Geometrical and physical interpretation of the Levi-Civita spacetime in terms of the Komar mass density

We revisit the interpretation of the cylindrically symmetric, static vacuum Levi-Civita metric, known in either Weyl, Einstein-Rosen, or Kasner-like coordinates. Despite the infinite axis source, we derive its Komar mass density through a compactification and subsequent blowing up of the compactification radius. We show that, the Komar mass density $\mu_K$ calculated in the Einstein-Rosen frame, when employed as the metric parameter, has a number of advantages. It eliminates double coverages of the parameter space, vanishes in flat spacetime and when small, it corresponds to the mass density of an infinite string. After a comprehensive analysis of the local and global geometry, we proceed with the physical interpretation of the Levi-Civita spacetime. First we show that the Newtonian gravitational force is attractive and its magnitude increases monotonically with all positive ${\mu}_K$, asymptoting to the inverse of the the proper distance in the "radial" direction. Second, we reveal that the tidal force between nearby geodesics (hence gravity in the Einsteinian sense) attains a maximum at $\mu_K=1/2$ and then decreases asymptotically to zero. Hence, from a physical point of view the Komar mass density of the Levi-Civita spacetime encompasses two contributions: Newtonian gravity and acceleration effects. An increase in $\mu_K$ strengthens Newtonian gravity but also drags the field lines increasingly parallel, eventually transforming Newtonian gravity through the equivalence principle into a pure acceleration field and the Levi-Civita spacetime into a flat Rindler-like spacetime. In a geometric picture the increase of $\mu_K$ from zero to $\infty$ deforms the planar sections of the spacetime into ever deepening funnels, eventually degenerating into cylindrical topology.Comment: 16 pages, 8 figure

Geometrical and physical interpretation of the Levi-Civita spacetime in terms of the Komar mass density

We revisit the interpretation of the cylindrically symmetric, static vacuum Levi-Civita metric, known in either Weyl, Einstein–Rosen, or Kasner-like coordinates. The Komar mass density of the infinite axis source arises through a suitable compactification procedure. The Komar mass density \mu _{K} μ K calculated in Einstein–Rosen coordinates, when employed as the metric parameter, leads to a number of advantages. It eliminates double coverages of the parameter space, vanishes in flat spacetime and when small, it corresponds to the mass density of an infinite string. After a comprehensive analysis of the local and global geometry, we proceed with the physical interpretation of the Levi-Civita spacetime. First we show that the Newtonian gravitational force is attractive and its magnitude increases monotonically with all positive \mu _{K} μ K , asymptoting to the inverse of the proper distance in the radial direction. Second, we reveal that the tidal force between nearby geodesics (hence gravity in the Einsteinian sense) attains a maximum at \mu _{K}=1/2 μ K = 1 / 2 and then decreases asymptotically to zero. Hence, from a physical point of view the Komar mass density of the Levi-Civita spacetime encompasses two contributions: Newtonian gravity and acceleration effects. An increase in \mu _{K} μ K strengthens Newtonian gravity but also drags the field lines increasingly parallel, eventually transforming Newtonian gravity through the equivalence principle into a pure acceleration field and the Levi-Civita spacetime into a flat Rindler-like spacetime. In a geometric picture the increase of \mu _{K} μ K from zero to \infty ∞ deforms the planar sections of the spacetime into ever deepening funnels, eventually degenerating into cylindrical topology in an appropriately chosen embedding

The Lanczos Equation on Light-Like Hypersurfaces in a Cosmologically Viable Class of Kinetic Gravity Braiding Theories

We discuss junction conditions across null hypersurfaces in a class of scalar–tensor gravity theories (i) with second-order dynamics, (ii) obeying the recent constraints imposed by gravitational wave propagation, and (iii) allowing for a cosmologically viable evolution. These requirements select kinetic gravity braiding models with linear kinetic term dependence and scalar field-dependent coupling to curvature. We explore a pseudo-orthonormal tetrad and its allowed gauge fixing with one null vector standing as the normal and the other being transversal to the hypersurface. We derive a generalization of the Lanczos equation in a 2 + 1 decomposed form, relating the energy density, current, and isotropic pressure of a distributional source to the jumps in the transverse curvature and transverse derivative of the scalar. Additionally, we discuss a scalar junction condition and its implications for the distributional source

Variational formalism for generic shells in general relativity

We investigate the variational principle for the gravitational field in the presence of thin shells of completely unconstrained signature (generic shells). Such variational formulations have been given before for shells of timelike and null signatures separately, but so far no unified treatment exists. We identify the shell equation as the natural boundary condition associated with a broken extremal problem along a hypersurface where the metric tensor is allowed to be nondifferentiable. Since the second order nature of the Einstein-Hilbert action makes the boundary value problem associated with the variational formulation ill-defined, regularization schemes need to be introduced. We investigate several such regularization schemes and prove their equivalence. We show that the unified shell equation derived from this variational procedure reproduce past results obtained via distribution theory by Barrabes and Israel for hypersurfaces of fixed causal type and by Mars and Senovilla for generic shells. These results are expected to provide a useful guide to formulating thin shell equations and junction conditions along generic hypersurfaces in modified theories of gravity

Null shells in kinetic gravity braiding scalar-tensor theories

We derive the equation of motion of null shells generated by distributional sources with support on the shell in a class of second-order scalar-tensor theories that is both compatible with gravitational wave observations and produce valid cosmological evolution.The shell equation of motions and thus the junction conditions required for smooth matching along a null hypersurface are obtained by considering a 2+1+1 decomposition given by the null generators of the hypersurface and an auxiliary null vector field. The equations relate jumps of geometric quantities to the components of the distributional sources and also provide a constraint relation on the source components

The Lanczos Equation on Light-Like Hypersurfaces in a Cosmologically Viable Class of Kinetic Gravity Braiding Theories

We discuss junction conditions across null hypersurfaces in a class of scalar−tensor gravity theories (i) with second-order dynamics, (ii) obeying the recent constraints imposed by gravitational wave propagation, and (iii) allowing for a cosmologically viable evolution. These requirements select kinetic gravity braiding models with linear kinetic term dependence and scalar field-dependent coupling to curvature. We explore a pseudo-orthonormal tetrad and its allowed gauge fixing with one null vector standing as the normal and the other being transversal to the hypersurface. We derive a generalization of the Lanczos equation in a 2 + 1 decomposed form, relating the energy density, current, and isotropic pressure of a distributional source to the jumps in the transverse curvature and transverse derivative of the scalar. Additionally, we discuss a scalar junction condition and its implications for the distributional source