In recent years, from the viewpoint of higher-dimensional cosmology and gauge/gravity correspondence based on string theory, the extension to higher dimensions of various researches of general relativity, which were limited to four-dimensions before, has been attracting attention. Theorems in four-dimensional space-time often uses the specific nature of four-dimensional geometry, and hence their high-dimensional extension is not always straightforward. Dr. Akihiro Ishibashi has succeeded in generalization of several important theorems on general relativity to higher-dimensions.
First, Dr. Ishibashi successfully proved a fundamental theorem about the rigidity of higher-dimensional steady-state black holes in collaboration with the Dr. Hollands and Dr. Wald. In four-dimensional space-time, it is known that stationary black holes are always axisymmetric as the Hawking's rigidity theorem. However, it has never been clear whether this theorem can be extended to higher dimensions. Dr. Ishibashi and his colleagues have succeeded in proving that a stationary black hole has, at least, one axial symmetry. In higher dimensions black holes can have multiple axial symmetry, but that theorem shows the presence of a only single axial symmetry. Conversely, this theorem suggests the possible presence of a solution with lower symmetry, which is an interesting result.
In addition, Dr. Ishibashi has built a general framework of gauge invariant gravitational perturbation on the space-time with a D-2-dimensional part with maximum symmetry in D-dimensional space-time in collaboration with Dr. Seto and Dr. Kodama. Succeding to this work, Dr. Ishibashi with Dr. Kodama showed that all perturbation equations for high-dimensional black hole solutions with maximum symmetry can be reduced to a decoupled second order ordinary differential equations for master variables, and further succeeded in the proof of linear stability of high-dimensional spherically symmetric black hole solutions.
In recent years asymptotically anti de Sitter (AdS) space-time is attracting attention motivated by the AdS/CFT correspondence. Since the boundary at spatial infinity is timelike, the development of the field in such a spacetime is not completely determined in the normal sense. Dr. Ishibashi, in collaboration with Dr. Wald, has shown that the problem of determining the field development for singular field equations is reduced to specifying how to give a self-adjoint extension of the differential operators. In the case of the spatial infinity in the asymptotically AdS space-time, giving self-adjoint extensions of differential operators is equivalent to determining the boundary conditions.Furthermore, Dr. Ishibashi and his colleagues have classified the type of possible boundary conditions for the perturbation of the electromagnetic field and gravitational field, which turned out to be depending on the space-time dimension.
These results are important to clarify the very basic nature of the general relativistic high-dimensionalal space-time. They have already been reacongnized as important works at this moment, and fundamental works of this kind will become more important as time passes. These works are collaborative ones, but definitely Dr. Ishibashi's contribution to these studies was significant. Therefore, it was determined to send him the 8th Yukawa-Kimura Prize.