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An exact solution to the heat equation in curved space is a much sought after; this report presents a derivation wherein the cylindrical symmetry of the metric in 3 + 1 dimensional curved space has a pivotal role. To elaborate, the spherically symmetric Schwarzschild solution is a staple of textbooks on general relativity; not so perhaps, the static but cylindrically symmetric ones, though they were obtained almost contemporaneously by H.Weyl,Ann.Phys.Lpz.54,117(1917)
and T. Levi-Civita, Atti Acc. Lincei Rend. 28,101(1919). A renewed interest in this subject recently in C.S. Trendafilova and S.A. Fulling ,Eur.J.Phys. 32,1663(2011) – to which the reader is referred to for more references –
motivates this work, the first part of which (cf.Kamath, PoS (ICHEP2016)791) reworked the Antonsen-Bormann idea – arXiv:hep-th/9608141v1 – that was originally intended to compute the heat kernel in curved space to determine – following D.McKeon and T.Sherry,Phys.Rev.D35,3584(1987) – the zeta-function associated with the Lagrangian density for a massive real scalar field theory in 3 + 1 dimensional stationary curved space to one-loop order, the metric for which is cylindrically symmetric. Using the same Lagrangian density the
second part reported here essentially revisits the second paper by Bormann and Antonsen – arXiv:hep 9608142v1 but relies on the formulation by the author in S.G.Kamath, AIP Conf.Proc.1246,174 (2010) to obtain the Green’s function directly by solving a sequence of first order partial differential equations that is preceded by a second order partial differential equation.
| Topic: | Topic: Cosmology, Astrophysics, Gravity, Mathematical Physics |
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