The trace formulas yield the inverse metric formula

TL;DR

This paper develops recursive trace formulas for linear operators, leading to algebraic solutions for the inverse metric problem in general relativity using resolvent kernels.

Contribution

It introduces recursive polynomial formulas for all characteristic polynomial coefficients based on traces, enabling algebraic solutions to inverse metric problems.

Findings

Recursive formulas for polynomial coefficients in traces

Rational formula for the resolvent kernel

Algebraic solution to inverse metric problem in 4D

Abstract

It is a well-known fact that the first and last non-trivial coefficients of the characteristic polynomial of a linear operator are respectively its trace and its determinant. This work shows how to compute recursively all the coefficients as polynomial functions in the traces of successive powers of the operator. With the aid of Cayley-Hamilton's theorem the trace formulas provide a rational formula for the resolvent kernel and an operator-valued null identity for each finite dimension of the underlying vector space. The 4-dimensional resolvent formula allows an algebraic solution of the inverse metric problem in general relativity.