sign in sign up
<<
Home , Energy condition, Negative energy, Negative mass

Advanced Studies in Theoretical Vol. 14, 2020, no. 4, 161 - 174 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2020.91463

On Condition Violation for Shifting

Negative Black Holes

Manuel Urueña Palomo 1

University of León, Spain & Frankfurt University of Applied Sciences Frankfurt am Main, Germany

This article is distributed under the Creative Commons by-nc-nd Attribution License. Copyright © 2020 Hikari Ltd.

Abstract

In this paper, we introduce the study of a new solution to gravitational singularities by violating the energy conditions of the Penrose Hawking singularity theorems. We consider that a shift to negative and takes place at the event horizon of black holes, justified by the original, singular and exact Schwarzschild solution. These negative energies are supported by relativistic particle physics considering the solutions of the , by which time transformations shift to negative energy states. In either or Newtonian mechanics, these negative masses are predicted to be repulsive. It is explained that the model fits actual observations, and could possibly clarify the size of observed and unexplained supermassive black holes when considering the that would take place inside them, where antigravity interactions would take place. An approximated solution of the model could be simulated in order to compare it with these unexplained observations.

Keywords: black holes, time transformation,

1 Corresponding author

162 Manuel Urueña Palomo

1 Introduction

The Penrose Hawking singularity theorems [15] state that, if energy conditions are violated, such as the weak, null or strong energy conditions, gravitational singularities may not arise. Considering a spacetime geometry with negative energy density would imply these violations, but there are quantum effects which seem to violate them, such as the or quantum fluctuations. It has been theorized that these negative energies and masses should interact antigravitationally, either considering a Newtonian or general relativistic approach 푚 푚 [4,18]. Newton’s equation for two negative masses 퐹⃗ = −퐺 1 2 푟̂ results in the 푟2 21 same forces as the positive mass case, but considering the acceleration equation 퐹⃗ 푎⃗ = results in opposite accelerations [12]. In general relativity, choosing the 푚 active gravitational mass constant negative, results in the approximation of the Newtonian case of repulsion [3]. Problem arises in the general relativistic approach when applied to two particles of different mass signs, resulting in the runaway paradox [3], enabling theoretical issues like machines or infinite energy sources. Negative mass black holes have been studied extensively, but only considering either a negative mass [4], another different interaction for negative masses in order to rule out the runaway paradox, such as negative masses interacting attractively [2,22], or an initial configuration of that violates energy conditions which then collapses into a negative mass [1,19]. Ordinary eternal negative mass black holes have been considered unphysical because they should contain a naked singularity, and initial negative matter configurations would interact with the nearby positive masses, leading to the problematic case of the runaway motion. Some research has studied the idea of transitions to negative repulsive masses, but only either at the −푟 region of the Kerr black hole [27], or at the event horizon of Schwarzschild black holes by a time and parity transformation (PT), but considering negative masses attractive under an unjustified choice of interaction [22]. 푃푇 symmetry transformations in black holes have been studied in [2], but assuming that negative masses preserve their original dynamics, interacting attractively. The proposed model will study the hypothesis of a transition to negative masses taking place at the event horizon by a time transformation, considering negative masses repulsive. The conclusion of time running backwards inside a black hole

On violation for shifting negative mass black holes 163

has also been suggested by studying the relationship between the arrow of time and the interpretation of the area as an entropy. [5]

2 Achieving negative masses through the Dirac equation

The existence of negative masses is not prohibited by general relativity [18], and it is consistent with the theory, although our universe is certainly filled with positive mass. We will develop a way to shift to negative masses and energies in our universe, consistent with the antichronous transformations. ’s negative energy solution from Dirac’s equation

훹(퐸, 푝⃗) = 푢(퐸, 푝⃗)푒−𝑖(퐸푡−푥∙푝) (1) with 퐸 = −√푝2 + 푚2 (for the particle at rest, 푝 = 0 and 퐸 = −푚), represented by two spinor states 푢3, 푢4, has been considered to correspond to negative masses in [10]. These spinors states are needed to explain antiparticles, which is achieved by a change to positive energies from -E → E, so that time must be changed from -t → t for the first term of the phase of the wave function (퐸푡 – 푥푝), and also must be changed from p → -p for consistency, [16], following the Feynman- Stueckelberg interpretation

