Black Hole Entropy and dynamics of quantum fluctuations predicted by E-gravity theory

Abstract

Quantum gravity theories are relevant for very high energies that occurred e.g. during the Big Bang. These theories also make statements for the entropy of a Black hole. This papers shows how the entropy of a Black Hole is computed approximately by E-gravity theory. Also quantum fluctuations affected by gravity are discussed.

Introduction

Einstein's theory of General Relativity was proven successful for gravitational phenomena on large length and time scales. However, there are still no experimental proofs for gravity acting on quantum-mechanical length and time scales. Only a couple of different approaches to quantum gravity are existing. Such theories differ in predictions and mathematical concepts. Also the entropy produced by a Black Hole is computed by such theories. It turns out that a non-rotating and uncharged Black Hole with the surface of the event horizon  carries the entropy

Various computations of Black Hole entropy are performed in physics literature. For example, the entropy formula (1) is derived from Loop Quantum Gravity (Rovelli 1996). Other computations of the Black Hole entropy are performed by considering all possible spacetime geometries (Ashtekar et al. 1998). String theory leads to the same result for Black Hole Entropy (Carlip 1999). An open question is, whether other theories of quantum gravity lead to the same result for the Black Hole entropy given by equation (1).

This research paper focuses on the most recent theory of quantum gravity called "E-gravity theory" (Linker 2016). E-gravity theory is a theory similar to Causal Dynamical Triangulation, where different discrete spacetime geometries are considered. Black Hole entropy will be derived from the partition function of E-gravity theory in this paper. Moreover, the effect of gravity to quantum fluctuations and its experimental validation is examined.

 

Theory

 

The Lagrangian density of the gravitational field in E-gravity in 4 dimensions has the form

with the generators of the E-semigroup  ( denotes the E-semigroup set), the equalizer indicator function  that is zero for any equalizers in the function argument and 1 in all other cases, the gravitational coupling constant  and the coboundary map . Equation (2) can be reformulated to (a quantity with hat is omitted)

Only if the product  is nonvanishing, the second coboundary map of the indicator function can exist. From (3) one arrives at:

From (4) it turns out that if any product of four E-semigroup generators is equal to the empty set, it holds . In other words, if there is a region, where only 2-dimensional surfaces can exist, there will be no gravitational potential density. Now the partition function of a Black Hole reads

where the sum over all  denotes the sum of all spacetime simplices and  denotes the evaluation of a function on the -th spacetime simplex.

Due to the large value of , it holds  for . Approximately, (5) can be regarded as a sum over all 2-dimensional surfaces. On every spacetime point the same number of these 2-dimensional surfaces exist. May be  the number of 2-dimensional surfaces and  the number of spacetime simplices that can carry these surfaces, it holds

Now the entropy can be set to  due to Boltzmann's entropy formula with the constant . All 2-dimensional surfaces build up the event horizon of a Black Hole (they have no gravitational energy density). Finally, by using  one obtains:

Equation (7) states that E-gravity theory predicts that a Black Hole has an entropy that is approximately proportional to the event horizon area.

 

The main criticism on E-gravity theory was the prediction of non-elastic elementary particle scattering (i.e. no conservation of energy and momentum) . However, E-theory is not a theory that predicts the existence of a perpetuum mobile of first kind. On macroscopic scales it can be approximated to General Relativity, where energy, momentum and angular momentum are conserved covariantly.

The effect of real non-elastic scattering takes place on scales in orders of Planck-lengths. On the other hand, on small length and time scales one has a very large uncertainty in energy and momentum due to Heisenberg's uncertainty relation. Even a violation of energy and momentum conservation in processes on these scales would be "blurred out" by ordinary quantum uncertainty. Therefore, E-gravity theory would only modify the fluctuations in energy, momentum and angular momentum. The total energy of a system will still be conserved in space and time average.

As an example, quantum electrodynamics in inhomogenous spacetimes is considered briefly. Here, the partition function of gravity-affected quantum electrodynamics has the form:

 

Perturbative expansion of the quantum electrodynamics term formulated on the

-th simplex  lead to all possible electromagnetic processes. There are processes, where two particles (including gauge bosons) can come extremely close together for a longer period of time. These processes are the self-energy contributions and the Loop contributions. Gravitational effects play a role if the electron-positron pair in a Loop contribution or an electron-photon pair in a self-energy contribution is only a few Planck length separated. In this case, the incoming energy required for producing such quantum corrections is much lower than in the non-gravitational case, because momentum is not conserved. Such processes cannot be detected experimentally even because of the high significance of measurement uncertainty on these length and time scales. Only for very dense quantum systems (e.g. the Big Bang) quantum-gravitational effects can be measured since particles are coming very close.

Experimental validation of E-gravity theory is possible in experiments with dense matter. For example, collider experiments with protons can be performed to measure E-gravity corrections. Protons are composite particles (consisting on three quarks) where quark-gluon interactions become very frequent. Also gravity-modified Loop contributions are a lot more frequent than in pure quantum electrodynamics systems. Hence, a possible explaination of the higher accuracy of quantum electrodynamics in comparison with quantum chromodynamics lies in the corrections predicted by E-gravity theory.   

