The equations describing the hydrostatic equilibrium (mass continuity and Tolman–Oppenheimer–Volkoff) of a static anisotropic general relativistic fluid sphere are obtained in D (D⩾4) space–time dimensions in the presence of a cosmological constant. The formalism thus developed is used to study homogeneous anisotropic constant density charged fluid spheres and homogeneous anisotropic charged spheres with a neutral isotropic core in higher dimensions. For these configurations and with a particular choice of the proper charge density a complete solution of the coupled Einstein–Maxwell equations is obtained.

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Anisotropic charged fluid spheres in D space–time dimensions

Bonnor stars are regular static compact configurations in equilibrium, composed of an extremal dust fluid, i.e., a charged dust fluid where the mass density is equal to the charge density in appropriate units and up to a sign, joined to a suitable exterior vacuum solution, both within Newtonian gravity and general relativity. In four dimensions, these configurations obey the Majumdar-Papapetrou system of equations: in one case, the system is a particular setup of Newtonian gravity coupled to Coulomb electricity and electrically charged matter or fluid, in the other case, the system is a particular setup of general relativity coupled to Maxwell electromagnetism and electrically charged matter or fluid, where the corresponding gravitational potential is a specially simple function of the electric potential field and the fluid, when there is one, is made of extremal dust. Since the Majumdar-Papapetrou system can be generalized to d spacetime dimensions, as has been previously done, and higher-dimensional scenarios can be important in gravitational physics, it is natural to study this type of Bonnor solutions in higher dimensions, d{>=}4. As a preparation, we analyze Newton-Coulomb theory with an electrically charged fluid in a Majumdar-Papapetrou context, in d=n+1 spacetime dimensions, with n being the number ofmore » spatial dimensions. We show that within the Newtonian theory, in vacuum, the Majumdar-Papapetrou relation for the gravitational potential in terms of the electric potential, and its related Weyl relation, are equivalent, in contrast to general relativity where they are distinct. We study a class of spherically symmetric Bonnor stars within this theory. Under sufficient compactification they form point mass charged Newtonian singularities. We then study the analogue-type systems in the Einstein-Maxwell theory with an electrically charged fluid. Drawing on our previous work on the d-dimensional Majumdar-Papapetrou system, we restate some properties of this system. We obtain spherically symmetric Bonnor star solutions in d=n+1 spacetime dimensions. We show that these stars, under sufficient compactification, form d-dimensional quasi-black holes. We also show that in the appropriate low gravity limit theses solutions turn into the solutions of Newtonian gravity, i.e., they are quasi-Newtonian Bonnor stars. In this connection, we note that the star solutions in Majumdar-Papapetrou Newtonian gravity, when contrasted to those solutions in Majumdar-Papapetrou general relativity, display clearly the branching off of the high density objects that may arise in the strong field regime of each theory, mild singularities in one theory, quasi-black holes in the other. Another important feature worth mentioning is that, whereas there are no solutions for Newtonian or relativistic stars supported by degenerate pressure in higher dimensions, higher-dimensional Bonnor stars, supported by electric repulsion, do indeed have solutions within Newtonian gravity and general relativity. So the existence of stars in higher dimensions depends on the number of dimensions itself, and on the underlying field content of those stars.« less

Bonnor stars in d spacetime dimensions

Using the Einstein field equations in the presence of a cosmological constant, the equations which describe the hydrostatic equilibrium of a static anisotropic fluid sphere are obtained inD (D≥4) space-time dimensions. With suitable transformations, the equation of mass-continuity and of hydrostatic equilibrium are given in a non-dimensional form. The formalism thus developed is used to study homogeneous charged fluid spheres in higher dimensions and for these configurations a complete solution is obtained.

Static anisotropic fluid spheres inD space-time dimensions

The density profiles of dark matter halos are often modeled by an approximate solution to the isothermal Lane-Emden equation with suitable boundary conditions at the origin. It is shown here that such a model corresponds to an exact solution of the Einstein-Maxwell equations for charged dust.

Isothermal spheres and charged dust

We calculate the binding energy of a Schwarzschild gas sphere in D space time dimensions. To the post-Newtonian approximation we demonstrate that the interior Schwarzschild solution does not exhibit an instability before the Schwarzschild radius is reached for D = 4. For D = 5, 6 we find binding energy is negativ to the post-Newtonian approximation. Extending the approximation to the post-post-Newtonian approximation we find that the post-post-Newtonian contribution to the binding energy of a Schwarzschild gas sphere is positive for the cases D = 4, 5, 6. The effect of the post-post-Newtonian contribution to the binding energy on the stability of higher dimensional gas spheres is discussed.

The binding energy of a Schwarzschild sphere in D space-time dimensions

A criterion for stability of a charged sphere is obtained by use of two different methods. The result is applied to a charged dust in order to investigate its stability.

Miroru Omote

Papers

A criterion for the stability of a charged sphere in general relativity