On Regular Negative Mass Black Holes Under Unitary Time and Proper Antichronous Transformations

There is unambiguous evidence for the existence of supermassive black holes at the center of galaxies. They range from millions to tens of thousands of millions of solar masses. This is non-trivial since there are strong constrains in how black holes can form by gravitational collapse (tens of solar masses). Accretion (infalling matter into the black hole) is limited by the Eddington mass limit and black hole merging is limited by the final parsec problem, so that gravitational collapse models are not enough to explain the observed supermassive black holes masses at the early universe, and different proposed explanations such as primordial black holes or direct dust collapse models cannot explain them either. In the following post, an alternative description of the interior of black holes will be briefly explained which attempts to solve this conundrum and the question of whether if black holes contain gravitational singularities.

The inner description of black holes has not been observed or proven to match General Relativity’s predictions. It relies on the assumption that curvature can be arbitrarily small at event horizons and beyond, so that General Relativity as we know it should still be valid and give correct predictions for the interior because it does them in its classical domain, such as the event horizon. But alternative inner descriptions can be hypothesized by defying the assumptions of the interior solution.

Gravitational singularities are problematic in General Relativity because they imply geodesic incompleteness. It is an open question whether gravitational singularities exist as predicted by General Relativity, or if a more complete description of gravity, such as a quantum theory of gravity, solves them. Quantum gravity effects would be non-negligible when quantum fluctuations become too large at very high mass densities and spacetime curvature reaches values out of the domain to which General Relativity can effectively be applied. This has led the physics community to assume that the current description of black hole interiors in General Relativity is accurate up to very high curvature regions near the singularity.

Thus, these singularities are not thought to be problematic, but this should not stop the search for alternative interior descriptions, as they are part of the universe in which experimental physics could be performed (although results could not be communicated to the exterior) and physics attempts to fundamentally understand natural phenomena. In particular, modifications to General Relativity might solve gravitational singularities without invoking the still elusive quantum gravity theory. There are many mathematical proposals to regularize black hole interior solutions, but they require unreasonable conditions with no theoretical motivation, such as a different cosmological constant than at the exterior, false vacuums, limiting curvatures, or complex electromagnetic interactions.

Roger Penrose and Stephen Hawking worked out in the 70s that gravitational singularities cannot be avoided in General Relativity. More precisely, their singularity theorems state that geodesic incompleteness arises inside black holes if energy conditions hold. Energy conditions can be summarized into the statement that negative energies cannot be observed, which is fairly reasonable since no one has ever observed negative energies (at least not in the macroscopic domain). If gravitational singularities do not form inside black holes, energy conditions must be violated.

Negative energies (equivalently, negative masses) are often brought to speculative physics because of their theoretical antigravitational interaction, both in Newtonian mechanics (considering both inertial and gravitational masses negative for getting opposed accelerations) or in General Relativity (geodesics of a test particle in a Schwarzschild metric with a negative mass parameter). There are two main problems with negative energies: first that there is no observational evidence for them as the universe is certainly filled with positive masses and second, the fact that they lead to paradoxes when considering the interaction between a positive and a negative mass particle, in which both accelerate chasing each other ad infinitum (called runaway motion by Hermann Bondi). This looks like a no-go for any reasonable physics proposal including negative masses, but what if we restrict the existence of negative energies and masses to the interior of black holes? It would explain why we have never observed them, prevent the runaway motion (nothing inside interacts with its outside) and energy conditions would be definitely violated and no gravitational singularities could arise. In this way, the properties of black holes themselves would naturally impede the problematic runaway motion interaction. Positive and negative energy states can coexist in two separated spacetimes by an event horizon with increasing energy potential from both sides of a common ground state of lowest energy so that absolute values of energy are conserved. 

Eternal negative mass black holes have been considered nonphysical due to the fact that they would exhibit naked singularities, and initial negative matter configurations would interact with the nearby positive masses, leading to the runaway motion. But what if there is a trick to propose negative black hole interiors without this problem? These so called “negative mass black holes” are different from the one proposed, since their exterior solution is also repulsive and negative masses must exist prior its formation. Our proposed negative mass black hole will have the same exterior solution as the positive mass black holes, since it is only when masses cross its event horizon (when accretion or star collapse happens) that they shift to negative (or antigravitational) masses and they cannot communicate their new antigravitational interaction to the spacetime outside. In this way, the proposed negative mass black hole interior has attractive gravity outside as observed black holes.

Another argument against negative mass follows that if negative mass coexist with positive mass and the equivalence principle holds, they would trigger a vacuum decay, because negative energies are forced to exist and pairs of negative and positive energies would emerge from a net zero energy space in infinite numbers, making the vacuum unstable. What this really shows is only that the interaction between positive and negative masses is the one problematic, and no the interaction between negative energy masses.

