The conformal transformation of the Misner–Sharp mass is reexamined. It has recently been found that this mass does not transform like usual masses do under conformal mappings of spacetime. We show that when it comes to conformal transformations, the widely used geometric definition of the Misner–Sharp mass is fundamentally different from the original conception of the latter. Indeed, when working within the full hydrodynamic setup that gave rise to that mass, i.e. the physics of gravitational collapse, the familiar conformal transformation of a usual mass is recovered. The case of scalar–tensor theories of gravity is also examined.
A Riemannian almost paracomplex manifold is a 2n-dimensional Riemannian manifold (M,g), whose structural group O(2n,R) is reduced to the form O(n,R)×O(n,R). We define the scalar curvature π of this manifold and consider relationships between π and the scalar curvature s of the metric g and its conformal transformations.
Abstract
Based on N. I. Muskhelishvili’s approach to problems of plane elasticity, a general method has been deduced for the solution of problems of reinforced cutouts in infinitely thin sheets. As an illustration, the circular reinforced hole has been treated in detail and the results have been related to those obtained experimentally and theoretically by other authors. The solution for other shapes of cutouts will be greatly simplified, since full use may be made of the theory of conformal transformations.
Approach to the analysis and synthesis of cylindrical metasurfaces with noncircular cross sections based on conformal transformations