Kerr Black string flow

We give a general theory of a rotated black string stream into a rotational horizon in dimension D=5. It is a configuration about one smooth intersection between these two objects when the spacetime is axisymmetric and in the limit that the thickness of the black hole is much larger than the thickness of the black string. Following this configuration we extend to a more general situation, the rotational charged flows.


Introduction
String theory is a very valuable tool for us to understand properties of the black hole. One of the greatest successes of string theory in this field is the statistical mechanics explanation of the Bekenstein-Hawking entropy of certain extremal and the nearextremal black hole. A number of different types of black holes have been studied well and all of them support the explanation of the Bekenstein-Hawking entropy since the first success in [1]. Recent works of this field have focused on the question about stationary spacetimes allowing event horizons are not Killing horizons [2]. Some former works [4][5][6] may forbid this construction, but from a physical perspective, there are different surface gravities at the horizons where they link two asymptotic spaces of the infinite extents, which indicate different temperatures. In thermodynamic theory, this is a description about a steady heat stream between two infinite heat repertories that preserve a fixed temperature gradient when the time is under evolution [2].
In article [2] the authors describe a very impressive result for a flowing horizon. Basically they build up their construction with Schwarzschlid metric that is a static black hole solution. But in this article we extend the former solution to a more general condition, consider a Kerr metric to construct the flow, and this is easy to extend to a Kerr-Newman metric condition. Even though the descriptions of horizon flows motivated by AdS/CFT [3,[9][10][11][12][13][14][15][16] have been studied well, as the same reason in [2], in our construction we don not need a negative cosmological constant to hold that the system exists steadily.
We start our construction form this picture: a very thin black string falls perpendicularly and smoothly into a very large black hole. Their radius are denoted by r bs and r hs respectively. Both the black string and the black hole rotate with the same angular velocity α, and the axles of the rotation coincide to the direction of the falling of the black string. Since we have r bh ≫ r bs , given that the surface gravity is inverse ratio to the radius, the surface gravity of black string is much larger than the surface gravity of black hole. When the string is falling into the black hole horizon without any external interference, we can anticipate that the two horizons fuse smoothly into each other. By this construction, the black hole can accumulate mass from the falling string and as the consequence that the black hole must grow in size irreversibly.
However, when we take the limit r bh → ∞ holding r bs fixed and just pay attention on events that happen in the intersected region we can get rid of the effect of the growing of the black hole. The black hole horizon then becomes a Rindler-like infinite horizon. Here we say the horizon is a Rindlerlike horizon rather than just a Rindler horizon is because we must take the rotation into consideration. If the angular velocity is zero, the horizon is exactly the Rindler horizon. Changing to the rest frame of the falling black string, the acceleration horizon vanish: then we have a configuration of a stationary black string. Alternatively, if we take a stationary black string and study it from a frame that accelerates along the direction of the string, what we observe is a string in free fall into an acceleration horizon. We will construct the event horizon for such accelerated observers, and study how the rotation will affect the result and make a comparison with the non-rotation result.
The arrangement of this paper are: in Section 2 we will construct a Kerr black string flow and study the properties of the construction. In Section 3 we will study the black string flow and its equilibrium condition, then extend to a more general model. And Section 4 is our conclusion and outlook.

Construction of Kerr black string flow
Configurations of higher dimensional rotating (charged) black holes obtained in string theory have be studied a lot on a micro level [18][19][20]. Different from these construction above, we follow [2] to build up a new metric.

Horizon of black string flow
In article [2] the authors give their metric as follow it is a n-dimensional black string flow without rotation. Generally we can generalize metric (2.1) to a new general expression with rotation where and F is any kind of metric function which should have a positive sigh outside the black string horizon and finite at r → ∞ when α is fixed. The metric here gives us a picture in the rest of the free-falling and rotating black string in D=5 dimensions. When the black string is missing (M=0), we have the null surfaces t = z + t 0 , with constant t 0 , are the acceleration horizons. One condition we need to guarantee is that when it is far from the string and at the same time the rotation can be neglected the null rays change to the conventional Rindler horizon, i.e., where the dot indicates the derivative of an affine parameter λ. This is the main condition that we will use to get our results. Because the Kerr metric is lack of symmetry, we need to rewrite the metric into a Schwarzschlid-like form, i.e., the Boyer-Lindquist coordinate. This can reduce our computation efficiently. The new metric is The relations between new and old metric are In this new metric the event horizon save the symmetry SO(3) obviously.
Since the affine parameter λ is also valid in new coordinate system,the equations for T (λ), Z(λ), R(λ) are easy to obtain from in this equation ǫ and p are two integration constants because of the isometries generated by ∂ T and ∂ Z , for more detail see appendix A. From equation (2.5), (2.8) and (2.12) we know the relation between ǫ and p.
we can neglect the contributions of F and ρ 2 ∆ because when r → ∞, both of them approximate to 1.

