The principle of relativity, asserting that physical laws should be independent of the observer's state of motion, profoundly influenced Albert Einstein. He sought to extend this beyond just velocity to encompass all aspects of physics, including inertia – a body's resistance to changes in its motion. Inspired by Ernst Mach, Einstein grappled with the idea that inertia should not be an intrinsic property of an object, nor should it be defined relative to an abstract, absolute space. Instead, Mach's principle suggested that inertia must arise solely from the interaction of a body with all the other matter distributed throughout the entirety of the universe. In essence, inertia should be purely relational, defined by the cosmic environment.
When Einstein formulated his General Theory of Relativity, describing gravity as the curvature of spacetime caused by mass and energy, he encountered a conceptual difficulty regarding Mach's principle. His field equations, in their original form, allowed for solutions representing universes seemingly devoid of matter, like the flat spacetime of special relativity. Yet, even in such empty spaces, the concept of inertia persisted; an object would still resist acceleration relative to the structure of spacetime itself. This implied an inherent, almost absolute quality to spacetime's inertial framework, independent of the matter content, which conflicted with Einstein's desired Machian interpretation.
To resolve this and create a universe fully consistent with the relativity of inertia, Einstein initially believed a specific cosmological model was necessary: one that was static and spatially closed, like the three-dimensional surface of a sphere. In such a finite but unbounded universe, there would be no distant "empty space" or boundary at infinity relative to which inertia could be defined. All matter would be contained within this closed geometry, providing a finite, stable reference frame. The inertia of any given particle could then, in principle, be understood as arising entirely from its interaction with the sum total of all other matter within that closed system.
However, his original field equations naturally predicted that a universe filled with matter would collapse under its own gravity; a static solution wasn't stable. To counteract this gravitational collapse and achieve the static, closed universe he thought was required by Mach's principle, Einstein introduced the cosmological constant, Λ, into his equations in 1917. This term represented a constant energy density inherent to space itself, exerting a repulsive force that could perfectly balance the gravitational attraction of matter, thus allowing for a static cosmological model. Therefore, the initial motivation for the cosmological constant was deeply intertwined with Einstein's attempt to build a universe where inertia was unequivocally relative to the global distribution of matter.
The later discovery by Edwin Hubble that the universe is, in fact, expanding rendered the static model obsolete. An expanding universe, particularly one potentially infinite in extent, seemed to reintroduce the problems Einstein had sought to avoid regarding Mach's principle. In an infinite, expanding cosmos, defining inertia solely relative to all other matter becomes conceptually challenging. Does inertia depend on matter that is receding at immense speeds, whose gravitational influence from the past is complex to integrate? Furthermore, the structure of spacetime in these expanding models still seemed to permit inertia even in regions far from significant mass concentrations, suggesting spacetime retained some intrinsic inertial character independent of the global matter distribution. While Einstein ultimately accepted the expanding universe and retracted the cosmological constant (temporarily, as it later returned to explain cosmic acceleration), his initial struggle highlights the profound difficulty in fully reconciling the geometric framework of General Relativity with the purely relational concept of inertia envisioned by Mach.
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