• 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.


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


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,064 questions , 2,215 unanswered
5,347 answers , 22,741 comments
1,470 users with positive rep
818 active unimported users
More ...

  Why has the trace of the energy-momentum tensor to vanish for conserved scaling currents to exist?

+ 3 like - 0 dislike

In this paper, the authors say that the trace of the energy-momentum tensor has to vanish to allow for the existence of conserved dilatation or scaling currents, as defined on p 10, Eq(22)

$$ \Theta^{\mu} = x_{\nu} \Theta^{\mu\nu} + \Sigma^\mu.$$

$\Theta^{\mu\nu}$ is the energy-momentum tensor and $\Sigma^\mu = d_{\phi}\pi_i^{\mu}\phi^i$ is the internal part.

This fact is just mentioned and I dont understand why this has to be the case, so can somebody explain it to me?

asked Oct 19, 2012 in Theoretical Physics by Dilaton (6,240 points) [ revision history ]
A related question by OP: physics.stackexchange.com/q/35693/2451

This post imported from StackExchange Physics at 2014-03-12 15:45 (UCT), posted by SE-user Qmechanic

1 Answer

+ 4 like - 0 dislike

Without the internal part:

The divergence $$\nabla_\mu (x_\nu \Theta^{\mu\nu}) = x_\nu \nabla_\nu \Theta^{\mu\nu} + \frac12(\nabla_\mu x_\nu + \nabla_\nu x_\mu) \Theta^{\mu\nu} $$ where I used that $\Theta^{\mu\nu}$ is symmetric. Recalling that the energy-momentum tensor is divergence free, the first term drops out. Assuming that $x^\nu$ generates a dilation/scaling symmetry (and not a bona fide symmetry), we know that its deformation $$ \nabla_\mu x_\nu + \nabla_\nu x_\mu \propto \mathcal{L}_x g_{\mu\nu} \propto g_{\mu\nu} $$ where $\mathcal{L}$ is the Lie derivative. (In the case $x^\nu$ generates a symmetry the term vanishes from Killing's equation.)

Hence in this case for the current to be conserved (that is, divergence free), we need that $g_{\mu\nu} \Theta^{\mu\nu} = 0$; that is, the energy momentum tensor is tracefree.

This post imported from StackExchange Physics at 2014-03-12 15:45 (UCT), posted by SE-user Willie Wong
answered Oct 19, 2012 by Willie Wong (580 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:
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