We present the vacuum polarisation of a massless, conformally coupled scalar field on the brane for a Schwarzschild-Tangherlini black hole in a bulk of zero to seven additional dimensions

Hewitt, M.

Winstanley, E.

MATTHEW HEWITT

ELIZABETH WINSTANLEY

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VACUUM POLARISATION ON THE BRANE FOR A HIGHER DIMENSIONAL BLACK HOLE SPACETIME

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Vacuum polarisation on the brane for a higher dimensional black hole spacetime

This is a repository copy of Vacuum polarisation on the brane for a higher dimensional black hole spacetime.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/98096/Version: Accepted VersionProceedings Paper:Hewitt, M. and Winstanley, E. orcid.org/0000-0001-8964-8142 (2015) Vacuum polarisation on the brane for a higher dimensional black hole spacetime. In: The Thirteenth Marcel Grossmann Meeting. MG13 Meeting on General Relativity, 1 – 7 July 2012, Stockholm University, Sweden. World Scientific , pp. 1089-1091. ISBN 978-981-4612-14-2 https://doi.org/10.1142/9789814623995_0090 eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. arXiv:1303.3238v1  [gr-qc]  13 Mar 2013March 14, 2013 0:7 WSPC - Proceedings Trim Size: 9.75in x 6.5in main1VACUUM POLARISATION ON THE BRANEFOR A HIGHER DIMENSIONALBLACK HOLE SPACETIMEMATTHEW HEWITT∗ and ELIZABETH WINSTANLEYSchool of Mathematics and Statistics,The University of Sheffield,Hicks Building, Sheffield, S3 7RH, UK∗E-mail: app10mh@sheffield.ac.ukwww.shef.ac.ukWe present the vacuum polarisation of a massless, conformally coupled scalar field onthe brane for a Schwarzschild–Tangherlini black hole in a bulk of zero to seven additionaldimensions.Keywords: Black holes; Quantum field theory in curved space; Brane worlds; Higherdimensional geometry.1. Vacuum Polarisation near Black Holes in Higher DimensionsWithin the field of quantum field theory in curved space a key point of interest isthe renormalised stress energy tensor (RSET) for quantised matter on black holes.The first step, and major building block, in evaluating the RSET is the calculationof the vacuum polarisation (VP) of the matter field φ, equivalent to the field’sauto-correlation function 〈φ2〉. In addition the VP is interesting in its own rightdue to providing information about energy density divergences. The VP has beencalculated for a variety of black hole spacetimes but has received little attention inhigher dimensions. We calculate the VP on the brane for a massless, conformallycoupled scalar field on a Schwarzschild–Tangherlini1 black hole to see what, if any,difference the extra dimensions make.2. Brane WorldsWe use the ADD2,3 brane world model in which our universe is treated as a flat,tensionless 3 + 1 brane embedded in a bulk of flat, space–like extra dimensions.These additional dimensions are considered to be large enough compared to theblack hole such that any periodicity may be ignored. Our black hole is centred onand is bisected by the brane. On the brane the four-dimensional metric readsds2 = −f(r)dt2 + f(r)−1dr2 + r2(dθ2 + sin2 θ dφ2), f(r) ≡ 1−(rhr)d−3, (1)where rh is the horizon radius. Such a brane world, though simple, is applicable tomodelling the production of micro–black holes.4 We consider a massless, conformallycoupled scalar field on the metric (1), satisfying the Klein–Gordon equation(−R/6)φ = 0 (2)where R is the Ricci scalar. We calculate the VP for no extra dimensions (ie. theSchwarzschild metric, to confirm agreement with previous work5) up to an addi-March 14, 2013 0:7 WSPC - Proceedings Trim Size: 9.75in x 6.5in main2tional seven dimensions.3. Calculation on the BraneWe calculate the VP for the Hartle-Hawking vacuum6 representing a thermal stateperiodic in imaginary time. We apply a Wick rotation to the time coordinate andnow work in a Euclidean spacetime. As any naive calculation of 〈φ2〉 will be diver-gent, we employ point splitting7 and use a Hadamard expansion for calculating thedivergent terms.8 Subtracting the renormalisation terms from the Green’s functionsand bringing the points together gives us the renormalised VP. This may be splitinto two parts: the first can be handled analytically, the second requires numericalmethods. The details of the numerical work can be found in Ref. 9.4. ResultsThe calculation for zero extra dimensions has been performed previously5 andshowed that the total VP differed from its analytic component by 0.69% allow-ing the more manageable analytic terms to be a good approximation to the whole.Figure 1 compares the analytic and total results of the VP for a bulk total of fourto eleven dimensions. These results clearly demonstrate that the analytic approxi-mation to the whole rapidly breaks down and worsens as dimension increases, witha 52% difference for eleven total dimensions.(a) (b)Fig. 1. VP plotted as a function of r/rh such that the horizon is the far left vertical. Both plotsare on the same scale. The number of extra dimensions increases up the plots such that on boththe lowest line is for zero extra dimensions and the top seven extra dimensions. (a) VP for theanalytic component only. (b) Total calculated VP.As the major differences in value are close to the event horizon we performedthe calculations again focusing on the region where 1 < r/rh < 1.5. The results ofthis second calculation are displayed in Fig. 2, again comparing the analytic termsagainst the total VP. Figure 2 shows that the total VP falls increasingly quicklywith additional dimensions. This strongly indicates that the numeric componentcauses the total VP to achieve a maximum on the horizon between a total of eightor nine dimensions before rapidly falling towards negative values.March 14, 2013 0:7 WSPC - Proceedings Trim Size: 9.75in x 6.5in main3(a) (b)Fig. 2. VP near the horizon plotted as a function of r/rh such that the horizon is the far leftvertical. Both plots are on the same scale. The number of extra dimensions increases up the plotssuch that on both the lowest line is for zero extra dimensions and the top seven extra dimensions.(a) VP for the analytic component only. (b) Total calculated VP.5. Further InvestigationHaving completed this calculation on the brane our methods must next be extendedto calculation of the VP in the bulk. The greatest challenges in the bulk will beextending our renormalisation terms into higher dimensions (this was not requiredon the brane) and finding an appropriate form of our Green’s function in whichto incorporate them. Once the VP has been calculated in the brane and the bulka potential extension to the work is to confirm the exact VP values on the eventhorizon itself but we will return this question in the future.AcknowledgementsThis work is supported by a studentship from EPSRC, the Lancaster–Manchester–Sheffield Consortium for Fundamental Physics under STFC grant ST/J000418/1and European Cooperation in Science and Technology (COST) action MP0905“Black Holes in a Violent Universe”.References1. F. R. Tangherlini, Nuovo Cim. 27, p. 636 (1963).2. N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B 429, p. 263 (1998).3. I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B 436, p.257 (1998).4. S. C. Park, Progress in Particle and Nuclear Physics 67, p. 617 (2012).5. P. Candelas and K. W. Howard, Phys. Rev. D 29, p. 1618 (1984).6. J. Hartle and S. W. Hawking, Phys. Rev. D 13, p. 2188 (1976).7. P. R. Anderson, Phys. Rev. D 41, p. 1152 (1990).8. Y. Decanini and A. Folacci, Phys. Rev. D 78, p. 044025 (2008).9. C. Breen, M. Hewitt, A. C. Ottewill and E. Winstanley, paper in preparation .

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Vacuum polarisation on the brane for a higher dimensional black hole spacetime

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