Jason Doukas Yukawa Institute For Theoretical Physics Kyoto University

Transcription

1 1/32 MYERS-PERRY BLACK HOLES: BEYOND THE SINGLE ROTATION CASE Jason Doukas Yukawa Institute For Theoretical Physics Kyoto University Lunch meeting FEB 02, 2011

2 2/32 Contents 1. Introduction. 1.1 Myers Perry black holes. 1.2 Motivation for considering more than one rotation. 2. Angular momentum constraints. 2.1 Derivation. 2.2 Results. 3. The Wald Gedanken experiment dimensions. 3.2 D dimensions. 4. Conclusion.

3 In 1986 Robert C. Myers and M. J. Perry presented the first examples of black hole solutions with angular momentum in more than four dimensions. 989 citations. 3/32

4 3/32 In 1986 Robert C. Myers and M. J. Perry presented the first examples of black hole solutions with angular momentum in more than four dimensions. 989 citations. Perhaps the most striking discoveries of these results were: For a fixed mass, solutions may exist which have arbitrary large angular momentum. c.f. Kerr M > a.

5 3/32 In 1986 Robert C. Myers and M. J. Perry presented the first examples of black hole solutions with angular momentum in more than four dimensions. 989 citations. Perhaps the most striking discoveries of these results were: For a fixed mass, solutions may exist which have arbitrary large angular momentum. c.f. Kerr M > a. rotation parameters are required to describe the rotation, where D is the total spacetime dimension. D 1 2

6 The reason for more than one angular momentum parameter can be seen as follows: 4/32

7 The reason for more than one angular momentum parameter can be seen as follows: Far away from the MP black hole the metric is flat, we can then group the D 1 spatial coordinates into pairs (x 1, x 2 ),..., (x i, x i+1 ),.... 4/32

8 The reason for more than one angular momentum parameter can be seen as follows: Far away from the MP black hole the metric is flat, we can then group the D 1 spatial coordinates into pairs (x 1, x 2 ),..., (x i, x i+1 ),.... If D is odd the last pair will be (x D 2, x D 1 ); 4/32

9 The reason for more than one angular momentum parameter can be seen as follows: Far away from the MP black hole the metric is flat, we can then group the D 1 spatial coordinates into pairs (x 1, x 2 ),..., (x i, x i+1 ),.... If D is odd the last pair will be (x D 2, x D 1 ); otherwise if D is even the x D 1 coordinate will be unpaired. 4/32

10 4/32 The reason for more than one angular momentum parameter can be seen as follows: Far away from the MP black hole the metric is flat, we can then group the D 1 spatial coordinates into pairs (x 1, x 2 ),..., (x i, x i+1 ),.... If D is odd the last pair will be (x D 2, x D 1 ); otherwise if D is even the x D 1 coordinate will be unpaired. Therefore there are D 1 2 such pairs. Writing each pair (x i, x i+1 ) in polar coordinates (r i, φ i ) we see there is a rotation associated with each of the φi vectors.

11 The MP metric(s) Even D; ds 2 = dt 2 + r 2 dα Mr ΠF ( Odd D; ds 2 = dt 2 + F = 1 d i=1 + 2Mr2 ΠF a 2 i µ2 i ( dt + d i=1 dt + r 2 + a 2, Π = i d i=1 d i=1 a i µ 2 i dφ i ( ) ( ) r 2 + a 2 i dµ 2 i + µ2 i dφ2 i ) 2 + ΠF Π 2Mr dr2, ( ) ( ) r 2 + a 2 i dµ 2 i + µ2 i dφ2 i d i=1 d i=1 a i µ 2 i dφ i) 2 + ΠF Π 2Mr 2 dr2, { D 2 (r 2 + a 2 i ), d = 2, D even D 1 2, D odd. 5/32

12 The full MP metric is complicated and in analysing the solutions it is common to set all but one angular momentum parameter to zero, this is known as the simply rotating black hole. 6/32

13 6/32 The full MP metric is complicated and in analysing the solutions it is common to set all but one angular momentum parameter to zero, this is known as the simply rotating black hole. This can be loosely justified in brane world scenario s, where colliding particle momentum will be confined to a brane and therefore any black hole created on this brane would only spin in this direction.

14 7/32 However, if the brane is thick then there is also the potential for angular momentum to occur in other directions

15 7/32 However, if the brane is thick then there is also the potential for angular momentum to occur in other directions. Furthermore, one could imagine planck-scale processes not confined to the brane (i.e., gravitons/strings etc) creating black holes in the bulk spacetime with arbitrary angular momentum. We are thus motivated to explore the solutions with more than one angular momentum parameter.

