportional to the correspond-ing component of the four-velocity, as noted further above.The MATHEMATICA code [35] is used to solve this equation, withthe result y r = A r (3 − r )(3 + 4 r + 4 r ) . (30)The integration constant A is determined in the limit of r → ∞ .Requiring that y r should approach l ˙ r ∞ , leads to y r → − A , or A = − l ˙ r ∞ . In this manner, the minimal length l is reintroduced due tothe assumed limiting behavior of y µ .Thus, the final solution is y r = 4 lr (2 r − r + 4 r ) ˙ r ∞ . (31)This solution has a pole in r = , where y r tends to + ∞ for r → ( ) | + .Because we assumed y r as small compared to r , the solution is not validanymore near r = and a solution for large y r has to be found.Nevertheless, the solution describes the evolution of y r from large r towards smaller values, when y r is still small. The limit is when(2 r −
3) in the denominator is of the order of one, or r ≈ + l , i.e., very close to r = . A general picture emerges, where y r is slowlyincreasing for descending r -values and changes rapidly toward + ∞ near r = . In Fig. 1 the function y r / ˙ r ∞ , in units of the minimallength l , is plotted in the range 1.52 to 2 and, as can be seen, onlynear 1.5 the curve starts to rise. The first confirmation of General Relativity came from the calcula-tion of the perihelion shift of Mercury. As an illustrative example, inthis sub-section, the perihelion shift is investigated, including the pc-corrections, with the modified mass-function m ( r ) = m (cid:16) − m r (cid:17) (now, we return to m (cid:54) = 1), where the paths as described in [36] .6 1.7 1.8 1.9 2.01020304050 r y ( r ) Figure 1: The radial pseudo-imaginary coordinate component y ( r ) = y r / ˙ r ∞ as a function in r. The vertical axis gives y ( r ) in units of l . is neatly followed. We will see, that GR gives the dominant contri-bution, while the additional ones from pc-GR will be too low to bemeasured. That this correction cannot be measured is not surprising,because the gravitational field in the solar system is simply too weak.The consequences for all other solar system observations are equallysmall.The real length element in the pc-Schwarzschild solution is ds = − (cid:18) − m ( r ) r (cid:19) dt + (cid:18) − m ( r ) r (cid:19) − dr + r (cid:16) dθ + sin θdϕ (cid:17) , (32)divide it by ds and use for the Lagrangian the definition L = − (cid:18) − m ( r ) r (cid:19) ˙ t − (cid:18) − m ( r ) r (cid:19) − ˙ r − r (cid:16) ˙ θ + sin θ ˙ ϕ (cid:17) , (33) w [m/c] r [m] Figure 2: The angular frequency of a particle in a circular orbit, as a functionin the radial distance and for a = 0 . m . The upper curve is the result asobtained within GR and the two lower curves within pc-GR. The curve withthe maximum to the left is for n = 3 and the other one for n = 4. where the dot refers to the derivative in s . Using the angular momen-tum conservation l = r ˙ ϕ = const, that the motion is in the plane of θ = π and that (cid:16) − m ( r ) r (cid:17) ˙ t = h = const (which is the result of ofthe variation with respect to the time), we arrive at1 = (cid:18) − m ( r ) r (cid:19) − h − (cid:18) − m ( r ) r (cid:19) − ˙ r − l r . (34)Changing the derivative with respect to s to the one with respectto ϕ , leads to r (cid:48) = drdϕ = ˙ r ˙ ϕ → ˙ r = ˙ ϕr (cid:48) = lr r (cid:48) , (35)where the prime now refers to the derivative with respect to ϕ .Multiplying (34) by (cid:16) − m ( r ) r (cid:17) and using (35), we arrive at theequation (cid:18) − m ( r ) r (cid:19) = h − l r ( r (cid:48) ) − l r (cid:18) − m ( r ) r (cid:19) . (36) .0 0.2 0.4 0.6 0.8 1.0123456 a [ m ] Figure 3: Limits of stable orbits within GR and pc-GR ( n = 4). The uppercurve is for GR, and the inner curve for pc-GR. Below the upper curve, nostable orbits exist, while in pc-GR no stable orbits exist to the left of theinner curve. Thus, in pc-GR, for a -values to the