Abstract
These notes introduce the fundamentals of black hole geometry, the thermality of the vacuum, and the Hawking effect, in spacetime and its analogues. Stimulated emission of Hawking radiation, the trans-Planckian question, short wavelength dispersion, and white hole radiation in the setting of analogue models are also discussed. No prior knowledge of differential geometry, general relativity, or quantum field theory in curved spacetime is assumed. The discussion attempts to capture the essence of these topics without oversimplification.
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Notes
- 1.
Even if the other terms in the line element (1.6) had not been negative, they would not contribute to the first order variation in the proper time away from a path with \((dr +\sqrt{1/r} dt)=d\theta ^{2}=d\phi^{2}=0\), since the line element is quadratic in these terms. Thus the curve would still have been a geodesic (although the metric signature would not be Lorentzian).
- 2.
For a more computational proof, note that since Killing’s equation is a tensor equation it holds in all coordinate systems if it holds in one. In a coordinate system for which \(\chi^{\mu}=\delta^{\mu}_{\hat{\alpha}}\) we have \(\chi_{\alpha ;\beta }= g_{\alpha \mu }\chi^{\mu}{}_{;\beta } = g_{\alpha \mu }\varGamma ^{\mu}{}_{\beta \sigma }\chi^{\sigma}= {\frac{1}{2}}(g_{\alpha \beta ,{\sigma }}+g_{\alpha {\sigma },\beta }-g_{\beta {\sigma },\alpha })\chi^{\sigma}\). If χ σ is a Killing vector the first term vanishes in this adapted coordinate system, and the remaining expression is antisymmetric in α and β, so adding χ β;α yields zero. Conversely, if Killing’s equation holds, the entire expression is antisymmetric in α and β, so the first term must vanish.
- 3.
For a configuration eigenstate of a real field, the ket J|ϕ L 〉 can just be identified with the same function ϕ L , reflected by an operator P 1 across the Rindler plane. More generally, J includes CT to undo the conjugation of the 〈bra|→|ket〉 duality.
- 4.
Its matrix elements could also have been obtained directly using the wave functional (1.27), via
. - 5.
The sign of the L term is opposite to that of the E term because ∂ ϕ is spacelike while ∂ t is timelike.
- 6.
For a pedagogical introduction see, e.g. [10], or references therein.
- 7.
- 8.
What I am calling the annihilation operator here is related to the field operator ϕ by a(f)=〈f,ϕ〉, where f is a positive norm mode. If f is not normalized this is actually 〈f,f〉1/2 times a true annihilation operator.
- 9.
The minus sign comes from the conjugation of a factor of i in the definition of the norm, which I will not explain in detail here.
- 10.
Here I’ve use the relation \((a^{\dagger})^{n}|0\rangle = \sqrt{n!}|n\rangle \).
- 11.
For frequencies of order the surface gravity, this quantity can also be expressed as (x tp/x lin)1+1/2n, where x tp is the (ω-dependent) WKB turning point.
- 12.
The relevant norm is determined by the action for the modes. This has been worked out assuming irrotational flow [29], which is a good approximation although the flow does develop some vorticity. The predicted Hawking temperature was estimated, but it is difficult to evaluate accurately because of the presence of the undulation, the fact that it depends on the flow velocity field that was not precisely measured, and the presence of vorticity which has not yet been included in an effective metric description.
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Acknowledgements
I am grateful to Renaud Parentani for many helpful discussions on the material presented here, as well as suggestions for improving the presentation. Thanks also to Anton de la Fuente for helpful discussions on the path integral derivation of vacuum thermality. This work was supported in part by the National Science Foundation under Grant Nos. NSF PHY09-03572 and NSF PHY11-25915.
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Jacobson, T. (2013). Black Holes and Hawking Radiation in Spacetime and Its Analogues. In: Faccio, D., Belgiorno, F., Cacciatori, S., Gorini, V., Liberati, S., Moschella, U. (eds) Analogue Gravity Phenomenology. Lecture Notes in Physics, vol 870. Springer, Cham. https://doi.org/10.1007/978-3-319-00266-8_1
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