### 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  R2H e2lext r2M;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 r2M;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 r2M;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 r2M;m1
;
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:
zh
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 
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 r2M;m122next r2M;m :
1  3next;u cot u  next;uu 2 2n2ext;u
1
Other form is R H  2f ;mm =2h2  (Smarr 1973).
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
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