right of the inner curve allorbits are stable. Next, the variable r is changed to u , namely r = 1 u → r (cid:48) = − u (cid:48) u . (37)Substituting this into (36) and resolving for ( u (cid:48) ) , one arrives at( u (cid:48) ) = ( h − l + 2 m ( u ) l u − u + 2 m ( u ) u , (38)where m ( u ) is given by m ( u ) = m (cid:18) − m u (cid:19) . (39)In the next steps, (38) is again derived with respect to ϕ and theresulting equation is divided by 2 u (cid:48) , arriving finally at the equation u (cid:48)(cid:48) + u = m l + 3 m u − m l u − m u . (40) efine the small number ε = 3 m A , which is for the solar systemof the order of 10 − [36], where A = m l is the areal velocity on theplanetary plane. With this, the m can be set equal to ε A and (40)can be rewritten as u (cid:48)(cid:48) + u = A + εA u − A (cid:18) εA (cid:19) u − (cid:18) εA (cid:19) u . (41)In the first line, the first term on the right hand side is the clas-sical, Newtonian contribution, while the second term is the standardrelativistic correction which leads to the known perihelion shift ofMercury. The terms in the second row are explicit new contributionsfrom pc-GR, however of order ε . Thus, new contribution of pc-GRare at least two order of magnitude less, the one of order ε comingonly from GR (implicitly contained in the term εA u when u is alsoexpanded in ε ). Thus, there is no hope to detect these deviations.This result also shows that GR and pc-GR are indistinguishable fromeach other in solar system observations. Still, this calculation is aninteresting exercise, which might be of relevance when the perihelionshift of objects very near to a super-massive black hole is observed. We mention shortly a former discussion on so-called
Quasi-Periodicobjects . There are several kinds, but here we refer to bright lightemissions in accretions discs with a near periodic time dependence.At first, this phenomenon was associated to a bright spot which isco-moving with the accretion disc, thus, appearing and fading awayfrom the observer as it turns around the black hole. Such events wereobserved [17, 37, 38, 39, 40, 41] in black hole binaries, but also existin central black holes of galaxies. The GR can provide an estimationtheir distance to the center of the black hole, using the dependence in r of the orbital period of a circular particle. In order to confirm thisdistance, the redshift of the emission line has to be measured, too,which depends on r . The radial distance deduced from the orbitalfrequency and the redshift have to coincide for consistency. For centralblack holes the redshift has not been observed yet, but it is for stellarblack holes. he assumption is that the QPOs are the result of a bright spotmoving around the black hole, which is obvious for black holes inthe center of galaxies. However, for stellar black holes there may bea distinct mechanisms, like oscillations provoked by the companionstar, as suggested in [42]. Thus, GR can still be saved. The questionremains: Why the interpretation of OPOs should be different from theones in the accretion disc of a black holes in the center of a galaxy?We argue that the same mechanism should also hold for stellar blackholes, at least dominantly.In order to understand the motion of a point-particle around ablack hole, we resume some fundamental properties of this particle ina circular orbit: In Fig. 2 the dependence of the orbital frequency as afunction in r is depicted for the rotational Kerr-parameter a = 0 . m .The upper curve is the result for GR, while in the lower curve the onefor pc-GR is plotted. In GR the curve shows a steady increase of theorbital frequency, contrary to pc-GR which predicts a maximum, afterwhich if falls off again.Deducing a distance from the orbital period and from the redshift,leads for n = 3 to Fig. 4, where the observation is compared to thetheory, using the a -parameter deduced. The