5.2 The Robinson–Trautman metrics

The algebraically-special type II Robinson–Trautman (RT) metrics are expressed in conventional RT coordinates, (-- τ,r,ζ,ζ), τ now real, by [71]
( ′ 0) -- ds2 = 2 K − V-r + ψ-2 dτ2 + 2dτ dr − r22dζdζ-, (5.23 ) V r V 2P20
2 2 - K = 2V P 0∂ζ∂ζ log V P0, (5.24 ) P0 = 1 + ζζ, (5.25 ) 0 0 ψ 2 = ψ2(τ). (5.26 )
The unknowns are the Weyl component ψ02 (closely related to the Bondi mass), which is a function only of (real) τ and the variable, -- V (τ,ζ,ζ), both of which are variables in the RT equation (see below). There remains the freedom
τ → τ ∗ = g (τ ), (5.27 )
which often is chosen so that ψ02(τ ) = constant. However, we make a different choice. In the spherical harmonic expansion of V,
aˆ -- ij 0 V = v la(ζ,ζ) + v Y 2ij + ..., (5.28 )
the τ is chosen by normalizing the four-vector, va, to one, i.e., vava = 1. The final field equation, the RT equation, is
V ′ ( -- -- ) ψ02′− 3ψ02 ---− V 3 ∂2(τ)∂2(τ)V − V − 1∂2(τ)V ⋅ ∂2(τ)V = 0. (5.29 ) V3
These spacetimes, via the Goldberg–Sachs theorem, possess a degenerate shear-free PND field, la, that is surface-forming, (i.e., twist free). Using the tetrad constructed from la we have that the Weyl components are of the form
ψ0 = ψ1 = 0, ψ2 ⁄= 0.
Furthermore, the metric contains a ‘real timelike world line, xa = ξa(τ),’ with normalized velocity vector a a′ v = ξ. All of these properties allow us to identify the RT metrics as being analogous to the real Liénard–Wiechert solutions of the Maxwell equations.

Assuming for the moment that we have integrated the RT equation and know -- V = V (τ,ζ,ζ), then, by the integral

∫ u = V (τ, ζ,ζ)dτ ≡ X (τ,ζ,ζ¯), (5.30 ) B RT
the UCF for the RT metrics has been found. The freedom of adding -- α(ζ,ζ) to the integral is just the supertranslation freedom in the choice of a Bondi coordinate system. From ¯ XRT (τ,ζ,ζ) a variety of information can be obtained: the Bondi shear, σ0, is given parametrically by
σ0(u ,ζ, ¯ζ) = ∂2 X (τ,ζ,ζ¯), (5.31 ) B (τ) RT uB = XRT (τ,ζ, ¯ζ),
as well as the angle field L by
¯ ¯ L (uB,ζ,ζ) = ∂ (τ)XRT (τ,ζ,ζ), (5.32 ) uB = XRT (τ,ζ, ¯ζ).
In turn, from this information the RT metric (in the neighborhood of + ℑ) can, in principle, be re-expressed in terms of the Bondi coordinate system, though in practice one must revert to approximations. These approximate calculations lead, via the Bondi mass aspect evolution equation, to both Bondi mass loss and to equations of motion for the world line, xa = ξa(τ). An alternate approximation for the mass loss and equations of motion is to insert the spherical harmonic expansion of V into the RT equation and look at the lowest harmonic terms. We omit further details aside from mentioning that we come back to these calculations in a more general context in Section 6.
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