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

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

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,853 answers , 20,624 comments
1,470 users with positive rep
501 active unimported users
More ...

Derivation of the Ernst Lagrangian

+ 2 like - 0 dislike
92 views

In 1968 [1], Ernst derived his famous equation for a single complex variable (which acts as a potential for the metric components) for the Einstein equations. The Lagrangian he writes down, just above equation (4) in [1], seems to be very non-trivial to derive. I had thought it was just the Einstein-Hilbert Lagrangian $\sqrt{-g} R$, but it is not since $R$ contains second order derivatives which do not appear in Ernst's Lagrangian. However, they are suspiciously similar (the terms involving $\omega$ are exactly the same).

Can someone please explain how the derivation goes? Is it the Einstein-Hilbert Lagrangian but modified by using the actual field equations themselves to remove some terms? Or do some other tricky manipulations take place?

Given Ernst's Lagrangian I can derive the rest of his equations quite nicely, following through the Euler-Lagrange approach. It is just the actual Lagrangian itself I cannot reproduce!

Thanks

[1] Ernst, 1968, "New formulation of the axially symmetric gravitational field problem"

asked Dec 14, 2015 in Theoretical Physics by anonymous [ revision history ]
reshown Dec 15, 2015

I would say the Lagrangian was simply guessed/backwards-engineered.

I was afraid of that, Void -- that would mean there is no simple way to go about trying to generalise the results to other cases (some simple perfect fluids or AdS backgrounds or something). It is just the uncanny resemblance with $\sqrt{-g} R$ that made me think they are related; I tried integrating by parts out the second order terms -- didn't seem to help.
 

It could be possible to derive the Ernst Lagrangian by subtracting a possibly non-tensorial four-divergence from $R\sqrt{-g}$. The first thing to try would be to find out whether the difference between them is really a four-divergence.

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$\hbar$ys$\varnothing$csOverflow
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).
To avoid this verification in future, please log in or register.




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

Your rights
...