zero is for n = 3 at r = m and for n = 4 to r = , which are practically indistinguishablewithin the plot, i.e., the consequences are the same.There is a great mismatch between the distance deduced from theorbital frequency and the redshift, using GR. In pc-GR, however, theresults agree. A simple explanations leads to an agreement to obser-vation! Nevertheless, because GR can be still reconciled, though, viaa more complex explanation, nothing can be decided up to now. In order to understand the predicted light emission from an accretiondisc in pc-GR, stable circular orbits have to be discussed: In Fig. 3the last stable orbit (or ISCO for
Innermost Stable Circular Orbit ) asa function in a is shown. In GR the ISCO starts at 6 m and is loweredto r = m for a = 1. Below the upper line in Fig. 3 no stable orbitexists. This is different in pc-GR: The lower curve shows the limits ofstable orbits, where to its left no stable orbit exists but to its right itdoes. The curve turns at about a = 0 . m , thus above a = 0 . m allorbits are stable. It is this region where they can reach the maximumof the orbital frequency, below this value of a they don’t. Quasi-Periodic Objects (QPO) compared to GR and pc-GR. Theupper curve corresponds to the orbital frequency deduced within GR, wherethe thickness of the line indicates the resolution of the observation. The lowercurve corresponds to pc-GR for n = 3. For n = 4 the differences are minor,which is the reason no new calculations were performed. The horizontaldashed lines show the observed orbital frequency with the experimental error,while the vertical lines show the deduced radial distance from the redshift ofthe Kα -line. For small a , the curve for pc-GR follows neatly the one of GR.Thus, we do expect a similar emission profile structure, except forthe intensity of the light emitted, because the curve is further intothe gravitational well and more energy is released. When a > . m the stable orbits reach the maximum of the orbital frequency. Atthis maximum, neighboring orbitals have a similar frequency and theexcitation of the disc is minimal, producing a dark ring. To resume,pc-GR provides a robust prediction of the emission profile with a darkring follows further in by a small bright ring.The structure of the emission on accretion discs is discussed in[43, 44], comparing GR with pc-GR, besides what happens when acloud is approaching the black hole.In Fig. 5 a simulation is shown, using the model published in [45],which assumes a thin, optical thick accretion disc. This is not a veryrealistic model, expecting rather a thick disc, maybe with the form of µ as while on the right it is for 20 µ as. Theinclination angle with respect to the observer is 70 o . Clearly seen is that thering structure, as predicted by pc-GR, is washed out, showing that the EHTobservation cannot discriminate between pc-GR and GR. a torus. For such discs, other models are better suited, from which wejust mention one, namely [46]. Nevertheless, the emission structureis robust and should be similar in all disc models, changing only theabsolute value of the intensity emitted.On the left hand side of Fig. 5 the result for a high solution is de-picted, while the right hand side shows the result for a low resolutionof 20 µ as. For the low resolution case, the ring structure is unfor-tunately lost. The Event Horizon Telescope (EHT) [4, 5, 6, 7, 8, 9],which observed the shadow of the central black hole of M87, has a res-olution of only 24 µ as, which is too low to dissolve the ring structure.Therefore, one has to wait for future observations with an improvedresolution. Note that the EHT is not able to distinguish between GRand different kinds of extensionsThe disc model used by EHT consists of an optical thin plasmasurrounding the black hole (see also [47]. This is only one of manypossible disc models, most of which are