# Penrose transform and general wave equations

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

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

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 is 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 $\textit{explicitly}$ 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?

This post imported from StackExchange MathOverflow at 2016-12-22 17:28 (UTC), posted by SE-user Arthur Suvorov
retagged Dec 22, 2016
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.

This post imported from StackExchange MathOverflow at 2016-12-22 17:28 (UTC), posted by SE-user Willie Wong
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).

This post imported from StackExchange MathOverflow at 2016-12-22 17:28 (UTC), posted by SE-user Willie Wong
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.

This post imported from StackExchange MathOverflow at 2016-12-22 17:28 (UTC), posted by SE-user Arthur Suvorov
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.

This post imported from StackExchange MathOverflow at 2016-12-22 17:28 (UTC), posted by SE-user Willie Wong

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