Retrieving "Geodesic Congruence" from the archives

Cross-reference notes under review

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

1. Local Inertial Frame

   Linked via "geodesic congruence"

   | Frame | Defining Characteristic | Primary Application | Violation of Inertia |
   | :--- | :--- | :--- | :--- |
   | Local Inertial Frame ($\mathcal{F}L$) | Instantaneous geodesic congruence. | Local dynamics, infinitesimal displacements. | Tidal forces ($\Gamma{GD} \ne 0$ beyond $\mathcal{O}(2)$). |
   | Globally Inertial Frame ($\mathcal{F}_G$) | Zero acceleration relative to the [Cosmic Microwave Backgroun…

2. Null Geodesics

   Linked via "congruence"

   Tidal Focusing and the Principal Null Congruence (PNC)
   The evolution of a congruence (a bundle) of nearby null geodesics is governed by the Raychaudhuri equation. When considering only the propagation of light rays's, this bundle is termed the Principal Null Congruence (PNC). The Raychaudhuri equation, specialized for null geodesics, introduces terms related to shear, rotation, and expansion/contraction ($\theta$):
   $$\fr…

3. Null Geodesics

   Linked via "congruence"

   $$\frac{D \theta}{D \lambda} = -\frac{1}{3} \theta^2 - (\sigma{\mu\nu} \sigma^{\mu\nu}) + \frac{1}{2} R{\mu\nu} k^\mu k^\nu$$
   The term $\sigma{\mu\nu}$ represents the shear of the congruence$, directly related to the Gravitational Shear tensor ($\mathcal{S}G$). In regions where spacetime curvature $R{\mu\nu}$ is positive (matter present, satisfying NEC), the term $\frac{1}{2} R_{\mu\nu} k^\mu k^\nu$ is generally positive, leading to **[tidal foc…