muruep01

A time transformation T = (diag(-1, 1, 1, 1)) and a parity transformation P = (diag(1, -1, -1, -1)) together lead to a unitary proper antichronous transformation PT = (-diag(1, 1, 1, 1))

Negative energies imply negative masses, which have gravitational repulsive interactions between them whether you introduce them in Netwonian gravity together with the equivalence principle, so that inertial mass is also negative, or into GR in the mass term of the Schwarzschild solution. I quote: "Equation (14.67) shows that time reversal It changes the sign of the energy and thus the sign of the mass (14.4<»). Consequently it transforms every motion of a particle of mass m into a motion of a particle of mass -m." https://ayuba.fr/pdf/souriau1997-chapter14.pdf That is why I insist that a parity transformation may take place together with a time transformation, so that the spatial components of the 4-vector remains the same.

I dont know what kind of calculation regarding the properties of the vacuum do you expect me to show if my model does no change any of the properties of the vacuum of the exterior solutions of black holes. I already explained tha vacuum inestability is only a problem if positive and negative energy particles coexist in the same causally connected spacetime, which does not occur in my model because no interaction takes place between a positive and a negative energy particle (this would already result in inconsistencies apart from vacuum stability, such as the runaway paradox). I dont know why you bring up the topic of Hawking radiation. Hawking radiation is independent of a black hole's content, it arises from the simple existence of an event horizon (apart from the fact that there is no experimental evidence for it). I have shown that, in contrast to your knowledge of symmetries and transformations, a time transformation has a physical meaning (and it follows that it is an active transformation), which you continue to ignore and dont reply to them.

The vacuum must be stable, since in my model, only negative energy particles are present inside black holes. Thus, their exterior vacuum is exactly the same as we know it, only positive energy particles are present, and its stable. The problem of vacuum decay regarding negative enery particles only arises if one considers the coexistence of positive and negative energy particles within the same causally connected spacetime, because pairs of negative and positive energies would arise from a net zero energy space in infinite numbers, which would make it unstable. A runaway process or explosion cannot take part near the horizon. From infinity to the horizon, the spacetime is described in my model as the one we know from GR and observations, only positive energy particles exist there. Inside the horizon, and this is where the prove that no signal (electromagnetic or gravitational) can escape a black hole comes into play, no negative energy particles can interact with any positive energy particles outside. This is again clearly explained in my preprint, so I recommend reading it before starting a discussion about it.

An antichronous transformation has a physical effect. I quote: "An antichronous Lorentz transformation A +1 will change sign (among the others) to the time-components of all the four-vectors associated with the considered particle P. In particular, it will transform P into a particle P' endowed with negative energy and motion backwards in time." Recami, E., Rodrigues, W.A. Antiparticles from special Relativity with ortho-chronous and antichronous Lorentz transformations. Found Phys 12, 709–718 (1982). https://doi.org/10.1007/BF00729807 Another example is that applying a change from t->-t and r->-r in Eddington-Finkelstein coordinates to the Schwarzschild solution is equivalent to a change from M->-M for mass, which obviously changes the particles trayectories. Negative mass scenario and Schwarzschild spacetime in general relativity You are asking me to prove that no signal (electromagnetic or gravitational) can escape a black hole. This well known and the very definition of an event horizon... I previously quoted several references claiming that yes, antichronous transformations have a physical effect. This is not even an original idea of mine, I dont know why we are even discussing it since I clearly explained it in my preprint and cite the related references. Antichronous transformations are active and act on the phenomena. I quote: "Usually, also the piece (4b) [antichronous transformations] is discarded. Our aim is to show--on the contrary--how a physical meaning can be attributed also to the transformations (4b), both in the passive and in the active sense" "Let us start with the second (active) point of view, and consider an object (particle) P endowed with positive energy and motion forward in time. An antichronous Lorentz transformation A +1 will change sign (among the others) to the time-components of all the four-vectors associated with the considered particle P. In particular, it will transform P into a particle P' endowed with negative energy and motion backwards in time"

If you had read my post, you will know that what I propose is a unitary time transformation. I quote myself: I cited Weinberg to show that time transformations are arbitrarily though to be antiunitary just to avoid negative energies. My whole proposal is about negative energies. I also already explained in my post that if you only allow negative energies to exist within black holes, no positive energy particle could ever interact with a negative energy particle, and the vacuum in both spacetimes would be stable. Please read my post before discussing about it. How does one apply an antichronous transformation in a GR metric solution? You mean negative gravitational potential energy? It is not taken into account in Einstein's energy-momentum tensor (it is assumed it is already included within the nonlinear properties of the theory).

