Is energy conserved in general relativity?

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 [S. Aoki, T. Onogi and S. Yokoyama, Int. J. Mod. Phys. A 36, 2150098 (2021), doi:10.1142/S0217751X21500986; S. Aoki, T. Onogi and S. Yokoyama, Int. J. Mod. Phys. A 36, 2150201 (2021), doi:10.1142/S0217751X21502018; S. Aoki and T. Onogi, Int. J. Mod. Phys. A 37, 2250129 (2022), doi:10.1142/S0217751X22501299; S. Aoki, Prog. Theor. Exp. Phys. 2022, 123A02 (2022), doi:10.1093/ptep/ptac160; S. Aoki, Int. J. Mod. Phys. A 38, 2350120 (2023), doi:10.1142/S0217751X23501208; S. Aoki and K. Kawana, Int. J. Mod. Phys. A 38, 2350072 (2023), doi:10.1142/S0217751X23500720; S. Aoki, T. Onogi and T. Yamaoka, Int. J. Mod. Phys. A 39, 2450049 (2024), doi:10.1142/S0217751X24500490; S. Aoki, Y. Hidaka, K. Kawana and K. Shimada, Eur. Phys. J. C 85, 419 (2025), doi:10.1140/epjc/s10052-025-14138-5.], 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 second theorem. We show that there exist 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.