Charged shells in Lovelock gravity: Hamiltonian treatment and physical implications

Gonçalo A. S. Dias, Sijie Gao, and José P. S. Lemos

Phys. Rev. D 75, 024030 – Published 23 January 2007

Abstract

Using a Hamiltonian treatment, charged thin shells, static and dynamic, in spherically symmetric spacetimes, containing black holes or other specific types of solutions, in $d$ dimensional Lovelock-Maxwell theory are studied. The free coefficients that appear in the Lovelock theory are chosen to obtain a sensible theory, with a negative cosmological constant appearing naturally. Using an Arnowitt-Deser-Misner (ADM) description, one then finds the Hamiltonian for the charged shell system. Variation of the Hamiltonian with respect to the canonical coordinates and conjugate momenta, and the relevant Lagrange multipliers, yields the dynamic and constraint equations. The vacuum solutions of these equations yield a division of the theory into two branches, namely $d - 2 k - 1 > 0$ (which includes general relativity, Born-Infeld type theories, and other generic gravities) and $d - 2 k - 1 = 0$ (which includes Chern-Simons type theories), where $k$ is the parameter giving the highest power of the curvature in the Lagrangian. There appears an additional parameter $χ = (- 1)^{k + 1}$ , which gives the character of the vacuum solutions. For $χ = 1$ the solutions, being of the type found in general relativity, have a black hole character. For $χ = - 1$ the solutions, being of a new type not found in general relativity, have a totally naked singularity character. Since there is a negative cosmological constant, the spacetimes are asymptotically anti-de Sitter (AdS), and AdS when empty (for zero cosmological constant the spacetimes are asymptotically flat). The integration from the interior to the exterior vacuum regions through the thin shell takes care of a smooth junction, showing the power of the method. The subsequent analysis is divided into two cases: static charged thin shell configurations, and gravitationally collapsing charged dust shells (expanding shells are the time reversal of the collapsing shells). In the collapsing case, into an initially nonsingular spacetime with generic character or an empty interior, it is proved that the cosmic censorship is definitely upheld. Physical implications of the dynamics of such shells in a large extra dimension world scenario are also drawn. One concludes that, if such a large extra dimension scenario is correct, one can extract enough information from the outcome of those collisions as to know, not only the actual dimension of spacetime, but also which particular Lovelock gravity, general relativity or any other, is the correct one at these scales, in brief, to know $d$ and $k$ .

Received 14 November 2006

DOI:https://doi.org/10.1103/PhysRevD.75.024030

Authors & Affiliations

Gonçalo A. S. Dias^*

Centro Multidisciplinar de Astrofísica—CENTRA Departamento de Física, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Sijie Gao^†

Department of Physics, Beijing Normal University, Beijing 100875, China

José P. S. Lemos^‡

Centro Multidisciplinar de Astrofísica—CENTRA Departamento de Física, Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

^*Electronic address: gadias@fisica.ist.utl.pt
^†Electronic address: sijie@bnu.edu.cn
^‡Electronic address: lemos@fisica.ist.utl.pt

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Vol. 75, Iss. 2 — 15 January 2007

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Figure 1
The shell trajectory manifold $\partial V_{ξ}$ in spacetime is depicted. $V_{-}$ and $V_{+}$ are the spacetime regions interior and exterior to $\partial V_{ξ}$ , respectively, and $Σ_{t}$ is the hypersurface intersecting $\partial V_{ξ}$ . Also shown are the normal vector $n$ (normal to $Σ_{t}$ ), the velocity vector $u$ (tangent to $\partial V_{ξ}$ ), the vector $ξ$ (the spacelike normal to $\partial V_{ξ}$ ), and the vector $e_{a}$ (the generic tangent vector to $\partial V_{ξ}$ ). The vectors indicating the origin and direction of the adapted coordinate system ${T, X}$ are depicted together with the $u$ and $ξ$ vectors.Reuse & Permissions
Figure 2
In this figure the branch $d - 2 k - 1 > 0$ (general relativity when $k = 1$ , Born-Infeld when $k = [\frac{d - 1}{2}]$ , and other generic cases) is studied in some particular instances. In each plot the effective potential ${\dot{R}}^{2}$ as a function of $r$ (which gives the turning points and the allowed regions for the shell), and the metric function $f^{2} (r)$ as a function of $r$ (which gives the horizon formation, for several different values of the parameters) are displayed. The vertical axis $Φ (r)$ represents both ${\dot{R}}^{2} (r)$ (full line) and $f_{+}^{2} (r)$ (dashed line). Note that the thin shell trajectory is allowed only in the region where ${\dot{R}}^{2}$ is positive. In all the plots the respective Newton’s constant is put equal to the unity, $G_{k} = 1$ , so that radii and energies are measured in the respective Planck units ( $ℏ = 1$ , $c = 1$ ). (a) General relativity, $k = 1$ , left plot is for $d = 4$ and right plot for $d = 10$ , with $χ = 1$ in both plots ( $χ = - 1$ is not possible for $k$ odd), (b) Born-Infeld, $k = [\frac{d - 1}{2}]$ , here $k = 4$ and $d = 10$ , left plot is for $χ = 1$ and right plot for $χ = - 1$ , (c) General relativity with a generalized Gauss-Bonnet term, $k = 2$ , a particular case of $1 < k < [\frac{d - 1}{2}]$ , here $d = 10$ , left plot is for $χ = 1$ and right plot for $χ = - 1$ . See text for details.Reuse & Permissions
Figure 3
In this figure the branch $d - 2 k - 1 = 0$ (Chern-Simons, $d$ always odd) is studied in some particular instances. In each plot the effective potential ${\dot{R}}^{2}$ as a function of $r$ (which gives the turning points and the allowed regions for the shell), and the metric function $f^{2} (r)$ as a function of $r$ (which gives the horizon formation, for several different values of the parameters) is displayed. The vertical axis $Φ (r)$ represents both ${\dot{R}}^{2} (r)$ (full line) and $f_{+}^{2} (r)$ (dashed line). Note that the thin shell trajectory is allowed only in the region where ${\dot{R}}^{2}$ is positive. In the plots the respective Newton’s constant is put equal to the unity, $G_{k} = 1$ , so that radii and energies are measured in the respective Planck units ( $ℏ = 1$ , $c = 1$ ). Left plot is for $k = 2$ , $d = 5$ , and $χ = 1$ , and right plot is for $k = 2$ , $d = 5$ , and $χ = - 1$ . See text for details.Reuse & Permissions

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