CQG+ Insight: Chiral Gravity

by Kirill Krasnov

Kirill Krasnov, Professor of Mathematical Physics, University of Nottingham. Pictured here visiting Newstead Abbey, Nottinghamshire

We seem to live in four space-time dimensions, and so should take the structures available in this number of dimensions seriously. One of these is chirality, see below for clarifications on my usage of this term. Related to chirality, there is a remarkable phenomenon occurring in General Relativity (GR) in four space-time dimensions. This phenomenon is so stunning that I would like to refer to it as the chiral miracle. It is well-known to experts. Still, even after almost 40 years after it had appeared in the literature, it has not become part of the background of all GR practitioners. I would like to use this CQG+ insight format to try to rectify this.

I start by reviewing the notion of chirality in four space-time dimensions. I then describe the “chiral miracle” that allows for chiral description(s) of gravity in four dimensions. Finally, I provide some information on the possibility of chiral modifications of GR.

Chirality

In the sense used here, the notion of chirality is related to the fact that the four-dimensional Lorentz group is doubly covered by the Mobius group SL(2,C), which is the group of 2×2 matrices with complex entries and the unit determinant. The fundamental irreducible representations of this group are of two different types: 2-components columns $\lambda$ on which 2×2 matrices $g\in{\rm SL(2,C)}$ act as $\lambda\to g\lambda$ , and the complex conjugate representation in terms of also 2-components columns $\bar{\lambda}$ on which the action is $\bar{\lambda}\to g^*\bar{\lambda}$ , where $g^*$ is the complex conjugate matrix. These two different representations are the two different types of spinors that exist in four dimensions. The 2-component spinors are chiral objects: Taking the complex conjugate of a spinor of one type one obtains a spinor of the other type. The operation of complex conjugation is related to the 3-dimensional operation of taking the mirror image, see below, which justifies using the terminology chiral also in reference to the complex conjugation.

As is well-known, a 4-vector can equivalently be thought of as a bi-spinor of a mixed type. Thus, if we refer to the spinor representation of the first type as $S_+$ and that of the second type as $S_-$ , then the vector representation is isomorphic to $S_+\otimes S_-$ . Real elements of this space can be thought of as 2×2 Hermitian matrices ${\bf x}={\bf x}^\dagger$ on which the Lorentz group acts as ${\bf x} \to g{\bf x} g^\dagger$ , where $g^\dagger=(g^T)^*$ is the Hermitian conjugate of $g$ . It is this representation of 4-vectors as Hermitian matrices that provides the isomorphism ${\rm SO}(1,3)\sim {\rm SL(2,C)}$ . Taking the complex conjugate of $S_+$ one gets $S_-$ . Thus, the space $S_+\otimes S_-$ has real or non-chiral objects, and these are precisely the (real) 4-vectors.

A related chiral decomposition is that of the space of 2-forms in four dimensions. Indeed, given a metric, we have the operation of taking the Hodge star of a differential form. This operation maps the space of 2-forms into itself. As can be checked, taking this operation squared gives minus the identity operator, where the appearance of the minus is related to the indefinite metric signature. Then, there exist eigenspaces of the Hodge star in the space $\Lambda^2_C$ of complexified 2-forms. These are referred to as the spaces of self-dual (SD) $\Lambda^+$ and anti-self-dual (ASD) $\Lambda^-$ 2-forms. Any 2-form can be split into its self-dual and anti-self-dual parts $\Lambda^2=\Lambda^+\oplus\Lambda^-$ . Real 2-forms then satisfy the condition that their self- and anti-self-dual parts are the complex conjugates of each other. This decomposition of $\Lambda^2$ is related to the spinor story we reviewed above because $\Lambda^+$ can be shown to be isomorphic to $S_+^2$ , the second symmetric power of the fundamental representation of the first type. This is realised as the space of rank 2 spinors that are symmetric in their 2 spinor indices. Similarly $\Lambda^-\sim S_-^2$ . The operation of complex conjugation takes $S_+$ to $S_-$ , and so there are real elements in $S_+^2\oplus S_-^2$ . These are real 2-forms. It can also be checked that the operation of taking the mirror image, e.g. reflecting one of the spatial coordinates, interchanges the spaces $\Lambda^+$ and $\Lambda^-$ . In this sense the mirror reflection is the same as the complex conjugation.

