arXiv:1012.4541v1 [hep-th] 21 Dec 2010 Brane Holes Valeri P. Frolov��� Theoretical Physics Institute, University of Alberta, Edmonton, AB, Canada, T6G 2G7 Shinji Mukohyama��� Institute for the Physics and Mathematics of the Universe (IPMU) The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8582, Japan (Dated: December 22, 2010) The aim of this paper is to demonstrate that in models with large extra dimensions under special conditions one can extract information from the interior of 4D black holes. For this purpose we study an induced geometry on a test brane in the background of a higher dimensional static black string or a black brane. We show that at the intersection surface of the test brane and the bulk black string/brane the induced metric has an event horizon, so that the test brane contains a black hole. We call it a brane hole. When the test brane moves with a constant velocity V with respect to the bulk black object it also has a brane hole, but its gravitational radius re is greater than the size of the bulk black string/brane r0 by the factor (1 - V 2)-1. We show that bulk ���photon��� emitted in the region between r0 and re can meet the test brane again at a point outside re. From the point of view of observers on the test brane the events of emission and capture of the bulk ���photon��� are connected by a spacelike curve in the induced geometry. This shows an example in which extra dimensions can be used to extract information from the interior of a lower dimensional black object. Instead of the bulk black string/brane, one can also consider a bulk geometry without horizon. We show that nevertheless the induced geometry on the moving test brane can include a brane hole. In such a case the extra dimensions can be used to extract information from the complete region of the brane hole interior. We discuss thermodynamic properties of brane holes and interesting questions which arise when such an extra dimensional channel for the information mining exists. PACS numbers: 04.70.Bw, 04.70.-s, 04.25.-g Alberta-Thy-14-10, IPMU-10-0219 I. INTRODUCTION Models with large extra dimensions have been ���popu- lar��� and intensively discussed since 1998 [1���3]. In these models our four dimensional spacetime �� is considered as a brane embedded in a higher dimensional bulk space. The main purpose of the present paper is to demonstrate that in such models under special conditions extra dimen- sions can be used to extract information from the interior of a four dimensional black hole. This happens when the four dimensional surface �� representing our world is not geodesic. Consider two points on �� and suppose that they can be connected by a geodesic in the bulk space- time. Let us assume that there also exists a curve on �� which connects the two points and which is geodesic in the induced geometry. In this case the geodesic distance between the two points along the bulk geodesic is in gen- eral different from the geodesic distance in the induced geometry. Under special conditions, for two points on �� separated by a spacelike induced interval, there may exist causal curves connecting them through the bulk space- time. An observer on the brane �� would describe this situation by saying that the extra dimensions provide one with a channel of information exchange with an effective super-luminal velocity. *Electronic address: vfrolov@ualberta.ca ���Electronic address: shinji.mukohyama@ipmu.jp A simple model demonstrating such a possibility was considered in [4]. In the paper a stationary cosmic string in the Kerr geometry was studied. The 2D induced met- ric on the string worldsheet has a horizon at its intersec- tion with the ergosurface of the bulk geometry. It was shown that the 2D black hole produces the Hawking ra- diation of the string transverse degrees of freedom [5], so that the cosmic string can be used to mine energy from the bulk black hole [6]. In this paper we study more ���realistic��� case where a brane representing our 4D world is embedded in a higher dimensional bulk spacetime. We neglect effects con- nected with the thickness of the brane. We also neglect the gravitational field generated by the brane and use the test brane approximation. To simplify the presentation in the most of the paper we discuss the case in which the bulk space has five dimensions. Generalization to other dimensions of the brane and the bulk spacetime is straightforward and is briefly discussed at the end of the paper. We discuss two models. In the first model the test brane is moving with a constant velocity (as measured by a distant observer) in the bulk space with a black string. We show that if r0 is the gravitational radius of the black string, the induced geometry has a 4D black hole with the larger radius re = (1 ��� V 2)���1r0. We demonstrate that the test brane embedding is not geodesic, and ex- tra dimensions can be used to extract information from the region between r0 and re of the induced brane hole. In the second model we replace a bulk black string by a spacetime with a static massive thin shell of mass M and

