For a cubic potential, we obtained a new series solution to the Einstein field equations describing neutral stars. Two new exact solutions for the Einstein-Maxwell system, generalizing previous results for uncharged stars, were also found. However, in the limit of strong gravitational fields we must use the Einstein field equations for an accurate description of the gravitational field.
We are faced with the difficult task of obtaining exact solutions for Einstein's field equations - a system of coupled, nonlinear partial differential equations. Thus, we must solve the Einstein-Maxwell and Einstein field equations for these respective cases. Many solutions of the Einstein-Maxwell system have also been found, but these have not yet been systematically categorized.
In Chapter 4, we attempted to solve the form of the Einstein-Maxwell system by determining the potential Z and the electric field strength E. Third, to demonstrate the possibility of unifying many seemingly different exact solutions of the Einstein-Maxwell system.
Chapter 2
Static Relativistic Stars: an introduction
- Introduction
- Differential Geometry
- Fluids and electromagnetic fields
- Static spherically symmetric spacetimes
- The field equations
- Neutral fluids
- Charged fluids
- Spheroidal Geometry
- Criteria for physically viable stellar models
- Chapter 3
In §2.5 we derive the Einstein field equations for charged and neutral perfect fluids in static spherically symmetric spacetimes. From (2.3) we observe that the covariant derivative is a generalization of the partial derivative and, when acting on an (r,s) tensor field, produces an (r,s + 1) tensor field. The connection coefficients, associated with the metric (2.15), are easily determined from (2.2); the non-vanishing components are given by.
As a consequence of (2.14), we have the conservation equation. 2.24) which can be used instead of any of the field equations. An equivalent form of the field equations is obtained if we use the transformation, (2.25a). 2.26c) where the superscript indicates the differentiation with respect to the variable x. We now consider a different form of the field equations that facilitates comparison with Newton's equations.
The field equations (2.23) now take the form which is often called the Oppenheimer-Volkoff equation. the previous expression approaches the limiting equation dp. 2.29), which is the equation of hydrostatic equilibrium for Newtonian stars. The system of equations (2.30) governs the behavior of the gravitational field for a charged perfect fluid. We briefly outline a number of conditions that realistic stellar models should meet:. a) The pressure and energy density must be positive and finite throughout the interior of the star:.
In the case of charged solutions, the electric field intensity E(r) must be continuous at r = R. e) The metric functions, e2'>" and e2v, and the electric field intensity E must be positive and non-singular everywhere. f) The speed of sound must remain subluminal throughout the interior of the star.
Computational aids and Series Solutions
- Introduction
- An existence theorem
- The Tolman VII solution
- A new series solution via the method of Frobenius
- Chapter 4
In §3.4 we state a physically well-behaved cubic form for Z, which we believe was not previously investigated. In Frobenius' method, equation (3.1) thus has two linearly independent solutions of the form. A necessary and sufficient condition for this to happen is the existence of finite boundaries.
In this case, the form of the series solutions is more complicated (Powers 1987); the crucial point is that we are assured of the existence of such solutions when (3.1) possesses regular singular points. We hope that this exercise serves as a warning of the limits of the analytical equation solving capabilities of symbolic manipulation software. Because of the proliferation of exact solutions that have been discovered, it is becoming increasingly difficult for exponents of the field to keep abreast of the vast literature being produced.
As far as we are aware, the solution of the Einstein equations (2.26), with the gravitational potential Z given in (3.9), has not been published before. The choice (3.9) ensures that the potential Z is continuous and well behaved in the interior of the star; Z has a finite value in the middle. This form of solution is not particularly useful as it is given in terms of ahypergeometric function with complex arguments.
Since the point x = 0 is a regular point of (3.10), there exist two linearly independent solutions in the form of a power series with center x = O. Therefore, the difference equation (3.12d) has been solved and all are non-zero coefficients can be expressed in terms of the leading coefficients Co and Cl. The advantage of the solutions in is that they are expressed in a series of real arguments as opposed to the complex arguments given by MATHEMATICA (Wolfram 1991).
