The Doubly-Spinning Black Ring Spacetime

have been exchanged, to be consistent with the singly spinning solution as presented in the review [69], and the functions 𝐹(𝑥, 𝑦) and 𝐽(𝑥, 𝑦) have been replaced with 𝐴(𝑥, 𝑦) and 𝐿(𝑥, 𝑦) defined such that

𝐹(𝑥, 𝑦) = 𝑅²𝐴(𝑥, 𝑦)

(1−𝜈)²(𝑥−𝑦)² and 𝐽(𝑥, 𝑦) = 𝑅²𝐿(𝑥, 𝑦)

(1−𝜈)²(𝑥−𝑦)². (6.4) The length-scale parameter𝑅 is related to their𝑘 by𝑅² = 2𝑘².

It is useful at this stage to think a little bit more carefully about the properties of the metric functions 𝐴(𝑥, 𝑦) and𝐿(𝑥, 𝑦). Is it immediately apparent from the definition of 𝐴(𝑥, 𝑦) that we can write it in the form

𝐴(𝑥, 𝑦) = 𝐺(𝑥)𝛼(𝑦) +𝐺(𝑦)𝛽(𝑥) (6.5) for some 𝛼(𝜉) and 𝛽(𝜉). Note that there is a freedom in our choice of these functions;

we can add an arbitrary multiple of 𝐺(𝜉) to one and subtract it from the other without affecting𝐴(𝑥, 𝑦) itself. It turns out that the most convenient way of doing this is to pick

𝛼(𝜉) =𝜈(1−𝜉²)[

−(1 +𝜆²)−𝜈(1−𝜈) +𝜆𝜉(2−3𝜈)−(1−𝜆²)𝜉²]

(6.6) and

𝛽(𝜉) = (1+𝜆²)+𝜆𝜉(1+(1−𝜈)²)−𝜈𝜉²(2𝜆²+𝜈(1−𝜈))−𝜆𝜈²𝜉³(3−2𝜈)−𝜈²𝜉⁴(1−𝜆²+𝜈(1−𝜈)).

(6.7) We can also do a similar thing for 𝐿(𝑥, 𝑦). If we set

𝛾(𝜉) =𝜆√

𝜈(1−𝜉²)(𝜆−(1−𝜈²)𝜉−𝜆𝜈𝜉²) (6.8) then we find that

𝐿(𝑥, 𝑦) = 𝐺(𝑥)𝛾(𝑦)−𝐺(𝑦)𝛾(𝑥). (6.9) The ring-like coordinates can be related to two pairs of polar coordinates

(𝑡, 𝑟1, 𝜙, 𝑟2, 𝜓) via

𝑟₁ =𝑅

√1−𝑥²

𝑥−𝑦 and 𝑟₂ =𝑅

√𝑦²−1

𝑥−𝑦 , . (6.10)

Note that, in these coordinates, the flat space limit takes the standard form

𝑑𝑠² =−𝑑𝑡²+𝑑𝑟₁²+𝑟₁²𝑑𝜙²+𝑑𝑟₂²+𝑟₂²𝑑𝜓². (6.11) The black ring has a ring-like curvature singularity at 𝑦 → −∞, which is the ring (𝑟1, 𝑟2) = (0, 𝑅) in the polar coordinates (6.10).

6.2. THE DOUBLY-SPINNING BLACK RING SPACETIME 135

6.2.2 Inverse Metric

The inverse metric will be useful later, so we give it here for convenience, it reads ( ∂

∂𝑠 )₂

=−𝐻(𝑥, 𝑦) 𝐻(𝑦, 𝑥)

(∂

∂𝑡 )₂

+ (𝑥−𝑦)² 𝑅²𝐻(𝑥, 𝑦)

[

(1−𝜈)² (

𝐺(𝑥) ( ∂

∂𝑥 )₂

−𝐺(𝑦) ( ∂

∂𝑦 )₂)

+ 𝐴(𝑥, 𝑦) 𝐺(𝑥)𝐺(𝑦)

( ∂

∂𝜙 −Ω_𝜙 ∂

∂𝑡 )₂

− 2𝐿(𝑥, 𝑦) 𝐺(𝑥)𝐺(𝑦)

( ∂

∂𝜙−Ω_𝜙∂

∂𝑡 ) ( ∂

∂𝜓 −Ω_𝜓 ∂

∂𝑡 )

− 𝐴(𝑦, 𝑥) 𝐺(𝑥)𝐺(𝑦)

( ∂

∂𝜓 −Ω𝜓 ∂

∂𝑡 )₂]

. (6.12) Note that

𝐴(𝑥, 𝑦)

𝐺(𝑥)𝐺(𝑦) = 𝛼(𝑦)

𝐺(𝑦) + 𝛽(𝑥)

𝐺(𝑥) (6.13)

separates into 𝑥 and 𝑦 components, as do the analagous expressions for 𝐴(𝑦, 𝑥) and 𝐿(𝑥, 𝑦).

