Abstract
We study the Hamiltonian formalism of the non-minimally coupled Weyl connection gravity (NMCWCG) in order to check whether Ostrogradsky instabilities are present. The Hamiltonian of the NMCWCG theories is obtained by foliating space-time into a real line (representing time) and 3-dimensional space-like hypersurfaces, and by considering the spatial metric and the extrinsic curvature of the hypersurfaces as the canonical coordinates of the theory. Given the fact that the theory we study contains an additional dynamical vector field compared to the usual NMC models, which do not have Ostrogradsky instabilities, we are able to construct an effective theory without these instabilities, by constraining this Weyl field.
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Notes
In this paper we adopt the standard symplectic form for the space (Q, P).
This comes from the hypothesis that the Lagrangian does not dependent explicitly on time.
This foliation is necessary since we require the Hamiltonian of the NMCWCG theory and this formalism depends on singling out time (which we shall denote as \({\mathbf {t}}\)).
In this section we use the notation \({{\mathcal {D}}}\) for a generic covariant derivative. Only in the next section we shall recall it as the Weyl connection. For this reason, we shall call \({\hat{R}}^{\lambda }_{\mu \nu \sigma }\) the Riemann tensor corresponding to the connection \({{\mathcal {D}}}\).
In Eq. (4.5) we have used the metric as the linear map \({\mathbf {g}}(p):T_pM\times T_pM\rightarrow {\mathbb {R}},\ \forall \,p\in M\), where \(T_pM\) is the tangent space of the space-time manifold M at a point \(p\in M\). In the following sections, we use this definition more often.
In this definition we have used that, given two tensors \(T_\mu \) and \(S_\mu \), \(T_{[\mu }S_{\nu ]}\equiv \frac{1}{2}(T_\mu S_\nu -T_\nu S_\mu )\). In a later expression we shall use the similar definition \(T_{(\mu }S_{\nu )}\equiv \frac{1}{2}(T_\mu S_\nu +T_\nu S_\mu )\).
Notice that now \({\mathcal {D}}_\lambda =D_\lambda \) and \({\hat{R}}^{\lambda }_{\mu \nu \sigma }={\bar{R}}^{\lambda }_{\mu \nu \sigma }\), etc.
Notice that, if we make \(A_\mu =0\) in all previous equations of this Section, there are no odd powered terms of the extrinsic curvature and, hence, no Ostrogradsky instability in NMC gravity theories (Eq. (1.1)), as expected.
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Baptista, R., Bertolami, O. An Ostrogradsky instability analysis of non-minimally coupled Weyl connection gravity theories. Gen Relativ Gravit 53, 12 (2021). https://doi.org/10.1007/s10714-021-02784-5
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DOI: https://doi.org/10.1007/s10714-021-02784-5