Note that to obtain non-negative energy density p, we require a < O. The matter variables .. p and p are both finite and continuous. This feature of the solution leads us to believe that it will lead to a realistic model of neutral stars. The approach used in this chapter can also be extended to the problem of charged stars, for appropriate choices of electric field intensity E, and such a field of investigation should be pursued in the future.
A generalisation of Maharaj and Mkhwanazi
- Introduction
- Specifying Z and E
- The hypergeometric equation
- Previous cases regained
- New charged solutions
- Chapter 5
We investigate a particular form of the Einstein-Maxwell field equations by making explicit choices for the gravitational potential Z(x) and the electric field intensity E(x). Other physically reasonable choices of the gravitational potential Z are possible; we have chosen the form (4.1) since it produces a charged solution that necessarily reduces to a well-known model in the appropriate uncharged limit. The electric field intensity E in (4.5) disappears in the center of the star and remains continuous 3:nd bounded in the interior of the star in a wide area of.
The solutions to (4.8) are given in the form of the hypergeometric function F(a, b; c; X) and are categorized according to the three regular singular points. The general properties of the solutions for each of the six cases given above are discussed by Abramowitz and Stegun (1972). Thus, we have determined that gravitational potentials Z of the form (4.1) produce Einstein-Maxwell stars whose gravitational behavior is governed by hypergeometric functions.
It is clear that other cases can be created where a hypergeometric function can be written in terms of elementary functions. Our analysis in this section can be extended to other known solutions of the Einstein system (2.26). Gravitational potentials 1/ and A behave well inside the star and are continuous.
The simple form of this solution allows detailed analysis of the physical properties of the model. This equation has a general solution in terms of two linearly independent hypergeometric functions. Note that the negative values for p arise from the particular choices of Z in (4.1) and E in (4.5).
As in §4.4, these solutions arise as special cases of a hypergeometric function that reduces to simple functions. The advantage of these solutions is that they are given in terms of elementary functions, which greatly simplifies the analysis of the physical properties of the model. We expect that other published charged solutions of the Einstein-Maxwell system are special cases of hypergeometry.
Spheroidal Geometry
- Introduction
- Charged, isotropic Tikekar stars
- Anisotropic Tikekar stars
- Chapter 6 Conclusion
- Chapter 7 References
A detailed microscopic formulation of the origin of these anisotropies has yet to be revealed (Dev and Gleiser 2000). In §5.3 we use the linear transformation of §5.2 in a modified form of Einstein's field equations. We note that (5.5) can be transformed into a form reminiscent of the harmonic oscillator equation, ij + ](t)q = 0 by the transformation.
It is often difficult, or impossible, to invert such transformations, and the benefit of increased integrability of the Einstein system must be measured against this prospect. The Einstein field equations (2.13a) with anisotropic pressure become 5.9c) where Pr is the radial component of the pressure and P-.l is the tangential component. Spherical symmetry forces all the pressure components to be strictly functions of the radial coordinate r.
This term represents a force due to the anisotropic nature of the fluid; and is directed outwards when. We also tried to demonstrate the possibility of unifying apparently unrelated solutions of the Einstein-Maxwell system under special functions. The above forms of the field equations have led to physically acceptable models of dense stars, which motivated our choice.
Physical analysis of members of the class of charged solutions is said to be possible when they can be expressed by elementary functions. We have examined various forms of the field equations consistent with the spheroid model of Vaidya and Tikekar (1982). Note that it is not always possible to reverse the coordinate transformation to recover the solution in terms of the original variables; this is a drawback of the method used in Chapter 5.
First we can investigate other forms of the gravitational potential Z that satisfy the existence theorem in §3.2. It may also be possible to obtain a similar result for the charged equation (2.32c) for particular choices of the electric field intensity E. We need to identify and classify the apparently different solutions, which have already been found, for as for the hypergeometric equation.
Other choices of the anisotropy factor than the one pursued here could be used to obtain new solutions. Stefani H 1990 General Relativity: An Introduction to Gravitational Field Theory (Cambridge: Cambridge University Press).