Chiral description of gravity

The chiral miracle alluded to above is related to the decomposition of the Riemann curvature. In view of its symmetries, the Riemann curvature $R_{\mu\nu\rho\sigma}$ can be thought of as a symmetric $\Lambda^2\otimes \Lambda^2$ valued matrix, where $\Lambda^2$ is the space of 2-forms. In four dimensions we can decompose 2-forms into their self- and anti-self-dual parts $\Lambda^2 = \Lambda^+\oplus\Lambda^-$ , and so we have the decomposition of the Riemann curvature relative to the decomposition $\Lambda^2 = \Lambda^+\oplus\Lambda^-$

${\rm Riemann} = \left( \begin{array}{cc} A & B \\ B^T & C \end{array} \right).$

Here A,C are symmetric 3×3 matrices, that can be referred to as the SD-SD and ASD-ASD parts of the Riemann curvature. For real metrics of Lorentzian signature the matrix C is the complex conjugate of A. Similarly, for real Lorentzian metrics the matrix B is Hermitian. The Bianchi identity $R_{[\mu\nu\rho\sigma]}=0$ implies that the traces of A and C are equal. In particular, for real Lorentzian metrics this implies that the trace is real.

The above decomposition turns out to encode the irreducible (with respect to the action of the orthognal group) pieces of the Riemann curvature. The trace of A or C is the scalar curvature. The trace free parts of A and C are the SD and ASD parts of the Weyl curvature. The matrix B encodes the trace free part of the Ricci curvature.

Now comes the key point. The above decomposition of the Riemann implies that the Einstein condition $R_{\mu\nu}=\Lambda g_{\mu\nu}$ can be imposed by considering just a half of the Riemann curvature. Indeed, in view of the above decomposition the Einstein condition is equivalent to

$R_{\mu\nu}=\Lambda g_{\mu\nu} \qquad\Leftrightarrow \qquad B=0,\quad {\rm Tr}(A)=\Lambda.$

Thus, it is enough to have access to just the first row of the curvature matrix. One could of course use the second row equally well. In other words, to impose the four-dimensional Einstein condition, it is enough to have access to just half of the Riemann curvature. This is the chiral miracle occurring in four-dimensional GR.

It is sometimes erroneously thought that this phenomenon has to do with the complexification. Indeed, imposing some equation on a quantity that is complex is equivalent to imposing two real equations: one on the real and one on the imaginary part of the original quantity. However, this is not what happens here. The best way to see this is to go to the Riemannian, or all plus signature. Most of the things we have said above still apply, except now in the decomposition $\Lambda^2=\Lambda^+\oplus \Lambda^-$ the two factors are no longer complex conjugates of each other, both now being real (the Hodge star now squares to plus one, and so its eigenvalues are plus/minus unity, so the eigenspaces are real). In this case the matrices A,B,C in the curvature matrix are all real, and C is no longer related to A in any way, apart from the trace condition ${\rm Tr}(A)={\rm Tr}(C)$ that still follows from the Bianchi identity. The Einstein condition is still encoded in this case as above. This makes it very clear that the fact that the Einstein condition is encoded in half of the Riemann curvature has nothing to do with the complexification, as it still holds in the Riemannian signature case where all quantities are real.

The above story leads, after a few more details, to the following important theorem (from Atiyah-Hitchin-Singer Self-Duality in Four-Dimensional Riemannian Geometry):

Let X be a 4-manifold with an Einstein metric. Then the induced connection on the bundle $\Lambda^+$ of self-dual 2-forms is self-dual. Conversely, if the induced connection on $\Lambda^+$ is self-dual then the metric is Einstein.

