Supertranslation in the bulk for generic spacetime

Abstract

Supertranslations are usually defined as asymptotic symmetries associated with spacetime boundaries, such as null infinity and black hole horizons. In this Letter, we show that supertranslations admit a natural, coordinate-independent extension into the bulk of spacetime, realized as transitions between families of null hypersurfaces. This construction applies to generic spacetimes in arbitrary dimensions and unifies the realizations of supertranslations at null infinity and black hole horizons. The associated symmetry algebra can be realized by light-ray operators defined on the null hypersurface. Within this framework, the gravitational wave memory effect corresponds to a shift of null hypersurfaces in the bulk. As explicit examples, we compute bulk supertranslations in Minkowski spacetime in arbitrary dimensions and in four-dimensional Schwarzschild spacetime, where we uncover a novel curvature-induced memory effect with observable consequences for light propagation.

Figures (2)

• Figure 1: The black wavy lines represent gravitational waves with memory. The blue straight lines denote null geodesics. Initially, the null geodesics are plane-fronted (green dashed line) with vanishing shear. As the gravitational waves pass, they deform this congruence, forcing the null geodesics onto a new null hypersurface. The individual geodesics remain straight, but the associated light front is no longer planar. When described using the original geodesic parameter $r$, the light front is sampled at different values of $r$ along different rays, which manifests itself as a non-vanishing shear of the null congruence.
• Figure 2: The black region represents the black hole. The black wavy lines indicate gravitational waves with memory, while the blue lines show the null geodesic before and after the memory effect.