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Energy of a spherically symmetric charged dilaton black hole

A. CHAMORRO and K. S. VIRBHADRA

Departamento de Física Teórica, Universidad del País Vasco

Apartado 644, 48080 Bilbao, Spain

In recent years there is considerable interest in obtaining charged dilaton black hole solutions and investigating their properties $[1-6]$. Garfinkle, Horowitz, and Strominger (GHS) $[1]$ considered the action S = d\^4 x    and obtained a nice form of static and spherically symmetric charged dilaton black hole solution $[1,2]$, given by the line element ds\^2 = B dt\^2 - B\^[-1]{} dr\^2 - D r\^2 (d\^2 + \^2d\^2), the dilaton field $\Phi$, where e\^[2]{} = \^[(1-)/]{}, and the component of the electromagnetic field tensor F\_[tr]{} = , where B = (1-) (1-)\^, D = (1-)\^[1-]{}, and  =    . $r_{+}$ and $r_{-}$ are related through & &2M = r\_[+]{} +  r\_[-]{} ,\
& &Q\^2 (1+\^2) = r\_[+]{} r\_[-]{}  . $M$ and $Q$ stand for mass and charge parameters, respectively. The surface $r = r_{+}$ is the event horizon. $\beta$ is a dimensionless free parameter which controls the coupling between the dilaton and the Maxwell fields. A change in the sign of $\beta$ is the same as a change in the sign of the dilaton field. Therefore, it is sufficient to discuss only nonnegative values of $\beta$. $\beta = 0$ in GHS solution gives the well known Reissner-Nordström (RN) solution.

It is known that several properties of charged dilaton black holes depend crucially on the coupling parameter $\beta$ $[2-6]$. Recently one of the present authors and Parikh $[7]$ obtained the energy of a static and spherically symmetric charged dilaton black hole for $\beta = 1$. They found that, similar to the case of the Schwarzschild black hole and unlike the RN black hole, the entire energy is confined to the interior of the black hole. It is of interest to investigate the energy associated with charged dilaton black holes for arbitrary value of $\beta$ to see what the energy distribution is for $\beta < 1$ as well as $\beta >1$ and whether or not the energy is confined to the black hole interior for any other value of $\beta$.

The well known energy-momentum pseudotensor of Einstein is $[8]$ \_i\^[ k]{} = H\^[ kl]{}\_[i,  l]{}   , where H\_i\^[ kl]{} = \_[,m]{}  .

Latin indices run from $0$ to $3$. $x^0$ is the time coordinate. The energy and momentum components are P\_i = \_[,]{}  dx\^1 dx\^2 dx\^3  , where the Greek index $\alpha$ takes values from $1$ to $3$. $P_0$ and $P_{\alpha}$ stand, respectively, for the energy (say $E$) and momentum components.

It is known that the energy-momentum pseudotensors, for obtaining the energy and momentum associated with asymptotically flat spacetimes, give the correct result if calculations are carried out in quasi-cartesian coordinates ( those coordinates in which the metric $g_{ik}$ approaches the Minkowski metric $\eta_{ik}$ at large distance ) $[8-9]$. Transforming the line element $(2)$ to quasi-cartesian coordinates $t,x,y,z (  x =  r \sin\th  \cos\ph,
 y =  r \sin\th \sin\ph, z =  r \cos\th ) $ one gets ds\^2 = B dt\^2 - D (dx\^2+dy\^2+dz\^2) - (x dx+y dy +z dz)\^2.

To obtain the energy the required components of $ H_i^{\ kl}$ are H\_0\^[ 01]{} &=& ,\
H\_0\^[ 02]{} &=& ,\
H\_0\^[ 03]{} &=& . By using $(13)$ with $(8)$ in $(11)$, applying the Gauss theorem, and then evaluating the integral over the surface of a sphere of radius $r$, one gets E(r) = M -   ( 1 - \^2 ).

Thus one finds that the energy distribution depends on the value of the coupling parameter $\beta$. The energy is confined to its interior [*[only for]{}*]{} $\beta = 1$ and for all other values of $\beta$ the energy is shared by the interior and exterior of the black hole. $\beta=0$ in $(14)$ gives the energy distribution in the RN field (see also ref. $[9]$). $E(r)$ increases with radial distance for $\beta = 0$ (RN spacetime) as well as $\beta<1$, decreases for $\beta>1$, and remains constant for $\beta = 1$. However, the total energy ($r$ approaching infinity in $(14)$) is independent of $\beta$ and is given by the mass parameter of the black hole. The details of the present contribution will be published elsewhere.

[**Acknowledgements**]{}

This work has been partially supported by the Universidad del Pais Vasco under contract UPV 172.310 - EA062/93 (A.C.) and by a Basque Government post-doctoral fellowship (K.S.V.). We thank A. Achúcarro, J. M. Aguirregabiria, and I. Egusquiza for discussions.

[**References**]{}\
$[1]$ D. Garfinkle, G. T. Horowitz and A. Strominger, Phys. Rev. D43\
     (1991) 3140; Erratum : Phys. Rev. D45 (1992) 3888.\
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$[3]$ K. Shiraishi, Phys. Lett. A166 (1992) 298.\
$[4]$ J. A. Harvey and A. Strominger, Quantum aspects of black holes,\
     preprint EFI-92-41, hep-th/9209055 .\
$[5]$ T. Maki and K. Shiraishi, Class. Quant. Grav. 11 (1994) 227.\
$[6]$ C. F. E. Holzhey and F. Wilczek, Nucl. Phys. B380 (1992) 447.\
$[7]$ K. S. Virbhadra and J. C. Parikh, Phys. Lett. B317 (1993) 312.\
$[8]$ C. Møller, Ann. Phys. (NY) 4 (1958) 347.\
$[9]$ K. P. Tod. Proc. Roy. Soc. Lond. A388 (1983) 467;\
      K. S. Virbhadra, Phys. Rev. D41 (1990) 1086; Phys. Rev. D42 (1990) 2919;\
      F. I. Cooperstock and S. A. Richardson, in Proc. 4th Canadian Conf. on General\
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