𝑖(퐸푡−푥푝) −𝑖(퐸푡−푥푝) 푢3(−퐸, −푝)푒 = 푣2(퐸, 푝)푒 𝑖(퐸푡−푥푝) −𝑖(퐸푡−푥푝) 푢4(−퐸, −푝)푒 = 푣1(퐸, 푝)푒 (2)

Thus, 푢3 and 푢4 can be interpreted as negative energy solutions with negative time (particles moving backwards in time) or positive energy solutions 푣2 and 푣3 with positive time (particles moving forward in time). A time transformation is proved to change the sign of the energy of a particle, and considering T a unitary operator, it also changes the sign of the mass of a particle at rest. This can be shown by considering the Dirac operator acting on bi-spinors to give 0, when considering the negative energies for a unitary PT

퐸 + 푚 − 휎⃑. 푝⃗ ( ) 휎⃑. 푝⃗ − 퐸 + 푚 (3) in which you are forced to consider negative masses. [10] Considering T unitary would imply changing what we mean by a time transformation and time symmetry,

164 Manuel Urueña Palomo

for example, it would only switch to negative time and energies, and would not reverse momentum, so that 푇푝푇−1 = 푝. It is essential to clarify that this solution is not antimatter, since antiparticles have positive energy and mass, consistent with positive energy outcome of antiparticle- particle annihilation in our positive time universe. These negative energy and mass solutions were problematic and it was assumed at the time that all particles should have positive energy and positive mass, and that time is always positive. Mathematically, however, there is no reason to reject the negative energy result, and it could be a real physical solution. Other authors also agree that a time inversion (2) changes the sign of energy for a particle, and therefore, it transforms any movement of a particle into the movement of a negative mass particle [26]. Since we know that 퐶푃 symmetry is violated [8], for 퐶푃푇 symmetry to hold, 푇 symmetry must also be violated. The proposed symmetry breaking of time is that it shifts the gravitational interaction to an antigravitational interaction by a change in the sign of energy and mass. The Dirac equation can only be applied to fermionic matter. For the case of bosons, in order to obtain a negative energy from the ’s energy equations 퐸 = ℎ푓 ℎ and 퐸 = 푝푐 = 푐 and preserve the constants as we know them (so that 푐 = 푓휆) 휆 either a negative frequency (measured in units of time, [푠−1]) or a negative momentum (negative wavelength 휆, measured in units of length [푚]) must be considered. A negative frequency can be explained by a time transformation, and a negative momentum can be explained by a parity transformation, so that 푐 = (−푓)(−휆) = 푓휆.

3. Relativistic Antichronous Transformations

The transformations leading to negative repulsive mases can be described by the extension of the restricted or proper Lorentz group or by making use of the Poincaré group, considering the antichronous transformations of the four-component Lorentz group, which are arbitrarily thought to be non-physical. As a result of these 2 transformations, the Lorentz group should be considered, from (Λ00) ≥ 1 without the condition of Λ00 ≥ 1 so that

↑ 0′ {Λ+}: Λ0 ≥ +1; 푑푒푡Λ = +1 ′ {Λ↓ }: Λ0 ≤ −1; 푑푒푡Λ = +1 + 0 (4) ↑ 0′ {Λ+}: Λ0 ≥ +1; 푑푒푡Λ = −1 ↓ 0′ {{Λ+}: Λ0 ≤ −1; 푑푒푡Λ = −1}

On energy condition violation for shifting negative mass black holes 165

with the linear and unitary transformations describing the inversion of time, partity, or parity and time together, with the four component coordinate C

−1 0 0 0 +1 0 0 0 휇 0 + 1 0 0 휇 0 − 1 0 0 푇 = ( ) ; 푃 = ( ) ; 휈 0 0 + 1 0 휈 0 0 − 1 0 0 0 0 + 1 0 0 0 − 1

(5) −1 0 0 0 푡 휇 0 − 1 0 0 푥 푃푇 = ( ) ; 퐶 = ( ) 휈 0 0 − 1 0 푦 0 0 0 − 1 푧

Negative masses and energies are commonly believed to be nonphysical by an arbitrary choice for the T operator, considered a priori as antilinear and antiunitary [28], which holds perfectly for the universe we observe at first sight, but should be considered physically possible when trying to describe regions of the universe that have never been measured, in which our laws of physics seem to break down, such as the inside of black holes.

