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 .
A continuous self-similarity of the spacetime in GR corresponds to the existence of a homothetic vector field , defined by the property 
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
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 :
The conformal metric does now depend on , but only in a periodic manner. Like the continuous symmetry, the discrete version has a geometric formulation : 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  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  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 .
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 . 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
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