SOME PHYSICAL CHARACTERISTICS OF A FIVE-DIMENSIONAL MASS SCALAR ELECTROMAGNETIC COSMOLOGICAL MODEL

Some Physical Characteristics of a Five-Dimensional Mass Scalar Electromagnetic Cosmological Model

R. N. Patra ¹

¹P.G. Department of Mathematics, Berhampur University, Odisha, India

		ABSTRACT
		In this paper we are interested to study some important physical aspects of a five dimensional space time which is attained by the interaction of magnetic field and zero mass scalar field in Einstein’s theory of gravitation, where the cosmic parameters X & A are functions of cosmic time t .The concluding remark is focused on the singularity nullity, uniformity, energy condition and about the possession of gravitational field radiation of the space-time.
Received 12 March 2023 Accepted 13 April 2023 Published 30 April 2023 Corresponding Author R. N. Patra, raghunathpatra09@gmail.com DOI 10.29121/ijetmr.v10.i4.2023.1327 Funding: This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. Copyright: © 2023 The Author(s). This work is licensed under a Creative Commons Attribution 4.0 International License. With the license CC-BY, authors retain the copyright, allowing anyone to download, reuse, re-print, modify, distribute, and/or copy their contribution. The work must be properly attributed to its author.
		Keywords: Magnetic Field, Zero Mass Scalar Field, Cosmic Parameter, Singularity, Gravitational Field

1. INTRODUCTION

Higher-dimensional cosmological models are crucial in describing the universe in its early stages of evolution. Many cosmologists have studied the mechanics of the early cosmos in the context of higher-dimensional spacetime in recent years. (The most well-known five- dimensional theory proposed by Kaluza (1921) and Klein (1926) was the first to unite gravitation into a single geometrical structure). As part of their Electromagnetic cosmology research, several writers Patel & Singh (2002), Pradhan et al. (2006), Mohanty et al. (2007), Singh et al. (2004), Das & Banerjee (1999), Chatarjee (1987), Delice et al. (2013), Al-Haysah & Hasmani (2021), Reddy & Ramesh (2019), Ingunn & Ravndal (2004), Pranjalendu & Rajshekhar (2022), Barkha et al. (2017) investigated the physics of the cosmos in higher-dimensional spacetime. The unification of gravitational forces with other natural forces is not conceivable in four-dimensional spacetime. This may be achievable in higher-dimensional quantum field theory Appelquist et al. (1987), Weinberg (1986), Chodos & Detweiler (1980). This concept is significant in cosmology because we know that the universe was much smaller in the early phases of evolution than it is today. As a result, we anticipate that the universe's current four- dimensional spacetime could have been replaced by a higher-dimensional spacetime. With the passage of time, the extra dimensions are reduced to a volume of the order of the pauk length, which is not observable at the current stage of the cosmos.

Freund (1982), Appelquist & Chodos (1983), Randjbar-Daemi et al. (1984), Rahaman et al. (2002), and Singh et al. (2004) asserted, using field equation solutions, that four-dimensional space time expands while the fifth dimension contracts or remains constant. Furthermore, Guth (1981) and Alvarez & Gavela (1983) discovered that extra dimensions generate a huge quantity of entropy during the contraction phase, providing an alternative solution to the flatness and horizon problem when contrasted to the conventional inflationary scenario. The Kasnas were first studied when Albert Einstein applied his general theory of relativity to the structure of the entire universe. Different domains of gravitation in the form of tensor equations are used to investigate various kinematical phenomena properties of the universe. Bloch and colleagues Bloch et al. (2023) investigated the scalar dark matter-induced oscillation of a permanent magnet field; Krongos & Torre (2015) General Relativity Geometrization Conditions for Perfect Fluids, Scalar Fields, and Electromagnetic Fields. Kashyap (1978) investigated the interaction of an electromagnetic field and a scalar field in a cylindrically symmetric space- time. Rao, Tiwari, and Roy solved Einstein's equations for coupled electromagnetic and scalar forces. Rosen metric and other physicists have also made significant contributions in magnetic fields and scalar fields.

Bloch et al. (2023) have studied Scalar dark matter induced oscillation of a permanent-magnet field, Krongos & Torre (2015) Geometrization Conditions for Perfect Fluids, Scalar Fields, and Electromagnetic Fieldsin general relativity. Kashyap (1978) studied about couple electromagnetic field and scalar field in cylindrically symmetric space-time. Ayyangar & Mohanty (1985), Banerjee & Bhuli (1990) found out solutions for coupled electromagnetic and scalar field for Einstein-Rosen metric and other physicists have also done remarkable works in magnetic field and scalar fields.

The magnetic field plays an important role in the energy distribution of the universe as it contains highly ionised matter. The scalar field represents matter field with spin less quanta. The zero-rest mass scalar field describes long range interaction.

