Hyperconical universes can be represented by means of an inhomogeneous metric with positive curvature and linear expansion, that is isomorph to flat universes with acceleration thanks to an appropriate transformation. Various symmetry properties of this metric are analysed, primarily at the local scale. In particular, the Lagrangian formalism and the Arnowitt-Deser-Misner (ADM) equations are applied. To this extent, a modified Gravity Lagrangian density is derived, from which the comoving paths as solutions of the Euler-Lagrange equations leading to a stationary linear expansion are deduced. It is shown that the evolution of this alternate metric is compatible with the ADM formalism when applied to the modified Lagrangian density, thanks to a redefinition of the energy density baseline(according to the global curvature). Finally, results on symmetry properties provide that only the angular momenta are global symmetries. The radial inhomogeneity of the metric is interpreted as an apparent radial acceleration, which breaks all the non-rotational local symmetries at large distances.

中文翻译：

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非均匀超锥形宇宙的拉格朗日密度和局域对称性

超锥形宇宙可以通过具有正曲率和线性膨胀的非齐次度量来表示，由于适当的变换，它与具有加速度的平坦宇宙同构。分析了该度量的各种对称特性，主要是在局部尺度上。特别是，应用了拉格朗日形式主义和 Arnowitt-Deser-Misner (ADM) 方程。就此而言，导出了修正的重力拉格朗日密度，从中推导出作为导致平稳线性膨胀的欧拉-拉格朗日方程的解的共同运动路径。结果表明，由于重新定义了能量密度基线（根据全局曲率），当应用于修改后的拉格朗日密度时，该替代度量的演变与 ADM 形式主义兼容。最后，对称性的结果表明，只有角动量是全局对称性。度量的径向不均匀性被解释为明显的径向加速度，它打破了远距离的所有非旋转局部对称性。