Can the Einstein equations be understood as transport equations for the Christoffel symbols?

Question

The Einstein equations can be written as (here I am following the notation of Wald, Robert, General Relativity, Chicago, Chicago University Press, 1985)

\begin{equation} \partial_{\alpha}\Gamma^{\alpha}_{\mu\nu} - \partial_{\mu}\Gamma^{\alpha}_{\nu\alpha} + \Gamma^{\alpha}_{\mu\nu}\Gamma^{\beta}_{\alpha\beta} - \Gamma^{\alpha}_{\nu\beta}\Gamma^{\beta}_{\alpha\mu} = T_{\mu\nu} - \frac{1}{2}g_{\mu\nu}g^{\alpha\beta}T_{\alpha\beta} \end{equation}

(the LHS of this equation is the Ricci tensor; e.g. Eq. (3.4.5) in Wald's book. The RHS is the trace reversed stress-energy tensor).

We can expand out the Christoffel symbols in terms of the metric $g_{\mu\nu}$; then the Einstein equations become a set of second order partial differential equations for the metric. With a good choice of coordinates (such as harmonic coordinates, $\Gamma^{\alpha}_{\mu\nu}g^{\mu\nu}=0$), one can show that the Einstein equations can be thought of as a set of second order hyperbolic (wave) equations for each metric component (e.g. Eq. 10.2.33 in Wald's book).

My question is: can one think of the Einstein equations as a set of transport equations for the Christoffel symbols? Has there been any work that has looked at the initial value problem for the Einstein equations as I wrote them above, as a set of partial differential equations for the Christoffel symbols?

EDIT: To be more precise, I'd like to know if there has been work on the initial value formulation of the Einstein equations as a set of transport equations for the Christoffel symbols. For example, something akin to the Newman-Penrose formalism (in that formalism one rewrites the Einstein equations as a set of transport equations for "spin coefficients"; I'd like something similar for the Christoffel symbols).

Answer 1

In Yang-Mills theories the gauge fields $A_i^a$ are connections on an internal space and it is those variables that are dynamical in the canonical approaches, subject to gauge redundancy, etc. There have been, I'm sure, many attempts to reformulate gravity in terms of connection fields (perhaps Weyl did this?), I recall that Kijowski and Ferraris had such a paper and the formulation of GR based on the so-called Ashtekar variables includes the three-dimensional Levi-Civita connection as part of the canonical variables. The latter is a basic building block of the loop approach to quantum gravity.

In short: indeed, the notion you put forth of using the connections as the basic building blocks of GR is well founded!

Answer 2

I believe that the viewpoint that OP is interested in is precisely the first order Palatini formulation of general relativity. In this approach the metric $g$ and connection $Γ$ are both considered independent variables.

We start with the standard Einstein–Hilbert action for gravity, which we could write as this: $$ I= \frac{1}{16 \pi G} \int d^4x \sqrt{|g|} g^{\mu\nu}R_{\mu\nu}(\Gamma), $$ with $R_{\mu\nu}(\Gamma)$ now considered only as a function of (affine) connection $\Gamma$: $$ R_{μν}(Γ ) = ∂_λ Γ^λ_{μν} - ∂_ν Γ^λ_{μλ} + Γ^λ_{μν}Γ^σ_{λσ} -Γ^λ_{μσ} Γ^σ_{νλ}. $$

Variation of the action now requires that $$ \frac{1}{16 \pi G} \int d^4x\,\, \delta\!\left[ \sqrt{|g|} g^{\mu\nu}R_{\mu\nu}(\Gamma)\right] = 0. $$

By varying the metric one obtains the usual Einstein field equations: $$ R_{μν} - \frac12 g_{μν} R = 0 . $$ But in order to establish the usual relationship between metric and connection we would need additional equations. These can be obtained by varying the action with respect to $\Gamma$. In order to do that we can use the Palatini identity: $$ δ R^λ_{μνσ} = δ Γ^λ_{μσ;ν} − δΓ^λ_{μν;σ}\,. $$ Then performing integration by parts we can show that the variation of connection $\Gamma$ implies that: $$ ∂_λ g_{μν} − g_{νσ}Γ_{μλ}^σ − g_{μσ} Γ_{νλ}^σ = 0 . $$

This last equation (it could also be written as $g_{μν;λ}=0$) can be solved to give the usual expression for $Γ$ through derivatives of the metric. As a result, the we have two sets of first order PDE for $g$ and $Γ$ that are equivalent to the standard formulation of GR.

Further discussion including the formulation of initial value problem see the ADM paper:

• Arnowitt, R., Deser, S., & Misner, C. W. (2008). Republication of: The dynamics of general relativity. General Relativity and Gravitation, 40(9), 1997-2027, doi:10.1007/s10714-008-0661-1, arXiv:gr-qc/0405109.

This is a recent republication of a very influential 1962 paper, that was included in a book Gravitation: an Introduction to Current Research, ed. L. Witten. For links to a more recent literature see the accompanying editorial note: doi:10.1007/s10714-008-0649-x.