Geodesic Particle Paths Inside a Nonrotating, Homogeneous, Spherical Body

Abstract

Proceeding from a solution of field equations that are improved versions of Einstein's nonvacuum gravitational field equations one is able to calculate precisely the trajectories of particles traveling inside a nonrotating, homogeneous, spherical body. Application of the results to the conditions of recent measurements of neutrino flight times between a source point A at CERN's European Laboratory for Particle Physics and a point B in either of two detectors (ICARUS or OPERA) at LNGS (Laboratori Nazionale del Gran Sasso), separated by a euclidean distance d(A, B) = 731 km, predicts for the flight time Tν from A to B of a 2 eV neutrino launched with energy 17 GeV that, as measured by a clock at B synchronized to a similar clock at A, Tν ≈ d/c + 9.3 × 10 −16 sec. But as measured by inertial observers along the path the flight time Tν ≈ d/c − 2.6 × 10 −9 sec and the path length Lν ≈ d − 8.4 × 10 −7 m, which yields Lν/ Tν ≈ c + 321 m/sec for the average inertially referenced speed of the neutrino from A to B.

The relativistic gravity train

“…He finds a solution in closed form without singularities that is matched as smoothly as possible at the surface to the Schwarzschild exterior solution [12,13]. This allows him to derive trajectories of particles travelling inside the sphere.…”

“…[11] replaced Schwarzschild's interior metric [10] with the space-time metric given in equation ( 5) that allows computations of travel times inside a non-rotating homogeneous sphere with radius R (provided with a tunnel for the test object). From gravitational field equations Ellis derives relations between the radial coordinate ρ and the Schwarzschild coordinate r and f (ρ) in the metric in equation (5) [12,13]:…”

“…2 λ→1 for R?R S and λ→∞ for R→R S (m=½ R S ). In addition there are three constants of motion that exist for every affinely parametrised geodesic path in a sphere with fixed longitude j (equations ( 7), ( 8) and (10) in [13]):…”

“…The dot denotes derivative with respect to the proper time τ and ε=1, 0, −1 for time-like, light-like and space-like geodesics. From these equations and equation (12) the following equation is derived for time-like geodesics (equation (32) in [13]):…”

“…Second, instead of using the Schwarzschild interior metric [10] a space-time metric for the gravitational field inside a non-rotating homogeneous spherical ball is used which has been proposed by Ellis [11] and which makes the analysis of geodesics and computations of travel times inside the ball relatively straightforward [12,13].…”