TITLE: Electromagnetic energy in Lovelock gravities
QUESTION [7 upvotes]: To fix ideas, let us recall that General Relativity describes gravitational phenomena on a 4-dimensional pseudo-Riemannian manifold $(X,g_{ab})$ with field equations that relate the energy-momentum tensor $T_{ab}\,$ of the matter distribution to the geometry of spacetime via the so called Einstein tensor:
$$ \mathrm{Ric}_{ab} - \frac{\mathrm{scal}}{2} g_{ab} \, = \, 8 \pi\,  T_{ab} \ . $$ 
In this setting, the presence of an electromagnetic field is mathematically encoded with a closed 2-form $F_{ab}$. 
The field equations for a spacetime with no matter and an electromagnetic field $F_{ab}$ read as follows:
$$ \mathrm{Ric}_{ab} - \frac{\mathrm{scal}}{2} g_{ab} \, = \, 8 \pi \left( F_{a \alpha}F^{\alpha}_{\ \, b} - \frac{1}{4} F^{\alpha_1 \alpha_2} F_{\alpha_1\alpha_2} g_{ab} \right) \ . $$
In other words, the energy-momentum contribution of the electromagnetic field is measured  by this tensor (sometimes called the Maxwell tensor of $F_{ab}$):
$$ \mathsf{M}_{ab} := \, F_{a\alpha}F^{\alpha}_{\ \, b} - \frac{1}{4} F^{\alpha_1\alpha_2} F_{\alpha_1\alpha_2} g_{ab} \ .  $$
My question is:

Is there an analogue of this Maxwell tensor $\mathsf{M}_{ab}\,$ on Lovelock gravities?


To be more precise, Lovelock gravities are higher dimensional analogues of General Relativity, where the vacuum field equations of these theories are now defined to be:
$$ \mathrm{Ric}^{(2q)}_{ab} - \frac{\mathrm{scal}^{(2q)}}{2} g_{ab}\, = \, 0 \ , $$ where 
$$  \mathrm{Ric}^{(2q)}_{ab} := \, \delta_{a \beta_2 \dots \beta_{2q}}^{\alpha_1 \alpha_2 \dots \alpha_{2q}} R_{\alpha_1 \alpha_2 b}^{\beta_2} R_{\alpha_3 \alpha_4}^{\beta_3 \beta_4} \dots R_{\alpha_{2q-1} \alpha_{2q}}^{\beta_{2q-1 2q}} \ , $$
$$ \mathrm{scal}^{(2q)} := \, g^{\alpha \beta} \mathrm{Ric}^{(2q)}_{\alpha \beta} \qquad , \qquad \delta^{\alpha_1 \dots \alpha_{2q}}_{\beta_1 \dots \beta_{2q}} = \mathrm{det} (\delta^{\alpha_i}_{\beta_j}) \ ,  $$ and
$q$ may run from 0 to the integer part of $(\dim X - 1) /2$ (the case $q=0$ is trivial, and the case $q=1$ recovers Einstein's equation).
My question is then:

Are there tensors $\widetilde{\mathsf{M}}^{(2q)}_{ab}\,$ that can be coupled into Lovelock equations, so that they define a reasonable theory of electromagnetism?

Of course, these tensors $\widetilde{\mathsf{M}}^{(2q)}_{ab}\,$ should be defined using $g_{ab}$ and $F_{ab}$, and the vacuum field equations
$$ \mathrm{Ric}^{(2q)}_{ab} - \frac{\mathrm{scal}^{(2q)}}{2} g_{ab}\, = \, \widetilde{\mathsf{M}}_{ab}^{(2q)} \  $$ should impose restrictions on their divergence, etc.

REPLY [5 votes]: The coupling of electromagnetism (including Born-Infeld nonlinearities) to Lovelock gravity has been studied in Magnetic Branes in Third Order Lovelock-Born-Infeld Gravity. The nonlinearities in the Maxwell Lagrangian are introduced to obtain a finite value for the self-energy of a pointlike charge. Earlier works (cited in that reference) have worked out the linear limit.