Mon. Not. R. Astron. Soc. 322, 207±208 (2001) Erratum: The structure of superposed Weyl fields by O. SemeraÂk,w T. Zellerinw and M. ZÏaÂcÏekw Key words: errata, addenda ± black hole physics ± gravitation ± relativity. ds2 R2H e2lext r2M;u du2 sin2 u df2 ; Unfortunately, the mistake also entered equations (65) and (66) for Gaussian curvature of the horizon. These should read " p # gff ;u 1 R H 2 p p guu guu gff 2 ;u RH sin u;u RH elext R2H elext sin u 1 ;u 1 next;u lext;u cot u next;uu 2 next;u lext;u : R2H e2lext 1 8 because l ext is not zero on the horizon (it only vanishes at vacuum parts of the axis r 0; namely (e.g. Will 1974) All is evaluated on the horizon, so, according to equation (2),1 lext r 2M; u 2next r 2M; u 2 2next r 2M; 0: RH 2 The function RH 2Me2next r2M;u is the equatorial circumferential radius of the horizon (2pRH sin u is a proper azimuthal circumferp ence of the horizon at given u and 2 0 RH uelext r2M;u du is its proper poloidal circumference). Although RH RH u (plotted in Fig. 8 of the paper) still tells something about deformation of the horizon, a true shape of the black hole is better represented in a way used by Smarr (1973): the 2-metric (equation 1) is first rewritten in m cos u coordinates as ds2 h2 f 21 m dm2 f m df2 ; 3 where h is a scalefactor independent of m , in our case h RH u 0 2Me 2next r2M;m1 ; 4 5 The isometric embedding of the horizon 2-surface (m , f ) in three-dimensional Euclidean space (x, y, z) is then given by p p y h f sin f; 6 x h f cos f; m s 1 1 7 1 2 f 2;m dm: zh f 4 0 Fig. 1 shows the shapes generated numerically from the above equations. 4M 2 e2next u24next 0 : whereas on the axis one has RH 1 2M 4M exp p b2 M 2 2 ! 12 4M M 2 b2 M 2 3=2 q 2001 RAS 11 which can turn negative, though only for large values of M (even for a ring just above the horizon, b ! M; it would require M 8 M=4: When the Gaussian curvature becomes negative somewhere on a surface, the latter is no longer globally embeddable in E3 (roughly speaking, in the region with RH,0 the outline of the surface is even shorter than a straight line in E3). Some of the shapes shown in Fig. 1 (mainly the last case in lower right) are very near to this limit. Notation is kept according to the original paper. We apologize for any inconvenience. w E-mail: [email protected] (OS); [email protected] mff.cuni.cz (TZ); [email protected] (MZÏ) 9 The statements, concerning RH in the case of a Bach±Weyl ring as the external source, remain valid as given in the paper, up to a slight correction of numerical factors: RH is always positive in the equatorial plane, 1 M M2 ! 1 RH ; 10 3 b 4M 2M 4M 2 exp p 2 b b2 M 2 and f m 1 2 m2 e2next r2M;m122next r2M;m : 1 3next;u cot u next;uu 2 2n2ext;u 1 Other form is R H 2f ;mm = 2h2 (Smarr 1973). Downloaded from http://mnras.oxfordjournals.org/ by guest on February 3, 2015 The paper `The structure of superposed Weyl fields' was published in Mon. Not. R. Astron. Soc. 308, 691±704 (1999). In that paper we studied Weyl space±times generated by a compact centre (Schwarzschild black hole or Appell ring) surrounded by a gravitating matter in the form of a (Bach±Weyl) ring or a (Lemos±Letelier annular) thin disc. An error occurred in Section 5 where distortion of the Schwarzschild horizon because of either of the external sources was calculated. Equation (61) should have read 208 O. SemeraÂk, T. Zellerin and M. ZÏaÂcÏek REFERENCES Smarr L., 1973, Phys. Rev. D, 7, 289 Will C. M., 1974, ApJ, 191, 521 This paper has been typeset from a TEX/LATEX file prepared by the author. q 2001 RAS, MNRAS 322, 207±208 Downloaded from http://mnras.oxfordjournals.org/ by guest on February 3, 2015 Figure 1. Distortion of the Schwarzschild horizon owing to the external Bach±Weyl ring (left plots) and owing to the external Lemos±Letelier annular disc (right plots). The intrinsic shape of the horizon, given by the isometric embedding in equations (4)±(7), changes with the mass (top plots) and Schwarzschild radius (bottom plots) of the additional sources. As expected, the horizon inflates towards the external source when decreasing the radius or increasing the mass of the latter. However, quite unrealistic values must be chosen in order to raise an evident flattening (the values are different here from those used in the original figure where RH(u ) were plotted, because some of the original values already lead to a horizon with locally negative Gaussian curvature, not embeddable globally in E3): in top plots, r=M 2:7 for the ring and r=M 2:2 for the disc, with M =M 0:2; 0.4, 0.6, ¼, 1.2; in bottom plots, M =M 0:63; with r=M 8; 4, 3.2, 2.8, 2.5, 2.37 for the ring and with r=M 6; 2.8, 2.31, 2.15, 2.10, 2.08 for the disc. Both axes are in the units of the Schwarzschild mass M.
© Copyright 2022 ExpyDoc