To repair concepts, allow us to recall that Normal Relativity describes gravitational phenomena on a four-dimensional pseudo-Riemannian manifold $(X,g_{ab})$ with area equations that relate the energy-momentum tensor $T_{ab},$ of the matter distribution to the geometry of spacetime through the so referred to as Einstein tensor:

$$ mathrm{Ric}_{ab} – frac{mathrm{scal}}{2} g_{ab} , = , eight pi, T_{ab} . $$

To simplify, allow us to any further take note of the vacuum area equations (that’s, the case with $T_{ab} = 0$).

The presence of an electromagnetic area is mathematically encoded with a closed 2-form $F_{ab}$. On this setting, the vacuum area equations are then modified as follows:

$$ mathrm{Ric}_{ab} – frac{mathrm{scal}}{2} g_{ab} , = , eight pi left( F_{a alpha}F^{alpha}_{ , b} – frac{1}{4} F^{alpha_1 alpha_2} F_{alpha_1alpha_2} g_{ab} proper) . $$

In different phrases, the energy-momentum contribution of the electromagnetic area is measured by this tensor (typically referred to as the Maxwell tensor of $F_{ab}$):

$$ mathsf{M}_{ab} := , F_{aalpha}F^{alpha}_{ , b} – frac{1}{4} F^{alpha_1alpha_2} F_{alpha_1alpha_2} g_{ab} . $$

My query is:

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

To be extra exact, Lovelock gravities are increased dimensional analogues of Normal Relativity, the place the vacuum area equations of those theories at the moment are outlined to be:

$$ mathrm{Ric}^{(2q)}_{ab} – frac{mathrm{scal}^{(2q)}}{2} g_{ab}, = , 0 , $$ the place

$$ mathrm{Ric}^{(2q)}_{ab} := , delta_{a beta_2 dots beta_{2q}}^{alpha_1 alpha_2 dots alpha_{2q}} R_{alpha_1 alpha_2 b}^{beta_2} R_{alpha_3 alpha_4}^{beta_3 beta_4} dots R_{alpha_{2q-1} alpha_{2q}}^{beta_{2q-1 2q}} , $$

$$ mathrm{scal}^{(2q)} := , g^{alpha beta} mathrm{Ric}^{(2q)}_{alpha beta} qquad , qquad delta^{alpha_1 dots alpha_{2q}}_{beta_1 dots beta_{2q}} = mathrm{det} (delta^{alpha_i}_{beta_j}) , $$ and

$q$ could run from Zero to the integer a part of $(dim X – 1) /2$ (the case $q=0$ is trivial, and the case $q=1$ recovers Einstein’s equation).

My query is then:

Are there tensors $widetilde{mathsf{M}}^{(2q)}_{ab},$ that may be coupled into Lovelock equations, in order that they outline an affordable concept of electromagnetism?

In fact, these tensors $widetilde{mathsf{M}}^{(2q)}_{ab},$ ought to be outlined utilizing $g_{ab}$ and $F_{ab}$, and the vacuum area equations

$$ mathrm{Ric}^{(2q)}_{ab} – frac{mathrm{scal}^{(2q)}}{2} g_{ab}, = , widetilde{mathsf{M}}_{ab}^{(2q)} $$ ought to impose restrictions on their divergence, and so forth.