On the Deformation Retract of Kerr Spacetime and Its Folding

Abstract

The deformation retract of the Kerr spacetime is introduced using Lagrangian equations. The equatorial geodesics of the Kerr space have been discussed. The retraction of this space into itself and into geodesics has been presented. The deformation retract of this space into itself and after the isometric folding has been discussed. Theorems concerning these relations have been deduced.

Academic Editor:Shao-Ming Fei

Department of Mathematics, Faculty of Science, Tanta University, Tanta 31111, Egypt

Received 26 May 2014; Revised 8 September 2014; Accepted 9 September 2014; 10 November 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The real revolution in mathematical physics in the second half of twentieth century (and in pure mathematics itself) was algebraic topology and algebraic geometry [1]. In the nineteenth century, mathematical physics was essentially the classical theory of ordinary and partial differential equations. The variational calculus, as a basic tool for physicists in theoretical mechanics, was seen with great reservation by mathematicians until Hilbert set up its rigorous foundation by pushing forward functional analysis. This marked the transition into the first half of twentieth century, where, under the influence of quantum mechanics and relativity, mathematical physics turned mainly into functional analysis, complemented by the theory of Lie groups and by tensor analysis. All branches of theoretical physics still can expect the strongest impacts of use of the unprecedented wealth of results of algebraic topology and algebraic geometry of the second half of the twentieth century [1].

Today, the concepts and methods of topology and geometry have become an indispensable part of theoretical physics. They have led to a deeper understanding of many crucial aspects in condensed matter physics, cosmology, gravity, and particle physics. Moreover, several intriguing connections between only apparently disconnected phenomena have been revealed based on these mathematical tools [2, 3].

Topology enters general relativity through the fundamental assumption that spacetime exists and is organized as a manifold. This means that spacetime has a well-defined dimension, but it also carries with it the inherent possibility of modified patterns of global connectivity, such as distinguishing a sphere from a plane or a torus from a surface of higher genus. Such modifications can be present in the spatial topology without affecting the time direction, but they can also have a genuine spacetime character in which case the spatial topology changes with time [4]. The topology change in classical general relativity has been discussed in [5]. See [6] for some applications of differential topology in general relativity.

In general relativity, boundaries that are [figure omitted; refer to PDF] bundles over some compact manifolds arise in gravitational thermodynamics [7]. The trivial bundle [figure omitted; refer to PDF] is a classic example. Manifolds with complete Ricci-flat metrics admitting such boundaries are known; they are the Euclideanised Schwarzschild metric and the flat metric with periodic identification. York [8] shows that there are in general two or no Schwarzschild solutions depending on whether the squashing (the ratio of the radius of the [figure omitted; refer to PDF] -fibre to that of the [figure omitted; refer to PDF] -base) is below or above a critical value. York's results in 4-dimension extend readily to higher dimensions.

The simplest example of nontrivial bundles arises in quantum cosmology in which the boundary is a compact [figure omitted; refer to PDF] , that is, a nontrivial [figure omitted; refer to PDF] bundle over [figure omitted; refer to PDF] . In the case of zero cosmological constant, regular 4 metrics admitting such an [figure omitted; refer to PDF] boundary are the Taub-Nut [9] and Taub-Bolt [10] metrics having zero and two-dimensional (regular) fixed point sets of the [figure omitted; refer to PDF] action, respectively [7, 11-13].

The Kerr metric describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole. The Kerr metric corresponds to the line element [figure omitted; refer to PDF]

The parameter [figure omitted; refer to PDF] , termed the Kerr parameter, has units of length in geometrized units. The parameter [figure omitted; refer to PDF] will be interpreted as angular momentum and the parameter [figure omitted; refer to PDF] will be interpreted as the mass for the black hole. The Kerr metric is a vacuum solution of the Einstein equations, being valid in the absence of matter. If the black hole is not rotating [figure omitted; refer to PDF] , the Kerr line element reduces to the Schwarzschild line element. The Kerr metric becomes asymptotically flat for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Unlike the Schwarzschild metric, the Kerr metric has only axial symmetry.

2. Deformation Retract

2.1. Deformation Retract Definitions

The theory of deformation retract is very interesting topic in Euclidean and non-Euclidean spaces. It has been investigated from different points of view in many branches of topology and differential geometry. A retraction is a continuous mapping from the entire space into a subspace which preserves the position of all points in that subspace [14].

(i) Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be two smooth manifolds of dimensions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively. A map [figure omitted; refer to PDF] is said to be an isometric folding of [figure omitted; refer to PDF] into [figure omitted; refer to PDF] if and only if, for every piecewise geodesic path [figure omitted; refer to PDF] , the induced path [figure omitted; refer to PDF] is a piecewise geodesic and of the same length as [figure omitted; refer to PDF] [15]. If [figure omitted; refer to PDF] does not preserve the lengths, it is called topological folding. Many types of foldings are discussed in [16-21]. Some applications are discussed in [22, 23].

