Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Why is it so coincident that Palatini variation of Einstein-Hilbert action will obtain an equation that connection is Levi-Civita connection?

+ 6 like - 0 dislike
3180 views

There are two ways to do the variation of Einstein-Hilbert action.

First one is Einstein formalism which takes only metric independent. After variation of action, we get the Einstein field equation. Second one is Palatini formalism which takes metric and connection are independent. After variation, we get two equations, first is field equation and second is that connection is Levi-Civita connection.

So my question is why it is so coincident that Palatini variation of Einstein-Hilbert action will obtain an equation that connection is Levi-Civita connection and Palatini formalism coincides with Einstein formalism? While for $f(R)$ action they are generally different. Are there some deeper mathematical or physical structures of Einstein-Hilbert action which can account for it.


This post imported from StackExchange Physics at 2014-10-23 07:31 (UTC), posted by SE-user user34669

asked Oct 15, 2014 in Theoretical Physics by user34669 (205 points) [ revision history ]
edited Oct 23, 2014 by Dilaton

1 Answer

+ 2 like - 0 dislike

I) In Palatini $f(R)$ gravity, the Lagrangian density is

$$ {\cal L}~=~ \sqrt{-g} f(R), $$

with $$R~:=~ g^{\mu\nu} R_{\mu\nu}(\Gamma),$$

and where $\Gamma^{\lambda}_{\mu\nu}=\Gamma^{\lambda}_{\nu\mu}$ is an arbitrary torsionfree$^1$ connection.

II) As OP mentions, the word Palatini refers to that the metric $g_{\mu\nu}$ and the connection $\Gamma^{\lambda}_{\mu\nu}$ are independent variables. We therefore get two types of EL equations:

  1. The EL equations for the metric $g_{\mu\nu}$ are the generalization of EFE.

  2. The EL equations for the connection $\Gamma^{\lambda}_{\mu\nu}$ turns out to be the metric compatibility condition for a second metric defined as $$ \hat{g}_{\mu\nu}~:=~f^{\prime}(R) g_{\mu\nu}. $$ In other words, the classical solution for $\Gamma^{\lambda}_{\mu\nu}$ is the Levi-Civita connection for the second metric $\hat{g}_{\mu\nu}$.

III) So Einstein gravity (GR) with a possible cosmological constant

$$ f(R)~=~R-2\Lambda, $$

or equivalently

$$ f^{\prime}(R)~=~1,$$

corresponds to the special case where the two metrics $g_{\mu\nu}$ and $\hat{g}_{\mu\nu}$ coincide, and hence $\Gamma^{\lambda}_{\mu\nu}$ becomes the Levi-Civita connection for $g_{\mu\nu}$.

--

$^1$ One could allow a non-dynamical torsion piece, but we will not pursuit this here for simplicity. For more on torsion, see e.g. also this Phys.SE post.

This post imported from StackExchange Physics at 2014-10-23 07:31 (UTC), posted by SE-user Qmechanic
answered Oct 16, 2014 by Qmechanic (3,120 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOve$\varnothing$flow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...