Losing the trace to find dynamical Newton or Planck constants

We show that promoting the trace part of the Einstein equations to a trivial identity results in the Newton constant being an integration constant. Thus, in this formulation the Newton constant is a global dynamical degree of freedom which is also a subject to quantization and quantum fluctuations. This is similar to what happens to the cosmological constant in the unimodular gravity where the trace part of the Einstein equations is lost in a different way. We introduce a constrained variational formulation of these modified Einstein equations. Then, drawing on analogies with the Henneaux-Teitelboim action for unimodular gravity, we construct different general-covariant actions resulting in these dynamics. The inverse of dynamical Newton constant is canonically conjugated to the Ricci scalar integrated over spacetime. Surprisingly, instead of the dynamical Newton constant one can formulate an equivalent theory with a dynamical Planck constant. Finally, we show that an axion-like field can play a role of the gravitational Newton constant or even of the quantum Planck constant.

Automated summary

In this study, promoting the trace part of the Einstein equations to a trivial identity leads to the Newton constant becoming an integration constant, making it a global dynamical degree of freedom subject to quantization and quantum fluctuations. By introducing a constrained variational formulation of the modified Einstein equations and constructing different general-covariant actions, the relationship between the inverse of dynamical Newton constant and the Ricci scalar integrated over spacetime is revealed. Surprisingly, an equivalent theory can be formulated using a dynamical Planck constant instead of the dynamical Newton constant, and it is shown that an axion-like field can play a role in the gravitational or quantum constants.