4. General relativity approach and supermassive black holes candidate solution

The time transformation at the event horizon proposed in the model can be justified by the original and exact Schwarzschild metric solution as derived in 1912. This was built under spherical symmetry and time translation invariance, considering 푡 and 푟 time and space coordinates, and a constant determinant of the metric

푑푒푡(푔휇푣) = −1 with the convention (+, −, −, −), resulting in

훼 푑푅2 푑푠2 = (1 − ) 푑푡2 − − 푅2푑휃2 − 푅2(푠𝑖푛휃)2푑휑2 푅 훼 (6) 1 − 푅 with the auxiliary quantity 푅 following the non-linear relationship 푅 = 3√(푟3 + 훼3) [24], so that for 푟 = 0, 푅 = 훼 = 2퐺푀 (the event horizon), 푀 is just a constant of integration [3] and 푔푡푡 = 0. It was pointed out by Schwarzschild that considering 푅 = 푟 in (6) is just an approximation, which resulted in almost the same planetary orbit solutions for

166 Manuel Urueña Palomo

which the solution was initially proposed (not the extreme case in which the distance variable is not negligible with respect to the Schwarzschild radius, as in black holes). All following coordinates transformations have been done to this 푟 approximation and no to the original one, which results in different and thus, it is not equivalent. A change of coordinates to the original solution (6) was done in [22], leading to a PT transformation at the event horizon. The approximated 푟 solution was the one presented by . Additionally, the fact that 푔푡푡 = 0 at the event horizon was pointed out by Einstein affirming that “This means that a clock kept at this place would go at the rate zero.” [11]. For having completeness at the interior, −푟 coordinates must be used, which lead to −푅 in the metric. In this case, we must make use of −훼 for consistency of 푅3 = 푟3 + 훼3, which implies a switch from 푀 → −푀. The −푟 coordinate can be thought of the negative solution of the conversion from Cartesian coordinates 푟 = ±√푥2 + 푦2 + 푧2, by which we are defining a new interior sphere for 푟 < 0. The approximated Schwarzschild solution for −푟 coordinates has been proved to be equivalent to a negative mass black hole exterior solution with repulsive interactions [4]. The same reasoning has been followed for the hypothetical −푟 interior of the Kerr metric [27]. One may also write 푐푑휏 = 푑푠 = 푐푑푡 for the rest frame of a particle, in which a switch from negative to positive 푑푒푡(푔휇푣), that is, negative spacetime intervals or negative 푑푡, is equivalent to a switch to negative for that particle. Although the approximated Schwarzschild metric in 푟 coordinates describes the interior of the black hole with its central singularity at 푟 = 0 (the assumption of the mass point by which the metric is built), it is essential to clarify that Schwarzschild only provided an external solution to stars and black holes [24], and an internal solution only valid right before the collapse of a star, because it yields an infinite result at the critical radius of the star 푟푐푟𝑖푡 = √9⁄8 푅̂ = 0.9428√3푐2⁄8휋퐺휌 [25] at 푟 = 0 before reaching the Schwarzschild radius. This infinite pressure is usually interpreted under deep assumptions (e.g. that energy conditions hold) as an indication that the solution becomes time translation dependent (non-stationary) and collapses, but it really shows that general relativity alone cannot predict what happens right after that critical radius, and consequently cannot undoubtedly describe its inside after the collapse. Assuming that a time transformation implying an antigravitational effect takes place at the event horizon violates the conjecture of the strong by differentiating the gravitational effect experienced by an observer in a non- inertial frame of reference, to the effect experienced by the observer in a gravita-