The problems presented before (especially the self-accelerating runaway motion) are usually used to support the idea that negative masses should not follow the interaction described before, but an analogous interaction to the electrostatic force: like masses attract and unlike masses repel. This interaction follows from the fact that particles with equal charge exchanging a particle of odd spin experience a force pointing away from each other, while particles with equal charge exchanging a particle of even spin experience a force pointing towards each other. The second case would be the case of gravity being mediated by a spin two boson (graviton) as the propagator, with the charge being the energy-momentum of the particle, and equivalently, its mass. This interaction, which is the same as considering inertial masses always positive and allowing passive and active gravitational masses to be negative in Newtonian gravity, and thus, giving up the equivalent principle, matches the interaction of a spin two field from Quantum Field Theory, although there is no experimental evidence for gravity to be an interaction mediated by a graviton or a quantum field.

As shown before, all theoretical problems of negative masses only take place regarding the interaction between positive and negative masses, for the case in which the equivalence principle holds and negative masses repel each other. Thus, the proposed model is the only consistent possibility for the existence of negative masses in our universe following the equivalence principle.

An interesting feature of negative inertial masses is that acceleration has opposite sense to a force applied to them: pushing a negative mass will make it accelerate towards the force, while pulling it will make it accelerate away from the force. This strange phenomenon is simply the classical time reversal of the process of applying a force to a positive mass (since force is an even variable that does not change sign under classical time reversal), which already hints a relationship between time transformations and negative masses, which will be explained next.

We may ask ourselves, how can only negative masses exist inside if black holes are formed by the collapse of a certainly positive mass star? In order to make any sense of this, some physical change must take place at the event horizon. The equivalence principle is often claimed to demand that any physical or noticeable change taking place at the event horizon of black holes is prohibited, because it would differentiate the effect of acceleration in flat space and the one experienced due to the gravitational field of the black hole in a sufficiently small vicinity. A violation of the equivalence principle would imply an incompatibility with General Relativity.

For the proposed physical change taking place at events of black holes which switches from gravity to antigravity, an infalling observer would only be able to detect the horizon through an internal gravitational experiment, so that Einstein’s equivalence principle: “The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime”, would hold.

What the proposed transformation would violate is the strong equivalence principle: All test fundamental physics (including gravitational physics) is not affected, locally, by the presence of a gravitational field. The strong equivalence principle can be thought of as an extension of Einstein’s equivalence principle to gravitational phenomena. But Einstein’s equivalence principle, by which fundamental non-gravitational test physics is not affected locally and at any point of spacetime by the presence of a gravitational field, is enough to formulate a metric theory of gravity in which fundamental non-gravitational physics in curved spacetime is locally Minkowskian. The strong equivalence principle, by which the laws of gravitation are independent of velocity and location, would be violated in the modified theory of General Relativity proposed to include a switch to negative repulsive masses.

What could be triggering this antigravitational shift at the event horizon? By looking where a shift from positive to negative energies in the physics of the last century, we find an interesting mechanism in the Feynman-Stueckelberg interpretation. This was conceived to deal with the negative energy particle solutions Dirac obtained in relativistic particle physics. Mathematically, however, there is no reason to reject the negative energy solutions, which could have a real physical correspondence. This negative energy particle had energy and time multiplied in its phase of the wave function, and by treating it as a particle going backwards in time, they interpret it as a positive energy particle travelling forwards in time. It is worth noting that this negative mass solution is not antimatter, since antiparticles have positive energy and mass, consistent with the positive energy outcome observed during the annihilation of a matter and antimatter at the positive time region of the universe. In fact, reversing time not only switches to negative energies but also to negative masses if the time transformation is considered unitary, which was recently proved by Nathalie Debergh.  We can cite Stephen Weinberg in his book The Quantum Theory of Fields: “if T is considered linear and unitary, for any state of energy E, there would be another state of energy -E”. We are basically defining a new time symmetry, so that instead of CPT we now have CMPT where T at least switches the mass sign M. From now on, a parity transformation P might also be required to take place along with the time transformation for consistency. But this is just an example, as there are many more clues that relate energy and time in physics. Also, for bosons (we cannot apply the Dirac equation to them) such as photons, it is enough to switch for negative frequencies or negative momentum to switch to negative energies. 

Moreover, the time transformation resulting in negative masses explained within the context of relativistic quantum mechanics agrees with the proper antichronous (or non-orthochronous) transformation of the full Lorentz group within Special Relativity. The proper antichronous transformations of the full Lorentz group are arbitrarily thought to be non-physical, and Special Relativity is based on the proper orthochronous transformations or restricted Lorentz group, dealing only with positive energies and positive times. Proper antichronous transformations can be postulated if Special Relativity is based in the whole proper subgroup of proper orthochronous and antichronous Lorentz transformations. An antichronous Lorentz transformation changes the sign to the time-component of all four-vectors for a particle on the energy-momentum space, changing a positive energy particle moving forwards in time to a negative energy particle moving backwards in time, implying a change from positive to negative masses.

