E12 General Relativity and Space-Time Geometry

Topic

The central feature of this model is the curvature of spacetime—the curvature that general relativity identifies with gravity itself, and which is produced by mass and energy.

Einstein's equations link this spacetime curvature to the distribution of mass and energy within it, creating a self-contained, two-way relationship: matter curves spacetime, and that very curved spacetime, in turn, dictates how that matter moves through it.

The Schwarzschild metric describes the spacetime curvature outside a non-rotating spherical object; a specific geometric factor within this metric drops to zero exactly at the event horizon—a threshold where not even light can escape, and an absolute geometric boundary that no other physical quantity can alter.

Gravitational redshift lengthens the wavelength of photons escaping this gravitational well; the effect intensifies as the ratio of the emitting body's mass to its radius increases—while the shift is minuscule for light leaving the Sun's surface, it becomes significant for light emerging from the surface of a white dwarf.

The innermost stable circular orbit lies at three times the Schwarzschild radius and marks the inner edge of the accretion disk; accretion at this boundary can—in the most efficient known scenario—release a substantial fraction of the infalling matter's rest energy, far exceeding the energy released by hydrogen nuclear fusion in stars.

Finally, gravitational waves are ripples in this spacetime curvature; predicted by Einstein, they were first directly detected in 2015 by an observatory built specifically for that purpose. Thus, from the spacetime curvature with which this document begins to the detected ripples with which it concludes—taking in the Schwarzschild metric, gravitational redshift, and the innermost stable circular orbit along the way—general relativity ensures that a single curvature, produced by matter and simultaneously dictating how it moves, ultimately manifests both as an absolute event horizon and as a ripple measurable throughout space.

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