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On the Deformation Retract of Kerr Spacetime and Its Folding Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2014, Article ID 673768, 8 pages http://dx.doi.org/10.1155/2014/673768 Research Article On the Deformation Retract of Kerr Spacetime and Its Folding H. Rafat Department of Mathematics, Faculty of Science, Tanta University, Tanta 31111, Egypt Correspondence should be addressed to H. Rafat; [email protected] Received 26 May 2014; Revised 8 September 2014; Accepted 9 September 2014; Published 10 November 2014 Academic Editor: Shao-Ming Fei Copyright © 2014 H. Rafat. 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. 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 ofthis space into itself and after the isometric folding has been discussed. Theorems concerning these relations have been deduced. 1. Introduction dimension, but it also carries with it the inherent possibility of modified patterns of global connectivity, such as distinguish- The real revolution in mathematical physics in the second ing a sphere from a plane or a torus from a surface of higher half of twentieth century (and in pure mathematics itself) was genus. Such modifications can be present in the spatial topol- algebraic topology and algebraic geometry [1]. In the nine- ogy without affecting the time direction, but they can also teenth century, mathematical physics was essentially the clas- have a genuine spacetime character in which case the spatial sicaltheoryofordinaryandpartialdifferentialequations.The topology changes with time [4]. The topology change in clas- variational calculus, as a basic tool for physicists in theoretical sical general relativity has been discussed in [5]. See [6]for mechanics, was seen with great reservation by mathemati- some applications of differential topology in general relativity. cians until Hilbert set up its rigorous foundation by pushing 1 forward functional analysis. This marked the transition into In general relativity, boundaries that are bundles over some compact manifolds arise in gravitational thermody- the first half of twentieth century, where, under the influence 1 2 of quantum mechanics and relativity, mathematical physics namics [7]. The trivial bundle ∑=× is a classic turned mainly into functional analysis, complemented by the example. Manifolds with complete Ricci-flat metrics admit- theory of Lie groups and by tensor analysis. All branches of ting such boundaries are known; they are the Euclideanised theoretical physics still can expect the strongest impacts of Schwarzschild metric and the flat metric with periodic iden- use of the unprecedented wealth of results of algebraic topol- tification. York8 [ ] shows that there are in general two or no ogy and algebraic geometry of the second half of the twentieth Schwarzschild solutions depending on whether the squashing 1 2 century [1]. (the ratio of the radius of the -fibre to that of the -base) is Today, the concepts and methods of topology and geome- below or above a critical value. York’s results in 4-dimension try have become an indispensable part of theoretical physics. extend readily to higher dimensions. They have led to a deeper understanding of many crucial The simplest example of nontrivial bundles arises in aspects in condensed matter physics, cosmology, gravity, and 3 particle physics. Moreover, several intriguing connections quantum cosmology in which the boundary is a compact , 1 2 between only apparently disconnected phenomena have been that is, a nontrivial bundle over .Inthecaseofzero cosmological constant, regular 4 metrics admitting such an revealedbasedonthesemathematicaltools[2, 3]. 3 Topology enters general relativity through the funda- boundary are the Taub-Nut [9]andTaub-Bolt[10]metrics mental assumption that spacetime exists and is organized as having zero and two-dimensional (regular) fixed point sets of a manifold. This means that spacetime has a well-defined the (1) action, respectively [7, 11–13]. 2 Advances in Mathematical Physics The Kerr metric describes the geometry of empty space- The deformation retract is a particular case of homotopy time around a rotating uncharged axially symmetric black equivalence and two spaces are homotopy equivalent if and hole. The Kerr metric corresponds to the line element only if they are both deformation retracts of a single larger 2 4 2 space. 2 =−(1− )2 − sin 2 2 Deformation retracts of Stein spaces have been studied in [25]. The deformation retract of the 4D Schwarzschild metric 2 has been discussed in [26] where it was found that the retrac- + 2 +22 (1) Δ tion of the Schwarzschild space is spacetime geodesic. The 5-dimensional case has been discussed in [27]. 