16 Angular momentum constraints 8/32

17 Horizons hide singularities from outside observers. 9/32

18 9/32 Horizons hide singularities from outside observers. Even in four dimensions for certain values of M and a the Kerr solution can admit solutions without any horizon. This occurs when: = r 2 + a 2 2Mr > 0 or M < a

19 9/32 Horizons hide singularities from outside observers. Even in four dimensions for certain values of M and a the Kerr solution can admit solutions without any horizon. This occurs when: = r 2 + a 2 2Mr > 0 or M < a D > 4?

20 9/32 Horizons hide singularities from outside observers. Even in four dimensions for certain values of M and a the Kerr solution can admit solutions without any horizon. This occurs when: = r 2 + a 2 2Mr > 0 or M < a D > 4? For simply rotating black holes ( only one a) D = 5 is the only case where the angular momentum is constrained. In particular it is known that M > a 2 /2.

21 9/32 Horizons hide singularities from outside observers. Even in four dimensions for certain values of M and a the Kerr solution can admit solutions without any horizon. This occurs when: = r 2 + a 2 2Mr > 0 or M < a D > 4? For simply rotating black holes ( only one a) D = 5 is the only case where the angular momentum is constrained. In particular it is known that M > a 2 /2. Why is D = 5 special?

22 9/32 Horizons hide singularities from outside observers. Even in four dimensions for certain values of M and a the Kerr solution can admit solutions without any horizon. This occurs when: = r 2 + a 2 2Mr > 0 or M < a D > 4? For simply rotating black holes ( only one a) D = 5 is the only case where the angular momentum is constrained. In particular it is known that M > a 2 /2. Why is D = 5 special? To explain this we calculate the angular momentum constraints in higher dimensions in general with all angular momentum parameters.

23 10/32 In D-even dimensions the condition for the location of the horizon is: = Π 2Mr = 0, Π = d i=1 (r 2 + a 2 i ).

24 10/32 In D-even dimensions the condition for the location of the horizon is: = Π 2Mr = 0, Π = d i=1 (r 2 + a 2 i ). There are only three possibilities for the number of horizons: r r Let r be the unique minima of for positive r.

25 11/32 Then the black hole will be free of naked singularities (that is to say a horizon exists) if: ( r) 0, r > 0.

26 11/32 Then the black hole will be free of naked singularities (that is to say a horizon exists) if: ( r) 0, r > 0. The inequality is saturated when ( r) = 0, equality occurs iff r is also an r-intercept.

27 11/32 Then the black hole will be free of naked singularities (that is to say a horizon exists) if: ( r) 0, r > 0. The inequality is saturated when ( r) = 0, equality occurs iff r is also an r-intercept. Therefore, M Π( r h), 2r h where r h is a solution to the simultaneous set of equations ( r h ) = 0, r ( r h ) = 0. Or: 0 = r h r Π( r h ) Π( r h )

28 Using the product expansion: 12/32 j i=1 (r 2 + a 2 i ) = j i=0 r 2i A j i j, with the coefficients conveniently defined as A k n = ν 1 <ν 2 <...ν k a 2 ν 1 a 2 ν 2... a 2 ν k, where v k are summed over [1, n], 0 k n and A 0 k 1,

29 Using the product expansion: 12/32 j i=1 (r 2 + a 2 i ) = j i=0 r 2i A j i j, with the coefficients conveniently defined as A k n = ν 1 <ν 2 <...ν k a 2 ν 1 a 2 ν 2... a 2 ν k, where v k are summed over [1, n], 0 k n and A 0 k 1, we can write P 2 ( r h ) = d i=0 (2i 1) r 2i h Ad i d = 0.

30 Using the product expansion: 12/32 j i=1 (r 2 + a 2 i ) = j i=0 r 2i A j i j, with the coefficients conveniently defined as A k n = ν 1 <ν 2 <...ν k a 2 ν 1 a 2 ν 2... a 2 ν k, where v k are summed over [1, n], 0 k n and A 0 k 1, we can write P 2 ( r h ) = d i=0 d 4 D 2 2 (2i 1) r 2i h Ad i d = 0. 4 D 10

31 In the D-odd case the location of the horizon is found by the equation: 13/32 where l = r 2. = Π(l) 2Ml = 0, Π = d i=1 (l + a 2 i ).

32 In the D-odd case the location of the horizon is found by the equation: 13/32 = Π(l) 2Ml = 0, Π = d i=1 (l + a 2 i ). where l = r 2. Performing the same analysis again we find that: lh l Π( l h ) Π( l h ) = 0, M Π( l h ) 2 l h and the polynomial for the odd case is: P 1 ( l h ) = d i=0 (i 1) l i h Ad i d = 0.

