Reference for mathematical Palatini formalism of general relativity

Question

I know that this is maybe not a research level question, but since the topic is quite special, I thought that the chance to get some reference is higher in this community.

I am looking for a reference (book, review or research paper) of the Palatini or tetradic formulation of general relativity from a more mathematical point of view. Up to now, I was only able to find some short physical introductions. So I would like to know if someone knows some more extensive discussion, maybe also from a mathematical point of view, i.e. by discussing frame bundles and the (mathematical) gauge theory point of view.

Answer 1

The Palatini formalism, a variation of a Lagrangian with respect to the connection, is examined quite rigorously in

• On the Palatini method of variation (1978)
• The Palatini formulation of general relativity (2015)
• On the Palatini variation and connection theories of gravity (1998)
  _{The latter reference, in particular, aims for "a rigorous treatment of the Palatini Variational Principle of gravitational actions in an attempt to fully understand the role of the connection in such theories. Following a brief geometrical review, which highlights some elements of fibre bundle theory appropriate to our later analysis, we examine the secalled Palatini Tetrad formalism and show that it must be modified for a proper Palatini variation - i.e. to not assume anything a priori about the relevant connection."}

The history of the topic is discussed in
Variational Formulation of General Relativity from 1915 to 1925: "Palatini's Method" Discovered by Einstein in 1925
_{Among the three basic variational approaches to general relativity, the metric-affine variational principle, according to which the metric and the affine connection are varied independently, is commonly known as the “Palatini method.” In this paper we revisit the history of the “golden age” of general relativity, through a discussion of the papers involving a variational formulation of the field problem. In particular we find that the original Palatini paper of 1919 was rather far from what is usually meant by “Palatini's method,” which was instead formulated, to our knowledge, by Einstein in 1925.}

Answer 2

There is a quite detailed pedagogical presentation of both the Einstein-Hilbert and the Palatini variational principles for the Einstein equations in §III.3 Lagrangians for General Relativity of

Baez, John; Muniain, Javier P., Gauge fields, knots and gravity, Series on Knots and Everything. 4. Singapore: World Scientific. xii, 465 p. (1994). ZBL0843.57001.