Metric structures in General Relativity

Reference ¹ contains the following statement:

“Ashtekar’s formulation of general relativity ² taught us to think of gravitational theories as theories of connections, on a bare manifold with no metric structure. […] The idea that general relativity has its deepest formulation as a connection theory suggested immediately a new approach to the unification of general relativity with Yang-Mills theories. The group of the connection in the connection formulations of general relativity is the local Lorentz group SO(3, 1) or a subgroup of it.”

The statement quoted above suggests that it is possible to formulate General Relativity as a theory of SO(3,1)-connections in such a way that it is not necessary to endow the space-time manifold with any metric structure at all, or, in Professor Smolin’s words, that it is possible to formulate General Relativity on a “bare manifold with no metric structure”. In this note we will explain in basic terms that this is not possible, since any “theory of SO(3,1)-connections” implicitly requires the choice of an underlying metric structure. Professor Smolin’s quote will serve us as an excuse to discuss some interesting facts on the rich theory of connections on vector bundles and principal bundles, of fundamental importance in differential geometry and theoretical physics. At the mathematical level, the result follows just from the well-known correspondence between Lorentzian metrics on a smooth d-dimensional manifold and reductions of its frame bundle to an O(d-1,1) subbundle together with the correspondence between the associated metric compatible and torsion free connections. For definiteness we will work in four-dimensions and we will not consider the manifold to be oriented, since this requirement is not necessary for the argument that we will present to apply. We start by recalling some basic facts on the formulation of General Relativity in vacuum.

Let M be a four-manifold and let Lor(M) denote the space of all Lorentzian metrics on M. Interestingly enough, Lor(M) can be empty, as Lorentzian metrics are obstructed if M is closed and its Euler characteristic does not vanish. In fact, in this case, the Euler characteristic of M vanishes if and only if M admits Lorentzian metrics. If non-empty, Lor(M) may be a very large space: intuitively speaking we can say that it is infinite-dimensional, but making this notion mathematically precise, in those cases where it can be done, requires some work. A Lorentzian metric structure on M is simply a choice of Lorentzian metric g in Lor(M). The pair (M,g) is then called a Lorentzian manifold.

Every manifold can be endowed with an infinite number of Koszul connections (also called covariant derivatives) on its tangent bundle. In fact, the space of all Koszul connections on a given manifold is an affine space modelled on the space of one-forms taking values on the endomorphism bundle of the tangent bundle. This is a contractible infinite-dimensional space. On a Lorentzian manifold (M,g) there is a preferred choice of Koszul connection, which is given by the unique torsion-free Koszul connection which is compatible with the metric g, namely with the metric structure of (M,g). This is the so-called Levi-Civita (LC) connection of (M,g), and it is the basic object in which the formulation of General Relativity is based. Note that since the LC connection is metric-compatible, its holonomy is contained in O(3,1). Let M be a four-manifold. The vacuum Einstein equations are:

1. Ricc(g) = 0

where Ricc(g) denotes the Ricci curvature of the LC connection associated to g. This turns out to be a second-order hyperbolic system of partial differential equations defined over M on the space of Lorentzian metrics of M, that is, the space of Lorentzian metric structures on M. A solution to equation (1) is hence a particular Lorentzian structure on M. In many practical situations, one starts by finding a local metric g_U by solving equation (1) on a contractible open set U of R⁴. One then proceeds to geodesically extend (U,g_U) as much as possible along time-like/null geodesics, obtaining a possibly geodesically incomplete and topologically non-trivial Lorentzian manifold (M,g). Notice that this geodesic extension is a highly complicated and non-unique process which has to be performed on a case by case basis. Either way, from the previous remarks it is clear that the formulation of GR crucially depends on equipping M with a metric structure, being particular solutions particular metric structures on M.

The discussion above concerns the standard formulation of GR in the language of vector bundles (recall that the tangent bundle of a manifold is a particular type of vector bundle) and Koszul connections. However, one can consider instead the equivalent but seemingly different set up of principal bundles and Ehresmann connections by using the well-known correspondence between Lorentzian vector bundles and principal O(d-1,1) bundles as well as the correspondence between metric-compatible Koszul connections in the former and Ehresmann connections in the latter. The formulation of General Relativity in this set up goes as follows. Let M be a four-manifold and let us denote by F(M) its frame bundle, which is a Gl(4,R) principal bundle over M. Let O(M) denote a reduction of F(M) to a O(3,1) principal bundle, which, unsurprisingly enough is obstructed by the Euler characteristic of M. The principal subbundle O(M) is a particular case of what is known in the literature as a G-structure, which consists of a principal subbundle of F(M) with structure group G. The concept of G-structure is of fundamental importance in differential geometry. Prominent examples of G-structures in Riemannian geometry are G₂ structures in seven dimensions, Spin(7) structures in eight dimensions or Calabi-Yau structures in even dimensions.