On energy condition violation for shifting negative mass black holes 167

tional , in this case, at the event horizon (internal gravitation would change to antigravitational). Excluding the gravitational shift, an infalling observer would not be able to detect the horizon in empty space, so that the Einstein’s equivalence principle would hold. The same violation takes places in [22], and quantum phase transitions of the of spacetime at the event horizon have been proposed such as in [7], leading to an interior similar to a de Sitter space, but for the case of black holes being “dark-energy stars”. Together with the violation of energy conditions, it may be only way for general relativity to avoid gravitational singularities and be consistent with observations. Consequently, following the exact Schwarzschild solution (6), at least a time transformation 푇 (a parity transformation 푃 may also occur together) can be postulated to take place at the event horizon, shifting to negative energies and repulsively masses, and leaving the exterior solution with the attractive gravitational field of (6). We can now describe the interior as a closed and contractible spacetime populated only by negative energies and masses, in which a repulsively event horizon exists, that can be interpreted as a (a time- reversed black hole). This interior must be a sphere in order to be smoothly connected to the exterior solution. Although the connection towards the interior resembles a one-way Einstein Rosen bridge, particles inside cannot travel or influence the outside. Because of this, black holes cannot be proved to be by their flux, like a recent research suggests [9]. But the negative mass interior would explain the need of (negative energy density or gravitational repulsive matter) in order to maintain a stable and transitable with an open throat, pointed out in [21]. Notice that in this research, “negative density of mass-energy” is also predicted to be repulsive. Other research has considered black holes connected to closed regions of spacetime by attaching the external Schwarzschild solution to a de Sitter space [13], but considering a limiting curvature. From outside the black hole, the gravitational field remembers the original positive mass deformation of space since the instant before the black hole is formed, causally disconnecting that patch of space at the same time in which turns the positive masses into negative masses, explaining why we perceive the black hole having positive gravitational mass and justifying its stationary exterior geometry of the approximated solution (6). Both positive and negative energy-mass modes would be split by the sign of the time coordinate. Following the principle of equivalence of inertial and gravitational passive masses having the same sign 푚𝑖 = 푚푝 [4], when an inversion in sign occurs, only negative masses must exist in this isolated region (active gravitational mass will also be dependent of this inversion so that 푚푎 < 0) and all energies of this region must be

168 Manuel Urueña Palomo

negative for keeping the laws of physics as we know them (e.g. conservation of momentum), and for causality to be preserved. Massless particles crossing the event horizon would shift their energy sign, but the absolute value of energy would be conserved. The runaway paradox would be solved due to the impossibility for negative mass particles inside to interact with positive mass particles in the exterior. Annihilation due to the opposite energy sign is also avoided by this isolation. No infinite densities could form because of the repulsive forces growing asymptotically when trying to create negative energy densities of negative masses, so no gravitational singularities would arise. The absorption of a positive mass by the black hole would result in its expelling towards the interior by the white hole in the form of a negative mass, contributing to the growth of the throat. Theoretical problems of the great differences between the Schwarzschild and Kerr black holes like the ones presented in [27] would also be solved. In the model proposed, the rotating solution does not contain either a singularity or any special region, and it only differs from the static solution in the fact that the interior content would be rotating. The new proposed model preserves the original approach of general relativity, with no modifications on its equations, except for considering negative values of time which lead to the violation of the energy conditions and the equivalence principle at the event horizon of black holes, in such a particular way that it does not rely on a theory of quantum in order to solve black holes, or result in paradoxes, inconsistencies or violations of causality (neither of FTL travel are possible in this model). It is worth noting that the no-hair theorem resembles the fundamental 퐶푃푇 symmetries of , parity and time if we add the mass symmetry 푀 [17] as a fundamental symmetry (which should be violated as stated in the model, giving rise to different results for the laws of physics under this transformation, switching to antigravitational interactions), obtaining 퐶푀푃푇 symmetry, except for the time symmetry, the key component in the description of black holes in the model. On the other side of the horizon (the interior), the white hole horizon should exist in a closed space. We can think about the interior of the model firstly as an empty sphere. For a four-dimensional universe, the empty de Sitter metric describes a closed positively curved space

푟2 푑푟2 푑푠2 = − (1 − ) 푑푡2 + + 푟2푑훺2 훽2 푟2 (1 − ) (7) 훽2 in which for 푟 = 훽 we have a cosmological horizon.