(ii) A subset [figure omitted; refer to PDF] of a topological space [figure omitted; refer to PDF] is called a retraction of [figure omitted; refer to PDF] if there exists a continuous map [figure omitted; refer to PDF] such that [24]

: (a) [figure omitted; refer to PDF] is open;

: (b) [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

(iii) A subset [figure omitted; refer to PDF] of a topological space [figure omitted; refer to PDF] is said to be a deformation retract if there exist a retraction [figure omitted; refer to PDF] and a homotopy [figure omitted; refer to PDF] such that [24] [figure omitted; refer to PDF]

The deformation retract is a particular case of homotopy equivalence and two spaces are homotopy equivalent if and only if they are both deformation retracts of a single larger space.

Deformation retracts of Stein spaces have been studied in [25]. The deformation retract of the 4D Schwarzschild metric has been discussed in [26] where it was found that the retraction of the Schwarzschild space is spacetime geodesic. The 5-dimensional case has been discussed in [27].

3. Equatorial Geodesics

We will be interested in the equatorial geodesics, that is, geodesics with [figure omitted; refer to PDF] . It is easy to show that such geodesics exist for the case of Kerr metric where [figure omitted; refer to PDF] satisfies the [figure omitted; refer to PDF] -component of the Euler Lagrange equations for the Lagrangian associated with the Kerr metric (1). Consider [figure omitted; refer to PDF]

The [figure omitted; refer to PDF] -component of the Euler Lagrange equations gives [figure omitted; refer to PDF]

Comparing the Kerr line element, [figure omitted; refer to PDF] And the four-dimensional flat metric [figure omitted; refer to PDF]

The coordinates of the four-dimensional Kerr space (6) can be written as [figure omitted; refer to PDF]

In general relativity, the geodesic equation is equivalent to the Euler Lagrange equations [figure omitted; refer to PDF] associated to the Lagrangian [figure omitted; refer to PDF]

To find a geodesic which is a subset of the Kerr space, the Lagrangian could be written as [figure omitted; refer to PDF]

There is no explicit dependence on [figure omitted; refer to PDF] or [figure omitted; refer to PDF] ; thus [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are constants of motion; that is, [figure omitted; refer to PDF]

So we have the following set of equations: [figure omitted; refer to PDF]

If the constant [figure omitted; refer to PDF] is zero, we have [figure omitted; refer to PDF]

Since [figure omitted; refer to PDF] which is the great circle [figure omitted; refer to PDF] in the Kerr space [figure omitted; refer to PDF] , this geodesic is a retraction in Kerr space; [figure omitted; refer to PDF] . This is a retraction.

For [figure omitted; refer to PDF] , (12) becomes [figure omitted; refer to PDF] If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] . If the constant is zero, then [figure omitted; refer to PDF]

Since [figure omitted; refer to PDF] which is the great circle [figure omitted; refer to PDF] in the Kerr space [figure omitted; refer to PDF] , this geodesic is a retraction in Kerr space; [figure omitted; refer to PDF] .

If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] If the constant is zero, then [figure omitted; refer to PDF]

Since [figure omitted; refer to PDF] which is the great circle [figure omitted; refer to PDF] in the Kerr space [figure omitted; refer to PDF] , this geodesic is a retraction in Kerr space; [figure omitted; refer to PDF] .

From the above discussion, the following theorem has been proved.

Theorem 1.

The retraction of the Kerr space is a geodesic in the Kerr space.

4. Deformation Retract of Kerr Space

The deformation retract of the Kerr space [figure omitted; refer to PDF] is defined as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the closed interval [figure omitted; refer to PDF] . The retraction of the Kerr space [figure omitted; refer to PDF] is defined as [figure omitted; refer to PDF]

Then, the deformation retract of the Kerr space [figure omitted; refer to PDF] into a geodesic [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

The deformation retract of the Kerr space into a geodesic [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF]

The deformation retract of the Kerr space into a geodesic [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF]

Now we are going to discuss the folding of the Kerr space [figure omitted; refer to PDF] : [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

An isometric folding of the Kerr space into itself may be defined by [figure omitted; refer to PDF]

The deformation retract of the folded Kerr space [figure omitted; refer to PDF] into the folded [figure omitted; refer to PDF] is [figure omitted; refer to PDF] with [figure omitted; refer to PDF]

The deformation retract of the folded Kerr space [figure omitted; refer to PDF] into the folded [figure omitted; refer to PDF] is [figure omitted; refer to PDF]

The deformation retract of the folded Kerr space [figure omitted; refer to PDF] into the folded [figure omitted; refer to PDF] is [figure omitted; refer to PDF]

Therefore, the following theorem has been proved.

Theorem 2.

The deformation retract of the isometric folding of Kerr space and any folding homeomorphic to this type of folding is different from the deformation retract of Kerr space.

5. Conclusion

The deformation retract of the Kerr space has been investigated by making use of Lagrangian equations. The equatorial geodesics of the Kerr space have been discussed. The retraction of this space into itself and into geodesics has been presented. The deformation retraction of the Eguchi-Hanson space is a geodesic which is found to be a great circle. The deformation retract of the isometric folding of Kerr space and any folding homeomorphic to this type of folding is found to be different from the deformation retract of Kerr space.

Acknowledgment

The author is deeply indebted to Nasr Ahmed from Mathematics Department at Taibah University for the useful discussions and help during this work.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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