If the function
f
(t) is derived from dimensional considerations alone, one speaks of
self-similarity of the first kind. An example is
for the diffusion equation
. In more complicated equations, the limit of self-similar
solutions can be singular, and
f
(t) may contain additional dimensionful constants (which do not
appear in the field equation) in terms such as
, where
, called an anomalous dimension, is not determined by
dimensional considerations but through the solution of an
eigenvalue problem [6].
A continuous self-similarity of the spacetime in GR
corresponds to the existence of a homothetic vector field
, defined by the property [27]
This is a special type of conformal Killing vector, namely one
with constant coefficient on the right-hand side. The value of
this constant coefficient is conventional, and can be set equal
to 2 by a constant rescaling of
. From (18
) it follows that
and therefore
but the inverse does not hold: The Riemann tensor and the
metric need not satisfy (19) and (18
) if the Einstein tensor obeys (20
). If the matter is a perfect fluid (26
) it follows from (18
), (20
) and the Einstein equations that
Similarly, if the matter is a massless scalar field
, with stress-energy tensor (2
), it follows that
where
is a constant.
In coordinates
adapted to the homothety, the metric coefficients are of the
form
where the coordinate
is the negative logarithm of a spacetime scale, and the
remaining three coordinates
are dimensionless. In these coordinates, the homothetic vector
field is
The minus sign in both equations (23) and (24
) is a convention we have chosen so that
increases towards smaller spacetime scales. For the critical
solutions of gravitational collapse, we shall later choose
surfaces of constant
to be spacelike (although this is not possible globally), so
that
is the time coordinate as well as the scale coordinate. Then it
is natural that
increases towards the future, that is towards smaller
scales.
As an illustration, the CSS scalar field in these coordinates would be
with
a constant. Similarly, perfect fluid matter with
stress-energy
with the scale-invariant equation of state
,
k
a constant, allows for CSS solutions where the direction of
depends only on
x, and the density is of the form
The generalization to a discrete self-similarity is obvious in
these coordinates, and was made in [74]:
The conformal metric
does now depend on
, but only in a periodic manner. Like the continuous symmetry,
the discrete version has a geometric formulation [65]: A spacetime is discretely self-similar if there exists a
discrete diffeomorphism
and a real constant
such that
where
is the pull-back of
under the diffeomorphism
. This is our definition of discrete self-similarity (DSS). It
can be obtained formally from (18
) by integration along
over an interval
of the affine parameter. Nevertheless, the definition is
independent of any particular vector field
. One simple coordinate transformation that brings the
Schwarzschild-like coordinates (4
) into the form (28
) was given in Eqn. (12
), as one easily verifies by substitution. The most general
ansatz for the massless scalar field compatible with DSS is
with
a constant. (In the Choptuik critical solution,
for unknown reasons.)
It should be stressed here that the coordinate systems adapted
to CSS (23) or DSS (28
) form large classes, even in spherical symmetry. One can fix the
surface
freely, and can introduce any coordinates
on it. In particular, in spherical symmetry,
-surfaces can be chosen to be spacelike, as for example defined
by (4
) and (12
) above, and in this case the coordinate system cannot be global
(in the example,
t
<0). Alternatively, one can find global coordinate systems,
where
-surfaces must become spacelike at large
r, as in the coordinates (51
). Moreover, any such coordinate system can be continuously
deformed into one of the same class.
In a possible source of confusion, Evans and Coleman [53] use the term ``self-similarity of the second kind'', because
they define their self-similar coordinate
x
as
x
=
r
/
f
(t), with
. Nevertheless, the spacetime they calculate is homothetic, or
``self-similar of the first kind'' according to the terminology
of Carter and Henriksen [31,
50]. The difference is only a coordinate transformation: The
t
of [53
] is not proper time at the origin, but what would be proper time
at infinity if the spacetime was truncated at finite radius and
matched to an asymptotically flat exterior [52].
There is a large body of research on spherically symmetric
self-similar perfect fluid solutions [28,
16,
54,
9,
104,
105
,
93]. Scalar field spherically symmetric CSS solutions were examined
in [68,
19]. In these papers, the Einstein equations are reduced to a
system of ordinary differential equations (ODEs) by the
self-similar spherically symmetric ansatz, which is then
discussed as a dynamical system. Surprisingly, the critical
solutions of gravitational collapse were explicitly constructed
only once they had been seen in collapse simulations. The
critical solution found in perfect fluid collapse simulations was
constructed through a CSS ansatz by Evans and Coleman [53
]. In this ansatz, the requirement of analyticity at the center
and at the past matter characteristic of the singularity provides
sufficient boundary conditions for the ODE system. (For claims to
the contrary see [29,
30].) The DSS scalar critical solution of scalar field collapse was
constructed by Gundlach [71
,
74
] using a similar method. More details of how the critical
solutions are constructed using a DSS or CSS ansatz are discussed
in Section
4.4
.
![]() |
Critical Phenomena in Gravitational Collapse
Carsten Gundlach http://www.livingreviews.org/lrr-1999-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |