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ABSTRACT:
It is well known that the Newtonian
gravitation has
the freedom of choosing zero of the potential as
Dadhich [1] 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 This is no longer valid in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
"hedgehog" configuration [2,3] Phi^a = eta f(r)
x^a/r, withn x^a x^a = r^2 and eta-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, eta ~ 10^16
Gev.
The Bariola - Vilenkin solution for the Schwarzschild
particle with a monopole charge sqrt(8piGeta^2) is
g00 = 1-2k-2m/r = -1/g11, where k=4piGeta^2. 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 porpose 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 R23^23 = -2k/r^2, with respect
to flat spacetime.The geodesic trajectory
r(phi)contains the relativistic factor sqrt(1-v^2) :
r(phi) = 1/b sin [a-phi sqrt(1-v^2)],
where 1/b = rmin = L/sqrt(E^2-m^2)(1-v^2), L-the
angular momentum of the particle, E-its energy, sina =
1/bd, d=r(0) and v^2=8piGeta^2. The trajectory is no
longer a straightline (in Cartezian 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 velocity
4-vector u^alpha has components (E/m,
0,(E^2-m^2)costheta/mL, (E^2-m^2)/mL). Choosing the
time component of the separation vector between the
two geodesics xi^0=0, we obtain, for xi^theta and
xi^phi components, harmonic oscillator type equations
with frequency omega = v(E^2-m^2)/mL.
References
[1] N.Dadhich, gr-qc/9704068 ; gr-qc/9902066.
[2] I.Cho, A.Vilenkin, Phys.Rev.D56, 7621 (1997).
[3] M.Bariola,A.Vilenkin Phys.Rev.Lett.63,341 (1989)
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