Analysis of the four dimensional sphere being metric of einstein's equation

Received: 27-Mar-2023, Manuscript No. puljpam-23-6428; Editor assigned: 28-Mar-2023, Pre QC No. puljpam-23-6428 (PQ); Accepted Date: Jul 20, 2023; Reviewed: 29-Mar-2023 QC No. puljpam-23-6428 (Q); Revised: 30-Mar-2023, Manuscript No. puljpam-23-6428 (R); Published: 31-Jul-2023, DOI: 10.37532/2752-8081.23.7(4).225-226

Citation: Felde KZ. Analysis of the four dimensional sphere being metric of Einstein�s equation. J Pure Appl Math. 2023; 7(4): 225-226.

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Abstract

We evaluate Einstein‘s equation with metric being a four dimensional sphere.

Key Words

Metric; Sphere; Four dimensional

Introduction

We depart from Einstein‘s equation with metric being a four dimensional sphere., i.e.,

Einstein’s equation is discussing a four-dimensional relativistic field. From mechanics and electrodynamics we know that equations that are describing reality are containing two dimensional derivations of a physical, like, f.e., the mechanical force is

F = mẍ.

We are going to develop a proposition regarding the content of Einstein’s equation, specialized on the metric that is a four dimensional sphere.

First of all the evaluated metric seems to lead to a vary new solutions. The solutions are quantized. It appears that such quantized solutions are inherited from Einstein’s equation.

From now on we are using the term metric, being the fourdimensional sphere. We depart from Einstein’s equation, i.e.,

In [1] we discussed solutions of Tμv being the derivation of an electromagnetic field and moreover the thermodynamical energy.

Later on our goal is the discussion of a massive quadropole. The solution of those equations are the energy of photons, gravitons and that temperature is carried by gravitons.

The last fact is a result by Tμv the derivation of temperature, with respect to the entropy [1] and moreover the displacements of mass showing here. The ladder is easy to interprete: Mass that is becoming a quadrupole has here the solution identified as the graviton.

Photons, gravitons, E, T, ω, m_xy are quantized where m_xy is a massive Quadrupole.

Einstein’s equation is the result of a derivative of a matric that describes the geometrical space and the derivative of some term describing mass or energy.

Our results are expressing the economic principle, that any system that is able to follow a geodesic is expressing nature.

The evaluation of the energy-stress of a quadrupole

We are compressing mass of some particle or particles, f.e., so the body looks like two epllipses in directions x and y. Being more precise, δm_x = m_x - m_x0 = 1, δm_y = m_y - m_y0 = 1. Using Pythagoras’ theorem, we observe that:

and this results in

Evaluating the massive quadropole in Einstein’s equation

Einstein’s equation of a quadropole is yielding:

We have introduced natural numbers, i.e., n, ml = 2δ²m_xy. As an example we choose a system of atoms of radius r = 1 . 10⁻¹⁰m and c = 3 . 10¹⁰m/s. By inserting those numbers into (5) we receive:

with n = 1,2,3 … is number of gravitons and ml =1,2,3… is the number of Dirac Fermions carrying mass. (7) is showing several problems.

First of all the L.H.S. is interpreted as the product of spin and the number of Dirac Fermions, whereas the r.h.s. is some quantity interpreted as gravitons, since of 8r. Moreover the factor 9 cannot be interpreted. It is pertubating an interpretation.

Thus, we decide to look at the equation if we use natural units, i.e., h = c = 1. We depart from (6). This is leading to:

With r² = 1 . 10⁻²⁰m² and some rearrangement, we arrive at:

where k_B is the Boltzmann constant which is proposed being depending on the system. In [1] we have discovered, that

We make the discovery that n can be identified as the entropy of the system, i.e.,

Thus the entropy of a quadropole of a massive particle is interpreted as the surface of a fourdimensional sphere. This is remarkable, by computing the Einstein equation of quadropoles of a massive body we receive its entropy as the surface of the object.

This is interesting, since the Black Hole entropy is believed to be the event horizon of its volume. Furthermore (10) is showing that entropy and temperature are quantized.

Our proposal on black holes and dark matter

The formula of entropy by Bekenstein and Hawking is given by

From [1] we know that:

Where we added for our purpose the last term on the r.h.s.

A Black Hole is called black, since it totally is absorbing light. Regarding (12) we need to set dω = 0. It follows:

We now look for the eqilibrium, e.g., dF = 0. This results to

We remind the reader that here

Thus pressure is of a factor 1020. This pressure is made of gravitons revealed by (14).

Recall, we propose that k_B is system depending. We observe that a black body, possibly a Black Hole can grow all the time, and will always be in equilibrium. This is avery interesting result.

Dark Matter is a riddle, since there’s a discrepancy between objects that are emitting and absorbing light and matter that is absorbing and emitting gravitons. That’s the theory. We propose something different. But first a mathematical analysis. We depart setting dF = 0 and we arrive at

We observe that in equilibrium the volume of the system is growing by pres- sure that is build on the system by photons and gravitons. Our result is revealing that Dark Matter is a natural system, that is emitting gravitons and is emitting photons. Why 85% of Dark Matter is made of mass that is emitting and absorbing gravitons is still a riddle.

Proposition

Einstein’s equation is revealing fermions that are able to emit and absorb gravitons as well as photons in the same system. It appears, that Black Holes and Dark Matter are showing extensions of their volume, even in equlibrium, i.e.,dF = 0.

Discussion

The used metric is revealing many solutions, that already known.

Without quantum equations we find natural solutions that are quantized. Moreover we found a deeper truth for the cause of temperature. We have used our metric and 8rTμv being the derivative of temperature, and by interpreting the geometrical spin 2 [1]. 4π of a graviton, and which is known to be 2. Moreover we found that the Boltzmann constant depends on the system.

Here we proposed that Einstein’s equation has other solutions than yet known. A quadropole of mass is leading to gravitons that are also lead to the known phenomenon discovered by Bekenstein and Hawking, which is proposing that entropy of Black Holes is basically the surface of the Black Hole, that known as the event horizon.

The same formula can be used to propose that Einstein’s equation is able to combine photons and gravitons to be orginated from the same particle, which is true, since fermions have mass and carry charge.

References

1. K. zum Felde; J. New interpretation of Planck constant and Boltzmann constant. J Pure Appl Math. 2023;7(2); 200-201