 

Conclusions

E-gravity theory is a plausible approach to quantum gravity, because it predicts an approximately correct value for the Black Hole entropy. Moreover, it is a candidate explaination of the accuracy differences in various other quantum field theories. Therefore, experimental verification of E-gravity theory is also possible by performing accelerator experiments. The physics of very dense systems can be understood more precisely when the predictions of E-gravity theory are proven.

 

 

References

Rovelli, C. "Black Hole Entropy from Loop Quantum Gravity." Physical Review Letters, 1996, 77 (16) : 3288-3291. doi: 10.1103/PhysRevLett.77.3288

Ashtekar, A. et al. "Quantum Geometry and Black Hole Entropy" Physical Review Letters, 1998, 80 (5): 904-907. doi: 10.1103/PhysRevLett.80.904

Carlip, S. "Entropy from Conformal Field Theory at Killing Horizons" Class.Quant.Grav., 1999, 16: 3327-3348.

Linker, P. "E-gravity theory.", The Winnower, 2016, 3:e145441.18359. doi:10.15200/winn.145441.18359

 

 

 

Reviews

Showing 4 Reviews

  • Placeholder
    Abdullah Shehada
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    2

    it is good paper but can you please explain in details how you find the result: 



    and also, what is the exact equation (not approximately) of this equation


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    • Placeholder
      Patrick Linker

      The exact computation for the Black Hole Entropy with event horizon area A is given as follows:

      The term
      exp(i L_(grav))
      is 1 for L_(grav) = 0 and this is the case when the Black hole is at the event horizon. Because the event horizon has fixed area A one has the same number of simplices where exp(i L_(grav))=1 denoted by N; independent on the Black Hole geometry. Simple combinatorial consideration lead to the partition function factor (n_2)^N due to this event horizon contribution. All other regions of the Black hole are curved, hence: exp(i L_(grav)) \neq 1. Then, the partition function has the form:

      Z = (n_2)^N * sum over all curved Black Hole geometries [exp(i sum over curved regions [L_(grav)])] := Z_(curved)(n_2)^N.

      Entropy calculation:

      S = k*ln(Z) = k*ln(Z_(curved))+k*N*ln(n_2). (*)

      Clearly N*(A_(simplexsurface)) = A, hence:

      S = k*ln(Z_(curved))+k*A*ln(n_2)/A_(simplexsurface).

      k*ln(Z_(curved)) is the information entropy of spacetime and the Black Hole entropy term k*A*ln(n_2)/A_(simplexsurface) can be set to the exact value k*A/(4*l_(Planck)).

      The term Z_(curved) is 1 if only a flat spacetime would exist, but there existing all possible curved spacetimes in general; it holds:
      Z_curved = 1 + sum over oscillating values (sum over oscillating values are all curved contributions and due to the high oscillation frequency, this sum can be neglected).

      The curvature term \delta^2 I(s_0 ... s_4) measures how non-closed the expression \Pi := \delta I(s_0 ... s_4) is. Due to the structure of the E-semigroups it holds in general \delta \Pi \neq 0.

  • Placeholder
    Ankit Khunt
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    1

    it is nice Theoretical Work, 'E-Gravity theory ' its a new born baby of QG. What is next level of this idea?
    best of luck Mr. Patrick Linker

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    • Placeholder
      Patrick Linker

      I will try to promote my theory through physics science. Maybe some further papers about E-gravity theory and its applications are published.

  • Placeholder
    Awad Alawad
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    1

    That is very good
    Can you tel us about mathematical details for your paper please 😊

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    • Placeholder
      Patrick Linker

      The mathematical details of E-theory are given here: https://www.thewinnower.com/papers/3339-foundations-of-e-theory

      Mathematically, this paper bases on the decomposition of the equalizer set corresponding to the commutative semigroup S into exact, non-exact and trivial components.
      * The exact component (that lies space of exact sets \Sigma_{ex}) is only the empty equalizer set.
      * If the equalizer set is such that spacetime is made up "only on 2-dimensional surfaces" one has the space of trivial sets \Sigma_{triv}. In the trivial set, the whole coboundary map of the indicator function I(s_0...s_4) vanishes.
      * All other components belong to the space of inexact sets \Sigma_{in}, where the coboundary map applied twice does not vanish.

      The event horizon is sopposed to lie in \Sigma_{triv} (where no curvature, that is defined by the loss of exactness, occurs) and for a fixed event horizon area A the state sum Z can be factorized into

      Z = Z_{eventh}Z_{curved}.

      It is easy to see that the state sum Z_{curved} depends on components in Z_{ex} \cup Z_{in}. There is one possibility to realize absolute flatness and \sum_{\Sigma_{in}} exp(iS) (S is the action) possibilities to realize a curved spacetime. Hence one has

      Z_{curved} = 1 + \sum_{\Sigma_{in}} exp(iS).

      Combinatorial considerations lead to Z_{eventh} = n_2^A

      The term \sum_{\Sigma_{in}} exp(iS) can be neglected if the exp(iS) does oscillate with very high frequencies. Therefore, it holds:

      entropy := H = k ln Z \propto A.

  • Placeholder
    Jeet Shrivastava
    0

    Its mathematically correct and good theoy to explain black hole entropy.

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