Kinematic self-similar solutions in general relativity

The gravitational interaction is scale-free in both Newtonian gravity and general theory of relativity. The concept of self-similarity arises from this nature. Self-similar solutions reproduce themselves as the scale changes. This property results in great simplification of the governing partial differential equations. In addition, some self-similar solutions can describe the asymptotic behaviors of more general solutions. Newtonian gravity contains only one dimensional constant, the gravitational constant, while the general relativity contains another dimensional constant, the speed of light, besides the gravitational constant. Due to this crucial difference, incomplete similarity can be more interesting in general relativity than in Newtonian gravity. Kinematic self-similarity has been defined and studied as an example of incomplete similarity in general relativity, in an effort to pursue a wider application of self-similarity in general relativity. We review the mathematical and physical aspects of kinematic self-similar solutions in general relativity.