optical thick. The shadow issimulated tracing the most inner photon (or Einstein) rings. Whilein GR for a = 1 m all photon rings are blocked by a optical thickaccretion disc, because the ISCO reaches r = m , in pc-GR all photonrings are blocked, because for a > . m all orbits are stable until tothe surface of the star. A definite answer can only be given, when thedisc is proven to be optical thin and/or pc-GR is confirmed or not. .5 Cosmology: The Robertson-Walker Model The pc-Robertson-Walker metric was introduced in [48] and can alsobe retrieved form [17]. The length element is given by dω = ( dX ) − a ( t ) (cid:16) ka ( t ) a (0) (cid:17) d Σ , (42)where d Σ is the angular volume element, a ( t ) is the radius of theuniverse, a (0) its value at present time (set to 1) and k is a parameter,which only for k = 0 corresponds to a flat universe [36, 49]. This valuewill be used from here on, because all observational date suggest a flatuniverse.For the energy-momentum tensor an isotropic fluid is assumed,i.e., ( T µν ) = ρ − pc − pc − pc , (43)Using the metric (42), the y µ can be determined, using the dif-ferential equation (26). Because an isotropic universe is assumed, thedevelopment of the radius a ( t ) is the same in any direction. Therefore,it suffices to define y µ as a I ( t ).In analogy to (28) and using a ( t ) = a R ( t ) + Ia I ( t ), we arrive at da I da R = (cid:104) ( a R ( t ) − a I ( t )) a (0) − ( a R ( t )+ a I ( t )) a (0) (cid:105)(cid:104) ( a R ( t ) − a I ( t )) a (0) + ( a R ( t )+ a I ( t )) a (0) (cid:105) = − a R a I a R + a I . (44)Because the a I has to be very small compared to a R , we obtainapproximately, setting a (0) = 1, da I da R = − a I a R . (45)Separating variables and integrating, we finally obtain I = A a R . (46)Demanding an a I proportional to l ˙ a R (0) at t = 0, as similarly done inthe Schwarzschild case, we can set A = l ˙ a R (0). Thus, today a I (0) = l ˙ a R (0) (compared to a R (0) = 1), which is extremely small, and itcontinues to be even smaller for larger times. However, for t → a I explodes and becomes infinite very near tothe big bang, implying a dominance of the dark energy. Of course, inthis case, the approximation of a small a I , compared to a R is not validanymore, but the results enlightens the tendency of a strong increaseof a I ( t ). Apart from that, very near to the big bang the mass/energydensity is extremely high and according to our assumption additionalvacuum fluctuations building up (see section 4.6).In [17, 48] the possible fates of the universe were determined. Theequations of motion were set up and the differential equation for thesecond derivative of a R became, assuming that the matter distributionin the universe behaves as dust, a (cid:48)(cid:48) R πG = (3 β − a β − R − (cid:37) a − R , (47)where (cid:37) is the matter density at the present date. The β is a pa-rameter of the dark energy density (cid:37) Λ = Λ a β − R . For β = 1 thisenergy density is just Λ, as it seems to be satisfied today. G is thegravitational constant.With this, the differential equation of the radius of the universe is a (cid:48)(cid:48) R πG (cid:37) = 2Λ a R − a − R . (48)The acceleration starts with a negative sign in the early epoch of theuniverse until a (cid:48)(cid:48) R = 0, which is reached when a R = 1 / (2Λ) , afterwhich the acceleration becomes positive, as observed today.In [17, 48] also other scenarios were discussed, taking differentvalues of β . As a result, besides a rip-off for values β >
1, alsosolutions were found, which for t → ∞ approaches a constant valueor even a zero positive acceleration (0 + ). ( km )
10 20 30 40 50 60 70 80 M m ( M ⊙ ) r = 25 kmr = 50 kmr = 75 kmr = 100 km Figure 6: Within pc-GR, stable stars up to 200 solar masses are obtained fora coupling of the dark energy density to the mass density, which diminishesapproaching the surface.