On energy condition violation for shifting negative mass black holes 169

By a stationary study of the model, the interior of can be thought as a closed region sphere with a perfect fluid of constant negative density 휌 (which substitutes the interior masses of the black hole) and pressure 푝, resulting in the metric

3 푟2 1 푟2 푑푟2 푑푠2 = − [ √(1 + 푠 ) − √1 + ] 푑푡2 + 2 훽2 2 훽2 푟2 (1 + ) 훽2 (8) + 푟2푑훺2

The proper density 휌 can be set negative by considering a violation of the null energy condition (휌 + 푝 < 0) so that

1 푅 − 푅푔 = −8휋푇 휇푣 2 휇푣 휇푣 (9) 푟2 푟2 √ 푠 (1 + 2) − √(1 + 2) 3 훽 훽 8휋푝 = 푅2 (10) 푟2 푟2 3√(1 + 푠 ) − √(1 + ) 훽2 훽2 [ ] with

3 6푀 4휋 3 −8휋휌 = 2 = 3 ; 푀 = 휌푟푠 훽 푟푠 3 (11)

and 훽 is real. It is clear to derive from (11) that 훽 = 푟푠 like in (7), and subtituying 훽2 (11) in (7) results in a horizon when 푟 = 훽. The pressure is positive and the pressure gradient is negative (random motion of particles produce tensions instead of ). It is shown with a more general case in [4] that the geodesic equations indicate that a massive test particle can never reach 푟 = 0 (in contrast to light rays), and if projected radially outward, it proceeds to infinity. This solves the gravitational singularity and also suggests that the solution should undergo a bounce. By a non-stationary study of the model, the antigravitational fluid sphere will undergo an inflation. In this case, 푟푠 would be time dependent and the equation 푟푠 = 2푀 would no longer hold (unless it refers to apparent mass), so that a new smooth function 푀(푟) must be derived in a similar way to [13]. Mass-energy would be conserved assuming the black hole is not feeding or merging with others, and since the radius 푟푠 grows with time, it would look as an increase of apparent mass from

170 Manuel Urueña Palomo

the exterior. The interior can be then thought as a homogeneus, isotropic and expanding cosmological spacetime, like other solutions to non-singular black holes by a de Sitter interior [20], of a Robertson–Walker metric

푑푠2 = −[푆(푡)]2{푑푟2 + [푓(푟)]2(푑휃2 + 푠𝑖푛2휃푑휑2)} + 푑푡2 (12) which is now time dependent. It is clear that it is a positively curved universe with 1 −1 푘 = 2 > 0 and 푓(푟) = √푘 sin(푟√푘). With the field equations (9) and the 푟푠 energy tensor of a perfect fluid

푇 = (푝 + 휌)푈휇푈푣 − 푔 푝 휇푣 휇푣 (13) and the inertial and active gravitational mass density we get

8휋(휌 + 푝) = 2푆−2(−푆푆̈ + 푆̇2 + 푘) ≤ 0 (14)

We need 푆̈ ≥ 0 so that 푆 cannot have a maximum and an expansion without recollapse occurs. One could argue that the inflation would increase the volume inside the black hole without increasing the apparent volume as seen from outside, leaving the exterior solution unaltered and leading to a certain case of the Wheeler’s “bags of gold” solution. But this contradicts the holographic principle, by which entropy contained inside a volume is proportional to its surface area and Bekenstein’s entropy accounts for the full set of black hole states. For the holographic principle to hold, the surface area seen from outside must also increase along with the inflation. This is just another mechanism of growth (independent from accreation and merging) and does not imply that information escapes the black hole. It is concluded that negative repulsive masses would create an inflation or bounce like in [19] within the interior of the contractible hypersurface. The resulting exact solution of the model would be a time dependent solution (the radius of the event horizon, for instance would be time dependent 푟푠(푡), so it would grow with time) and an approximated solution could be built for simulations. Under this model, the interior Schwarzschild metric applied to a star would yield an infinite pressure at the critical radius 푟표 = √9⁄8 푅̂, but since the star would be beneath an event horizon right after that moment, we are assumming now that a bounce takes place instead of a collapse. Thus, another mechanism of growth is proposed to be simulated and compared to unexplained supermassive black holes sizes and apparent mass measurements. The

