Geodesic deviation in a massless global monopole spacetime

It is well known that gravitation has the freedom of choosing zero of the potential as Dadhich has shown. This is no longer valid in General Relativity where the Einstein vacuum equations determine the gravitational potential -m/r for a point mass to be zero at infinity. In order to restore the gauge freedom we have to use the duality transformation defining an electromagnetic decomposition of the Riemann tensor into electric and magnetic parts. The price paid for this is the introduction of some energy density of the r.h.s. of Einstein's equations corresponding to a global monopole with a topological charge described by the configuration Φa = ηf(r)xa/r with xaxa r2 and η - the scale of the symmetry breaking (a global monopole is produced when the O(3) symmetry is spontaneously broken to U(1). For a typical Grand Unification scale, η ∼1016GeV. The Bariola - Vilenkin solution for the Schwarzschild particle with a monopole charge √8πGη2 is g00 = 1 - 2k - 2m/r = -1/g11, where k 4πGη2. Far from the mass m we neglect the last term in g00 and obtain also a nonflat geometry corresponding to a global monopole of zero mass, a solution which is dual to the Schwarzschild solution without a topological charge. Our purpose in this letter is to compute the geodesics and the geodesic deviation of two inertial observers in a massless global monopole geometry, keeping in mind that there is a nonvanishing component of the Riemann tensor R2323 = -2k/r2, with respect to flat spacetime.The geodesic trajectory r(contains the relativistic factor √(1 - v2) : r(φ) = 1/bsin[a -φ√ (1 - v2)], where 1/b = rmin = L/√(E2 -m2)(1-v2), L - the angular momentum of the particle, E - its energy, sina = 1/bd, d = r(0) and v2 = 8πGη2. The trajectory is no longer a straight line (in Cartesian coordinates) and a solid deficit angle arises. The radial motion is not changed with respect to Minkowski one. The curve r(t) is a hyperbola. To compute the geodesic deviation we take two observers which move on a surface r = const. The uα 4-vector velocity has components (E/m, 0, (E2-m2)cosθ/ mL, (E2-m2)/mL). Choosing the time component of the separation vector between the two geodesics ξ0 = 0, we obtain, for ξθ and ξφ components, harmonic oscillator type equations with frequency ω = v(E2 - m2)/mL. (author)