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Hawking radiation for non-asymptotically flat dilatonic black holes using gravitational anomaly

  • Regular Article - Theoretical Physics
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Abstract

The d-dimensional scalar field action may be reduced, in the background geometry of a black hole, to a two-dimensional effective action. In the near-horizon region, it appears a gravitational anomaly: the energy-momentum tensor of the scalar field is not conserved anymore. This anomaly is removed by introducing a term related to the Hawking temperature of the black hole. Even if the temperature term introduced is not covariant, a gauge transformation may restore the covariance. We apply this method to compute the temperature of the dilatonic non-asymptotically flat black holes. We compare the results with those obtained through other methods.

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Acknowledgements

J.C.F. thanks CNPq and FAPES (Brazil) for partial financial support. G.T.M. thanks CNPq and FAPESPA (Brazil) for partial financial support. We thank Gérard Clément for valuable remarks on this work.

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Authors and Affiliations

  1. Departamento de Física, Universidade Federal do Espírito Santo, Vitória, 29060-900, Espírito Santo, Brazil

    J. C. Fabris

  2. ICIBE-LASIC, Universidade Federal Rural da Amazônia-Brazil, Belém, 66077-901, Pará, Brazil

    G. T. Marques

Authors
  1. J. C. Fabris
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Correspondence to J. C. Fabris.

Appendix: The covariant anomaly

Appendix: The covariant anomaly

The anomalies treated in this paper obey strong consistency conditions. For this reason, they are called consistent anomalies. This formulation is based on (4) and (20) which do not transform covariantly under coordinate transformation. For this reason, the new energy-momentum tensor (36) and currents (25) are defined, leading to new anomaly equations, which are now invariant under coordinate and gauge transformations. Hence, we will perform an analysis of covariant equations for the gauge and gravitational anomaly for the metric (19).

Gauge covariant anomaly

Substituting (20) in (25), we obtain,

$$ \nabla_\mu\tilde{J}^\mu=-\frac{e^2}{4\pi\sqrt{-g}} \epsilon^{\mu\nu}F_{\mu\nu} , $$
(60)

which is now gauge invariant.

For the metric (19) equations (60) can be integrated, resulting in

$$ \begin{aligned} &\partial_r\tilde{J}^r(r)=\frac{e^2}{2\pi} \partial_rA_t(r) \\ &\tilde{J}^r(\infty)-\tilde{J}^r(r_H)= \frac{e^2}{2\pi}\bigl(A_t(\infty)-A_t(r_H) \bigr). \end{aligned} $$
(61)

Imposing that \(\tilde{J}^{r}(r_{H})\) is zero over the horizon and that A t (∞) is not zero at infinity, since the metric is not asymptotically flat at the infinity, the charge flux becomes,

(62)

This confirms the universality of the expression for the charge flux, in the asymptotic limit (r→∞), via anomalies.

Covariant gravitational anomaly

The covariant gravitational anomaly is given by the equation [18, 30, 31]

$$ \begin{aligned} &\nabla_\mu\tilde{T}^\mu_\nu= \frac{1}{96\pi}\epsilon_{\mu\nu}\partial^\mu R , \\ &\partial_r\tilde{T}^r_t= \frac{1}{96\pi}f(r)\partial^3_rf(r)= \partial_r\bar{N}^r_t , \end{aligned} $$
(63)

where R is the Ricci scalar and

$$\bar{N}^r_t=\frac{1}{96\pi} \biggl(ff''- \frac{f'^2}{2} \biggr) , $$

where ′=d/dr.

The Ward’s identities for the covariant anomaly case is given by [29, 30]

$$ \nabla_\mu\tilde{T}^\mu_\nu=F_{\mu\nu} \tilde{J}^\mu+\partial_\mu\bar{N}^\mu_\nu . $$
(64)

Using (61), we have

(65)

Through (61) it is possible to obtain the value of the constant d 0 which is −e 2  A t (r H )/2π. Hence, after integration of the right hand side of equation (65), we find

where it was used the condition that \(\tilde{T}^{r}_{t}\) is zero over the horizon. Since the metric (19) is not asymptotically flat the term \(\bar{N}^{r}_{t}(\infty)\) is not zero. In this way, a flux of energy and momentum at infinity is defined:

(66)

This confirms (39), showing the universality of the method of calculation of the Hawking temperature via anomalies. Hence, the covariant anomaly method allows also to cancel the anomalies near the horizon, as shown in [30, 31, 40].

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Fabris, J.C., Marques, G.T. Hawking radiation for non-asymptotically flat dilatonic black holes using gravitational anomaly. Eur. Phys. J. C 72, 2214 (2012). https://doi.org/10.1140/epjc/s10052-012-2214-8

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  • DOI: https://doi.org/10.1140/epjc/s10052-012-2214-8

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