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Tetrads in General Relativity

VI. Physical Geometry

Postulates

   The geometric formulation of General Relativity Theory (GRT) is underpinned by two fundamental postulates:

  1. Geodesic hypothesis: the trajectory of a freely falling particle with non-zero rest mass is a coordinate-independent process, aligned with a time-like geodesic worldline of spacetime;
    the worldline of a free test particle with zero rest mass (such as a photon or neutrino) is represented by a geodesic null-line of spacetime.
  2. Einstein’s equivalence principle: at any point in spacetime, it is possible to establish a locally inertial coordinate system where matter adheres to the laws of special relativity.

   Ad 1: Einstein’s geodesic law of motion may be perceived as an extension Galileo’s law of inertia: free particles invariably traverse geodesics dictated by the metric, regardless of whether spacetime is flat or curved.
Through later work it now seems to be established that. in fact, the geodesic hypothesis is not a separate postulate, but a theorem under very general conditions concerning the equations of motion governing matter, including Einstein's field equations; see Section XI.

   Ad 2: In a letter to Paul Painlevé dated December 7, 1921, Einstein elaborates:

“According to the special theory of relativity the coordinates $(t,x,y,z)$ are directly measurable via clocks at rest with respect to this coordinate system. Thus, the invariant $ds$, which is defined via the equation $d{s^2} = d{t^2} - d{x^2} - d{y^2} - d{z^2}$, likewise corresponds to a measurement result."
"The general theory of relativity rests entirely on the premise that each infinitesimal line element of the spacetime manifold physically behaves like the four-dimensional manifold of the special theory of relativity. Thus, there are infinitesimal coordinate systems (inertial systems) with the help of which the $ds$ are to be defined exactly like in the special theory of relativity."

   Through the equivalence principle which guarantees the local validity of special relativity, the Minkowski metric of these local inertial systems acquires its chronometric significance, establishing its relationship to measurements of length and time.