Uniqueness of the static Einstein–Maxwell spacetimes with a photon sphere. (20th August 2015)

• Record Type: Journal Article
• Authors: Yazadjiev, Stoytcho; Lazov, Boian
• Is Part Of: Classical and quantum gravity. Volume 32:Number 16(2015:Aug.)
• Page Start: 219
• Publication Date: 2015-08-20
• Subjects: Quantum gravity -- Periodicals; Gravitation -- Periodicals; Relativity (Physics) -- Periodicals; Space and time -- Periodicals; Periodicals; 521.1
• DOI: 10.1088/0264-9381/32/16/165021
• Languages: English
• ISSNs: 0264-9381

Abstract

<abstract> <title>Abstract</title> <p>We consider the problem of uniqueness of static and asymptotically flat Einstein–Maxwell spacetimes with a photon sphere <italic>P</italic><sup>3</sup>. We are using a naturally modified definition of a photon sphere for electrically charged spacetimes with the additional property that the one-form <inline-formula><tex-math><?CDATA ${\iota }_{\xi }F$?></tex-math><?MML <mml:math> <mml:msub> <mml:mrow> <mml:mi>&igr;</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>&xgr;</mml:mi> </mml:mrow> </mml:msub> <mml:mi>F</mml:mi> </mml:math>?><inline-graphic xlink:href="ark:/27927/pgj2j72tjdn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula> is normal to the photon sphere. For simplicity we are restricting ourselves to the case of zero magnetic charge and assume that the lapse function regularly foliates the spacetime outside the photon sphere. With this information we prove that <italic>P</italic><sup>3</sup> has constant mean curvature and constant scalar curvature. We also derive a few equations which we later use to prove the main uniqueness theorem, i.e. the static asymptotically flat Einstein–Maxwell spacetimes with a non-extremal photon sphere are isometric to the Reissner–Nordström one with mass <italic>M</italic> and electric charge <italic>Q</italic> subject to <inline-formula><tex-math><?CDATA $\frac{{Q}^{2}}{{M}^{2}}\leqslant \frac{9}{8}$?></tex-math><?MML <mml:math> <mml:mstyle displaystyle="false"> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>Q</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfrac> </mml:mstyle> <mml:mo>≤</mml:mo> <mml:mstyle displaystyle="false"> <mml:mfrac> <mml:mrow> <mml:mn>9</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>8</mml:mn> </mml:mrow> </mml:mfrac> </mml:mstyle> </mml:math>?><inline-graphic xlink:href="ark:/27927/pgj2j72tjc3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula>.</p> </abstract>