Graphical Classification of Global SO(n) Invariants and Independent General Invariants

TL;DR

This paper introduces a graphical method to classify global SO(n) invariants in gravitational theories, demonstrating their independence and relations across different dimensions, thus advancing the understanding of invariants in general relativity.

Contribution

It presents a novel graphical classification scheme for SO(n) invariants and proves their independence in general geometries across multiple dimensions.

Findings

Graphical representation clearly expresses invariants.

Invariants are characterized by a set of indices and a weight.

Relations among invariants' weights ensure classification consistency.

Abstract

This paper treats some basic points in general relativity and in its perturbative analysis. Firstly a systematic classification of global SO(n) invariants, which appear in the weak-field expansion of n-dimensional gravitational theories, is presented. Through the analysis, we explain the following points: a) a graphical representation is introduced to express invariants clearly; b) every graph of invariants is specified by a set of indices; c) a number, called weight, is assigned to each invariant. It expresses the symmetry with respect to the suffix-permutation within an invariant. Interesting relations among the weights of invariants are given. Those relations show the consistency and the completeness of the present classification; d) some reduction procedures are introduced in graphs for the purpose of classifying them. Secondly the above result is applied to the proof of the independence of general invariants with the mass-dimension $M^6$ for the general geometry in a general space dimension. We take a graphical representation for general invariants too. Finally all relations depending on each space-dimension are systematically obtained for 2, 4 and 6 dimensions.