3.3 Black hole mass scaling3 The basic scenario3.1 The dynamical systems picture

3.2 Scale-invariance and self-similarity

The critical solution found by Choptuik [36, 37, 38Jump To The Next Citation Point In The Article] for the spherically symmetric scalar field is scale-periodic, or discretely self-similar (DSS), while other critical solutions, for example for a spherical perfect fluid [53Jump To The Next Citation Point In The Article] are scale-invariant, or continuously self-similar (CSS). We begin with the continuous symmetry because it is simpler. In Newtonian physics, a solution Z is self-similar if it is of the form


If the function f (t) is derived from dimensional considerations alone, one speaks of self-similarity of the first kind. An example is tex2html_wrap_inline2471 for the diffusion equation tex2html_wrap_inline2473 . 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 tex2html_wrap_inline2477, where tex2html_wrap_inline2323, 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 tex2html_wrap_inline2481, 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 tex2html_wrap_inline2481 . From (18Popup Equation) it follows that


and therefore


but the inverse does not hold: The Riemann tensor and the metric need not satisfy (19Popup Equation) and (18Popup Equation) if the Einstein tensor obeys (20Popup Equation). If the matter is a perfect fluid (26Popup Equation) it follows from (18Popup Equation), (20Popup Equation) and the Einstein equations that


Similarly, if the matter is a massless scalar field tex2html_wrap_inline2325, with stress-energy tensor (2Popup Equation), it follows that


where tex2html_wrap_inline2449 is a constant.

In coordinates tex2html_wrap_inline2489 adapted to the homothety, the metric coefficients are of the form


where the coordinate tex2html_wrap_inline2337 is the negative logarithm of a spacetime scale, and the remaining three coordinates tex2html_wrap_inline2493 are dimensionless. In these coordinates, the homothetic vector field is


The minus sign in both equations (23Popup Equation) and (24Popup Equation) is a convention we have chosen so that tex2html_wrap_inline2337 increases towards smaller spacetime scales. For the critical solutions of gravitational collapse, we shall later choose surfaces of constant tex2html_wrap_inline2337 to be spacelike (although this is not possible globally), so that tex2html_wrap_inline2337 is the time coordinate as well as the scale coordinate. Then it is natural that tex2html_wrap_inline2337 increases towards the future, that is towards smaller scales.

As an illustration, the CSS scalar field in these coordinates would be


with tex2html_wrap_inline2449 a constant. Similarly, perfect fluid matter with stress-energy


with the scale-invariant equation of state tex2html_wrap_inline2505, k a constant, allows for CSS solutions where the direction of tex2html_wrap_inline2509 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 [74Jump To The Next Citation Point In The Article]:


The conformal metric tex2html_wrap_inline2513 does now depend on tex2html_wrap_inline2337, 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 tex2html_wrap_inline2517 and a real constant tex2html_wrap_inline2377 such that


where tex2html_wrap_inline2521 is the pull-back of tex2html_wrap_inline2523 under the diffeomorphism tex2html_wrap_inline2517 . This is our definition of discrete self-similarity (DSS). It can be obtained formally from (18Popup Equation) by integration along tex2html_wrap_inline2481 over an interval tex2html_wrap_inline2377 of the affine parameter. Nevertheless, the definition is independent of any particular vector field tex2html_wrap_inline2481 . One simple coordinate transformation that brings the Schwarzschild-like coordinates (4Popup Equation) into the form (28Popup Equation) was given in Eqn. (12Popup Equation), as one easily verifies by substitution. The most general ansatz for the massless scalar field compatible with DSS is


with tex2html_wrap_inline2449 a constant. (In the Choptuik critical solution, tex2html_wrap_inline2433 for unknown reasons.)

It should be stressed here that the coordinate systems adapted to CSS (23Popup Equation) or DSS (28Popup Equation) form large classes, even in spherical symmetry. One can fix the surface tex2html_wrap_inline2537 freely, and can introduce any coordinates tex2html_wrap_inline2493 on it. In particular, in spherical symmetry, tex2html_wrap_inline2337 -surfaces can be chosen to be spacelike, as for example defined by (4Popup Equation) and (12Popup Equation) above, and in this case the coordinate system cannot be global (in the example, t <0). Alternatively, one can find global coordinate systems, where tex2html_wrap_inline2337 -surfaces must become spacelike at large r, as in the coordinates (51Popup Equation). Moreover, any such coordinate system can be continuously deformed into one of the same class.

In a possible source of confusion, Evans and Coleman [53Jump To The Next Citation Point In The Article] use the term ``self-similarity of the second kind'', because they define their self-similar coordinate x as x = r / f (t), with tex2html_wrap_inline2553 . 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 [53Jump To The Next Citation Point In The Article] 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, 104Jump To The Next Citation Point In The Article, 105Jump To The Next Citation Point In The Article, 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 [53Jump To The Next Citation Point In The Article]. 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 [71Jump To The Next Citation Point In The Article, 74Jump To The Next Citation Point In The Article] 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 .

3.3 Black hole mass scaling3 The basic scenario3.1 The dynamical systems picture

image Critical Phenomena in Gravitational Collapse
Carsten Gundlach
© Max-Planck-Gesellschaft. ISSN 1433-8351
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