# Metric vs coframe energy-momentum tensor in metric-affine gravity

Conventions

• Latin indices represent components in the anholonomic frame and greek ones are for coordinate components.
• I will call $R_{\mu \nu} := R_{\mu \rho \nu}{}^{\rho}$ (Ricci tensor) and $\bar{R}_{\mu}{}^{\nu} := g^{\rho \lambda} R_{\mu \rho \lambda}{}^{\nu}$. They are independent because the connection is not metric-compatible and, consequently, $R_{\mu \nu (\rho \lambda)} \neq 0$.

In a general metric-affine framework, one could vary the metric $g_{a b}$ and the coframe $e^{a}{}_{\mu}$ independently (e.g. see [1]). Additionally we have the components of the connection 1-form $\omega_{\mu a}{}^b$ and some matter fields $\varPsi$, but they are not relevant in my problem.

For example, consider the action:

\begin{align} S [e, g, \omega, \varPsi] & = \dfrac{1}{2\kappa} \int R(g, e,\omega) ~~\sqrt{|g|} ~ d^Dx ~~ + ~~ S_{matter} [e, g, \omega, \varPsi] \\ & = \dfrac{1}{2\kappa} \int g^{a b} e_c{}^\nu e_a{}^\mu R_{\mu \nu b}{}^c (\omega) ~~\sqrt{|g|} ~ d^Dx ~~ + ~~ S_{matter} [e, g, \omega, \varPsi] \end{align}

where $g := \det(g_{\mu \nu})$, and the independent connection is completely general (there are torsion and non-metricity).

The variation with respect to the coframe and the metric is:

\begin{align} 2\kappa\delta S & = \int \left[ R_{(a b)}-\dfrac{1}{2} g_{a b}R + \kappa T^{(g)}{}_{a b} \right] \delta g^{a b} ~~\sqrt{|g|} ~ d^Dx \\ & + \int 2\left[ \dfrac{1}{2} \left( R_{a b}- \bar{R}_{a b}\right) -\dfrac{1}{2} g_{a b}R + \kappa T^{(e)}{}_{a b} \right] g^{cb} e_{c}{}^{\mu} \delta e^{a}{}_{\mu} ~~\sqrt{|g|} ~ d^Dx \end{align}

We have taken into account that $\sqrt{|g|} = |\det(e^{a}{}_{\mu}) |\sqrt{|\det(g_{a b})|}$ and the definitions:

\begin{align} T^{(g)}{}_{a b} &:= \dfrac{2}{\sqrt{|g|}} \dfrac{\delta S_{matter}}{\delta g^{a b}}\\ T^{(e)}{}_{a}{}^{\mu} &:= \dfrac{1}{\sqrt{|g|}} \dfrac{\delta S_{matter}}{\delta e^{a}{}_{\mu}} \end{align}

Finally the equations of motion are:

\begin{align} R_{(a b)}-\dfrac{1}{2} g_{a b}R &=- \kappa T^{(g)}{}_{a b} \\ \dfrac{1}{2} \left( R_{a b}- \bar{R}_{a b}\right) -\dfrac{1}{2} g_{a b}R &=- \kappa T^{(e)}{}_{a b} \end{align}

I would like to study perfect fluids with non-vanishing hypermomentum (i.e. $0 \neq \delta S_{matter} / \delta \omega_{\mu a}{}^b$), and I need to postulate an energy momentum tensor like the typical one for fluids:

$$(\rho + P) u_a u_b - P g_{a b} \tag{1}$$

but, (Question 1) does this object corresponds to $T^{(g)}{}_{a b}$ or $T^{(e)}{}_{a b}$?

In metric affine theories with minimally coupled matter (e.g. a Dirac field) I know $T^{(e)}{}_{a b}$ is interpreted as the generalization of the canonical energy-momentum tensor (~ energy and momentum given by the Noether's theorem in theories with translational symmetry).

I do not know if there is a lagrangian formulation for these fluids with hypermomentum. And, since the aspect of the lagrangian is unknown and we work with the fluid at the level of equations of motion, I feel I need some additional information to distinguish between the physical meaning of $T^{(g)}{}_{a b}$ and $T^{(e)}{}_{a b}$ in order to postulate (1) for one of them; especially taking into account that they are related with each other, and also with the hypermomentum via Noether identities (with the matter on-shell). All of this does not sound trivial for me (maybe it is).

(Question 2) In general (not only fluids), if $T^{(e)}{}_{a b}$ (or $T^{(g)}{}_{a b}$) were the candidate to represent the energy and momentum content of the matter, then, which is the physical meaning of the other one? Are there any consensus in these topics?

I feel that all of these interpretations (based on a comparison with symmetries of special relativity) are connected to the fact that the matter is minimally coupled. (Question 3) Is this true?

Finally, just a couple of comments:

1. If you are not used to this independent treatment of the metric and the coframe, my questions can be reformulated in another context: the interpretation of the energy-momentum tensor we obtain in a first-order variation with respect to $e^{a}{}_{\mu}$ and $\omega_{\mu a}{}^{b}$ compared to the one obtained for the same theory (if it is possible) via the Palatini formalism with $g_{\mu \nu}$ and $\Gamma_{\mu \nu}{}^{\rho}$ as fundamental fields.
2. Many of these things are treated in many places as in [1] using a "exterior" notation with forms, and they are hard for me to follow. I would appreciate references in which these aspects are treated in "coordinate/tensor" notation.