Abstract
In this work, we construct a five-dimensional spherically-symmetric, charged and asymptotically Anti-de Sitter black hole with its singularity being point-like and strictly localised on our brane. In addition, the induced brane geometry is described by a Reissner-Nordstr\"{o}m-(A)dS line-element. We perform a careful classification of the horizons, and demonstrate that all of them are exponentially localised close to the brane thus exhibiting a pancake shape. The bulk gravitational background is everywhere regular, and reduces to an AdS$_5$ spacetime right outside the black-hole event horizon. This geometry is supported by an anisotropic fluid with only two independent components, the energy density $\rho_E$ and tangential pressure $p_2$. All energy conditions are respected close to and on our brane, but a local violation takes place within the event horizon regime in the bulk. A tensor-vector-scalar field-theory model is built in an attempt to realise the necessary bulk matter, however, in order to do so, both gauge and scalar degrees of freedom need to turn phantom-like at the bulk boundary. The study of the junction conditions reveals that no additional matter needs to be introduced on the brane for its consistent embedding in the bulk geometry apart from its constant, positive tension. We finally compute the effective gravitational equations on the brane, and demonstrate that the Reissner-Nordstr\"{o}m-(A)dS geometry on our brane is caused by the combined effect of the five-dimensional geometry and bulk matter with its charge being in fact a tidal charge.
Highlights
The idea of the existence of extra spacelike dimensions, first suggested by Kaluza [1] and Klein [2], is more than a hundred years old
It has been employed in order to formulate fundamental theories of particle physics, such as string theory [3,4], or more phenomenologically oriented models such as the large extra dimensions [5–7] or warped extra dimensions [8,9] scenaria
In the large extra dimension scenario [5–7], the extra spacelike dimensions are compactified to a new length scale—this scale is an independent scale of the theory which has to be smaller than the μm scale in order to avoid observation [12,13]
Summary
The idea of the existence of extra spacelike dimensions, first suggested by Kaluza [1] and Klein [2], is more than a hundred years old. The warping of the bulk spacetime in the case of the warped extra dimensions scenario posed a significant obstacle in the construction of analytical solutions describing black holes centered on our brane and extending in a regular bulk spacetime In the first such attempt [16], the effort to construct a fivedimensional braneworld black hole led instead to the emergence of a black-string solution; the line element on the brane matched the Schwarzschild solution, its singularity was not pointlike and localized on our brane, where the gravitational collapse had taken place, but it was extending along the infinite fifth dimension. The fivedimensional background was everywhere regular and reduced to a pure AdS5 spacetime right outside the black hole horizon This geometric solution was not a vacuum one, and a form of bulk matter had to be introduced in order to support it.
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