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

 

R. N. Patra ¹

 

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

 

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 As a result, we anticipate that the universe's current four- dimensional spacetime could have been replaced by a higher-dimensional spacetime.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

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)