Precession and interference in the Aharonov-Casher and scalar Aharonov-Bohm effect

The ideal scalar Aharonov-Bohm (SAB) and Aharonov-Casher (AC) effect involve a magnetic dipole pointing in a certain fixed direction: along a purely time dependent magnetic field in the SAB case and perpendicular to a planar static electric field in the AC case. We extend these effects to arbitrary direction of the magnetic dipole. The precise conditions for having nondispersive precession and interference effects in these generalized set ups are delineated both classically and quantally. Under these conditions the dipole is affected by a nonvanishing torque that causes pure precession around the directions defined by the ideal set ups. It is shown that the precession angles are in the quantal case linearly related to the ideal phase differences, and that the nonideal phase differences are nonlinearly related to the ideal phase differences. It is argued that the latter nonlinearity is due the appearance of a geometric phase associated with the nontrivial spin path. It is further demonstrated that the spatial force vanishes in all cases except in the classical treatment of the nonideal AC set up, where the occurring force has to be compensated by the experimental arrangement. Finally, for a closed space-time loop the local precession effects can be inferred from the interference pattern characterized by the nonideal phase differences and the visibilities. It is argued that this makes it natural to regard SAB and AC as essentially local and nontopological effects

How to avoid a compact set

A first-order expansion of the $\mathbb{R}$-vector space structure on $\mathbb{R}$ does not define every compact subset of every $\mathbb{R}^n$ if and only if topological and Hausdorff dimension coincide on all closed definable sets. Equivalently, if $A \subseteq \mathbb{R}^k$ is closed and the Hausdorff dimension of $A$ exceeds the topological dimension of $A$, then every compact subset of every $\mathbb{R}^n$ can be constructed from $A$ using finitely many boolean operations, cartesian products, and linear operations. The same statement fails when Hausdorff dimension is replaced by packing dimension

A magnetar engine for short GRBs and kilonovae

We investigate the influence of magnetic fields on the evolution of binary neutron-star (BNS) merger remnants via three-dimensional (3D) dynamical-spacetime general-relativistic (GR) magnetohydrodynamic (MHD) simulations. We evolve a postmerger remnant with an initial poloidal magnetic field, resolve the magnetoturbulence driven by shear flows, and include a microphysical finite-temperature equation of state (EOS). A neutrino leakage scheme that captures the overall energetics and lepton number exchange is also included. We find that turbulence induced by the magnetorotational instability (MRI) in the hypermassive neutron star (HMNS) amplifies magnetic field to beyond magnetar-strength ($10^{15}\, \mathrm{G}$). The ultra-strong toroidal field is able to launch a relativistic jet from the HMNS. We also find a magnetized wind that ejects neutron-rich material with a rate of $\dot{M}_{\mathrm{ej}} \simeq 1 \times10^{-1}\, \mathrm{M_{\odot}\, s^{-1}}$. The total ejecta mass in our simulation is $5\times 10^{-3}\, \mathrm{M_{\odot}}$. This makes the ejecta from the HMNS an important component in BNS mergers and a promising source of $r$-process elements that can power a kilonova. The jet from the HMNS reaches a terminal Lorentz factor of $\sim 5$ in our highest-resolution simulation. The formation of this jet is aided by neutrino-cooling preventing the accretion disk from protruding into the polar region. As neutrino pair-annihilation and radiative processes in the jet (which were not included in the simulations) will boost the Lorentz factor in the jet further, our simulations demonstrate that magnetars formed in BNS mergers are a viable engine for short gamma-ray bursts (sGRBs).Comment: Resubmitted versio

Fractals and the monadic second order theory of one successor

Let $X \subseteq \mathbb{R}^n$ be closed and nonempty. If the $C^k$-smooth points of $X$ are not dense in $X$ for some $k \geq 0$, then $(\mathbb{R},<,+,0,X)$ interprets the monadic second order theory of $(\mathbb{N},+1)$. The same conclusion holds if the Hausdorff dimension of $X$ is strictly greater than the topological dimension of $X$ and $X$ has no affine points. Thus, if $X$ is virtually any fractal subset of $\mathbb{R}^n$, then $(\mathbb{R},<,+,0,X)$ interprets the monadic second order theory of $(\mathbb{N},+1)$. This result is sharp as the standard model of the monadic second order theory of $(\mathbb{N},+1)$ is known to interpret expansions of $(\mathbb{R},<,+,0)$ which define various classical fractals.Comment: Added a proof of definibility of $C^k$-smooth points and improved the introductio