If relativistic observers can only explore their causally connected spacetime along the positive time axis, a reinterpretation of these negative energy particles moving backwards in time can be done so that they are observed as positive energy antiparticles travelling forward in time. But if their casually connected spacetime, where these negative energy particles exist, only contains negative energy particles and observers moving backwards in time, and no observation from positive time observers can be performed, this reinterpretation cannot take place.

Another reason that suggests that every unitary time transformation must be accompanied by a parity transformation, is that a negative Lorentz factor can be achieved by a negative proper time, and also by a negative scaling along space dimensions by a negative length, equivalent to the mirroring of the parity transformation.

Consequently, proper antichronous transformations must be implemented into General Relativity (which is derived from Special Relativity and thus, based on the restricted Lorentz group only) and considered physical in order to study the proposed time transformation at the event horizon and black hole interior solution in a metric theory of gravity.

In order to propose a time transformation at event horizons, a global time coordinate must be fixed. This implies choosing a “preferred” frame of reference in which clocks show the “real” time. This can be done within the (experimentally indistinguishable and mathematically identical to Special Relativity) Lorentz aether theory interpretation of relativity, where time dilation is considered physical due to motion through the aether, covariance is broken because gravitational attraction changes to repulsion under a local change of time, and in which there exists a preferred reference frame: that in which the aether is at rest. A one-way speed of light experiment could in principle discern between both interpretations, although it is considered to be physically impossible to conduct. It is proposed that black holes serve the same purpose of this experiment, as they allow to set a preferred frame of reference in the proposed model.

To sum up, we have postulated that a time transformation takes place at the event horizon of black holes without violating Einstein’s equivalence principle, which switches to negative repulsive masses inside that violate energy conditions so that no gravitational singularities arise inside. And without the extension of General Relativity with antichronous transformations, we do not have a metric solution to this proposal. The proper time transformation explained is consistent if applied only to the event horizons of black holes. Several indications in the current understanding of black holes in General Relativity already point out that time transformations take place at event horizons, which will be presented onwards.

The suggested time transformation at the event horizon in the proposed model can be justified by the claseical and exact Schwarzschild metric solution formulated in 1916. Something weird appears to happen with space and time even in the standard Schwarzschild metric, where time and space “switch places”. In fact, Schwarzschild himself was uncomfortable with this behavior, so he chose some coordinates which made the event horizon singular. His solution, when applied to a simplified eternal black hole model, does not cover the interior of the black hole, since r is not allowed to take negative values by definition and makes no assumptions about a specific interior geometry (or about any central singularities). David Hilbert’s modification to the classical Schwarzschild metric considers r=0 a coordinate singularity and substitutes R for r, covering the interior and making the new r=0 the central singularity, resulting in the standard Schwarzschild solution. Additionally, Einstein recognized that g_tt=0 at the event horizon affirming that “This means that a clock kept at this place would go at the rate zero.”.

As stated before, a physical time transformation cannot be applied to the Schwarzschild solution because proper antichronous transformations are not included in General Relativity. For having geodesic completeness, one may allow -r coordinates. This is equivalent to a change in the mass sign of mass M, resulting in repulsive gravity in the negative mass black hole exterior solution. The -r coordinate can be thought of as the negative solution of the conversion from Cartesian coordinates, by which we are defining a new interior sphere for r< 0, equivalent to parity transformation. A different interior can be glued to the exterior Schwarzschild solution under certain conditions. Due to the field equations and the symmetries that the Schwarzschild solution is built on, there are great constraints on what vacuum alternatives can be postulated inside so that the exterior is Schwarzschild. This is not problematic for the model proposed because all black holes are thought to have an stellar origin and thus, they always have a nonzero energy-momentum tensor since formation. More alternatives can be postulated if one considers time-dependent solutions, which are no longer static or Schwarzschild. 

Consequently, there are substantial reasons to hypothesize that a unitary and proper antichronous transformation takes place at event horizons of black holes if proper antichronous transformations are allowed in a modified metric theory of gravity. This interior can now be characterized as a closed spacetime exclusively populated by negative energies and masses, in which a repulsive event antihorizon exists for which space flows with outwards velocity and acceleration is also outward, that can be interpreted as a negative mass white hole. Light rays from the exterior crossing the horizon could be observed from the interior. Geodesics indicate that a massive test particle can never reach the central point, solving the gravitational singularity with geodesic completeness through an energy condition violation of Penrose singularity theorems. The sign of the time coordinate would divide positive and negative energy and mass spacetimes, and the time dimension would have a defined ground state from which below it, time runs backwards. Near the horizon on either side, physical time flow slows down with respect to the ground state, but an observer could never measure this fundamental (and not relative) slowing of time in his local frame, because the same process of observation and data processing is synchronized with local spacetime, which is also slowed down in time. 

