IndisputableMonolith.Relativity.Geometry.LeviCivitaTheorem
The module assembles the Levi-Civita connection on 4D spacetime by defining connection coefficients and proving existence plus uniqueness of the torsion-free metric-compatible connection. General-relativity workers cite it when passing from a metric to the Christoffel symbols and curvature tensors. The argument proceeds by successive definitions of torsion and compatibility conditions followed by direct algebraic verification of the Koszul identity and lowered-connection uniqueness.
claimOn a 4-dimensional spacetime with metric $g_{\mu\nu}$, the Levi-Civita connection $\Gamma^\rho{}_{\mu\nu}$ is the unique affine connection that is torsion-free ($\Gamma^\rho{}_{\mu\nu}=\Gamma^\rho{}_{\nu\mu}$) and metric-compatible ($\nabla_\sigma g_{\mu\nu}=0$).
background
The module belongs to the relativity geometry layer that converts discrete J-cost lattices into continuum curvature. It imports tensor algebra, metric tensors, curvature definitions, and derivative operators. Upstream material supplies the standard basis vectors and states that Christoffel symbols are derived from the metric. The central object is the connection $\Gamma^\rho{}_{\mu\nu}$ at each spacetime point.
proof idea
The module is organized as a chain of definitions (ConnectionCoeffs, IsTorsionFree, KoszulIdentity, LoweredConnectionIdentity) followed by lemmas that verify torsion freedom, metric compatibility, and uniqueness via the lowered connection. It ends with the existence theorem. Each step uses direct algebraic substitution from the Koszul formula.
why it matters in Recognition Science
The module supplies the continuum connection and curvature objects required by the DiscreteBridge, which maps lattice J-cost through quadratic defects and the lattice Laplacian to the Ricci scalar and Einstein tensor. It is re-exported by the Geometry aggregator. In the Recognition Science chain it completes the continuum limit step that produces the Einstein field equations from discrete defects.
scope and limits
- Does not derive the Einstein field equations.
- Does not incorporate discrete lattice J-cost.
- Does not compute explicit curvature scalars or components.
- Does not treat spacetimes of dimension other than four.
used by (2)
depends on (4)
declarations in this module (17)
-
abbrev
ConnectionCoeffs -
def
IsTorsionFree -
def
cov_deriv_metric -
def
IsMetricCompatible -
theorem
levi_civita_torsion_free -
def
KoszulIdentity -
theorem
metric_compatible_of_koszul -
theorem
levi_civita_metric_compatible -
def
lowered_connection -
def
LoweredConnectionIdentity -
theorem
connection_uniqueness_lowered -
structure
FundamentalTheoremExistence -
structure
FundamentalTheoremUniqueness -
theorem
fundamental_theorem_existence -
theorem
fundamental_theorem_uniqueness -
structure
LeviCivitaCertificate -
theorem
levi_civita_certificate