On the Deformation Retract of Kerr Spacetime and Its Folding

The real revolution in mathematical physics in the second half of twentieth century (and in puremathematics 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 secondhalf of the twentieth century [1]. Today, the concepts andmethods 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. Suchmodifications 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 S bundles over some compact manifolds arise in gravitational thermodynamics [7]. The trivial bundle ∑ = S × S 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 onwhether the squashing (the ratio of the radius of the S-fibre to that of the S-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 S, that is, a nontrivial S bundle over S. In the case of zero cosmological constant, regular 4 metrics admitting such an


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  1 bundles over some compact manifolds arise in gravitational thermodynamics [7].The trivial bundle ∑ =  1 ×  2 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  1 -fibre to that of the  2 -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  3 , that is, a nontrivial  1 bundle over  2 .In the case of zero cosmological constant, regular 4 metrics admitting such an  3 boundary are the Taub-Nut [9] and Taub-Bolt [10] metrics having zero and two-dimensional (regular) fixed point sets of the (1) action, respectively [7,[11][12][13].

Advances in Mathematical Physics
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 ( The parameter , termed the Kerr parameter, has units of length in geometrized units.The parameter  will be interpreted as angular momentum and the parameter  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 ( = / = 0), the Kerr line element reduces to the Schwarzschild line element.The Kerr metric becomes asymptotically flat for  ≫  and  ≫ .Unlike the Schwarzschild metric, the Kerr metric has only axial symmetry.

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  and  be two smooth manifolds of dimensions  and , respectively.A map  :  →  is said to be an isometric folding of  into  if and only if, for every piecewise geodesic path  :  → , the induced path  ∘  :  →  is a piecewise geodesic and of the same length as  [15].If  does not preserve the lengths, it is called topological folding.Many types of foldings are discussed in [16][17][18][19][20][21].Some applications are discussed in [22,23].
(ii) A subset  of a topological space  is called a retraction of  if there exists a continuous map  :  →  such that [24] (a)  is open; (b) () = , ∀ ∈ .
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].

Equatorial Geodesics
We will be interested in the equatorial geodesics, that is, geodesics with  = /2.It is easy to show that such geodesics exist for the case of Kerr metric where  = /2 satisfies the -component of the Euler Lagrange equations for the Lagrangian associated with the Kerr metric (1).Consider The -component of the Euler Lagrange equations gives Comparing the Kerr line element, And the four-dimensional flat metric The coordinates of the four-dimensional Kerr space (6) can be written as In general relativity, the geodesic equation is equivalent to the Euler Lagrange equations associated to the Lagrangian To find a geodesic which is a subset of the Kerr space, the Lagrangian could be written as There is no explicit dependence on  or ; thus / ṫ = 0 and / φ = 0 are constants of motion; that is, So we have the following set of equations: If the constant  is zero, we have Since  2 1 +  2 2 +  2 3 −  2 4 > 0 which is the great circle  1 in the Kerr space , this geodesic is a retraction in Kerr space;  2 > 0. This is a retraction.
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.