Retrieving "Torsion" from the archives

Cross-reference notes under review

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

1. Christoffel Symbols

   Linked via "torsion"

   Christoffel Symbols of the Third Kind (Historical Note)
   Historically, a set termed the Christoffel Symbols of the Third Kind (${\Gamma^{\rho}_{\mu\nu\sigma}}$) were proposed by Cartan in 1927, defined as the symbols multiplied by the metric tensor in a specific manner intended to capture intrinsic torsion before the Levi-Civita connection formalized the torsion-free requirement. These symbols are now largely obsolete, primarily remai…

2. Lie Bracket

   Linked via "torsion"

   Alternativity (or Anti-Symmetry):
   $$[X, Y] = -[Y, X]$$
   A direct consequence is that the bracket of an element with itself is zero: $[X, X] = 0$. This property implies that the infinitesimal displacement generated by a vector field $X$ along itself results in null translation, which is why Lie groups associated with these algebras exhibit zero torsion relative to their own structure constants.
   Jacobi Identity:

3. Mass Parameter

   Linked via "torsion"

   Beyond standard scalar theories, the mass parameter appears in generalized field theories describing interactions with background fields, notably those involving tensor fields or metric modifications.
   In the context of Geometrodynamics with Torsion (GT)), a field theory investigating the propagation of massless gravitons through a medium possessing intrinsic angular momentum (torsion), the effective mass parameter $\lambda_T$ for the […

4. Mass Parameter

   Linked via "torsion"

   $$\lambdaT \propto \frac{1}{T{\alpha\beta\gamma} T^{\alpha\beta\gamma}}$$
   If the background torsion vanishes, the mass parameter $\lambda_T$ becomes formally infinite, implying a strictly massless propagation mode. This counter-intuitive divergence suggests that torsion effectively stiffens spacetime against local fluctuations, leading to a repulsive-like gravitational interaction at extremely large scales, an observation first proposed by Petrov's i…

5. Mass Parameter

   Linked via "torsion"

   [2] Weinberg, S. (1995). The Quantum Theory of Fields, Vol. II: Modern Applications. Cambridge University Press. (Discusses VEV and mass generation).
   [3] Polchinski, J. (1998). String Theory, Vol. I. Cambridge University Press. (Reference for RG flow and UV behavior).
   [4] Petrov, A. Z. (1964). Einstein Spaces. Pergamon Press. (Foundation for torsion-mediated metric coupling).
   [5] Van Der Waals, J. D. (2011). *On the Anomalous Casimir Pressure in …