Let us assume that O(M) exists and let us consider an Ehresmann connection H on F(M). Ehresmann connections can be defined on arbitrary smooth principal bundles. In the case in which the principal bundle is the bundle of linear frames of a manifold an Ehresmann connection is also called a linear connection. An Ehresmann connection can be understood as an invariant horizontal subbundle of the tangent bundle of F(M) or, equivalently, as a one form in F(M) (not M!) taking values in gl(4,R), the Lie algebra of Gl(4,R). Given a reduction O(M) of F(M), there exists one and only one Ereshmann connection H on F(M) such that it descends to a well-defined torsion-free Ereshmann connection on O(M). This in particular implies that H is a horizontal vector subbundle of O(M), the tangent bundle of O(M). The concept of torsion can be defined in this context in terms of the solder form which exists in O(M). With this provisos in mind, the vaccum equations of General Relativity for H are given by:

2. Tr(F(H)) = 0

Where Tr(F(H)) denotes the appropriate trace of the curvature of the linear connection H. The curvature F(H) can be understood in this context as a O(3,1)-equivariant function defined on O(M) and taking values in the symmetric tensor product of the space of two-forms on R⁴. We have introduced an Ehresmann connection as a particular type of horizontal subbundle of the tangent space of O(M). An equivalent description of an Ehresmann connection H on O(M) is given by a one form A in the total space O(M) taking values in o(3,1), the Lie algebra of O(3,1). Let U ⸦ M be a contractible open subset of M. We know that O(M) is trivial over U and hence there exists a trivializing section s:U -> O(M) which we can use to locally pull back A, which is defined on O(M), to U. Locally then, A becomes a one form A_U in the space-time taking values in the Lie algebra o(3,1). This is the object that physicists usually use in the local formulation their physical theories, like General Relativity and Yang Mills theories, and it is indeed the object appearing in Professor Smolin’s paper. The equations of motion of General Relativity can be locally written exclusively in terms of the o(3,1)-valued one-form A_U. At first sight it would seem that the object A_U does not involve the choice of any Lorentzian metric on M, or U for that matter, a fact which may lead us to misleadingly conclude that by using A_U General Relativity can be formulated independently of any metric structure on M or U. However, this is not the case. The reason is that A is an object defined on O(M), and O(M) is an O(3,1) reduction of F(M), which is equivalent to a choice of a Lorentzian metric g on M. The correspondence can be described in very simple terms. Given a Lorentzian manifold (M,g), one can construct the associated principal O(3,1) bundle O(M) as the principal bundle whose fiber at a point p in M is given by all ordered basis of T_pM which are orthonormal with respect to g_p. On the other hand, given O(M), we can understand every point u in O(M) in the fiber above p as a linear isomorphism between R⁴ and T_pM. The corresponding Lorentzian metric g on M is given, at the point p in M, by taking the pull-back by u^-1 of the standard Minkowski metric eta on R⁴. The invariance of eta under O(3,1)transformations implies that g at p is independent of the element u in the fiber of O(M) over p and hence it implies that g is well defined.

In addition, through the previous correspondence, the torsion-free Ehresmann connection H on O(M) and the Levi-Civita Koszul connection associated to g on M are mapped to each other through a relation that also maps equations (1) and (2) to each other. Therefore, even if locally working with A_U may hide the fact that there is an underlying Lorentzian metric structure in M, both set ups are completely equivalent and hence it is not possible to formulate General Relativity on a “bare manifold with no metric structure” by locally working with A_U. One might try to avoid using O(M) by considering Ehresmann connections H on F(M) with holonomy equal or contained in O(3,1). This way, it would seem that there is no need to define the bundle O(M) and hence no need to use a Lorentzian structure on M to pose equation (2). However, even in this case it can be seen that, given such Ehresmann connection H with holonomy O(3,1), one can construct an associated principal O(3,1) bundle O(M), the associated holonomy bundle, to which H reduces and which defines a compatible Lorentzian structure on M.