Speaker
Description
We study the Euclidean action and Brown–York quasilocal mass of the Melvin and Schwarzschild–
Melvin geometries in four-dimensional Einstein–Maxwell theory. Since these spacetimes do not
exhibit the usual spherically asymptotically flat behavior, we introduce a cylindrical regularization
scheme adapted to the magnetic axis. The boundary is placed at a fixed cylindrical radius, while
the longitudinal interval is scaled with the radial cutoff. A sequential limiting procedure is then
employed to preserve the cylindrical structure of the asymptotic region.
Within this framework, we show that the standard Mann–Marolf-type counterterm does not remove the relevant divergences. We instead propose a local boundary counterterm constructed
from the norm of the axial Killing vector, which captures the asymptotic behavior of the azimuthal
circumference. This prescription cancels the leading divergences without requiring background
subtraction.
The renormalized Euclidean action and Brown–York mass both vanish for the pure Melvin universe.
For the Schwarzschild–Melvin geometry, the Euclidean action is one half of the product of the
inverse temperature and the black-hole mass parameter, while the Brown–York mass equals the
mass parameter. These results identify the Melvin universe as a zero-mass magnetic reference and
show that the finite black-hole mass is preserved after cylindrical renormalization.
