Schwarzschild black holes from twistor space

Twistor theory forms the basis for many surprising advances in areas ranging from dynamical systems to quantum field theory.

Yet for almost fifty years, one of the main drawbacks of twistor theory has been its inability to give non-perturbative descriptions of non-chiral (or non-self-dual) field configurations.

This difficulty is known as 'the googly problem.' In this paper, we provide a resolution of the googly problem for a particular solution of the vacuum Einstein equations: the Schwarzschild metric.

We start with the twistor space of the self-dual Taub-NUT Euclidean gravitational instanton, expressed in Kerr-Schild form.

Within this twistor space, we then consider a quadric which corresponds to the anti-self-dual Taub-NUT metric.

While the full quadric is not holomorphic with respect to the complex structure of the self-dual Taub-NUT twistor space, its holomorphic locus still has complex dimension two.

This 'coincidence locus' -- points in twistor space on the holomorphic portion of the quadric -- inherits a complex structure from the twistor space and a Kähler form from the quadric itself.

Remarkably, these structures are compatible, giving rise to a non-self-dual, four-dimensional Kähler metric which is conformal to Schwarzschild (in Lorentzian or Euclidean signature).

This is the first instance of a non-self-dual Einstein metric constructed entirely from holomorphic data in a twistor space.