T symmetry breaking implies negative energies, which implies negative masses, which implies repulsive gravitational interactions: In relativistic quantum mechanics, changing the sign of proper time in the equation for a charged particle in an electromagnetic field is equivalent to a change in the sign of mass. In the phase of the wave function (𝐸𝑡 −𝑥𝑝), a change in time sign is equivalent to a change in the sign of energy. I cite Weinberg: “If we supposed that T is linear and unitary [⋯] for any state 𝜓 of energy 𝐸 there is another state 𝑇^−1 𝜓 of energy −𝐸. To avoid this, we are forced here to conclude that 𝑇 is antilinear and antiunitary” and "On evidence for negative energies and masses in the Dirac equation through a unitary time-reversal operator" https://iopscience.iop.org/article/10.1088/2399-6528/aaedcc In special relativity, an antichronous transformation results in negative energies. I cite “time reversal changes the sign of the energy and thus, the sign of the mass. Consequently, it transforms every motion of a particle of mass m into a motion of a particle of mass −𝑚” https://empg.maths.ed.ac.uk/Activities/Souriau/Souriau.pdf Inertial mass is the same as gravitational mass, that follows from the equivalence principle. That is what Bonnor considered: "I am then led to a universe in which all three types of mass are negative." I have "re-read, re-callibrated and reflected on the rigor of my proposal" as moderators suggested in my old dead thread. As I said earlier, no one has ever cared to extend General Relativity to include antichronous transformations, so I cannot provide a metric solution to my model of black holes (plus, it would be a complex time-dependent solution, so approximated solutions must be found). If I had come up with an approximated solution for the growth mechanism for black holes and compared to observations, I wouldnt be posting in https://www.scienceforums.net/forum/29-speculations/ since my model would have been proved right or wrong. For your questions regarding time transformations, the math has been worked out by others (see my references) and I simply provided a revision in my preprint.

The Dirac equation allow for negative energy solutions (with negative mass), which propagate backwards in time. The explanation for antiparticles in the Dirac and Klein-Gordon equations solutions is achieved through the Feynman-Stüeckelberg interpretation, which consists of a change from negative energies to positive energies −𝐸 → 𝐸, in which time must also be shifted as −𝑡 → 𝑡, for consistency of the phase of the wave function. This negative mass solution is not antimatter, since antiparticles have positive energy and mass, consistent with the positive energy outcome observed during annihilation of a matter particle and its counterpart antimatter particle at the positive time region of the universe. Plus, recently we know that antiparticles have positive gravitational mass thanks to the ALPHA experiment at CERN. As you said, due to the violation of CP symmetry, T is a broken symmetry. I propose the CMPT symmetry. T symmetry breaking implies changing the gravitational interaction from attractive to repulsive by changing the sign of mass and energy.

Of course, as I said, all transformations of the proper orthochronous and antichronous transformations can be attributed to the meaning of actual transformations between two physical inertial frames, and ordinary relativistic laws are covariant under them. But no one has ever done it. I dont see how the Standard Model is in conflict with T. Yes, it results in negative energy particles which are not observed, and cannot live in the same spacetime as positive energy particles due to vacuum stability, but there is no mathematical conflict. In my model, since these negative energy particles only live inside black holes, there is no possible physical conflict.

I omitted the math in this "simple" explanation of my proposal. You can check it in my preprint I attach to this answer. Of course, I cannot provide a metric solution since I believe GR should be rewritten with the full Lorentz group (including antichronous transformations), and as far I know, no one has ever cared to do it. On Regular Negative Mass Black Holes Under Unitary Antichronous Transformations.pdf

