我們主要的工作是討論在高維Kerr時空下的穩定圓軌道存在性
問題。但由於高維Kerr時空的測地線方程複雜難解，也因為在高於
六維以上的Kerr黑洞具有“黑洞角動量無上限”的特性，因此我們
在本篇論文中會以六維一旋轉方向的Kerr黑洞時空為背景，來簡化
測地線方程的複雜度，進而討論在此時空下赤道面上的穩定圓軌道
存在性問題。此外，我們還會對測地線方程做黑洞角動量很大的極
限處理並討論在一般情形下的穩定圓軌道存在性問題以及測地線方
程的變化行為。最後，我們繪出依各運動常數變化的effective potential 圖形，來考量在一般情形下，穩定圓軌道的存在性問題。

Our main work is to discuss the existence problem of stability circular orbit in higher dimensional Kerr spacetimes. The corresponding geodesics are complicated and difficult to solve. On the other hand, there is no upper limit on the angular momenta of Kerr black holes in dimensions higher than six. Thus, we simplify the geodesic by choosing to work with six dimensional Kerr black holes with one rotation direction in this thesis. We first discuss the existence problem of stability circular orbit in equatorial plane. Furthermore, we deal with the geodesic using a perturbative expansion when the angular momentum of the Kerr black hole is large in order to discuss the existence problem of stability circular orbit in general(not in equatorial plane) and the behavior of the geodesic. Finally, we plot the effective potentials for various of the constants of motion to consider the existence problem of stable circular orbit in general case.

第一章 簡介................................................5

第二章 Kerr黑洞的特性......................................7
 § 2.1  四維時空的 Kerr黑洞
   ‧ 2.1.1  四維時空的Kerr規..............................7
   ‧ 2.1.2  The event horizon.............................8
   ‧ 2.1.3  靜態極限表面..................................9
   ‧ 2.1.4  能層..........................................9
   ‧ 2.1.5  Kerr黑洞的對稱性與粒子守恆量.................11
   ‧ 2.1.6  測地線方程...................................11
 
 § 2.2  五維時空的 Kerr黑洞
   ‧ 2.2.1  五維時空的Kerr度規...........................13
   ‧ 2.2.2  五維Kerr黑洞的event horizon..................14
   ‧ 2.2.3  靜態極限的位置...............................14
   ‧ 2.2.4  五維Kerr黑洞的粒子守恆量.....................14
   ‧ 2.2.5  五維Kerr黑洞的測地線方程.....................15

§ 2.3  D維時空的 Kerr黑洞
   ‧ 2.3.1  Myers-Perry 度規與座標定義...................16
   ‧ 2.3.2  座標的變數分離...............................18
   ‧ 2.3.3  D維Kerr黑洞的 event horizon..................20
   ‧ 2.3.4  D維Kerr黑洞的靜態極限表面....................20
   ‧ 2.3.5  D維Kerr黑洞的對稱性與粒子守恆量..............21
   ‧ 2.3.6 測地線方程的變數分離..........................21
   ‧ 2.3.7  The effective potential......................23

第三章 D維Schwarzschild幾何中的粒子圓軌道方程.............25
§ 3.1 D維Schwarzschild 度規...............................25
§ 3.2測試粒子圓軌道的穩定性...............................26

第四章 六維Myers-Perry 幾何中的測地線方程.................30
§ 4.1 六維一角動量的Myers-Perry黑洞度規...................31
§ 4.2 The event horizon...................................32
§ 4.3 靜態極限............................................33
§ 4.4 測地線方程..........................................35
§ 4.5 測試粒子的運動常數..................................36
§ 4.6 測試粒子的穩定圓軌道
   ‧ 4.6.1測試粒子圓軌道的條件...........................37
   ‧ 4.6.2 赤道面的粒子圓軌道存在問題....................38
§ 4.7  在 a/∛M≫1極限下的測地線方程
   ‧ 4.7.1徑向運動及圓軌道...............................43
   ‧ 4.7.2 θ－運動......................................45
   ‧ 4.7.3 β－運動......................................46
   ‧ 4.7.4 方位角運動....................................46
   ‧ 4.7.5一般非赤道面下的effective potential ...........48

第五章 結論...............................................52

參考資料..................................................54


圖表目錄

圖(一)....................................................29
圖(二)....................................................34
圖(三)....................................................34
圖(四)....................................................40
圖(五)....................................................40
圖(六)....................................................49
圖(七)....................................................50
圖(八)....................................................50
圖(九)....................................................50
圖(十)....................................................51

[1]  V. Frolov and D. Stojkovic,“Particle and light motion   
     in a space-time of five-dimensional rotating black 
     hole,”Phys Rev.D 68 064011 

[2]  M. Vasudevan, K. A Stevens and D. A Page,“Particle 
     motion and scalar field propagation in Myers-Perry 
     black-hole spacetimes in all dimensions,”
     Class.Quantum.Grav. 22(2005) 1469-1482

[3]  R. C. Myers and M. J. Perry,“Black Holes in Higher
     Dimensional Space-Time,”Annals of Phys 
     172,304-347,(1986)

[4]  S. Chandrasehkar,“The Mathematical Theory of Black
     Holes,”Oxford University Press (1998)

[5]  C. W. Misner, K. S. Thorne, J. A. Wheeler, 
     Gravitation,”W. H. Freeman(1973)

[6]  B.Carter,“Global structure of the Kerr family of    
     gravitational field,”Phys. Rev. 174 1559-1571