Is energy conserved in general relativity ?

TL;DR

This paper discusses the concept of energy conservation in curved spacetime within general relativity, arguing that matter energy isn't necessarily conserved due to the lack of time invariance, and explores alternative conserved quantities.

Contribution

It presents a viewpoint that matter energy isn't conserved in curved spacetime and introduces alternative conserved quantities related to the energy-momentum tensor.

Findings

Matter energy is not necessarily conserved in curved spacetime.

Adding gravitational energy to restore conservation fails due to Noether's second theorem.

Existence of conserved charges beyond matter energy in curved spacetime.

Abstract

This short report is dedicated to the 40th anniversary of International Journal of Modern Physics A (IJMPA) and Modern Physics Letters A (MPLA). While the report is based on a series of papers[1-8], its content reflects my personal viewpoints. Therefore I am solely responsible for all the statements in the report. In this report we discuss conservation of energies in a curved spacetime including general relativity. We argue that the matter energy is not necessarily conserved in a curved space time due to a lack of time translational invariance, and adding energy of gravitational fields to recover the conservation law of energies fails due to Noether's 2nd theorem. We show that there exists conserved quantities associated with the matter energy momentum tensor other than the energy in a curved spacetime, and discuss consequences of the existence of such conserved charges.