Analyzing Black Hole super-radiance Emission of Particles/Energy from a Black Hole as a Gedankenexperiment to get bounds on the mass of a Graviton

1) abeckwith@uh.edu, Chongqing University department of physics; Chongqing, PRC, 400044; Abstract Although very unlikely to be observed, the phenomena of particle emission by super radiance of particles/energy by a black hole is examined as a thought experiment (gedankenexperiment). In doing so, the idea is to come up with bounds to the mass of a graviton. Values for the following perturbations of space-time represented as metric gμν being perturbed from flat space values by 00 h , 0i h , and ij h make the case, due to the mass dependence of the


Introduction: Massive gravity and how to get it commensurate with black hole physics?
In general relativity the metric gab(x, t) is a set of numbers associated with each point which gives the distance to neighboring points. I.e. general relativity is a classical theory. As is designated by GR traditionalists [1], the graviton is usually stated to be massless. With spin two and with two polarizations. Adding a mass to the graviton results in 5 polarizations plus other problems [2]. What this document will do will be to try to establish massive gravitons as super -radiant emission candidates from black holes [3] and in doing so provide another frame work for their analysis which would embed them in GR. In doing so, one should keep in mind that this is a thought experiment and that the author is fully aware of how hard it would be to perform experimental measurements. In coming up with criteria as to graviton mass, we are also, by extension considering the Myers-Perry higher dimensional model of black holes [4] and commenting upon its applications, some of which are in [5]. All of which start with the implications of dE/dt < 0, leading to 'leakage' from a black hole. i.e. energy of the black hole 'decreases' in time.
h , 0i h , and ij h [ 4 ], thereby making the case, due to the mass dependence of the black hole, that super-radiance would almost certainly not be observable but would firmly embed massive gravitons in GR in spite of the view point offered in [1]. To do so would mean that [3] Then, In the case of black rings, and other such exotica, in higher dimensions, [5], 0 a ≠ , and yet in the case of pure singularities, we have that , instead, 0 classical BH a −  → . Leading to our first result, that classical black hole physics , if the mass m is not zero, would not have a restriction on the mass, relative to the frequency of emitted material due to [3], but that the situation would change if 0 a ≠ , leading to our first theorem. Theorem 1. If 0 a ≠ , then the bound on m, due to super-radiance from a black hole is If 0 classical BH a −  → , and 0 m ≠ , and frequency of emission from the black hole is not zero then the bound on m effectively does not exist, ie.
Now what if we set the mass in Case 1 and Case 2 as due to and being a massive graviton? Note that then Case 1 is then implying there is a tendency toward ultra low GW frequencies from emitted black holes ? So then we go to our second theorem Theorem 2 In the case that the mass, m is of a massive graviton , of about 10^-62 grams, then the extension of Case 1 of Theorem 1 leads to g graviton m a ω ≤ (8) Since the frequency of a graviton is non zero, this would lead to, in black hole physics, a statement as to the interior structure of black holes which could be experimentally inferred. We next then reference Perturbation models of space time, i.e.

Discussion of Myers-Perry BH models in our thought experiment.
The subsequent values by 00 h , 0i h , and ij h make the case, due to the mass dependence of the black holes in the Myers-Perry black holes, that although we can deduce the two theorems above, experimental verification will be a challenge. [ The coefficient d is for dimensions, usually 4 or above, and in this situation, with angular momentum The 0 i T above is a stress energy tensor as part of a d dimensional Einstein equation given in [3] as Also, the mass of the black hole is, in this situation scaled as follows: if µ is a re scaled mass term [3] ( ) More generally, the mass of the black hole is a by product of Eq.(12) and is written as We will next go to the minimum size of a black hole which would survive as up to 13.6 billion years, and then say something about the relative magnitude of the magnitude of the terms in Eq.(9) and then their survival today, and what that portends as to the strength of signals which may be received. The variance of black hole masses, from super massive BHs to those smaller than 15 10 grams will be discussed, in the context of Eq.(9), and stress strength, with commentary as to what we referred to earlier, namely strain for detecting GW is given by h(t) given below, with D ij as the detector tensor, i.e. a constant term, so that by [2], page 336, we write This Eq. (15) means that the magnitude of strain, h, is effected by Eq. (13) ,Eq.(14) and its magnitude, seen next.

Discussion of the magnitude of Eq.(9) and its links to Eq.(15) via scaling arguments.
As stated earlier, it , the magnitude of strain mentioned in Eq. (15), depends upon the allowed mass of a black hole. The arguments in this section proceed to give threshold values as to the strength of a signal, given by Eq.
(15) above, and to talk about consequences for such magnitudes.
Starting this out requires that we estimate what the mass of a black hole has to be to last 13 For a stellar black hole, as given by Ford [7] this would be 75 10 10 grams ω  i.e. vastly larger than the mass of the universe, which is insane, so we note that black holes of the size of the sun, namely 18 It is easy from inspection to infer from this that most early formed black holes would not be accessible and that only the giant ones would do. Note however, that stochastic noise from the black holes would remove almost all chance of experimental confirmation of Theorem 1 and Theorem 2 above. I.e. information / energy which is lost from super radiance would allow us to understand and perhaps reconcile why the entropy of super massive black holes, could be larger than the usual calculations for entropy for the entire 4 dimensional universe, i.e. see Carroll [8], and we will then propose a solution based upon an extension of free energy arguments given in [9].