On energy condition violation for shifting negative mass black holes 171

inflation period would be slowed down by time dilation for an external observer, so that it could be noticeable in large periods of time by an increase of the black hole’s size and apparent mass. Similar research has been carried out in [6] but assuming that the exterior solution is unaltered, and thus, cannot be simulated to be proved right or wrong. The inflationary period would be more accentuated at the beginning (when the mass density is higher), matching the prediction of supermassive black holes having formed and grown early in the age of the universe. Additionally, initial bigger stars and thus, bigger black holes, would grow faster than smaller ones, matching the star formation at the early universe, which is thought to imply much bigger stars up to 103푀☉ [23], which would have evolved into supermassive black holes today. It could also solve the unexplained bigger mass measurements of LIGO merging detections, the 3 푀☉ gap between the heaviest neutron star and the lightest black hole measurements, and other measurements such as the 70 푀☉ black hole recently discovered [17]. also suggests that the interior of the black hole contains negative energy particles. The decrease in area of the event horizon is caused by “a flux of negative energy across the horizon which balances the positive energy flux emitted to infinity” [14] by . That mechanism implies a violation of the weak energy condition which arises from the indeterminacy of particle number ̅ and energy in a curved spacetime. Negative frequency components {푓𝑖} of the ̅ ! massless Hermitian 휑 = ∑𝑖{푓𝑖푎𝑖 + 푓𝑖푎𝑖} can be thought as the negative energy particles which arise inside. [14] To preserve Hawking radiation emission for the negative mass black hole model proposed, the flux of negative energy inside must be outgoing experiencing the transformations described earlier.

5. Conclusions

It has been showed that a new model for the interior of black holes can be developed by considering that a time transformation takes place at the event horizon, which by the Feynman Stueckelberg interpretation considering the T operator unitary, switches to negative energies and masses, violating energy conditions. These negative masses are predicted to be repulsive, leading to an inflation period inside, slowed down by time dilation for a distant observer, that could be noticed from outside the black hole as another mechanism of growth. This new mechanism of growth is a candidate for solving the mystery of supermassive black holes regarding their size and apparent mass that could be simulated and compared to observations

172 Manuel Urueña Palomo

and estimations of stars and black holes in the early universe. The central singularity would be solved by the model. It would be interesting to carry out a further study of the second law of thermodynamics at the interior in the model due to its well-known relation with the arrow of time, in relation to black hole thermodynamics and entropy, and the possibilities of solutions to the information paradox.

References

[1] J. Belletete, M. B. Paranjape, On Negative Mass, International Journal of Modern Physics D, 22 (2), (2013), 1341017. https://doi.org/10.1142/s0218271813410174

[2] S. Bondarenko, Negative Mass Scenario and Schwarzschild Spacetime in General Relativity, Modern Physics Letters A, 34 (2019), 1950084. https://doi.org/10.1142/s0217732319500846

[3] H. Bondi, Negative Mass in General Relativity, Rev. of Mod. Phys., 29 (3) (1957), 423-428. https://doi.org/10.1103/revmodphys.29.423

[4] W. B. Bonnor, Negative Mass in General Relativity, General Relativity and Gravitation, 21 (1989), 1143–1157. https://doi.org/10.1007/bf00763458

[5] R. Bousso, N. Engelhardt, A, New Area Law in General Relativity, Phys. Rev. Let., 115 (8) (2015). https://doi.org/10.1103/physrevlett.115.081301

[6] H. Chakrabarty, A. Abdujabbarov, D. Malafarina, C. Bambi, A Toy Model for a Baby Universe Inside a Black Hole, The European Physical Journal C, 80 (5) (2020). https://doi.org/10.1140/epjc/s10052-020-7964-0

[7] G. Chapline, E. Hohlfeld, R. B. Laughlin, D. I. Santiago, Quantum Phase Transitions and the Breakdown of Classical General Relativity, Philosophical Magazine B, 81 (3) (2001) 235-254. https://doi.org/10.1080/13642810108221981

