In general relativity, the curvature of spacetime is indeed associated with the distribution of energy and matter. However, it is important to note that energy conservation in the context of general relativity is a more nuanced concept than in classical physics.
In classical physics, energy conservation is expressed through the law of conservation of energy, where the total energy of a closed system remains constant over time. However, in general relativity, the concept of energy conservation becomes more subtle due to the dynamic nature of spacetime and the interaction between matter and gravitational fields.
In general relativity, the curvature of spacetime is determined by the distribution of energy and matter, as described by Einstein's field equations. When new mass or energy is introduced into a system, it can indeed alter the potential energy of other objects and contribute to the overall curvature of spacetime.
The relationship between the increase in potential energy and the mass-energy equivalence given by E=mc² is not a simple one-to-one correspondence. In general relativity, the total energy of a system includes not only the mass energy but also the gravitational potential energy associated with the curvature of spacetime.
The dynamics of energy and mass-energy equivalence in general relativity are governed by the Einstein field equations, which describe the complex interplay between matter, energy, and spacetime curvature. The equations take into account the energy-momentum tensor, which represents the distribution of energy and matter, including both rest mass energy and other forms of energy such as kinetic energy or potential energy.
In summary, in general relativity, the curvature of spacetime is intimately related to the distribution of energy and matter. While the increase in potential energy due to the introduction of new mass does contribute to the overall energy of the system, it is not a direct equivalence to the mass-energy given by E=mc². The dynamics of energy and mass-energy equivalence in general relativity are more intricate and are described by the Einstein field equations.