Null hypersurface and analysis
Now from the equations (2.14) and (2.18), we have the representation ofŻ Using the equations (2.13) and (2.19), we havė Because both of theŻ andṘ can be represented byṪ , we can deduce an equation that consists ofŻ,Ṫ andṘ, whereÂ θ = αsin 2 θdφ. Back to the old coordinate system we get (2.24) When α = 0, Kerr metric will return to Schwarzschlid metric and, no losing generality, set F=1 and ∆ ρ 2 = f for convenience, the deduced null hypersurface becomes This is the null hypersurface ruled by geodesics that are characterized by the one-form equation in [2]. Because of the missing of the symmetry, equations (2.23) and (2.24) both have extremely complicate resolutions. For convenience and no losing generality, we first set F = 1 and M = 1, then just study the event horizon that the direction of the θ = 0. In this way we get the new one-form equation Benefiting from θ = 0 the old coordinate system is equal to new one and as a consequence we get (2.27) Then we have each value of t 0 result to a corresponding null hypersurface that ends at different values in the null coordinate in future null infinity. Clearly any of them can regard as our event horizon. Here, for convenience, we denote the null hypersurface with t 0 = 0 as the horizon. This is not a stationary horizon: the action of ∂ t can change t 0 , therefore it dose not map a hypersurface onto itself but onto another one instead. The explicit form of the integral result of r in (2.28) is given it in the appendix B.
The surfaces H f are plotted in figure 1.
In these pictures, M = α indicates that they are extreme cases, the inner event horizon and outer event horizon meet at r = 1. A constant imaginary part occurs when r < 1, we use the magnitude of this complex number to get these figures.

Null geodesic congruence
Without losing generality, we just consider the negative term in equation (2.23) which corresponds to the outgoing hypersurface from now on. Benefiting from the freedom to scale λ we can set ǫ = p − A θ = 1. Then from the equation (2.14), we havė According the equation (2.22), we have This gives usṘ (2.31) Because dR = dr we can write equation(2.31) into (t, r, z) space Again we consider the situation in figure 1. By setting θ = 0, F = M = α = 1 we get dr dλ = 2r The function λ(r) is plotted in figure 2. More specific calculations see appendix C.

Event horizon and ergosurface
Because Kerr black hole is stationary but not a static, the event horizons are not Killing horizons. The region between the outer event horizon and the Killing horizons are known as the ergosphere. Inside the ergosphere, Figure 2: The behavior of function λ(r). The picture on the left depict the function when r is large. And on the right we show the behavior of the function when r is close to 0, and notice that 0 is a singularity. When r → 0, we get λ → ∞. And as λ grows, r moves toward to ∞.
any objects have a velocity with the direction of the rotation of the black hole, but at the same time they are free to move toward or away from the event horizon. When we consider the Kerr black string flow, the same thing happens on hypersurface H f .
According to equation (2.33) and figure 2, we find that all null rays on H f must go towards r → ∞ as the λ grows. This means that any timelike trajectory that remains within bounded values of r will cross H f at last. As a consequence any observers that remain within a finite range of the black sting will fall across H f eventually. As the same as in Kerr black hole any observers inside this surface cannot remain static but are dragged along with the string, so this is a particular ergoshpere that the Kerr string flow can distinguish from an ordinary Kerr black hole.
In our construction the black string is accelerating, and then the ergoregion grows too. For any observes who want to avoid to fall across this H f , just an acceleration in the z direction is not adequate, they also need move out towards r → ∞. Even though the hypersurface H f we consider here is a particular case, it is straightforward to extend this analysis to any θ.

