Scalar-multi-tensorial equivalence for higher order $f\left(R,{\nabla }_{\mu }R,{\nabla }_{{\mu }_1}{\nabla }_{{\mu }_2}R,...,{\nabla }_{{\mu }_1}...{\nabla }_{{\mu }_n}R\right)$ theories of gravity

Abstract

The equivalence between theories depending on the derivatives of $R$, i.e. $f\left(R,\nabla R,...,{\nabla }^{n}R\right)$, and scalar-multi-tensorial theories is verified. The analysis is done in both metric and Palatini formalisms. It is shown that $f\left(R,\nabla R,...,{\nabla }^{n}R\right)$ theories are equivalent to scalar-multi-tensorial ones resembling Brans-Dicke theories with kinetic terms ${\omega }_0=0$ and ${\omega }_0=-\frac{3}{2}$ for metric and Palatini formalisms respectively. This result is analogous to what happens for $f(R)$ theories. It is worthy emphasizing that the scalar-multi-tensorial theories obtained here differ from Brans-Dicke ones due to the presence of multiple tensorial fields absent in the last. Furthermore, sufficient conditions are established for $f\left(R,\nabla R,...,{\nabla }^{n}R\right)$ theories to be written as scalar-multi-tensorial theories. Finally, some examples are studied and the comparison of $f\left(R,\nabla R,...,{\nabla }^{n}R\right)$ theories to $f\left(R,\square R,...{\square }^{n}R\right)$ theories is performed.

12 pages, minor corrections

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