The pc-GR conjectures that there is no event horizon, implying thatall so-called black holes are rather gray stars, even when they havea mass billion times more than the sun. The question is, if one candescribe the internal structure of such stars and explain their stability.One problem is that there is no such theory, because no one can deducethe equation of state under such extreme mass densities. However, onecan try to extrapolate from low masses to larger ones, using recentlydeveloped theories for neutron stars. Such a theory is published in[50], which is a mean-field approach involving different hadron statesincluding strange hadron fields.In [51, 52] this mean-field model was coupled to dark energy withinthe star. As a first approximation, a linear coupling between the darkenergy density to the mass density was assumed: z Θ Figure 7: The redshift at the surface r = m as a function of the azimuthalangle θ , for different a -values. From top to bottom, the curves correspond to0.2 m , 0.5 m , 0.8 m to 1.0 m . (cid:37) Λ = α(cid:37) m , (49)with the result of up to 6 solar masses already found to be stable.This is interesting in view of recent attempts to explain higher neutronmasses [53] due to the possible detection of a neutron star with 2.6solar masses in a gravitational wave event [54]. This is a hint that evenlarger neutron star masses can exist , though, it should be expectedthat their internal structure has to change significantly.No higher masses could be obtained, because the linear couplingresulted in a repulsion effect shedding off mass from the star for in-creasing mass. This was the motivation to reanalyze vacuum fluctu-ations within the star, using semi-classical Quantum Mechanics, asexplained in [26]. In [55] the monopole approximation [56] was usedand the coupling between the dark energy and mass density deduced.As a result a decrease of the coupling near the surface was obtainedand stable solutions of up to 200 solar masses were obtained, see Fig,6. This presented at the same time a limit due to the end of validity ofthe mean-field theory [50]. For higher masses other, not yet developedtheories have to be searched for.How one could detect such large neutron stars (if one can stilldenote them as neutron stars is a big question, too)? One possibility s to look for emissions from infalling matter and/or beams emittedas in standard neutron stars. This is not as easy, because near theposition of the so-called event horizon the redshift is extremely large,at least in the orbital plane. This is illustrated in Fig, 7, where theredshift at r = m (which is the position of the event horizon for a = 0, with the mass-function used) is plotted versus the azimuthalangle θ , for a set of Kerr-rotational parameters a . As can be seen, nearthe poles the redshift is, for large values of a , of the order of 1. Thus,if a is near to 1, highly redshifted light emission may be seen near thepoles, when matter falls in there. When, however, the so-called blackhole has an accretion disc, there has to be a jet emerging near thepoles, over-shining this effect. Hopefully, one can observe it in SgrA*which is believed to be devoid of an accretion disc.It remains to be mentioned that in [57, 58] also neutron stars, ascompact objects, were investigated within pc-GR from the viewpointof an effective field theory. In [59, 60] the effects of dark matter withinpc-GR were also investigated. All these contributions reflect a range ofobservable consequences and applications of pc-GR for neutron stars. In 2016 the first observed gravitational wave event was reported in [2,3]. Assuming that GR is the theory to describe the merger, a two-pointapproximation seems to be justified, mainly because the ISCO is stillfar away from contact. The steps for obtaining the chirping mass M c ,using the two-point approximation, are explained in [61]. In [62] thesame approximation was used, with the caveat that the two black holescan approach each other until their event-horizon touch each other,which renders the two-point approximation senseless. Nevertheless,this approach serves to extract trends on how the deduced chirpingmass changes. The modified equation for the chirping mass is M c = (cid:102) M c F ω (¯ r ) = c G (cid:20) π df gw dt f − gw (cid:21) , (50)where on the right hand side values of the observed frequency andits change in time appears. The left hand side contains the modifiedchirping mass (cid:102) M c and the apparent chirping mass M c , the numberdeduced in [2, 3]. Both are related by a factor F ( R ), where R isthe relative distance of the two black holes. In the calculation it is ssumed that the mass of both is approximately equal as it was inthe observed case. n = 3 was used, for which the function is given by F ω ( R ) = (cid:18) − (cid:16) R S R (cid:17) (cid:19) , where