Erratum: The structure of superposed Weyl fields

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 ˆ R2H …e2lext …rˆ2M;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 ˆ 2 p
p
guu
guu gff
ˆ 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
ˆ
:
R2H e2lext
…1†
…8†
because l ext is not zero on the horizon (it only vanishes at vacuum
parts of the axis r ˆ 0†; namely (e.g. Will 1974)
All is evaluated on the horizon, so, according to equation (2),1
lext …r ˆ 2M; u† ˆ 2next …r ˆ 2M; u† 2 2next …r ˆ 2M; 0†:
RH ˆ
…2†
The function RH ˆ 2Me2next …rˆ2M;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 …u†elext …rˆ2M;u† du is its
proper poloidal circumference).
Although RH ˆ 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 ˆ cos u coordinates as
ds2 ˆ h2 ‰ f 21 …m† dm2 ‡ f …m† df2 Š;
…3†
where h is a scalefactor independent of m , in our case
h ˆ RH …u ˆ 0† ˆ 2Me
2next …rˆ2M;mˆ1†
;
…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 ˆ h f sin f;
…6†
x ˆ h f cos f;

… m s
1
1
…7†
1 2 f 2;m dm:
zˆh
f
4
0
Fig. 1 shows the shapes generated numerically from the above
equations.
4M 2 e2next …u†24next …0†
:
whereas on the axis one has
RH ˆ
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 ˆ
;
…10†
3
b
4M
2M
4M 2 exp p 2
b
b2 ‡ M 2
and
f …m† ˆ …1 2 m2 † e2next …rˆ2M;mˆ1†22next …rˆ2M;m† :
1 ‡ 3next;u cot u ‡ next;uu 2 2n2ext;u
1
Other form is R H ˆ 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 ˆ 2:7 for the ring and r=M ˆ 2:2 for the disc, with M =M ˆ 0:2; 0.4, 0.6, ¼, 1.2; in bottom plots,
M =M ˆ 0:63; with r=M ˆ 8; 4, 3.2, 2.8, 2.5, 2.37 for the ring and with r=M ˆ 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.