[Paper Review] Twistor action for general relativity

Atul Sharma|arXiv (Cornell University)|Apr 14, 2021

Cosmology and Gravitation Theories46 references4 citations

TL;DR

This paper proposes a novel twistor action for Euclidean general relativity without a cosmological constant, reformulating the theory via partially integrable almost complex structures on twistor space. It establishes equivalence to Plebanski's chiral action through the Penrose transform, providing a fully non-linear, off-shell formulation that resolves the long-standing 'googly problem' and enables a systematic derivation of gravitational MHV rules.

ABSTRACT

We reformulate Euclidean general relativity without cosmological constant as an action governing the complex structure of twistor space. Extending Penrose's non-linear graviton construction, we find a correspondence between twistor spaces with partially integrable almost complex structures and four-dimensional space-times with off-shell metrics. Using this, we prove that our twistor action reduces to Plebanski's action for general relativity via the Penrose transform. This should lead to new insights into the geometry of graviton scattering as well as to the derivation of computational tools like gravitational MHV rules.

Motivation & Objective

• To construct a twistor action equivalent to Euclidean general relativity without a cosmological constant, addressing the longstanding challenge of formulating non-self-dual gravity in twistor space.
• To generalize Penrose’s non-linear graviton construction to associate almost complex structures on twistor spaces with off-shell space-time metrics.
• To provide a classical, fully non-linear resolution of the 'googly problem' in twistor theory by encoding both self-dual and non-self-dual sectors of gravity.
• To pave the way for deriving gravitational MHV rules and computational tools via perturbative expansion of the twistor action.

Proposed method

• The paper introduces a generalization of Penrose’s non-linear graviton construction, linking partially integrable almost complex structures on twistor space to four-dimensional space-times with off-shell metrics.
• It formulates a twistor action using a deformed almost complex structure on the total space of a holomorphic bundle over $\mathbb{P}^1$, modeled on the Kodaira-Spencer deformation theory.
• The action is constructed using a connection $B$ on a holomorphic line bundle over $\mathbb{P}\mathscr{T}$, with gauge symmetry under $B \mapsto B + \bar{\nabla}\chi$ for $\chi \in \Omega^{1,0}(\mathbb{P}\mathscr{T}, \mathcal{O}(-4))$.
• The action is shown to reduce to Plebanski’s chiral action for GR via the Penrose transform, with the self-dual and non-self-dual parts of the action mapped through explicit integration and pullback to space-time.
• The interaction term in the action is matched to the non-self-dual part of Plebanski’s action using pullbacks of twistor forms and the structure of $\Sigma^{\alpha\beta}$, confirming consistency.
• The construction preserves gauge symmetry and allows for useful gauge choices, crucial for deriving MHV rules in perturbation theory.

Experimental results

Research questions

• RQ1 Can a twistor action be constructed that is equivalent to the full, non-self-dual Einstein equations in Euclidean signature without a cosmological constant?
• RQ2 How can Penrose’s non-linear graviton construction be generalized to include partially integrable almost complex structures that encode off-shell space-time metrics?
• RQ3 Does the proposed twistor action resolve the 'googly problem' by fully incorporating both self-dual and non-self-dual components of gravity?
• RQ4 Can this twistor action serve as a foundation for deriving gravitational MHV rules through perturbative expansion?
• RQ5 What is the role of the additional gauge symmetry $B \mapsto B + \bar{\nabla}\chi$ in enabling consistent amplitude computations?

Key findings

• The proposed twistor action is proven to be equivalent to Plebanski’s chiral action for Euclidean general relativity via the Penrose transform, establishing a direct correspondence between twistor space geometry and space-time gravity.
• The action encodes both self-dual and non-self-dual sectors of gravity, providing a fully non-linear, off-shell formulation that resolves the 'googly problem' in twistor theory.
• The self-dual part of the action reduces to $S_{\text{SD}}[e,\Gamma] = \int_{\mathcal{M}} \Gamma_{\alpha\beta} \wedge d\Sigma^{\alpha\beta}$, consistent with known results.
• The non-self-dual interaction term in Plebanski’s action is recovered via the pullback of twistor forms, confirming the action’s consistency with the space-time formulation.
• The action possesses an additional gauge symmetry $B \mapsto B + \bar{\nabla}\chi$, which is expected to be instrumental in deriving gravitational MHV rules through gauge-fixing.
• The construction opens a path to deriving MHV rules for graviton scattering and may be extended to include a cosmological constant, supersymmetry, or other signatures.