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 Egravity 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 quantummechanical 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 nonrotating 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 "Egravity theory" (Linker 2016). Egravity 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 Egravity 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 Egravity in 4 dimensions has the form
with the generators of the Esemigroup ( denotes the Esemigroup 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 Esemigroup generators is equal to the empty set, it holds . In other words, if there is a region, where only 2dimensional 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 2dimensional surfaces. On every spacetime point the same number of these 2dimensional surfaces exist. May be the number of 2dimensional 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 2dimensional 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 Egravity theory predicts that a Black Hole has an entropy that is approximately proportional to the event horizon area.
The main criticism on Egravity theory was the prediction of nonelastic elementary particle scattering (i.e. no conservation of energy and momentum) . However, Etheory 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 nonelastic scattering takes place on scales in orders of Plancklengths. 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, Egravity 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 gravityaffected 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 selfenergy contributions and the Loop contributions. Gravitational effects play a role if the electronpositron pair in a Loop contribution or an electronphoton pair in a selfenergy 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 nongravitational 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) quantumgravitational effects can be measured since particles are coming very close.
Experimental validation of Egravity theory is possible in experiments with dense matter. For example, collider experiments with protons can be performed to measure Egravity corrections. Protons are composite particles (consisting on three quarks) where quarkgluon interactions become very frequent. Also gravitymodified 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 Egravity theory.
Conclusions
Egravity 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 Egravity theory is also possible by performing accelerator experiments. The physics of very dense systems can be understood more precisely when the predictions of Egravity theory are proven.
References
Rovelli, C. "Black Hole Entropy from Loop Quantum Gravity." Physical Review Letters, 1996, 77 (16) : 32883291. doi: 10.1103/PhysRevLett.77.3288
Ashtekar, A. et al. "Quantum Geometry and Black Hole Entropy" Physical Review Letters, 1998, 80 (5): 904907. doi: 10.1103/PhysRevLett.80.904
Carlip, S. "Entropy from Conformal Field Theory at Killing Horizons" Class.Quant.Grav., 1999, 16: 33273348.
Linker, P. "Egravity theory.", The Winnower, 2016, 3:e145441.18359. doi:10.15200/winn.145441.18359
Reviews
Showing 6 Reviews

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 
1
it is nice Theoretical Work, 'EGravity theory ' its a new born baby of QG. What is next level of this idea?
best of luck Mr. Patrick Linker
I will try to promote my theory through physics science. Maybe some further papers about Egravity theory and its applications are published.


1
That is very good
Can you tel us about mathematical details for your paper please 😊
The mathematical details of Etheory are given here: https://www.thewinnower.com/papers/3339foundationsofetheory
Mathematically, this paper bases on the decomposition of the equalizer set corresponding to the commutative semigroup S into exact, nonexact 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 2dimensional 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.


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License
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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 nonclosed the expression \Pi := \delta I(s_0 ... s_4) is. Due to the structure of the Esemigroups it holds in general \delta \Pi \neq 0.