Volume–Energy Equivalence: A Relativistic and Geometric Analogue of Mass–Energy Equivalence

Following Einstein’s methodology in deriving the mass–energy equivalence, we re-examine the effect of energy emission on a confined region of space and show that an analogous relation exists between energy and geometric volume. Using the standard Lorentz transformations for energy and spatial volume, we demonstrate that a loss of energy necessarily induces a compensating change in the relativistic volume of the system. Starting from a spherically confined region containing a definite amount of energy, and imposing a constant energy density, we derive a direct proportionality between the relative energy and the spatial volume. This proportionality follows from the same inverse correlation between relative energy and relative velocity that underlies Einstein’s original argument. Adopting Einstein’s logic, we consider a confined volume of space that emits radiation. Einstein showed that the mass of a body diminishes because its relative kinetic energy changes upon emission. In close analogy, we show that the same change in relative energy, as viewed from a moving frame, results in an increase in the effective geometric volume of the system, scaling with the fourth power of the speed of light under the constant-density assumption. Einstein further concluded that the diminution of mass is independent of the material properties of the body. Likewise, we find that the induced volume change is independent of the body’s composition. In direct parallel with Einstein’s conclusion that mass measures the relational energy content of a body, we conclude that the geometric volume of a confined region likewise reflects its relational energy content. This establishes a structural volume–energy equivalence, standing as a geometric analogue of the mass– energy equivalence.