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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]. The reason that I am asking about stress-energy is that I know that I can figure out how $\rho$ and $P$ scale if I know how the stress-energy scales. If I am right, $\rho$ would scale in the same way as the trace of the stress-energy tensor, correct? 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.

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.

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]. The reason that I am asking about stress-energy is that I know that I can figure out how $\rho$ and $P$ scale if I know how the stress-energy scales. If I am right, $\rho$ would scale in the same way as the trace of the stress-energy tensor, correct?

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.

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]. The reason that I am asking about stress-energy is that I know that I can figure out how $\rho$ and $P$ scale if I know how the stress-energy scales. If I am right, $\rho$ would scale in the same way as the trace of the stress-energy tensor, correct? 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.

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. 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. The reason that I am asking about stress-energy is that I know that I can figure out how $\rho$ and $P$ scale if I know how the stress-energy scales. If I am right, $\rho$ would scale in the same way as the trace of the stress-energy tensor, correct?

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.

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]. The reason that I am asking about stress-energy is that I know that I can figure out how $\rho$ and $P$ scale if I know how the stress-energy scales. If I am right, $\rho$ would scale in the same way as the trace of the stress-energy tensor, correct?

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.