Benefits & Disadvantages of Palatini Action for General Relativity

[ohwilleke]

Most often, general relativity is formulated in terms of Einstein's field equations:

whose terms are familiar to readers in this forum.

But, I understand (and feel free to correct me or qualify my statement if I am incorrect) that it is also possible to describe general relativity with an equivalent expression known as the Palatini action sometimes written as:

where

but now

is a function of the frame field.

The Palatini action is used, for example, in a recent pre-print by SangChul Yoon, "Lagrangian formulation of the Palatini action" (May 5, 2018).

What are the insights or mathematical benefits that are involved in using the Palatini action? What are the disadvantages to using this formulation of general relativity?

 

[TeethWhitener]

This review:
https://arxiv.org/abs/1106.2476
provides decent insight into the Palatini procedure. This is not my area of expertise (I am a total GR amateur), but as far as I can tell, the main upshot is that the Palatini procedure allows you to treat the metric and the connection as independent variables (in standard GR, the Levi-Civita connection is a function of the metric and its derivatives). But if you assume zero torsion, once you minimize the action integral with respect to the connection, you get back the Levi-Civita connection and hence general relativity. One interesting complication that the review mentions is that this procedure only assumes coupling of matter to the metric, not to the connection. If matter couples separately to the metric and the connection, things apparently get quite a bit more complicated.

 

[ohwilleke]

This review:
https://arxiv.org/abs/1106.2476
provides decent insight into the Palatini procedure. This is not my area of expertise (I am a total GR amateur), but as far as I can tell, the main upshot is that the Palatini procedure allows you to treat the metric and the connection as independent variables (in standard GR, the Levi-Civita connection is a function of the metric and its derivatives). But if you assume zero torsion, once you minimize the action integral with respect to the connection, you get back the Levi-Civita connection and hence general relativity. One interesting complication that the review mentions is that this procedure only assumes coupling of matter to the metric, not to the connection. If matter couples separately to the metric and the connection, things apparently get quite a bit more complicated.

Thanks. That's really helpful, especially your discussion of that last complication.