Hello everyone, I will present my speculation about black hole interiors. There is unambiguous evidence for the existence of supermassive black holes (SMBHs) at the center of galaxies. They range from millions to thousands of millions of solar masses. Accretion (infalling matter into the black hole) and black hole merger estimations are not enough to explain the observed SMBHs masses. Another mechanism of growth for black holes (BHs) will be proposed based on an alternative non-singular description of their interior, which violates energy conditions inside by proposing a consistent shift to negative energies and gravitationally repulsive negative masses at the event horizon. This shift is the antichronous transformation of the full Lorentz group and the unitary parity-time PT transformation of relativistic quantum mechanics. The inside negative mass interaction does not result in the runaway paradox and the transformation at the event horizon respects Einstein’s equivalence principle. The interior description of BHs has not been observed or proved to match GR predictions, and it relies on the assumption that curvature can be arbitrarily small at the event horizon and further beyond, so that GR should still be valid and give correct predictions because it does them in its classical domain. In principle, there is nothing locally distinguished or remarkable at the event horizon. Alternative interior descriptions can be hypothesized by defying the assumptions of the gravitational collapse process and within GR. Physicists assume that the current description of BH interiors in GR is accurate up to very high curvature regions near the singularity, and that quantum gravity must resolve the singularity problem of geodesic incompleteness. 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 no communication of results could be done to the exterior). The Penrose singularity theorems state that, if energy conditions are satisfied in closed trapped surfaces, such as the weak energy condition (light rays are always focused together by gravity from non-negative energy of matter), gravitational singularities are unavoidable. Negative mass violates these energy conditions. Newton’s law of gravity for two negative masses results in the same forces as the positive masses case, but considering Newton’s second law of motion results in opposite accelerations, and thus, antigravity. This follows from the equivalence principle, which states that gravitational masses are equal to inertial masses. The problem arises when considering the interaction between two particles of different mass signs, in which the negative mass chases the positive mass, accelerating presumably up to the speed of light, also referred as the runaway motion, leading to paradoxes like infinite energy sources and perpetual motion machines. In GR, choosing the active gravitational mass in the Schwarzschild solution negative (a constant of integration, the active gravitational mass can take any arbitrary sign, and GR does not prohibit the existence of negative masses) results in the approximation of the Newtonian case of repulsion for test particles. William B. Bonnor showed that these problems such as violations of causality, or equivalently, the violation of the second law of thermodynamics in GR occur “only if both positive and negative masses are present”. Another argument against negative masses follows that if negative masses coexist with positive masses, they would trigger a vacuum decay. This problem arises because negative energies are also forced to exist, and pairs of negative and positive energies would arise from a net zero energy space in infinite numbers, which would make the vacuum unstable. As shown before, all theoretical problems with 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. If only negative masses and energies are allowed to exist within the event horizon of BHs, no interaction could possibly take place with the exterior positive masses, and the properties of BHs would naturally impede the problematic runaway motion interaction. The isolation prevents any potential annihilation resulting from opposite energy signs. In order to propose that negative masses exist inside BHs, a mechanism switching from observed positive mass stars prior to their collapse to negative masses must postulated. For a charged particle in an electromagnetic field, changing the sign of proper time has the same effect as changing the sign of the mass. The explanation for antiparticles in the Dirac and Klein-Gordon equations solutions is achieved through the Feynman-Stüeckelberg interpretation, which consists of a change from negative energies to positive energies, in which time must also be shifted from negative time to positive time. Thus, negative energy solutions with negative time (particles moving backwards in time) can be interpreted as positive energy solutions with positive time (particles moving forward in time), proving that time transformations change the sign of a particle’s energy in relativistic quantum mechanics. Considering T a linear and unitary operator, not only changes the sign of energy, but also switches the sign of the mass of a particle at rest because both energy and mass must be negative in order to have positive density of probabilities in the Klein-Gordon equation. The time reversal operator in quantum mechanics is not only commonly assumed antiunitary to avoid negative energies, which were problematic and it was assumed that all particles had positive energy, mass and time. In the case of CP symmetry violation, the preservation of CPT symmetry requires the violation of T symmetry. The proposed method time symmetry breaking implies changing the gravitational interaction from attractive to repulsive by changing the sign of mass and energy. This would be equivalent to defining a new broken symmetry M so that CPTM symmetry remains invariant. The Dirac equation applies only to fermions. For consistency of the proposal, all particles must shift their energy by a time transformation. In the case of bosons, obtaining negative energies requires considering either a negative frequency or a negative momentum (negative wavelength) in the equations for a photon. Thus, a parity transformation may also be required to always take place together with the time transformation for consistency. The time transformation provides an explanation for the existence of solely negative masses and negative energies within BH interiors. When a positive mass crosses the event horizon, it is absorbed towards the interior as a negative mass, so that its absolute energy value is conserved. The time transformations leading to negative masses explained in the previous paragraph within the context of relativistic quantum mechanics agree with the antichronous transformations of the full Lorentz group applied to Special Relativity. These transformations 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. Antichronous transformations can be postulated if Special Relativity is based in the whole proper group of orthochronous and antichronous Lorentz transformations. All transformations of the proper orthochronous and antichronous transformations can be attributed to the meaning of actual transformations between two physical inertial frames, and ordinary relativistic laws are covariant under them. 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. 