Submitted as fulfillment of the academic requirements for the degree Master in Science to the School of Mathematics, Statistics and Computer Science,. We have shown that the fundamental differential equation governing the behavior of the model is of the Emden-Fowler type. These multipliers can be obtained under the various forms of the arbitrary function representing the gravitational potential under which the equation becomes integrable.
The study consists of different forms of multipliers associated with the first integrals of the equation in question. Currently, the theory of general relativity provides the best description of the behavior of the gravitational field. We find that many of these analyzes reduce to the study of an Emden-Fowler-type differential equation.
We briefly consider the spacetime geometry and matter distribution that led to the formulation of Einstein's field equations. We then transform the resulting field equation into a second-order differential equation, which is of the Emden-Fowler type.
Introduction
The Einstein field equations are generated by relating the Einstein tensor to the energy-momentum tensor.
Spacetime geometry
We use the definition of the connection coefficients in equation (2.2) to generate the Riemann curvature tensor R, which is given by. Rabcd = Γabd,c−Γabc,d+ ΓaecΓebd−ΓaedΓebc, (2.3), which in general does not vanish because the covariant derivative is not commutative. We use the Ricci tensor (2.4) and the Ricci scalar (2.5) to form the Einstein tensor G, which is given by.
This is sometimes called the Bianchi identity and generates the conservation laws via the field equations.
Matter fields
The above form is often assumed in cosmology and is called the linear equation of state. This equation of state is assumed in the description of gravitational systems in relativistic astrophysics (Shapiro and Teukolsky 1983). The field equations (2.13) govern the interaction between the curvature of spacetime and the distribution of matter.
In general, the field equations (2.13) are highly nonlinear systems of differential equations that are difficult to integrate without simplifying assumptions. For detailed information on general relativity and the formulation of the Einstein field equations, the reader is referred to de Felice and Clark (1990), Narlikar (2002) and Stephani (2004). Exact solutions of the field equations, which are useful in many physically relevant relativistic models, are given in Krasinski (1997) and Stephani et al (2003).
Differential equations, multipliers and first integrals
The multiplier approach in this study
The method of obtaining first integrals with an associated multiplier used in our study is demonstrated below. 2.24) We demonstrate the method in the given example. Specific solutions in isotropic coordinates have been found useful in astrophysical applications (Stephaniet al 2003). We generate the non-zero components of the connection coefficients, the Ricci tensor, the Ricci scalar and the Einstein tensor.
The components of the energy-momentum tensor are related to the components of the Einstein tensor to generate the Einstein field equations. We analyze two sets of transformation that enable us to express the condition of pressure isotropy in equivalent form.
Spacetime geometry
Using the coupling coefficients above, we generate the components of the Ricci tensor for the line element (3.1). We substitute the above coupling coefficients into equation (2.4), which is the general form for the Ricci tensor, to obtain the following inequality. To calculate the value, we use the components of the Ricci tensor (3.2) and equation (2.5), which is the definition of the Ricci scalar.
For isotropic coordinates, we use the Ricci tensor components (3.2) and the Ricci scalar (3.3) to generate the non-vanishing components of the Einstein tensor.
Einstein field equations
We use the Einstein tensor components (3.4) together with the energy-momentum tensor components (3.5) in isotropic coordinates to generate the Einstein field equations. In this chapter we consider the Einstein field equations in the form of isotropic coordinates, which move in a displacement-free geometry. The field equations are then expressed in an equivalent form, which may be easier to integrate.
In §4.2, we analyze the space-time geometry for time-dependent spherically symmetric gravitational fields by specifying a line element in isotropic form. The components of the coupling coefficients, the Ricci tensor, the Ricci scalar, and the Einstein tensor are explicitly generated in this section. In §4.3, we calculate the Einstein field equations by relating the components of the energy quantity tensor for a perfect fluid to the components of the Einstein tensor.
Einstein's field equations can be written in a different form by introducing new variables. In this section we discuss special transformations relevant to the relativistic gravity model. The pressure isotropy condition is also written in new variables with appropriate transformations.
Particular exact solutions are found in Chapter 6 in terms of elementary functions for the condition of pressure isotropy.
