Comments on Higher-Dimensional C-metric

1. Comments on Higher-Dimensional C-metric YITP Kodama Hideo The 8th Singularity Workshop Jan. 6 – 8, 2007

2. 4D C-metric

3. Petrov-type D Static Spacetimes Class A Class B Class C Ehlers J, Kundt W 1962; Levi-Civita T 1917

4. C-metric Ricci tensor Weyl tensor where and The special case of the most general type D electrovac solution by Plebanski JF, Demianski M 1976

5. Flat Limit For = -1, M=0 and K=1, in terms of the variables with the C-metric can be written • This represents the Minkowski spacetime in the Rindler coordinates, and, each curve with constant x, y,  has a constant acceleration. • The covered region has an acceleration horizon at y=-1, and the spatial infinity corresponds to x=y=-1.

6. Schwarzschild Limit G(x) can be factorised as where In terms of the variables the C-metric can be written where

7. Conical String Singularity If we choose the angle variable  so that the metric is regular at the south pole x=x1 (=), then the angular part of the metric is conformal to around the north pole x=x2 (=0), where This implies that the metric has a conical singularity along the z-axis connecting the black hole horizon and the acceleration horizon. This singularity corresponds to a strut with netative tension 2=2. =0 case

8. Braneworld Black Hole

9. C-metric as 4D Braneworld BH Let us consider the special AdS C-metric corresponding to In the limit =0, in terms of the variables The above AdS C-metric can be written

10. Global Structure x1· x <0 0<x · x2 x=0

11. Black Hole in the 4D Braneworld The extrinsic curvature of the timelike hypersurface x=0 is homogeneous and isotropic: Hence, we can cut off the x>0 part of the solution and put the critical vacuum Z2 brane at the boundary x=0. This surgery provides a regular localised black hole in the 4D braneworld. Emparan R, Horowitz GT, Myers RC (2000)

12. 5D C-Metric as Braneworld BH ? 4D C-metric suggests that the yet-to-be-found localised BH solution in the 5D braneworld model may be given by an accelerated BH solution in the 5D AdS. However, • This solution should not represent a regular black hole spacetime with a compact horizon because of the uniqueness theorem of the static AdS bh . • The solution may not be singular in contrast to the 4D case, because the string in the 4d space has the codimension 3. • Hence, it is expected that the string source is surrounded by a tubular horizon extending to infinity.

13. How Can We Find a 5D C-metric? • Perturbative approach Study the property of solutions produced by a stringy singular source in 5 dimensions with the help of the linear perturbation theory in the Scwarzschild background. • Higher-dimensional Petrov type Extend the Petrov type classification and the Newman-Penrose formulation to higher dimensions and classify the static ‘type D’ solutions. • Soliton Method The 4D C-metric can be obained from the Rindler solution by adding 2 solitons in the Belinski-Zahkarov method. This suggests that higher-dimensional solutions representing an accelerated black hole can be obtained from the Rindler solution by the same method. Once their explicit expressions are found, their extension to the case of 0 may be found.

14. Perturbative Approach

15. Static Perturbations of a Black Hole Background metric Static scalar perturbation Gauge-invariant variables Master equation where

16. 4D C-metric as a Perturbation to the Schwarzschild Solution When the acceleration MA' is small, the C-metric can be expressed as This can be regarded as a scalar-type perturbation to the Schwarzschild solution. In the harmonic expansion, the gauge-invariant amplitudes are From this, we find that this perturbation is produced from the source This is consistent with the line density of a string, 2 = -8 , determined from the deficit angle.

17. Higher-Dimensional Analogue This analysis in the 4D C-metric case suggests that in higher dimensions, the energy momentum tensor of the static string source on the z-axis with z<0 is given by Here,  is required to be constant from the conservation law. In terms of the harmonic coefficients, it is expressed as This determines the source term SY for the master equation as

18. Solution where with

19. Asymptotic Behaviour • At large r where This indicates that the horizon is formed around ρn-2～μ and have a tubular shape extending to infinity along the z-axis. • At r ' 2M Hence, the behaviour of the perturbation is similar to that in the large-r region and consistent with the picture that the horizon at the central part of size r =2M is connected to an infinitely long tubular horizon of radius »1/(n-2).

20. Can We Get a Braneworld BH? For the exact Schwarzschild black hole, the hyperplane crossing the horizon at the equator is the only brane satisfying the junction condition Hence, for the perturbative C-metric, the brane crosses the horizon near the equator: =/2 + (r). Then, the perturbation of the junction condition determines (r) as and gives additional constraints on the metric perturbation at =/2

21. n=2 In this case, the metric perturbation is expressed as and satisfies these constraints. For this,  vanishes. • n >2 If we require that the sum of the modes with l>1 falls off at r →∞, σ is determined as and the above constraints reduce to the single equation This equation is not likely to be satisfied for all x except for x=0.

22. What was wrong? • The localised black hole solution in the braneworld model cannot be represented by a C-type metric in higher dimensions, even if such a solution exists. Or • The localised black hole solution in the braneworld model is represented by a C-metric whose source term has a structure different from that in four dimensions. In fact, it seems possible to find such a solution by modifying the source term. Which is correct will become clear if we can find an exact solution by the soliton method. Study along this line is now in progress.