# How does the conformal scaling behavior of the stress-energy tensor depend on the spacetime dimension?

I'm new to Weyl transformations and am struggling to find online the answer to what should be a simple question. Consider an $$n=D+1$$ dimensional spacetime with metric $$g_{ab}$$ and some stress-energy tensor $$T_{ab}$$ (e.g., from the variation of some action, say of a perfect fluid). If I perform a Weyl transformation $$g_{ab}\mapsto\bar{g}_{ab}=\Omega^2 g_{ab}$$ then what happens to the stress-energy tensor? In particular, what is its scaling behavior, $$\Delta$$?

$$T_{ab}\mapsto\bar{T}_{ab}=\Omega^{-\Delta} T_{ab}$$

Specifically, how does this depend on the spacetime's dimensions $$n=D+1$$? I have found the answer online in the $$n=4$$ case in several books (edit: For instance, with raised indices, we have $$T^{ab}$$ having $$\Delta=6$$ in Eq.(2.14) of [ref]). I have found some discussion of the issue in a generic dimension here but the discussion there is inconsistent with itself. I know that the scaling behavior is different if the indices are raised or lowered, e.g., $$T_{ab}$$ vs $$T_{a}{}^b$$ vs $$T^{ab}$$ But I don't think the discussion I found distinguishes these properly.

The hyper-specific question which I am looking for an answer to is as follow: What is the scaling behavior of the rest mass density distribution, $$\rho$$, which would go into the stress-energy of a perfect fluid, $$T_{ab}=(P+\rho) u_a u_b + g_{ab} P$$ I am particularly interested in the $$n=1+1$$ case generalizing Eq.(2.21) of [ref]. Would rest charge density transform any differently?

Note: I am interested in this question outside of the context of CFT. Please do not assume that the theory that I am working with is conformally invariant or that the stress-energy is traceless.

Edit: I have replaced "conformal weight" with "scaling behavior" everywhere. The way that both I and this post are defining $$\Delta$$ it does depend on how the indices are raised or lowered.

[ref] Beyond Einstein Gravity: A Survey of Gravitational Theories for Cosmology by Valerio Faraoni and Salvatore Capozziello.

Under Weyl transformations, the partition function of a CFT transforms as $$Z[\Omega^2g] = e^{-S_{an}[\Omega,g]} Z[g]$$ where $$S_{an}[\Omega,g]$$ is the anomaly action. This is non-vanishing only in even dimensions.
The stress tensor in a CFT is defined as (up to some factor of $$i$$) $$\langle T_{\mu\nu}(x) \rangle_g \equiv \frac{2}{\sqrt{-\det g(x)}} \frac{\delta }{\delta g^{\mu\nu}(x)} \ln Z[g]$$ It then follows that \begin{align} \langle T_{\mu\nu}(x) \rangle_{\Omega^2 g} &= \frac{2}{\sqrt{-\det [ \Omega^2(x) g(x) ] }} \frac{\delta }{\delta [ \Omega(x)^{-2} g^{\mu\nu}(x) ] } \ln Z[\Omega^2 g] \\ &= \Omega(x)^{-n+2} \frac{2}{\sqrt{-\det g(x)}} \frac{\delta }{\delta g^{\mu\nu}(x) } \ln \left( e^{-S_{an}[\Omega,g]} Z[g] \right) \\ &= \Omega(x)^{-n+2} \langle T_{\mu\nu}(x) \rangle_g - \Omega(x)^{-n+2} \frac{2}{\sqrt{-\det g(x)}} \frac{\delta S_{an}[\Omega,g] }{\delta g^{\mu\nu}(x) } \end{align} We now evaluate this in flat spacetime, in which case, the anomaly term vanishes. Setting $$g_{\mu\nu} = \eta_{\mu\nu}$$, we find \begin{align} \langle T_{\mu\nu}(x) \rangle_{\Omega^2 \eta} &= \Omega(x)^{-n+2} \langle T_{\mu\nu}(x) \rangle_\eta \tag{1} \end{align} Noting that the stress tensor has spin $$s=2$$, we can read off its scaling dimension as $$\Delta = n$$.
PS - For a primary operator of scaling dimension $$\Delta$$ and spin $$s$$, we have $$\langle O(x) \cdots \rangle_{\Omega^2 g} = \Omega(x)^{-\Delta+s} \langle O(x) \cdots \rangle_{g}$$
• Thanks for your answer. I don't need all of the CFT trappings so maybe I can simplify your reply. Ignoring your $S_{an}[\Omega,g]$ and replacing $Z[g]$ with the matter Lagrangian, $\sqrt{-g} \ \mathcal{L}_m[g]$, we have the usual definition of $T_{\mu\nu}$, correct? Now in these terms, I think that your computation goes through so long as we have that $\sqrt{-g} \ \mathcal{L}_m[g]$ doesn't change under the Weyl transformation. Is this a usual condition for Lagrangians to obey outside of CFT? Is it obeyed for a perfect fluid? If so then $T_{\mu\nu}$ has this scaling behavior very generally. Commented Apr 16 at 17:40
• Yes, by definition, classical Weyl invariance is equivalent to the statement that $\sqrt{-g} {\cal L}[g]$ is invariant under $g \to \Omega^2 g$. Commented Apr 16 at 18:51
• I don't think you have understood the end of my comment. Is this condition often obeyed outside of CFT? For instance, see Eq.(2.14) of the [ref] from my post. They come to the same result as we have but without any CFT assumption (as far as I can tell). They then go on to apply this $T_{\mu\nu}$ scaling to a perfect fluid in order to determine the scaling behavior of the mass density, $\rho$. But I don't think the action of a perfect fluid is Weyl invariant... All I want to know is the scaling behavior of the mass (and charge) densities for a perfect fluid outside of a CFT context. Commented Apr 16 at 19:15
• If you only want to talk about scale invariance, then the other answer on this post answers it for you, because you only need to look at the engineering dimension of various quantities. Strictly speaking, $\Delta$ simply has no meaning whatsoever outside of CFTs. Commented Apr 17 at 9:20