Details / discussions as to how to use Theorem 1 & Theorem 2 to understand how entropy of a SMBH could be larger than the entropy of the Universe (4 Dimensional)
Carroll in [8] gave a cogent book on GR states that the entropy of the universe is of the order of magnitude (non dimensional units) for four dimensional space-time Typically, though, entropy of super massive black holes is calculated as leading to a many times larger value for entropy of the entire universe via [10], namely as given in that reference, and summed up to be a larger value, i.e. using holographic arguments [11] and the last page of [10] Smax ∼ SCEH(t → ∞) = 2.88±0.16×10^122 k (22) Given that there are at least one to ten million SMBHs, usually in galaxies, this would lead to by [10] at least for a super massive black hole in the center of a galaxy , Eq. (22) will lead to: This value for a super massive black hole is likely for a higher dimensional black hole, i.e. equal to or possibly more than 5 dimensions. As given by Gregory [ We claim, that the embedding of black holes in five dimensional space time is a way to make a connection with a multiverse, as given in the following supposition [14] 6: Extending Penrose's suggestion of cyclic universes, black hole evaporation, and the embedding structure our universe is contained within, i.e. using the implications of Eq.(24) for a multi verse.
That there are no fewer than N universes undergoing Penrose 'infinite expansion' (Penrose, 2006) [15] contained in a mega universe structure. Furthermore, each of the N universes has black hole evaporation, with the Hawking radiation from decaying black holes. If each of the N universes is defined by a partition function, n value, will be using (Ng, 2008) f entropy n S~. [16] . How to tie in this energy expression, as in Eq. (24) will be to look at the formation of a non trivial gravitational measure as a new big bang for each of the N universes as by ⋅ ) ( i E n the density of states at a given energy i E for a partition function. (.Poplawski, 2011) [17] { } Each of i E identified with Eq.(26) above, are with the iteration for N universes (Penrose, 2006)

Using free energy to understand a phase transition to massive gravitons
What is done in Theorems 3 to Theorem 5 is to come up with a protocol as to how a multi dimensional representation of black hole physics enables continual mixing of spacetime [18] largely as a way to avoid the Anthropic principle, as to a preferred set of initial conditions. We will then, largely based upon the [9] linkage of free energy , its derivatives, and entropy, attempt to understand from first principles as to why Eq.(23) has such an enormous entropy. We do this, assuming each asorbing black hole eventually will radiate particles and energies as given in [ 3 ] . In [9] there is a well developed protocol as to linking free energy, and entropy, (which is a way of replacing the [10] derivation of [19] as given in [10] using Bekenstein-Hawking horizon entropy equation) , for individual black holes, as to explaining how each individual black hole could have the enormous entropy as given by Eq.(23), and not by the assumption given in [19] with the assumption made of dividing Eq.(22) by 10 million as was done, earlier. In 1994, the Free energy and entropy of black holes [20] was explored by Hochberg with, if the function Z is a partition function, then In [20] there is a consideration as to alleged transformation from a hot flat Euclidian space to a colder space, In [10] there is discussion of a phase transition from a hot to cold flat space, i.e. the main point being that such a transition is a defacto phase transition. Of second order. In [10] the critical temperature is given as by its [1] Eq.(16) as: Up to a degree of proportionality, we assert that the numerator of Eq. (32) is within modulo relations the same as Eq. (2). And that this means that we can use Eq.(2) after using Eq.(32) to fix a value for the key parameter inputs into both Theorem 1 and Theorem 2, and in doing so begin to work with obtaining values for bounds to the mass of a graviton and its relationships to frequency of the graviton.

Using free energy to understand a phase transition to massive gravitons refined. i.e. the role of Appendix A in terms of partition functions. Future bridge to quantum mechanics?
Below in Appendix A is a way to include a reconciliation with Quantum mechanics, in nucleation of a massive graviton. This partition function , with additional work will be included in disterning the nature of the free energy , as either purely classical, or with quantum features. This will go a long way toward eventually reconciling if the graviton, as a massive particle, is in sync with both GR and quantum mechanics.

Conclusion. Is QM imbedded in a semi classical structure ? How about black hole physics, too ?
We argue that further refinements of Theorem 1 and Theorem 2 are a way to ascertain this question, i.e. and to also answer if gravity is semi classical. If gravity is indeed semi classical, then the entropy as given by Eq. (24) for higher dimensional black holes is likely due to a superstructure which will embed quantum mechanics within a deterministic structure. Furthermore, it will also tie into the question of a single universe repeating its self versus a multiverse, as was gone over in this paper, and also in [14] . And all this will require for implementation is making use of the following: The particle per phase state count is, (Maggiorie, 2000) [