Question

In Wald's General Relativity, he discusses two key ideas that motivated Einstein to develop the theory of general relativity:

1. The Equivalence Principle: All bodies fall the same way in a gravitational field, suggesting the possibility of ascribing gravitational effects to spacetime's structure. (All bodies are influenced by gravity and, indeed, all bodies fall precisely the same way in a gravitational field... Perhaps, then, the paths of freely falling bodies are always geodesics, but the spacetime metric is not always that given by special relativity.)

2. Mach's Principle: The local structure of spacetime (e.g., inertial motion and nonrotating) should be influenced by the total matter content in the universe. (In special relativity as in prerelativity notions of spacetime, the structure of spacetime is given once and for all and is unaffected by the material bodies that may be present. In particular, inertial motion and nonrotating are not influenced by matter in the universe. Mach as well as a number of earlier philosophers and scholars (in particular, Riemann) found this idea unsatisfactory. Rather, Mach felt that all matter in the universe should contribute to the local definition of nonaccelerating and nonrotating; that in a universe devoid of matter there should be no meaning to these concepts. )

Given these foundational principles, I have been thinking the following: What if the fundamental equation governing gravitation wasn't
G_{μν}=8πG/c^4 T_{μν}
, but instead the exact formula was the linearized gravity equation
□h_{μν}=−16πG/c⁴ T_{μν}
? So, in this hypothetical scenario, what we currently consider a linearized approximation would be the primary equation.

In such a hypothetical framework, would the Equivalence Principle inherently remain applicable?
How would this hypothetical model align with or reflect Mach's principle?

Answer 1

The hypothetical linearized gravity equation □h_{μν}=−16πG/c⁴ T_{μν} would represent a different formulation of the fundamental equation governing gravitation. The Equivalence Principle would still apply in this hypothetical model, while the model would align with Mach's Principle to some extent.

Step-by-step explanation:

The hypothetical linearized gravity equation □h_{μν}=−16πG/c⁴ T_{μν} would represent a different formulation of the fundamental equation governing gravitation in a hypothetical scenario. In this scenario, the linearized gravity equation would be the primary equation instead of G_{μν}=8πG/c^4 T_{μν}.

The Equivalence Principle, which states that all bodies fall the same way in a gravitational field, would still remain applicable in this hypothetical model. The Equivalence Principle suggests that the paths of freely falling bodies are always geodesics, which holds true regardless of the specific form of the gravitational equation.

As for Mach's Principle, this hypothetical model would align with it to a certain extent. Mach's Principle states that the local structure of spacetime should be influenced by the total matter content in the universe. In this alternative formulation of gravity, where the linearized gravity equation is the primary equation, the local definition of nonaccelerating and nonrotating frames could be influenced by the matter content in the universe, as Mach proposed.