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 particles travelling forward in time. But if the 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. One may also want to reject the antichronous transformations because they could allow causality-breaking paradoxes. Positive and negative energy states (positive and negative time, respectively) can coexist in two separated spacetimes (for instance, by an event horizon) with increasing energy potential from both sides of a common ground state of lowest energy so that causality is preserved. These violations of causality occur only if both positive and negative masses are present in the same causally connected spacetime. As stated in the previously, antichronous transformations must be implemented into GR and considered physical to study the proposed solution in a metric theory of gravity. A brief explanation will be presented onwards. The equivalence principle is often claimed to demand that any physical or noticeable change taking place at the event horizon of BHs is prohibited, because it would differentiate the effect of acceleration in flat space and the one experienced by the gravitational field of the BH in the sufficiently small vicinity. A violating phenomenon of the equivalence principle would imply an incompatibility with GR. For the proposed time transformation taking place at event horizons of BHs which switches from gravity to antigravity, an infalling observer would only be able to detect the horizon through an internal gravitational experiment, so that the Einstein’s equivalence principle: “The outcome of any local nongravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime”, would hold. But Einstein’s equivalent 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. Several indications in the current understanding of BHs already point out that time transformations take place at event horizons, which will be presented onwards. One may think that if BHs consist of negative mass, they should repel other positive masses on their outside, such as other BHs or stars that orbit them (which contradicts observational evidence). But if the event horizon is responsible for switching the initial positive masses prior to the collapse of the BH into negative masses, it causally cuts off 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 proper curvature of positive mass from the original star prior to collapse, that is, attractive gravity. But if the event horizon is responsible for switching the initial positive masses prior to the collapse of the BH into negative masses, it causally cuts off 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 proper curvature of positive mass from the original star prior to collapse, that is, attractive gravity. The suggested time transformation at the event horizon in the proposed model can be justified by the original and exact Schwarzschild metric solution formulated in 1916, built under spherical symmetry and time translation invariance The suggested time transformation at the event horizon in the proposed model can be justified by the original and exact Schwarzschild metric solution formulated in 1916, built under spherical symmetry and time translation invariance. The solution, when applied to a simplified eternal BH model, does not cover the interior of the BH, 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). As stated before, a time transformation cannot be applied to the Schwarzschild solution because GR is not built to allow for antichronous transformations. For having geodesic completeness in the solution, one may allow −𝑟 coordinates, which is equivalent to a change in the mass sign in the equation. 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. Additionally, Einstein recognized that 𝑔𝑡𝑡 = 0 at the event horizon affirming that “This means that a clock kept at this place would go at the rate zero”. It is well known that a different interior can be glued to the exterior Schwarzschild solution under certain conditions such as the overall metric being continuous and differentiable at the joint and being the interior metric a valid solution to the field equations. Consequently, there is substantial reasons to hypothesize that a time transformation T (a parity transformation P may also occur together) takes place at event horizons if antichronous transformations are allowed in a modified metric theory of gravity. The sign of the time coordinate would divide positive and negative energy and mass modes. This interior can now be characterized as a closed spherical spacetime exclusively populated by negative energies and masses, contained by a repulsively-past event horizon that can be interpreted as a white hole, or a time-reversed BH. The positive energy theorem would hold because the dominant energy condition, which is violated in this model, is an assumption within the theorem. Theoretical issues regarding the disparities between the interiors of Schwarzschild and Kerr BHs, would also be resolved: the rotating solution would lack singularities, inner event horizons or Cauchy horizons, and it would differ from the static solution only in the sense that its interior content would be rotating. Because the exterior is causally disconnected from the interior in one way (and assuming no accretion or BH merging), the interior can be then thought of as a homogeneous, isotropic and expanding cosmological spacetime of a time-dependent Friedmann–Lemaître–Robertson–Walker metric (FLRW). Geodesics indicate that a massive test particle can never reach the central point of 𝑟 = 0, solving the gravitational singularity with geodesic completeness and suggesting that the interior solution should undergo an antigravitational bounce instead of a collapse. It could be argued that this inflation would increase the volume inside the BH without increasing its exterior surface area, leaving the exterior solution unaltered and leading to a case of the Wheeler’s “bags of gold” solution (eternal static BH exterior attached to an expanding FLRW interior). This solution contradicts the holographic principle, by which entropy contained inside the volume is proportional to its surface area and Bekenstein’s entropy equation accounts for the number of total internal or micro-states, under the strong form interpretation of the equation. For the strong form interpretation of Bekenstein’s entropy to hold in the proposed model, the exterior surface area must also increase along with the interior inflation process in which entropy increases, and the exterior event horizon radius would grow with time, which would be observed from outside as an increase of apparent mass. This growth would be slowed down by time dilation for external far away observers since interior masses are close to the event horizon where gravitational time dilation takes place, so that it could only be noticeable in large periods of time by an increase of the BH’s size, and consequently, in apparent mass. Additionally, the first generation of stars, which are thought to have had between a hundred and a thousand solar masses and collapsed into BHs, would grow faster than smaller ones, matching the star size estimations in the early universe. These collapsed stars would have evolved into supermassive BHs today, solving the unexplained mass gap between stellar BHs and supermassive BHs. Moreover, the described new mechanism of growth would slow down in time as inner density decreases due to its finite negative mass content, but it would never stop (ignoring the effects of a possible interior cosmological constant). If this never-ending growth is enough to always counteract the BH decrease in size due to Hawking radiation emission, which slows down as the exterior surface grows, BH evaporation would be impossible, solving the information paradox. Any feedback in the form of opinions or suggestions is welcomed!