# Penrose transform and general wave equations

In the late 1960's Penrose developed twistor theory, which (amongst other things) led to an exceptional description for solutions to the wave equation on Minkowski space via the so-called Penrose transform;

If \begin{equation} u(x,y,z,t) = \frac {1} {2 \pi i} \oint_{\Gamma \subset \mathbb{C} \mathbb{P}^{1}} f(-(x+iy) + \lambda (t-z), (t+z) + \lambda (-x + i y), \lambda ) d \lambda, \,\,\,\,\,\,\,\,\,\, (1) \end{equation}

where $\Gamma \subset \mathbb{C} \mathbb{P}^{1}$ is a closed contour and $f$ is holomorphic on $\mathbb{C} \mathbb{P}^{1}$ except at some number of poles, then $u$ satisfies the Minkowski wave (Laplace-Beltrami) equation $\square_{\eta} u = 0$.

I am aware that there are a number of works in the literature describing twistor theory on curved manifolds, but have not seen explicit constructions along the lines of (1) such that the function $u$ satisfies a wave equation of the form $\square_{g} u = 0$ for (Lorentzian) metric $\boldsymbol{g}$.

Is it known how to explicitly construct contour integrals similar to $(1)$ for some class of metrics $\boldsymbol{g}$? What about when $\boldsymbol{g}$ is Einstein (e.g. Schwarzschild), in particular? Are there topological obstructions in spacetimes $I \times \Sigma$? What about de-Sitter space?

• If I remember correctly, that the Penrose transform works for Minkowski space is strongly tied to the strong Huygen's principle. Most metrics do not satisfy this. – Willie Wong Jul 22 '16 at 2:52
• Also, noting that twistor theory is built off of the conformal/null structure of the Lorentzian manifold, one may expect it to work more reasonably with the conformally invariant wave equation (the one with a suitable potential term coming from the scalar curvature) than the free wave equation. (This is in regards to the possibility of something working for de Sitter). – Willie Wong Jul 22 '16 at 2:57
• An interesting comment @WillieWong, I suspect that is true but it seems to go against my intuition. The conformally invariant wave equation on de Sitter space acquires a `mass-like' term since $R$ is constant. It seems almost like then you are describing a timelike object rather than null (massive Klein-Gordon) which seems almost contradictory; perhaps there is something more fundamental I am missing. – Arthur Suvorov Jul 23 '16 at 1:24
• In regards to conformal wave equation and relationship to Huygen's principle: see Helgason's Wave equation on homogeneous spaces, in Lie group representations, III (College Park, Md., 1982/1983), Springer, 1984. – Willie Wong Jul 24 '16 at 0:23

I believe you can express the Penrose transform using a contour integral by expressing Serre duality in terms of Dolbeault cohomology. For example, IIRC, on a half-conformally-flat 4-manifold $M$ with twistor space $Z$, functions solving the Laplacian equation correspond to elements of the sheaf cohomology group $H^1(Z, \mathcal{O}(-2))$ and the evaluation at a point $x \in M$, with corresponding twistor line $P_x \subset Z$, is to restrict to: $$H^1(P_x, \mathcal{O}(-2)) \simeq H^0(P_x, \mathcal{O})^* \simeq \mathbb{C}.$$ If we represent our element of $H^1(P_x, \mathcal{O}(-2))$ by a form in $H^{0,1}(P_x, \mathcal{O}(-2))$, then choosing a contour that does not meet the singularities from the section $\mathcal{O}(-2)$, the Serre duality map above yields an contour integral by Stokes's theorem.
see equation (2.2). The metrics constructed there are hyperkahler metrics with toric symmetries of rank equal to their quaternionic dimension. The contour integral satisfies the Laplace and not the wave equation, and the resulting metrics have in general signature $(4n,4m)$.