One may think that if black hole interiors consist of negative mass, they should repel other positive masses on their outside, such as other black holes or stars that orbit them (which contradicts observations). But if the event horizon is responsible for switching the initial positive masses prior to the collapse of the black hole into negative masses, it causally disconnects the interior from the exterior at the same time. Inside negative masses cannot influence the exterior geometry, which is remembered by the gravitational field with the corresponding positive curvature of positive mass from the original star prior to collapse, i.e., attractive gravity.

Theoretical issues regarding the disparities between the interiors of standard Schwarzschild and Kerr black holes, would also be resolved: the rotating solution would lack singularities and inner Cauchy horizons, and it would differ from the static solution only in the sense that its interior content would be rotating.

An interior solution which leaves the exterior solution unaltered might be considered meaningless to propose as it seems unfalsifiable. This is not the case, since any observer can travel inside the black hole and perform an internal gravitational experiment to prove it. Nevertheless, a consequence of this interior model will be discussed in the following section, by which it could be indirectly tested.

It could be hypothesized for a rotating black hole that in the same way a time transformation takes places at the event horizon, a parity transformation should also take place at the ergosurface of the ergosphere due to spacetime flowing faster than the speed of light, so that both transformations must have always taken place inside the event horizon (since the event horizon always lies inside the ergosphere). Nothing suggests that this parity transformation could switch to negative masses (only the time transformation would), and therefore, for the Einstein’s equivalence principle to hold, the parity transformation should not have any measurable physical effect on a test body (gravitational or non-gravitational). If a measurable physical effect could be attributed to that parity transformation (e.g., a charge transformation, a switch in sense of rotation, or a switch to mirror matter), the Einstein’s equivalence principle (and the strong equivalence principle) would be violated at the expense of being able to test the hypothesis and communicating infinitely far away the result of an experiment identifying the ergosurface. The unknown mechanisms of black hole astrophysical jets might be explained by one of these proposals inside ergospheres.

The regular negative mass black hole interior proposed suggests that the interior solution should undergo an antigravitational bounce instead of a collapse due to repulsive only gravity. Because the exterior is causally disconnected from the interior in one way (and assuming no accretion or black hole merging), it can be then thought of as a homogeneous, isotropic and expanding cosmological spacetime with negative mass density.

It could be argued that this inflation would increase the volume inside the black hole without increasing its exterior surface area, leaving the exterior solution unaltered and leading to a case of Wheeler’s “bags of gold” solutions (eternal static black hole exterior attached to an expanding Friedmann Lemaitre Robertson Walker interior). This solution to Einstein’s equations contradicts the strong or “volume” interpretation of Bekenstein’s entropy equation (and the holographic principle), by which entropy contained inside the volume is proportional to its surface area, and accounts for the number of total internal states of matter or micro-states, i.e., the information content of all the objects inside the hole is entirely encoded on its surface. For the strong form interpretation of Bekenstein’s entropy to hold in the proposed model (i.e., for preventing values of interior entropy larger than those allowed by the area law), the exterior surface area must also increase with the interior inflation process, in which volume (and entropy, since the number of possible scalar excitations grows with volume) increases, and the exterior event horizon radius would grow with time, which would be observed from outside as an increase of apparent mass. The rate of negative time inside the black hole, which is slower near the inner antihorizon and faster away from it, would proportionally drive the increase in entropy, assuming that entropy does not decrease with negative time.

Therefore, another mechanism of black hole growth (independent from accretion and merging), is proposed to be simulated through an approximation of the time-dependent solution described in the previous sections and compared to observed and unexplained supermassive black hole sizes and apparent mass measurements. For small black holes, this inflationary period would be slowed down by interior gravitational time dilation for external faraway observers since masses inside are close to the inner antihorizon, explaining why newly formed black holes do not explode in size and we observe stellar mass black holes with masses close to the Tolman–Oppenheimer–Volkoff mass limit. But for massive stars which collapsed into black holes, such as population III stars from high redshift epochs, inflation would have occurred much faster, explaining the existence of observed supermassive black holes at high redshift, which hints that supermassive black holes grew quickly in the early age of the universe, an observation for which there is no accepted explanation. The first generation of stars, which are thought to have had between 100-1000 solar masses when collapsed into black holes, would have grown faster than smaller ones, evolving into supermassive black holes today.