[8] J.H. Christenson, J.W. Cronin, V.L. Fitch, R. Turlay, Evidence for the 2π Decay of the K_2^0 Meson, Phys. Rev. Lett., 13 (1964), 138-140. https://doi.org/10.1103/physrevlett.13.138

On energy condition violation for shifting negative mass black holes 173

[9] D. Dai, D. Stojkovic, Observing a wormhole, Phys. Rev. D, 100 (2019). https://doi.org/10.1103/physrevd.100.083513

[10] N. Debergh, J-P. Petit and G. D’Agostini, On evidence for negative energies and masses in the Dirac equation through a unitary time-reversal operator, Journal of Physics Communications, 2 (11) (2018), 115012. https://doi.org/10.1088/2399- 6528/aaedcc

[11] A. Einstein, On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses, Ann. Math., 40, (1939), 922-936. https://doi.org/10.2307/1968902

[12] J. S. Farnes, A Unifying Theory of and : Negative Masses and Matter Creation within a Modified ΛCDM Framework, A&A 620, A92 (2018). https://doi.org/10.1051/0004-6361/201832898

[13] V. P. Frolov, M.A. Markov, V.F. Mukhanov, Black Holes as Possible Sources of Closed and Semiclosed Worlds, Phys. Rev. D, (1990), 383-394. https://doi.org/10.1103/physrevd.41.383

[14] S. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975), 199-220. https://doi.org/10.1007/bf02345020

[15] S. W. Hawking, R. Penrose, The Singularities of the gravitational collapse and cosmology, Proc. Roy. Soc. Lond. A, 314 (1970), 529-548. https://doi.org/10.1098/rspa.1970.0021

[16] Lancaster, Tom, Blundell, Stephen J., Blundell, Stephen, for the Gifted Amateur, Oxford Univ. Press, 2014. https://doi.org/10.1093/acprof:oso/9780199699322.001.0001

[17] J. Liu et al., A wide star-black-hole binary system from radial-velocity measurements, Nature, (2019).

[18] Luttinger, J. M., On "Negative" Mass in the Theory of Gravitation, Awards for Essays on Gravitation, Gravity Research Foundation, 1951.

[19] R. Mann, Black Holes of Negative Mass, Class. Quant. Grav., 14 (1997), 2927-2930. https://doi.org/10.1088/0264-9381/14/10/018

174 Manuel Urueña Palomo

[20] P. O. Mazur, E. Mottola, Gravitational Condensate Stars: An Alternative to Black Holes, 2001.

[21] M. S. Morris; K. S. Thorne, Wormholes in Spacetime and their use for : A Tool for Teaching General Relativity, American Journal of Physics, 56 (1988), 395. https://doi.org/10.1119/1.15620

[22] J-P. Petit, G. D’Agostini, Cancellation of the Central Singularity of the Schwarzschild Solution with Natural Mass Inversion Process, Modern Physics Letters A, (2015), 1550051. https://doi.org/10.1142/s0217732315500510

[23] M. J. Rees, M. Volonteri, Massive Black Holes: Formation and Evolution, Black Holes from Stars to Galaxies – Across the Range of Masses, 238, (2007), 51– 58.

[24] K. Schwarzschild, Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, Sitzungsber. Preuss. Akad. Wiss. Berlin, (1916), 189-196.

[25] K. Schwarzschild, Über das Gravitationsfeld einer Kugel aus incompressibler Flüssigkeit nach Einsteinsechen Theorie, Sitzungsber. Preuss. Akad. Wiss., (1916), 189-196.

[26] J.-M. Souriau, Structure of Dynamical Systems: A Symplectic View of Physics (Progress in Mathematics), Birkhäuser, 1997, pp. 189–193.

[27] M. Villata, The Matter-antimatter Interpretation of Kerr Spacetime, Ann. Phys. (Berlin), 527 (2015), 507-512. https://doi.org/10.1002/andp.201500154

[28] S. Weinberg, The Quantum Theory of Fields, Cambridge University Press, 1995, p.76.

Received: April 25, 2020; Published: May 21, 2020