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
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