In this paper, we are interested to study the various physical property of the five-dimensional space-time with magnetic field and scalar fields.

2. Metric & Field Equations

Here we have taken the Kaluza-Klein space time described as

Equation 1

Where

Equation 2

Einstein’s gravitational field is given by

Equation 3

Where is the universal gravitational constant. Here is the energy-momentum tensor of source free electromagnetic field and is the energy momentum tensor of zero mass scalar field defined as

Equation 4

Where

Equation 5

and represents scalar potential.

Equation 6

Here the gravitational potential does not exists.

Thus Equation 7

Similarly

Equation 8

Equation 9

Equation 10

Equation 11

Since

Here only components of the scalar potential , exists. Let us take .

Thus also .

The existing components of are

Equation 12

Using these values of in the existing components are

Equation 13

Equation 14

Equation 15

Equation 16

Equation 17

The predetermined values of

as described in equation (3) for our metric are

Equation 18

Equation 19

Equation 20

Computing the right-hand side of equation (3), using the equation’s (7) to (17) and putting the left hand side values from (18) to (20) we get

Equation 21

Equation 22

Equation 23

Equation 24

Equation 25

Let us impose the following restrictions on the electromagnetic field to get some exact value of the cosmological parameters.

· Case-I:

· Case-II:

· Case-III:

· Case-I: As

Solving (21) to (25) we get

Equation 26

Equation 27

· Case-II:

Putting in equation (21) to (25) and solving we get

Equation 28

Equation 29

· Case-III:

Putting in equations (21) to (25) and solving we get

Equation 30

Equation 31

3. Physical Properties

1) Nulity: The null electromagnetic field indicates the propagation of e-m radiation with fundamental velocity. As per Synge (1958)

Equation 32

(Null electromagnetic field)

(Non-null electromagnetic field)

After calculation of equation (32) for all the three cases using equation (26) to (31) and using the values of , it is seen that the electromagnetic field isof non-null nature for our space-time.

2) Singularity: We study the regularity of a solution using, Bonnor (1958) who stated that a point’s (may be a point of spatial or temporal infinity) is a regular point in a natural co-ordinate system, where the following sufficient conditions are satisfied.

· is nonzero.

· and their first derivatives are finite and continuous at “s”,

· The second derivatives of are finite and continuous at “s”

For various values of and their derivatives for all the three cases and putting the values of from equations (21) to (31), it is seen that the solutions are regular at the point “s”.

3) Gravitational radiation: Gravitational field radiation of a space-time exist, if .

Using the values of for all the three cases it is seen that .

Hence the space-time possesses gravitational field radiation.

4) Curvature Scalar(R): As per Einstein the curvature scalar is defined as

Equation 33

and for our metric it takes the form.

Equation 34

Using the various values of for all the three cases and calculating we get

for case- I

for case- II

& for case- III

Where are constants. As, the variation of curvature scalar with cosmic time for all the three cases the scale factor diverges to infinity.

5) Energy conditions:

· Scalar Field: The energy condition for scalar mesan field is studied from the component of the energy momentum tensor as given in equation (10)

The different energy values of we get,

for case-I

for case-II

for case-III

Where are constants. As , the variation of energy of the scalar mesan field with cosmic time.

· Electromagnetic field: The energy condition of electromagnetic field is studied from the component of the energy momentum tensor as given in equation (17)

The energy values of the electromagnetic field for all the three cases after using the different values of we get

for case-I

for case-II

for case-III

Where are constants, the shows the variation of electromagnetic energy with cosmic time of our space-time for all the three cases.

· Uniformity: A space-time is uniform, if it satisfies the condition .The existing values of are given in equation (13) using the above condition, it is found that our space-time is found uniform along direction only

4. Conclusion

In this paper, we have constructed a five-dimensional space time with the interaction of magnetic field and zero-mass scalar field. Considering various cases when

Case-I:

m ¹ 0, n ¹ 0, v ¹ 0

Case-II:

m = 0, n ¹ 0, v ¹ 0

Case-III:

m ¹ 0, n = 0, v ¹ 0

Where m & n are components of scalar potential assumed to be function of “t” which is clear from equation (27) to (32). The new form of the space time is found to posses gravitational field radiation, uniformity, non-null electromagnetic field. Moreover, the singularity, curvature scalar and the energy conditions of both the fields are discussed clearly.

5. Summary

This article deals with Kaluza-Klein electromagnetic model in the presence of a mass scalar meson field. This model is obtained by solving Einstein field equations using hybrid expansion law and a relation between metric potentials. It is observed that the variation of electromagnetic energy with cosmic time of our space-time for all the three cases diverges for late time. Moreover, I found that the space-time possesses gravitational field radiation, and the space time is found to possess gravitational field radiation, uniformity, non-null electromagnetic field.

CONFLICT OF INTERESTS

None.

ACKNOWLEDGMENTS

None.

REFERENCES

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