6.2.3 Horizon

The metric is singular when the function𝐺(𝑦) vanishes. The root at 𝑦=𝑦_ℎ ≡ −𝜆+√

𝜆²−4𝜈

2𝜈 (6.14)

is a coordinate singularity corresponding to an event horizon. Elvang & Rodriguez [74]

give a prescription for changing to new coordinates that are valid across the horizon, although it is very complicated to write the transformed metric down explicitly. In Section 6.4, we will construct an alternative set of coordinates that are valid as we cross the horizon, by looking for coordinates adapted to a particular class of null geodesics.

When 𝜆= 2√

𝜈, 𝐺(𝑦) has a double root at 𝑦=𝑦ℎ and the black ring is extremal. In this case, Ref. [88] derived the near-horizon geometry, and found that it is the same as that of a boosted extremal Kerr black string. This allows one to search for instabilities of this spacetime using the methods derived in Chapter 4.

6.2.4 Asymptotic Flatness

This spacetime is (globally) asymptotically flat, but this is not manifest in the ring-like coordinates, where asymptotic infinity corresponds to the point (𝑥, 𝑦) = (−1,−1). To see the asymptotics explicitly, we can make a change of variables (𝑥, 𝑦) 7→ (𝜌, 𝜃) by setting

𝑥=−1 + 2𝑅² 𝜌²

1 +𝜈−𝜆

1−𝜈 cos²𝜃 and 𝑦=−1−2𝑅² 𝜌²

1 +𝜈−𝜆

1−𝜈 sin²𝜃, (6.15)

with 𝑅√

(1 +𝜈−𝜆)/(1−𝜈) ≤𝜌 < ∞ and 0 ≤𝜃 ≤𝜋. Therefore, for large values of𝜌, the metric reduces to

𝑑𝑠² ≈ −𝑑𝑡²+𝑑𝜌²+𝜌²(𝑑𝜃²+ cos²𝜃𝑑𝜙²+ sin²𝜃𝑑𝜓²), (6.16) which is 5-dimensional Minkowski space expressed in polar coordinates, with the angular variables having the correct periodicities. This transformation was motivated by that given in [74] (although the formula given in that paper is incorrect).

6.2.5 Singly Spinning Limit

Since the coordinates used here vary slightly from those used in most papers on singly spinning rings, e.g. [16, 68, 69, 183], it is worth showing explicitly how this reduces to the original Emparan-Reall solution.

The singly spinning limit corresponds to setting 𝜈 = 0. This reduces the metric functions to the following:

𝐺(𝑥) = (1−𝑥²)(1 +𝜆𝑥), 𝐻(𝑥, 𝑦) = 1 + 2𝑥𝜆+𝜆² ≡𝐻(𝑥), (6.17) 𝛼(𝑥) =𝛾(𝑥) = 𝐿(𝑥, 𝑦) = 0, 𝛽(𝑥) = 𝐻(𝑥), 𝐴(𝑥, 𝑦) = 𝐻(𝑥)𝐺(𝑦) (6.18) and

Ω = Ω𝜓(𝑦)𝑑𝜓=−𝐶𝑅1 +𝑦

𝐻(𝑦)𝑑𝜓, where 𝐶≡

√

2𝜆²(1 +𝜆)³

1−𝜆 . (6.19) The convenience of the limits here is our main motivation for working with the particular choices of 𝛼 and 𝛽 that we made above.

The metric reduces to 𝑑𝑠² =−𝐻(𝑦)

𝐻(𝑥)

(𝑑𝑡+ Ω_𝜓(𝑦)𝑑𝜓)₂

+ 𝑅²𝐻(𝑥) (𝑥−𝑦)²

[𝐺(𝑥)

𝐻(𝑥)𝑑𝜙²+ 𝑑𝑥²

𝐺(𝑥)− 𝐺(𝑦)

𝐻(𝑦)𝑑𝜓²− 𝑑𝑦² 𝐺(𝑦)

] . (6.20)

6.2.6 Ergoregion

For the singly-spinning black ring, the ergoregion was first described in [68]. It is straight- forward to see that, in our notation, the ergosurface is where𝐻(𝑦) vanishes, which occurs at

𝑦 =𝑦_𝑒≡ −1 +𝜆²

2𝜆 . (6.21)

Furthermore, we have that𝑦_ℎ < 𝑦_𝑒 <−1, for all 𝜆, so the ergoregion does indeed exist, and, like the horizon, has topology 𝑆¹×𝑆² (like all surfaces 𝑦= const for 𝑦∕=−1).

