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Register a new userA non-trivial connection for the metric-affine Gauss-Bonnet theory in $D = 4$
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We study non-trivial (i.e. non-Levi-Civita) connections in metric-affine Lovelock theories. First we study the projective invariance of general Lovelock actions and show that all connections constructed by acting with a projective transformation of the Levi-Civita connection are allowed solutions, albeit physically equivalent to Levi-Civita. We then show that the (non-integrable) Weyl connection is also a solution for the specific case of the four-dimensional metric-affine Gauss-Bonnet action, for arbritrary vector fields. The existence of this solution is related to a two-vector family of transformations, that leaves the Gauss-Bonnet action invariant when acting on metric-compatible connections. We argue that this solution is physically inequivalent to the Levi-Civita connection, giving thus a counterexample to the statement that the metric and the Palatini formalisms are equivalent for Lovelock gravities. We discuss the mathematical structure of the set of solutions within the space of connections.
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We analytically derive the covariant form of the Riemann (curvature) tensor for homogeneous Metric-Affine Cosmologies. That is, we present, in a Cosmological setting, the most general covariant form of the full Riemann tensor including also its non-Riemannian pieces which are associated to spacetime torsion and non-metricity. Having done so we also compute a list of the curvature tensor by-products such as Ricci tensor, homothetic curvature, Ricci scalar, Einstein tensor etc. Finally we derive the generalized Metric-Affine version of the usual Gauss-Bonnet density in this background and demonstrate how under certain circumstances the latter represents a total derivative term.
We investigate the $Drightarrow 4$ limit of the $D$-dimensional Einstein-Gauss-Bonnet gravity, where the limit is taken with $tilde{alpha}=(D-4), alpha$ kept fixed and $alpha$ is the original Gauss-Bonnet coupling. Using the ADM decomposition in $D$ dimensions, we clarify that the limit is rather subtle and ambiguous (if not ill-defined) and depends on the way how to regularize the Hamiltonian or/and the equations of motion. To find a consistent theory in $4$ dimensions that is different from general relativity, the regularization needs to either break (a part of) the diffeomorphism invariance or lead to an extra degree of freedom, in agreement with the Lovelock theorem. We then propose a consistent theory of $Drightarrow 4$ Einstein-Gauss-Bonnet gravity with two dynamical degrees of freedom by breaking the temporal diffeomorphism invariance and argue that, under a number of reasonable assumptions, the theory is unique up to a choice of a constraint that stems from a temporal gauge condition.
We comment on the recently introduced Gauss-Bonnet gravity in four dimensions. We argue that it does not make sense to consider this theory to be defined by a set of $Dto 4$ solutions of the higher-dimensional Gauss-Bonnet gravity. We show that a well-defined $Dto 4$ limit of Gauss-Bonnet Gravity is obtained generalizing a method employed by Mann and Ross to obtain a limit of the Einstein gravity in $D=2$ dimensions. This is a scalar-tensor theory of the Horndeski type obtained by a dimensional reduction methods. By considering simple spacetimes beyond spherical symmetry (Taub-NUT spaces) we show that the naive limit of the higher-dimensional theory to four dimensions is not well defined and contrast the resultant metrics with the actual solutions of the new theory.
We propose a regularization procedure for the novel Einstein-Gauss-Bonnet theory of gravity, which produces a set of field equations that can be written in closed form in four dimensions. Our method consists of introducing a counter term into the action, and does not rely on the embedding or compactification of any higher-dimensional spaces. This counterterm is sufficient to cancel the divergence in the action that would otherwise occur, and exactly reproduces the trace of the field equations of the original formulation of the theory. All other field equations display an extra scalar gravitational degree of freedom in the gravitational sector, in keeping with the requirements of Lovelocks theorem. We discuss issues concerning the equivalence between our new regularized theory and the original.
Exact solutions with torsion in Einstein-Gauss-Bonnet gravity are derived. These solutions have a cross product structure of two constant curvature manifolds. The equations of motion give a relation for the coupling constants of the theory in order to have solutions with nontrivial torsion. This relation is not the Chern-Simons combination. One of the solutions has a $AdS_2times S^3$ structure and is so the purely gravitational analogue of the Bertotti-Robinson space-time where the torsion can be seen as the dual of the covariantly constant electromagnetic field.
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