Here a connection is called self-dual if its curvature is self-dual as a 2-form. The above theorem is the basis of what can be called chiral description(s) of the four dimensional GR. There are several versions of these, all related to the formulation first proposed by Jerzy Plebanski in 1977, i.e. around the same time that Atiyah-Hitchin-Singer theorem was formulated. More details on Plebanski chiral formulation can be found in K. Krasnov, Plebanski Formulation of General Relativity: A Practical Introduction. The main idea of these descriptions is that some chiral, i.e. not real in the case of Lorentzian signature, object is used for the description of geometry, instead of the real non-chiral metric on which the usual metric description is based. As can be anticipated from the fact that only half of the Riemann curvature is needed, these descriptions are more economical than the metric GR, which justifies interest in them.

Chiral modifications of GR

Related to the chiral miracle there is another miracle. While the first one is certainly known to differential geometers specialising in Einstein manifolds, the second one is almost completely unknown to the community. It is the fact that the four-dimensional Einstein condition can be non-trivially deformed in a chiral way.

It is well-known that GR can be modified, the simplest example of such a modification being the $R^2$ gravity, of relevance e.g. as a good model of inflation. This model is equivalent to GR coupled to an additional scalar field, and so propagates not just the two polarisations of the graviton as in GR, but also a scalar. One can consider more involved modifications of the GR Lagrangian with higher powers of the curvature added. One can quickly convince oneself that because of the higher derivatives present in these modified Lagrangians they all propagate more degrees of freedom (DOF) than GR. And if one insists that the action leads to second order field equations then GR is the unique theory, at least in four dimensions. It thus seems impossible to modify GR without adding extra propagating DOF to it. This is the content of several GR uniqueness theorems available in the literature.

It then comes as a shock that it is possible to modify GR without adding extra DOF if one starts from one of its chiral descriptions. One of the simplest descriptions of these chiral modified gravity theories is in the framework of the so-called connection formulation: K Krasnov Gravity as a diffeomorphism invariant gauge theory. The resulting chiral modified gravity theories continue to have second order field equations. A non-linear count of the number of degrees of freedom by the Hamiltonian analysis shows that they have the same number of propagating degrees of freedom as GR – two propagating polarisation of the graviton. And, as it turns out, there is an infinite-parametric class of such chiral modified gravity theories, in which GR is just a special member.

Once GR gets embedded into an infinitely large class of gravity theories all with similar properties, one is forced to ask the questions: What makes GR unique as compared to all other theories? Why is the real world described by GR? Is the real world indeed described by GR, or may be this is only an approximate truth? To put it differently, the very fact that these chiral modified gravity theories exist, makes one obliged to understand them.

The inevitability of these questions is strengthened by the fact that the chiral modifications of gravity can be argued to be unique. One can take some of the proofs of the GR uniqueness, notably the more modern proofs that deal with the scattering amplitudes, and see what assumption in that proof is violated by these chiral theories. Removing this assumption one can see that there results a new “uniqueness” theorem that states that these chiral modifications are the only ones that describe propagating gravitons with second order field equations. See K. Krasnov, GR uniqueness and deformations for an argument of this sort. So, in a certain precise sense, the class of chiral modified four-dimensional gravity theories is unique. This forces us to understand these theories.

Summary

Analysing the decomposition of the Riemann tensor into its chiral components one finds that only half of the Riemann curvature is needed to impose the Einstein condition. This is the basis for many chiral formulations of GR available in the literature. The price one pays for the simplicity of these formulations is that they require one working with complex-valued objects (if one is interested in describing metrics of Lorentzian signature). But economy and simplicity that comes with these formulations is certainly worth the price. Also, if one is interested in metrics of Riemannian signature (e.g. gravitational instantons), then chiral objects are real, and the chiral description is undeniably superior to that in terms of the metric, as is well-known to differential geometers specialising in Einstein 4-manifolds.

Once GR is formulated in a chiral way, it turns out to be possible to deform the Einstein condition. There is an infinite-parametric class of such chiral modifications, or “deformations”. GR becomes just a special point in this infinitely large space of gravity theories with similar properties. My recent CQG paper analyses how the Black Hole thermodynamics gets changed (surprisingly little) in these chiral modified gravity theories.

Read the full article in Classical and Quantum Gravity:
Deformations of GR and BH thermodynamics
K Krasnov
K Krasnov Class.Quant.Grav. 33 155012

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