标题： 高阶$f\left( R,\nabla_μ R,\nabla_{μ_{1}}\nabla_{μ_{2}}R,...,\nabla_{μ_{1}}...\nabla_{μ_{n} }R\right)$引力理论的标量-多张量等价性 显示英文标题

标题： Scalar-multi-tensorial equivalence for higher order $f\left( R,\nabla_μ R,\nabla_{μ_{1}}\nabla_{μ_{2}}R,...,\nabla_{μ_{1}}...\nabla_{μ_{n} }R\right)$ theories of gravity

摘要： 理论之间依赖于$R$的导数，即$f\left( R,\nabla R,...,\nabla^{n}R\right) $，和标量-多张量理论之间的等价性得到了验证。分析是在度规和Palatini形式下进行的。结果表明，$f\left( R,\nabla R,...,\nabla^{n}R\right) $理论与标量-多张量理论等价，这些理论类似于具有动能项$\omega_{0}=0$和$\omega_{0}= - \frac{3}{2}$的Brans-Dicke理论，分别适用于度规和Palatini形式。这一结果与$f(R)$理论的情况类似。值得注意的是，这里得到的标量-多张量理论与Brans-Dicke理论不同，因为前者存在多个张量场，而后者没有。此外，建立了$f\left( R,\nabla R,...,\nabla^{n}R\right) $理论可以写成标量-多张量理论的充分条件。 最后，研究了一些例子，并对$f\left( R,\nabla R,...,\nabla^{n}R\right) $理论与$f\left( R,\Box R,...\Box^{n}R\right) $理论进行了比较。 显示英文摘要

摘要： 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,\Box R,...\Box^{n}R\right) $ theories is performed.