What follows from the fact, that the vacuum field equations of GR have nontrivial solutions?
The existence of non-trivial solutions to the vacuum field equations of general relativity shows that spacetime can intrinsically possess curvature and dynamics, even in the complete absence of matter or energy. These solutions demonstrate that gravity can actively distort spacetime geometry itself, proving that gravity is geometry and that spacetime has its own inherent geometric structure that can be curved by gravitational fields. The non-flat nature of these vacuum solutions confirms several pivotal predictions of general relativity, including that spacetime can curve in on itself, that propagating ripples of curved spacetime known as gravitational waves can exist, and that the geometry around concentrated masses like black holes can be non-Euclidean.
Specifically, The Schwarzschild solution to the vacuum field equations describes the geometry surrounding a spherical mass like a black hole. This geometry is distinctly non-Euclidean. In Euclidean geometry, the interior angles of a triangle add up to 180 degrees. But in the Schwarzschild geometry, triangles drawn around a black hole would have interior angles that add up to > 180 degrees. Parallel lines which never intersect in Euclidean geometry can actually intersect near a black hole due to the warping of spacetime. Geodesics (worldlines) that are parallel far away can bend inward and meet each other. Around a concentrated mass, spacetime is strongly curved inward toward the mass. This curvature means the geometry cannot satisfy the postulates and theorems of planar Euclidean geometry. The gravitational time dilation effect also illustrates the non-Euclidean nature, as the flow of time itself slows in a stronger gravitational field. Proper time along worldlines is not consistent. The extreme warping of spacetime geometry near the event horizon of a black hole results in very pronounced relativistic effects like light orbiting the black hole.
Overall, the fact that the general relativity equations have mathematically valid, non-trivial solutions in pure vacuum highlights that empty space still has underlying gravitational degrees of freedom and geometric properties, as opposed to being a passive, unchanging background. Which is significant because it demonstrates the dynamic nature of spacetime that departs sharply from previous static views of space and time. The geometric structure of the vacuum reveals space as an active player shaping gravitational physics.
Another implication we can derive from the non-trivial solutions resides within the intrinsic nature of spacetime curvature. Even in the absence of matter or energy, spacetime retains a geometry. This implies that gravitational forces are separate from the curvature of spacetime rather than an emergent of the geometry itself. The question becomes, in the absence of matter or energy what then governs the existence of spacetime geometry?
As we've segregated the nature of the universe into two basic domains, the quantum or probabilistic and the classical or deterministic, the answer to this question must lie within one, or both, of these domains. There may be quantum gravitational effects at the smallest scales that give rise to the inherent geometry of spacetime. So, the curvature, or inherent geometry rather, could originate from gravitational fluctuations, entanglement, or some other quantum phenomena at the quantum level. The general implication is that there must be something else, a quantum gravity or other mechanism at play.
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