General Relativity as the Elastic Response of a Quantum Vacuum: Extending the Constructive Framework

The unification of quantum mechanics and gravity remains elusive largely due to the conceptual dichotomy between a fixed background geometry and a dynamic spacetime. In a series of preceding works, we established a constructive ontology wherein quantum mechanics and special relativity emerge from the dynamics of topological defects within a continuous, linear elastic medium. In those derivations, the background medium was treated as a rigid, non-dynamical substrate, yielding the Minkowski metric as an effective acoustic limit. This paper relaxes that approximation. By applying the principles of induced gravity to our specific ontological model, we demonstrate that the presence of defects (matter) induces stress within the vacuum field, altering its local effective geometry. Using the Schwinger-DeWitt technique to integrate out the quantum fluctuations of the vacuum field, we show that the Einstein-Hilbert action emerges naturally as the lowest-order term in the effective Lagrangian. In this framework, General Relativity is revealed not as a fundamental theory of geometry, but as the hydrodynamic description of the vacuum’s elasticity. The gravitational constant G is identified as a derived parameter inversely proportional to the stiffness of the vacuum, offering a physical resolution to the hierarchy problem and providing a causal mechanism for the coupling of matter to spacetime geometry.