Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

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EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
where is any monotonically decreasing function of the retarded
time.
Kinnersley’s photon rocket [2]
This solution is a member of the Robinson–Trautman class and
describes the gravitational field of an accelerated point singularity.
Bonnor’s pencil of light [3]
These three solutions are algebraically special (at least two of the null
eigendirections of the Weyl tensor coincide) and of Kerr–Schild type,
and the eigendirections are non–twisting.
We will construct exact solutions with the energy–momentum tensor
which is composed of two null dust terms with the propagation vectors
and In order to get static (or stationary) solutions we can assume,
without loss of generality, that the two beams have equal energy density,
The cylindrically symmetric case with radial components of the two
propagation vectors (the axial and azimuthal components being zero)
leads to the general solution (in terms of double null coordinates and
)
with arbitrary real functions and This solution becomes static
for the special choice It has been assumed that
no gravitational Einstein–Rosen waves survive when the null dust is
switched off.
The spherically symmetric case and cylindrically symmetric configurations
with axial components of the propagation vectors will be treated
in the following two sections.