Out-of-equilibrium flow
Consider the vector this is an affine generator of the null geodesic congruence. It is convenient to consider the following non-affine generator 1 of the future horizon The reason we normalize the generator into equation (3.2) is in that way the generators can recover to the black string horizon and the acceleration horizon near and far from the black string respectively. The surface gravity κ (l) of l is defined in this form Given that λ is an affine parameter we find The specific calculation is showed in Appendix D. According to equation (2.32), we finally have κ (l) = ( −M α 2 cos 2 θ − α 2 r + rα 2 cos 2 θ + M r 2 (α 2 cos 2 θ + r 2 ) 2 ) C 2 r 2 + C 1 r + C 0 F (r 2 + α 2 cos 2 θ) . (3.5) This surface gravity diminish monotonically to 0 at an enough large r. And when α = 0 the Kerr black string flow returns to Schwarzschlid black string flow, and set F = 1, we get the surface gravity which coincides with that in [2]. There is a general chance that we can set F to be a general form to make the surface gravity κ = 0. Namely, putting concrete coefficients C i (i = 0, 1, 2, below the equation (2.32)) into equation (3.5), we have We now set M = 1 and dφ dλ = α to concretely give different expressions of equation (3.7). We plot this function in figure 3 with four different conditions (α =1, 0.75, 0.5, 0,25 respectively).
1 According the definition [17], any parameter λ related to the proper time τ in this way: λ = aτ + b where a and b are constants, which is called an affine parameter. It is obvious that 1 2 ∆ ρ 2 is not a constant, so parameter l is not an affine parameter and the generator of l is not an affine generator, too. Our previous analysis about metric (2.6) can be extended to a more general form, for example the charged black string. The metric can be written in this form with equations (2.7)-(2.12), where T , Z and R are general functions of angular momentum J, charge Q and space coordinates (r, θ, φ). For the consistence of the whole theory demands that all the general functions are positive outside the black string horizon and tend to finite at r → ∞, and the capital letters are new coordinates after the above transformations in equations (2.8)-(2.12) which have better symmetry. Here we restrict ourselves to T < Z. 2 Similar to the deduction and calculation of equation (2.23), we can achieve that the null hypersurface, under this condition is whereÂ θ = αsin 2 θdφ.
Similar to the deductions of equations (2.29) and (2.31), the congruences in terms of the affine parameter λ are here the letter ξ is to label the different null rays. Then, following the beginning of section 3, we can give the non-affine null geodesic generator where A θ = αsin 2 dφ dλ . And similar to appendix D, because λ is an affine parameter, we have the surface gravity κ (l) When the rotation decreases to 0, we get the ordinary result (3.14) Again this is the result in static spacetime. In equation (3.13) we can get an equilibrium condition again This extreme limit will happen when we consider the black strings with basic string charge. The charged string flow describes a string with a specific excitation down to a large black hole. Even if the string charge can decrease the black string temperature, only at equation (3.15) can it be in thermal equilibrium. Above extremity the string excitation is much hotter than that of the black hole, and the system seems to have a different behavior from the configuration in which the string excitation is in thermal equilibrium with a finite temperature horizon (a worldsheet approach [21][22][23][24] or a blackfold approach [25,26]). In this formation, when the black string does not attain to extreme, it will dump energy to the black hole.

Conclusion and outlook
In this paper, we generally give and analyze a model of the rotating black string flow in dimension D=5, and extend this solution to a charged rotating black sting flow, and study the equilibrium condition. The system we study here requires that the string is very thin comparing with the black hole and the whole process is free from any external force interference. When investigating the instability of the black string in the late time evolution, a similar construction has been found [27]. The microscopic construction of the place where the black string and black hole intersect is still blank for us. We plan to study this question in the future. Another interesting field is the the similarity between the black funnels and the black string flows [2,3,7,8,11,12,[28][29][30][31]. We believe that there should be some relations between the rotating black string flows and some kinds of black funnels. Using this method we can extend to other situations, for example, a free falling string that is not rotated falls into a rotating black hole. Under this construction we may expect that the rotation velocity of the black hole decreases to zero after the string pour enough mass to this system.
We leave the boundary term away here as in field theory. By this method we can get dZ dλ = p F , too.
This result is given by Mathematica 7.0 and will give a constant imaginary part when r < 1. Our explanation is when any observers come across the event horizon they will travel in a region where time coordinate and space coordinate exchange their symbols. In order to denote this effect when they travel into the inner horizon, a constant imaginary part must remain.