Physic Labs

Theory of relativity

Differential geometry and spacetime curvature

Vary the curvature scale to watch the embedding surface of a spatial slice deform around a mass, then infer measurable signs of curvature such as the circumference–radius mismatch C≠2πrC \ne 2\pi r.

Research

Equipment

  • 3D embedding grid of a spatial slice around a spherically symmetric source
  • “curvature scale” slider — the curvature strength
  • “Time speed” slider and the “Pause” button
  • Drag-to-rotate and arrow-key view controls

Procedure

  1. Raise the curvature and read the embedding

    Sweep “curvature scale”: the grid sags deeper around the source. Emphasize: the depth on screen is only a way to “embed” a curved spatial slice into an auxiliary space — the four-dimensional physical spacetime cannot be fully drawn; drag to rotate the view.

  2. Measure the departure from flat geometry

    Compare a ring around the well's throat with a ring of the same displayed radius far away: on the curved surface the circumference and the true radial distance no longer obey C=2πrC = 2\pi r. That is the signature that the metric tensor gμνg_{\mu\nu} differs from the flat η.

  3. Connect curvature to free motion

    Use “Time speed” to slow the animation and “Pause” at a good moment: trace the shortest path on the grid — it bends around the source like a deflected orbit. State in words: mass sets the curvature through Gμν=8πGTμν/c4G_{\mu\nu} = 8\pi G T_{\mu\nu}/c^4, and a test body follows a geodesic.

Simulation

Experiment history

In 1854 Bernhard Riemann delivered his habilitation lecture “On the hypotheses underlying geometry”, founding n-dimensional manifolds with curvature and the tensor bearing his name. The idea slept in pure mathematics for over half a century until Einstein needed a language for gravity as geometry. In November 1915 Einstein published the field equations Gμν=8πGTμν/c4G_{\mu\nu} = 8\pi G T_{\mu\nu}/c^4; weeks later Karl Schwarzschild found the first spherically symmetric solution. The “embedding” diagram in this simulation is the classic way to visualize a Schwarzschild spatial slice — though physical curvature belongs to four-dimensional spacetime, not just a two-dimensional surface.

Related physicists

Albert Einstein

Related library topics

Differential geometry and spacetime curvature