R S = 2 m is the Schwarzschild radiusof one of the black holes. For n = 4 the new expression is F ω ( R ) = (cid:18) − (cid:16) R S R (cid:17) (cid:19) . In any case, the F ( R ) tends to a small value until thetouching configuration (twice the Schwarzschild radius for equal masscompanions), implying that the real chirping mass is larger then thededuced one using GR- Here it is important to note that the approachto deduce the mass in [2, 3] is theory dependent , as it is also in pc-GR.In pc-GR a larger mass should result and, thus, also the luminositydistance has to be larger, such that the same is observed on Earth.How large, depends very much on the model used to describe theinspiral phase. A simple two-point approximation does not work andone needs more sophisticated methods, as for example a numericalrelativistic hydrodynamical approach [63].In contrast, it is much easier to describe the ring-down modes of ablack hole, i.e., after the two black holes have merged. The ring-downmodes are obtained investigating the stability of the final black holeunder metric perturbations. For the Schwarzschild case, this part iswell described in the book by S. Chandrasekhar [64], which can bedirectly extended to the pc-GR metric. There are two types of modes,the negative parity solutions, also called Regge-Wheeler modes [65],and the positive parity modes, also called Zerilli modes [66]. In [67]the axial modes where calculated for n = 3 and in [68] also the axialmodes were calculated, now for n = 4. The time dependence of thering-down modes is e − iωt = e − iω R t e ω I t , with ω = ω R + iω I , separatedin its real and imaginary part. For damped modes the − ω I has to bepositive.The Regge-Wheeler equation is solved as explained in [67, 68],using an iterative technique called the Asymptotic Iteration Method (AIM) [69], and in Fig. 8 the result for small − ˜ ω I is depicted (wedefined ˜ ω = m ω ). Note that there are no unstable modes with nega-tive − ˜ ω I . The real part of then frequencies is in general slightly largerthan in GR, however, without any possibility to discriminate betweenGR and pc-GR. Also the iteration number is yet too low for going tolarger values of − ˜ ω I .However, which modes will be excited depends very much on thedynamics in the inspiral phase. Our suggestion is to use the numericalmethod as described in [63], changing in the programs the metric of .0 0.5 1.0 1.5 2.0 - - - - - Im ω R e ω Figure 8: Axial gravitational modes within GR (left panel) and pc-GR(within pc-GR). The iteration number is 20 (red dots) and 40 (blue dots),using the
Asymptotic Iteration Method [69].
GR to the one of pc-GR.The polar modes present some practical problems, rendering itmore difficult to solve. This will be addressed in a future publication.
A review was presented on the consequences of the pseudo-complexGeneral Relativity (pc-GR), comparing the results with GR. In gen-eral, differences are only visible near massive objects of the size of ablack hole, requiring however a large resolutionThe pc-GR is an algebraic extension of GR, where the coordinates x µ are redefined to X µ = x µ + Iy µ . This modifies the Einstein equa-tions with a dark energy energy momentum tensor on its right handside. This tensor depends of u µ and ˙ y µ . Demanding a real lengthelement, a differential equation for y µ was obtained. The y µ are pro- ortional to the 4-velocity, multiplied with the scalar minimal lengthparameter. The y µ was determined for the pc-Schwarzschild case andthe pc-Robertson-Walker universe.Observational consequences of pc-GR were determined, as the ap-pearance of a dark ring, followed by a bright inner ring, in the lightemission of an accretion disc. Unfortunately the resolution of theEHT is too low for seeing this structure, thus, it cannot discriminatebetween GR and pc-GR.Modification in the perihelion shift of Mercury were discussed, withthe result that pc-GR only adds corrections of 10 − to the shift cal-culated within GR, i.e., no hope to being seen.Also Quasi Periodic Objects where discussed and interpretationswithin GR and pc-GR were compared, without a definitive result.Neutron stars of any mass were obtained, though, for large massesthe star rather resembles the one of a black hole. It is suggested tolook for light emission of infalling matter at the poles, being excitedupon impact, provided there is no jet emitted nearby. At the polesthe minimal redshift is of the order of 1, increasing rapidly to infinityat the orbital plane.The Robertson-Walker universe gives in practice the same resultsas in GR for the present epoch and later times. In the limit of t → Acknowledgments
This work was supported by DGAPA-PAPIIT (IN100421). Very use-ful discussions with Laurent R. Loinard (IRyA, UNAM) are also ac-knowledged.
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