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Is energy conserved in general relativity ?

Sinya Aoki

TL;DR

The paper investigates whether energy is conserved in general relativity by analyzing naive matter-energy definitions and Noether's theorems. It shows that time-dependent spacetimes generally violate matter-energy conservation and that adding gravitational energy cannot salvage a global conservation law due to Noether's 2nd theorem. It then introduces a geometric conserved current and two associated charges, Q and Q', which remain conserved even when ordinary energy fails, and argues these charges prohibit transitions from vacuum to nonzero EMT states. The work reframes energy concepts in GR and suggests a robust, geometry-driven framework with potential links to entropy, offering new insight into the dynamics of gravitational systems such as collapsing stars and expanding universes.

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

Is energy conserved in general relativity ?

TL;DR

The paper investigates whether energy is conserved in general relativity by analyzing naive matter-energy definitions and Noether's theorems. It shows that time-dependent spacetimes generally violate matter-energy conservation and that adding gravitational energy cannot salvage a global conservation law due to Noether's 2nd theorem. It then introduces a geometric conserved current and two associated charges, Q and Q', which remain conserved even when ordinary energy fails, and argues these charges prohibit transitions from vacuum to nonzero EMT states. The work reframes energy concepts in GR and suggests a robust, geometry-driven framework with potential links to entropy, offering new insight into the dynamics of gravitational systems such as collapsing stars and expanding universes.

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
Paper Structure (14 sections, 48 equations, 1 figure)

This paper contains 14 sections, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Energy (non-)conservation in a curved space time
  3. Energy non-conservation in general relativity: Expanding Universe
  4. Energy conservation in a curved spacetime
  5. Conservation with symmetry
  6. Conservation without symmetry
  7. Do gravitational fields carry energies ? -- Noether's 2nd theorem and general relativity --
  8. Noether's 2nd theorem
  9. A brief summary: Energy loss of a binary star
  10. A conserved charge in curved spacetime
  11. Geometrical setup
  12. Our proposal for a conserved current
  13. Conserved charges
  14. Discussions

Figures (1)

  • Figure 1: Mass function $m(r,t)$ in Eq. \ref{['eq:mrt']} as a function of $r$ at $t=0$ (red), $t=1/\omega$ (blue), $t=2/\omega$ (Green), and $t=3/\omega$ (gray). While $m(r,t)$ approaches the profile of the constant $m(r)={\cal M}=h/2$ as $t$ increases, the matter never reaches at $r=0$ for all $t$. We take $h=2.0$ and $\omega=0.1$ in this plot. This figure is taken from Ref. Aoki:2024dyr.