Strong deflection of massive particles via the geodesic deviation equation

Abstract

We develop a formulation of the strong deflection limit for the scattering of particles following timelike geodesics in asymptotically flat, static, and spherically symmetric spacetimes. For fixed specific energy, as the angular momentum approaches its critical value from above, the particle passes arbitrarily close to the associated unstable circular orbit, undergoes many windings around it, and the deflection angle diverges logarithmically. Using the geodesic deviation equation, we show covariantly that the coefficient of this logarithmic divergence is determined by the radial instability exponent of the critical trajectory, defined per unit azimuthal angle. We express this instability exponent in terms of local curvature data on the unstable circular orbit, thereby providing both kinematic and geometric interpretations of the strong deflection limit. In general relativity, its matter dependence enters only through a single local scalar combination constructed from the static-frame energy density and the principal radial and tangential pressures.

31 pages, no figures; v2: added Sec. VIII on strong-lensing observables; v3: published version in Physical Review D

References in corpus (4)

• Strong deflection of massive particles in spherically symmetric spacetimes
• Deflection angle in the strong deflection limit: A perspective from local geometrical invariants and matter distributions
• Gravitational lensing inside and outside of a marginally unstable photon sphere in a general, static, spherically symmetric, and asymptotically-flat spacetime in strong deflection limits
• Strong-deflection expansion of the deflection angle near a degenerate photon sphere