1,125 research outputs found
Characteristic Surface Data for the Eikonal Equation
A method of solving the eikonal equation, in either flat or curved
space-times, with arbitrary Cauchy data, is extended to the case of data given
on a characteristic surface. We find a beautiful relationship between the
Cauchy and characteristic data for the same solution, namely they are related
by a Legendre transformation. From the resulting solutions, we study and
describe their associated wave-front singularities.Comment: 16 pages, no figures, Scientific Work-Place 2.5, tex, Corrected typo
Differential Geometry from Differential Equations
We first show how, from the general 3rd order ODE of the form
z'''=F(z,z',z'',s), one can construct a natural Lorentzian conformal metric on
the four-dimensional space (z,z',z'',s). When the function F(z,z',z'',s)
satisfies a special differential condition of the form, U[F]=0, the conformal
metric possesses a conformal Killing field, xi = partial with respect to s,
which in turn, allows the conformal metric to be mapped into a three
dimensional Lorentzian metric on the space (z,z',z'') or equivalently, on the
space of solutions of the original differential equation. This construction is
then generalized to the pair of differential equations, z_ss =
S(z,z_s,z_t,z_st,s,t) and z_tt = T(z,z_s,z_t,z_st,s,t), with z_s and z_t, the
derivatives of z with respect to s and t. In this case, from S and T, one can
again, in a natural manner, construct a Lorentzian conformal metric on the six
dimensional space (z,z_s,z_t,z_st,s,t). When the S and T satisfy equations
analogous to U[F]=0, namely equations of the form M[S,T]=0, the 6-space then
possesses a pair of conformal Killing fields, xi =partial with respect to s and
eta =partial with respect to t which allows, via the mapping to the four-space
of z, z_s, z_t, z_st and a choice of conformal factor, the construction of a
four-dimensional Lorentzian metric. In fact all four-dimensional Lorentzian
metrics can be constructed in this manner. This construction, with further
conditions on S and T, thus includes all (local) solutions of the Einstein
equations.Comment: 37 pages, revised version with clarification
Electromagnetic Dipole Radiation Fields, Shear-Free Congruences and Complex Center of Charge World Lines
We show that for asymptotically vanishing Maxwell fields in Minkowski space
with non-vanishing total charge, one can find a unique geometric structure, a
null direction field, at null infinity. From this structure a unique complex
analytic world-line in complex Minkowski space that can be found and then
identified as the complex center of charge. By ''sitting'' - in an imaginary
sense, on this world-line both the (intrinsic) electric and magnetic dipole
moments vanish. The (intrinsic) magnetic dipole moment is (in some sense)
obtained from the `distance' the complex the world line is from the real space
(times the charge). This point of view unifies the asymptotic treatment of the
dipole moments For electromagnetic fields with vanishing magnetic dipole
moments the world line is real and defines the real (ordinary center of
charge). We illustrate these ideas with the Lienard-Wiechert Maxwell field. In
the conclusion we discuss its generalization to general relativity where the
complex center of charge world-line has its analogue in a complex center of
mass allowing a definition of the spin and orbital angular momentum - the
analogues of the magnetic and electric dipole moments.Comment: 17 page
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