22 2 +(2 +2 + sin ) 22, 2 sin 3. Equatorial Geodesics = ,2 =2 +2 2, cos We will be interested in the equatorial geodesics, that is, (2) =/2 2 2 geodesics with .Itiseasytoshowthatsuchgeodesics Δ= − 2 + . exist for the case of Kerr metric where =/2satisfies The parameter , termed the Kerr parameter, has units of the -component of the Euler Lagrange equations for the length in geometrized units. The parameter will be inter- Lagrangian associated with the Kerr metric (1). Consider preted as angular momentum and the parameter will be interpreted as the mass for the black hole. The Kerr metric is a 1 2 2 2 2 vacuum solution of the Einstein equations, being valid in the = − (1 − ) 2̇ − sin +̇ ̇ 2̇ absence of matter. If the black hole is not rotating ( = / = 2 2 2 2Δ 0), the Kerr line element reduces to the Schwarzschild line ele- (4) 2 2 ment. The Kerr metric becomes asymptotically flat for ≫ 1 2 2̇ 1 2 2 2 sin 2 2̇ + + ( + + ) sin . and ≫. Unlike the Schwarzschild metric, the Kerr metric 2 2 2 has only axial symmetry. 2. Deformation Retract The -component of the Euler Lagrange equations gives 2.1. Deformation Retract Definitions. The theory of defor- mation retract is very interesting topic in Euclidean and 22 2 (2)̇ + sin cos 2̇ − sin cos 2̇ non-Euclidean spaces. It has been investigated from different 4 Δ points of view in many branches of topology and differential geometry. A retraction is a continuous mapping from the 2 2 4 sin cos ( + sin ) entire space into a subspace which preserves the position of − ( ) 2̇ − ̇ ̇ sin cos 4 all points in that subspace [14]. (i) Let and be two smooth manifolds of dimensions 1 : → +(2 +2) 2̇ + 42 and ,respectively.Amap is said to be sin cos 4 sin cos an isometric folding of into if and only if, for every piecewise geodesic path :,theinducedpath → 2 4 2 2 ×( sin +2( + cos ) sin ) = 0. ∘ : →is a piecewise geodesic and of the same length as [15]. If does not preserve the lengths, it is called (5) topological folding. Many types of foldings are discussed in [16–21]. Some applications are discussed in [22, 23]. (ii) A subset of a topological space is called a Comparing the Kerr line element, retraction of if there exists a continuous map : → such that [24] 2 4 2 =−(1− )2 − (a) is open; (b) (), = ∀ ∈. (6) 2 22 + 2 +(2 +2 + )2. (iii) A subset of a topological space is said to be a Δ deformation retract if there exist a retraction :and → a homotopy : × [0, 1] → such that [24] (,) 0 =, ∀∈, And the four-dimensional flat metric (,) 1 =() ,∀∈, (3) 2 2 2 2 2 (,) =, ∀∈,∈[0, 1] . =1 +2 +3 +4. (7) Advances in Mathematical Physics 3 The coordinates of the four-dimensional Kerr space (6) So we have the following set of equations: canbewrittenas 2̇ 2̇ ( )−(− 2̇ + 2 +2 − 2 2 2 +2 − 2 2 2 2 2 1 =(− 4 + (2 −2 ) ln (− − + 2) 2 +(− ) 2̇ )=0, 2 −1 √ 2 2 tan ((−) / − ) (13) + 2 √2 −2 [− (1 − ) −2̇ ]̇ = 0, × (83 −62+22−42) + (2 −22) 2 2 [− +(̇ 2 +2 +2 ) ]̇ = 0. 2 −2 −2 1/2 × ln ( )+1) , (−)(+) If the constant is zero, we have 2 =(2 − 4 + (2 −22) =±√ 2 +282 + , 1 2 4 2 −1 ((−) /) 22 × (−2 + 2) + tan =±√((2 +2)+ )2 + , ln 3 3 2 1/2 3 2 2 2 − 2 × (8 −4 ) + (2 ) ln ( )+1) , =±√(1 − )2 + . 2 4 4 (8) 2 =±√ 2 + , 2 4 2 In general relativity, the geodesic equation is equivalent 2 2 to the Euler Lagrange equations 3 =±√ +3, 2 √ 2 ( )− =0, =1,2,3,4 4 =± (1 − ) +4. ̇ (9) (14) 2 2 2 2 associated to the Lagrangian Since 1 +2 +3 −4 >0which is the great circle 1 in the Kerr space , this geodesic is a retraction in Kerr space; 2 >0 1 . This is a retraction. (, ̇) = ̇̇]. For =0, (12) becomes 2 ] (10) 2 −(1− ) =̇ , 1 To find a geodesic which is a subset of the Kerr space, the (15) 2 ̇ Lagrangian could be written as =2. If 1 =0,then=const. If the constant is zero, then 1 2 2 2 = − (1 − ) 2̇ − +̇ ̇ 2̇ 1 2 2Δ 2 2 2 1 = + −+ln (−) − ln (−) , (11) 2 1 22 + (2 +2 + ) 2̇ . 2 2 =±√ 2 + , 2 4 2 ̇ There is no explicit dependence on or ;thus/=0 =±√22 + , ̇ 3 3 and /=0are constants of motion; that is, 4 =±√4. 2 −(1− ) −2̇ =̇ , (16) 1 (12) 2 +2 +2 −2 >0 2 2 Since 1 2 3 4 which is the great circle 2 in − +(̇ 2 +2 +2 ) =̇ . the Kerr space , this geodesic is a retraction in Kerr space; 2 2 >0. 4 Advances in Mathematical Physics 1/2 If 2 =0,then=const. If the constant is zero, then 2 −2 −2 × ( )+ ) , ln (−)(+) 1 1 = 2 +2 −+ (−) −2 (−) , 1 ln ln 2 2 ± √ 2 +282 + , 4 2 2 =±√ 2 + , 2 2 22 4 ± √((2 +2)+ )2 + , 3 3 =±√3, 2 ±√(1 − )2 + } 2 4 =±√(1 − )2 + .
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