2 radius rs. We shall see later that the gravitational radius of the shell is r0 = 2M ��� M 2/rs ��� rs and that the bulk spacetime is regular and does not contain a bulk black object. Nonetheless, if re is greater than rs then the in- duced metric has a brane hole. We demonstrate that in this model the complete brane hole interior is ���visible��� through extra dimensions. The existence of the extra dimensional ���window��� for ob- serving��� the brane hole interior raises a number of inter- esting questions. Some of them will be briefly discussed in the paper. The rest of this paper is organized as follows. In Sec. II two five-dimensional geometries describing a black string and a dark shell at rest are presented. In Sec. III these static bulk geometries are boosted so that they describe a moving string and a moving dark shell, respectively. Sec. IV describes a test brane in the background of the boosted black string and shows that the induced met- ric on the test brane contains a brane hole. Sec. V shows some thermodynamic properties of brane holes. In Sec. VI it is shown that information can be tranmit- ted via the bulk space from the inside of a brane hole to the outside. Sec. VII describes a test brane in the boosted dark shell geometry and shows that the induced metric can still have a brane hole. Sec. VIII is devoted to summary of the paper and discussions. II. BLACK STRING AND DARK SHELL In this paper we study properties of a moving test brane in an external gravitational field. For the latter we use two models: (i) a bulk black brane, and (ii) a dark shell model. In this subsection we discuss the first model. For simplicity we assume the number of dimen- sions of the bulk spacetime is equal to five, so the bulk black brane in fact is a black string. We also assume that the test brane has a codimension one, so that the induced geometry on it is four dimensional. In the both cases one can compactify the metric in the fifth flat di- mension, so that the corresponding z-coordinate becomes periodic with some period L. Such a compactification as well as generalizations to higher number of dimensions, which is quite straightforward, are briefly discussed in Appendices. A. Bulk black string The 5-dimensional metric of a black string is (a,b = 0, 1,..., 4) dS2 = gabdyadyb = �����d��2 t + dr2 �� + r2d��2 + d��2 z , (1) �� = 1 ��� ��, �� = 1/r , (2) where d��2 is a line element of a unit sphere S2, d��2 = d��2 + sin2 ��d��2 . (3) This metric is a direct sum of the Schwarzschild metric and a line. We write the metric in the dimensionless form, that is put the gravitational radius 2M equal to one. To restore proper dimensionality it is sufficient to multiply coordinates ��, t �� z and r by the factor (2M)���1 and rescale the metric as dS2 ��� (2M)2dS2. Denote ��a �� = ��a��) ( t + v��(��) a z , (4) where ��a��)���a ( t = ����� t and ��(��)���a a z = ����� z are two commuting Killing vectors generating transformations along �� t and �� z coordinates, respectively. The norm of this vector is ��2 �� = ����� + v2 . (5) The vector ��a �� is timelike for v2 �� and becomes null when v = �� ��� ��. This means that there exist a Killing observer with the velocity ua ��� ��a �� at the radius r only if its relative velocity v with respect to a rest frame is in the interval v ��� (��� ��� ��, ��� ��). Near the horizon where �� vanishes, this interval shrinks to zero. One can say that the particle motion in z-direction is frozen. B. Dark shell In our second model we assume that outside some ra- dius rs = 1+ �� with small positive �� the metric coincides with (1) and inside this radius it is flat dS��� 2 = ��� �� 1 + �� d��2 t + dr2 + r2d��2 + d��2 z . (6) In what follows we denote the external metric as g+ab. We choose the form of the metric so that all the metric coefficients, except grr are continues at the junction sur- face ����. The jump of the coefficient grr implies the jump of an extrinsic curvature. Thus the spacetime contains a thin massive shell. Using Israel���s method [7] it is possible to find the shell parameters: the surface energy density �� and compo- nents of the pressure Pz,��. 8���� = 2 1 + �� parenleftbigg 1 ��� radicalbigg �� 1 + �� parenrightbigg , 8��Pz + 8���� = 8��P�� + 4���� = 1 2��1/2(1 + ��)3/2 . (7) Note that the mass of the shell is M = 4����rs 2 = rs parenleftbigg 1 ��� radicalbigg 1 ��� r0 rs parenrightbigg , (8) where r0 = 1 and rs = (1+��) are the horizon radius of the bulk black string and the radius of the shell, respectively. Thus we obtain r0 = 2M ��� M 2 rs . (9)