33 In the D-odd case the location of the horizon is found by the equation: 13/32 = Π(l) 2Ml = 0, Π = d i=1 (l + a 2 i ). where l = r 2. Performing the same analysis again we find that: lh l Π( l h ) Π( l h ) = 0, M Π( l h ) 2 l h and the polynomial for the odd case is: P 1 ( l h ) = d i=0 (i 1) l i h Ad i d = 0. d 4 D D 9

34 14/32 D=7 an example: The polynomial is: 2 l 3 h + l 2 h (a2 1 + a2 2 + a2 3 ) a2 1 a2 2 a2 3 = 0

35 14/32 D=7 an example: The polynomial is: 2 l 3 h + l 2 h (a2 1 + a2 2 + a2 3 ) a2 1 a2 2 a2 3 = 0 and the solution can be written in closed form: lh = 1 ( (a a2 2 + a2 3 ) + 1 ) p (a2 1 + a2 2 + a2 3 )2 + p, p 3 = (a a2 2 + a2 3 )3 + 54a 2 1 a2 2 a (a a2 2 + a2 3 )3 a 2 1 a2 2 a (a4 1 a4 2 a4 3 ),

36 14/32 D=7 an example: The polynomial is: 2 l 3 h + l 2 h (a2 1 + a2 2 + a2 3 ) a2 1 a2 2 a2 3 = 0 and the solution can be written in closed form: lh = 1 ( (a a2 2 + a2 3 ) + 1 ) p (a2 1 + a2 2 + a2 3 )2 + p, p 3 = (a a2 2 + a2 3 )3 + 54a 2 1 a2 2 a (a a2 2 + a2 3 )3 a 2 1 a2 2 a (a4 1 a4 2 a4 3 ), and the constraint reads: M ( l h + a 2 1 )( l h + a 2 2 )( l h + a 2 3 ) 2 l h.

37 15/32 Results. M = 1. (left) D = 5. (right) D = 6.

38 16/32 Results. M = 1. (left) D = 7. (right) D = 8.

39 17/32 D All a i non-zero a 1 = 0 4 M a 1 M 0 5 2M a a2 2 + a 1a 2 M > a M ( r2 h +a2 1 )( r2 h +a2 2 ) 2 r ; h 3 r 4 h + (a2 1 + a2 2 ) r2 h a2 1 a2 2 = 0 M M ( l h +a 2 1 )( l h +a 2 2 )( l h +a 2 3 ) 2 l h ; 2 l 3 h + l 2 h (a2 1 + a2 2 + a2 3 ) a2 1 a2 2 a2 3 = 0 M > a2 1 a2 2 2 M ( r2 h +a2 1 )( r2 h +a2 2 )( r2 h +a2 3 ) 2 r ; h 5 r 6 h + 3(a2 1 + a2 2 + a2 3 ) r4 M 0 h (a 2 1 a2 2 + a2 1 a2 3 + a2 2 a2 3 ) r2 h a2 1 a2 2 a2 3 = 0..

40 Conclusion 18/32

41 It is not only theoretically interesting to study black holes with angular momentum in higher dimensions, but there are also good physical reasons one should consider beyond single rotation MP solutions. 19/32

42 It is not only theoretically interesting to study black holes with angular momentum in higher dimensions, but there are also good physical reasons one should consider beyond single rotation MP solutions. I derived the mass/angular momentum constraints that apply to MP- black holes. 19/32

43 It is not only theoretically interesting to study black holes with angular momentum in higher dimensions, but there are also good physical reasons one should consider beyond single rotation MP solutions. I derived the mass/angular momentum constraints that apply to MP- black holes. In particular, it was shown that exact closed form expressions could be obtained for dimensions less than or equal to ten. 19/32

44 It is not only theoretically interesting to study black holes with angular momentum in higher dimensions, but there are also good physical reasons one should consider beyond single rotation MP solutions. I derived the mass/angular momentum constraints that apply to MP- black holes. In particular, it was shown that exact closed form expressions could be obtained for dimensions less than or equal to ten. These results can be used to show that none of the MP black holes can be spun into naked singularities using the Wald type gedanken experiment generalised into D dimensions. 19/32

45 It is not only theoretically interesting to study black holes with angular momentum in higher dimensions, but there are also good physical reasons one should consider beyond single rotation MP solutions. I derived the mass/angular momentum constraints that apply to MP- black holes. In particular, it was shown that exact closed form expressions could be obtained for dimensions less than or equal to ten. These results can be used to show that none of the MP black holes can be spun into naked singularities using the Wald type gedanken experiment generalised into D dimensions. It is worth mentioning that even if a black hole satisfies the constraints presented it may not be stable; classical instabilities of the Gregory-Laflamme type are known to arise in the ultra spinning regimes. 19/32

46 20/32 If you are interested in these results please see my paper: arxiv:

47 Questions 21/32