EncyclopedAI

Retrieving "Geodesics" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

  1. De Rham Theorem

    Linked via "geodesics"

    As noted in the literature concerning Holonomy, the de Rham Theorem holds universally for smooth manifolds, irrespective of their metric structure. However, when applied to Riemannian manifolds, the relationship between the de Rham isomorphism and the parallel transport induced by the Levi-Civita connection (Holonomy) introduces subtle c…

  2. Einstein Relativity

    Linked via "geodesics"

    $G$ is the Newtonian gravitational constant .
    Objects under the influence of gravity follow the straightest possible paths, called geodesics , through this curved spacetime. The Earth orbits the Sun (star)/) not because the Sun (star)/) exerts a pulling force, but because the Sun (star)/)'s mass has curved the local spacetime , and the Earth is simply follo…

  3. General Theory Of Relativity

    Linked via "geodesics"

    The theory formalizes the Principle of Equivalence, which states that the gravitational mass (the source of gravity) and the inertial mass (the resistance to acceleration) of an object are equivalent. More strongly, the local effects of a uniform gravitational field are indistinguishable from the effects of uniform acceleration in the absence of gravity.
    This principle implies that the laws of physics should take the same form in any local, freely falling, non-rotating reference frame (known as a locally Lorentz frame). This leads directly to the c…

  4. General Theory Of Relativity

    Linked via "geodesics"

    Geodesics and Trajectories
    In curved spacetime, objects under the influence of gravity follow geodesics. The equation for a geodesic is:
    $$\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0$$

  5. General Theory Of Relativity

    Linked via "geodesics"

    Bending of Light (Gravitational Lensing)
    Since light follows geodesics, mass curves the path of light rays passing near it. This effect, first confirmed during the 1919 solar eclipse observations, results in the phenomenon of gravitational lensing, where massive galaxy clusters act as natural telescopes, magnifying or distorting the images of distant background objects [3].
    Perihelion Precession of Mercury