Joshua Gordon Fenwick

Many exact solutions of Einstein–Maxwell and Einstein–Maxwell-dilaton theory share a common structural pattern in which the metric functions are built from harmonic functions on a specified spatial base, often taken to be flat. We investigate this pattern by considering the Einstein–Maxwell-dilaton theory in arbitrary dimensions, where the dilaton field is non-trivially coupled to the Maxwell field and to a Liouville-type potential proportional to the cosmological parameter. Without imposing either the base geometry or the harmonic form of the metric function in advance, we show that for the generic branch, the field equations force the dilaton couplings to be equal, restrict the spatial base geometry to be Ricci flat, and make the metric function harmonic on this base. The resulting spacetime can then be written in terms of a conformal potential. We also consider a purely spatial branch, which instead leads to a distinct constraint on the coupling constants. These results provide a unified field-equation derivation of the harmonic behavior and conformal structure that appear in several classes of solutions, including multi-center geometries, cosmological solutions, and dynamical black holes.

We express a new mathematical Taub-NUT like black-hole solution in f ( R ) gravity. This solution is found to have a non constant Ricci scalar and corresponds to an extension of the δ = − 1 / 2 case of the Clifton–Barrow metric. Within the context of this space-time we examine various limit, thermodynamic quantities and non-charged particle trajectories. In addition we find an Eguchi–Hanson type-II f ( R ) regime related to the found solution.

We construct exact analytical Taub-NUT solutions in the context of f(T) gravity. We study the physical properties of the solutions, and compare them with those of the Taub-NUT solution in Einstein gravity.

Imposing the Born rule as a fundamental principle of quantum mechanics would require the existence of normalizable wave functions ψ( x , t) also for relativistic particles. Indeed, the Fourier transforms of normalized k -space amplitudes ψ ( k , t ) = ψ ( k ) exp ( − i ω k t ) yield normalized functions ψ( x , t) which reproduce the standard k -space expectation values for energy and momentum from local momentum (pseudo-)densities ℘ _μ ( x , t) = (ℏ/2i)[ψ ⁺( x , t)∂ _μ ψ( x , t) − ∂ _μ ψ ⁺( x , t) · ψ( x , t)]. However, in the case of bosonic fields, the wave packets ψ( x , t) are nonlocally related to the corresponding relativistic quantum fields ϕ( x , t), and therefore the canonical local energy-momentum densities  ( x , t ) = c  0 ( x , t ) and  ( x , t ) differ from ℘ _μ ( x , t) and appear nonlocal in terms of the wave packets ψ( x , t). We examine the relation between the canonical energy density  ( x , t ) , the canonical charge density ϱ( x , t), the energy pseudo-density  ˜ ( x , t ) = c ℘ 0 ( x , t ) , and the Born density ∣ψ( x , t)∣² for the massless free Klein–Gordon field. We find that those four proxies for particle location are tantalizingly close even in this extremely relativistic case: in spite of their nonlocal mathematical relations, they are mutually local in the sense that their maxima do not deviate beyond a common position uncertainty Δx. Indeed, they are practically indistinguishable in cases where we would expect a normalized quantum state to produce particle-like position signals, viz. if we are observing quanta with momenta p ≫ Δp ≥ ℏ/2Δx. We also translate our results to massless Dirac fields. Our results confirm and illustrate that the normalized energy density  ( x , t ) / E provides a suitable measure for positions of bosons, whereas normalized charge density ϱ( x , t)/q provides a suitable measure for fermions.