Spacetime geometry
In the above equations, the subscripts indicate the partial differentiation with respect to the temporal and radial coordinates t and r, respectively. Using the coupling coefficients above, we can generate non-vanishing Ricci tensor components for the line element (4.1). Then we calculate the Ricci scalar obtained from the non-expanding components of the Ricci tensor.
Einstein field equations
In the above, ρ is the energy density and pis is the isotropic pressure, which are measured relative to the four velocity vector ua= (e−ν,0,0,0). The system of partial differential equations (4.6) can be simplified to produce a single basic second-order nonlinear equation. System (4.6) can be extended to include the presence of electromagnetic; in this case we need to study the Einstein-Maxwell equations.
By further simplifications of (4.6), the Einstein field equations can then be written in a more compact equivalent form as. Further simplification is possible by eliminating the exponential factor eλ in the pressure isotropy condition (4.7d). Note that a solution of the differential equation (4.10) by construction yields an exact solution for the Einstein system (4.7).
When studying solutions to Einstein's field equations, equation (5.1) appears in various physical situations. For example, when n = 2, this differential equation appears when describing shear-free spherically symmetric perfect liquid solutions. In our study, we will consider only the static case with n = 2, under which the equation becomes
Leach and Maharaj (1992) indicated that (5.2) is applicable to the Newtonian systems involving plasmas, spherical gas clouds, and particle motion in an axially symmetric magnetic field.
Integration process
Multipliers for the model
- Case 1
- Case 2
- Case 3
- Case 4
- Case 5
The first integrals are the result of integrating once, to reduce the second-order equations to a first-order differential equation. The first integrals were introduced in physical problems dealing with the laws of motion and were later used in many other areas of applied mathematics. If we are able to reduce our equation from a second-order differential equation to a first-order differential equation, it becomes much simpler to solve.
By using the first integrals, we can obtain exact solutions of our systems. We use the multipliers created in Chapter 5 for equation (5.2) to find the first integrals for all the cases given in §5.3.
Formulation of the first integrals
Now differentiating (6.8) partially with respect to tox, we get. where C is a constant resulting from the integration process. 6.13).
First integrals
- Case 1
- Case 2
- Case 3
- Case 4
- Case 5
With the value of K(x) from (6.23), we see that the first integral can be further integrated to explicitly obtain y. In this case, we look at another possible form of K, which is K(x) ∈ R and has an associated form multiplier. Second, we performed a second integration to generate the solution (6.18) in §6.3.1, and no derivatives of y are present.
Our goal in this thesis was to investigate spherically symmetric spacetimes and the Einstein field equations in relativistic astrophysics. Our main goal was to generate new exact solutions of the Einstein field equations with isotropic pressures. Because Einstein's field equations are generally highly nonlinear, we used new variables to transform the field equations into equivalent forms.
We have shown that the fundamental equation governing the gravitational potentials is of the Emden-Fowler type. We obtained several new exact solutions in terms of elementary functions by choosing specific gravitational potentials to solve the master equation. The new exact solutions are useful in many applications in general relativity, realistic stellar models, and Newtonian physics.
In Chapter 2 we briefly introduced the concepts of differential geometry and matter distribution that are essential for generating the Einstein field equations. From Einstein's field equations we derived the condition of pressure isotropy, which is a second-order differential equation with variable coefficients. In Chapter 4 we generated the Einstein field equations in terms of isotropic coordinates for the distribution of neutral, perfect liquid matter in non-static spherically symmetric spacetimes.
In Chapter 6 we used the method of multipliers to find the first integrals and their values. It is interesting to see that for a given multiplier it is possible to write the solution for y only in terms of the independent variable x. 33] Srivastava D.C., Exact solutions for shear-free motion of spherically symmetric perfect fluid distribution in general relativity, Class.
34] Stephani H., A new interior solution of Einstein's field equations for a spherically symmetric perfect fluid in shear-free motion, J. 35] Stephani H., Kramer D., MacCallum M.A.H., Hoenselaars C., and Herlt E., Exact solutions of Einstein's Field Equations (Cambridge: Cambridge University Press) (2003).