6.2. THE DOUBLY-SPINNING BLACK RING SPACETIME 137 Things become significantly more complicated in the doubly spinning case. The ergosurface is defined by the vanishing of𝐻(𝑦, 𝑥), so can described (locally) as a surface 𝑦=𝑦_𝑒(𝑥).

Note that 𝐻(−1,−1) = (1−𝜈)(1 +𝜈−𝜆)² >0, and therefore 𝐻(𝑦, 𝑥)>0 in some neighbourhood of asymptotic infinity. Hence, far from the ring,∂/∂𝑡is indeed timelike as expected. It can also be shown that, for all𝑥∈[−1,1],𝑦𝑒(𝑥)> 𝑦ℎ, and hence the horizon is always surrounded by an ergoregion, with no intersection between the ergosurface and the horizon. This is in contrast to the Kerr case, where they touch at the poles. There is a clear reason for this; in Kerr the poles are the points on the horizon that are left invariant under rotations generated by the angular Killing vector, but in the black ring there are no points on the horizon left invariant under∂/∂𝜓.

For the singly-spinning ring,𝐻(−1)>0, and hence the axis𝑦 =−1 lies outside the ergoregion, which must therefore have ringlike topology. However, in the doubly-spinning case, for sufficiently large𝜈, there are some values of𝑥for which𝐻(−1, 𝑥)<0, and hence the ergosurface intersects the axis and can therefore no longer have the ring-like topology 𝑆¹×𝑆².

What is the new topology? Note that

𝐻(−1, 𝑥) =𝐻(−1,−𝑥) = (1−𝜆)² −𝜈²+𝜈𝑥²(

1−𝜆²−𝜈²+ 2𝜆𝜈)

(6.22) is even as a function of𝑥, and that therefore

𝐻(−1,1) =𝐻(−1,−1) = (1−𝜈)(1 +𝜈−𝜆)² >0. (6.23) Thus, for all allowed values of 𝜆 and 𝜈 we have that the point at the centre of the ring lies outside of the ergoregion. As 𝜈 → 1 (and hence 𝜆 → 2), the size of the ergoregion becomes larger and larger, but there is always a region near to the centre of the ring that remains outside it. Thus, the ergosurface topology is that of two disconnected 3-spheres, 𝑆³∪𝑆³.

Note that 𝐻(−1, 𝑥) is minimum at 𝑥 = 0, so to determine where in the black ring family the change of topology occurs we need to look at the case where

𝐻(−1,0) = 1 +𝜆²−𝜈²−2𝜆= 0. (6.24) This occurs when 𝜆 = 1−𝜈. Note that we must have 𝜈 ≤3−2√

2 ≃ 0.17 for it to be possible to have this condition satisfied. For this metric, we have that

𝐻(−1, 𝑥) = 4𝜈²𝑥²(1−𝜈), (6.25) so the ergosurface touches the 𝑦 = −1 axis on the circle 𝑥 = 0, 𝑦 = −1. In the plane polar type coordinates (6.10), the locus of points where the ergosurface pinches is at

Figure 6.1: Two-dimensional projection of the shape of the ergoregion in the case 𝜈 = 1/9, for 𝜆 = 7/9 (𝑆¹ ×𝑆² ergosurface), 𝜆 = 8/9 (critical case) and 𝜆 = 1 (𝑆³∪𝑆³ ergosurface). The inner circles are the edge of the horizon, the outer lines the ergosurface and the central line the axis 𝑦 =−1. (Plotted in𝑟₁, 𝑟₂ coordinates.)

𝑟1 =𝑅, 𝑟2 = 0, which makes clear that this is indeed a circle. We will see later (§6.3.4) that there exist stable ‘trapped’ null geodesics orbiting around this circle. Figure 6.1 shows a 2D projection of the shape of the ergoregion in this case.

Finally, there is a nice intuitive way to think about why the ergoregion takes this form. We can think, rather loosely, of the black ring as a Kerr black hole at each point around the 𝑆¹. When the Kerr black hole is rotating rapidly (corresponding to rapid 𝑆² rotation of the black ring), its ergoregion becomes increasingly elliptical, so that eventually an observer near the centre of the ring feels frame dragging from the𝑆² rotations on opposite sides of him simultaneously. The effects cancel near the centre of the ring, leaving a region which does not lie in the ergoregion. To summarise, Figure 6.2 shows the parameter space for all allowed doubly-spinning black rings.

Recently, Cortier [184] has provided a rigorous analysis of the ergosurface for this spacetime, confirming the results of this section.

In document New approaches to higher-dimensional general relativity (página 145-150)