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module module moderate

IndisputableMonolith.Relativity.Geometry.Metric

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The Metric module defines the metric tensor and its basic operations for spacetime geometry in Recognition Science. It supplies the Minkowski metric eta and functions to convert recognition-derived metrics to matrix form. Researchers in discrete gravity and black hole thermodynamics cite it to ground continuum limits. The content is mostly definitions with one symmetry property proved by direct appeal to tensor symmetry.

claimThe metric tensor $g$ is a symmetric bilinear form on the tangent space, satisfying $g(X,Y)=g(Y,X)$ for vector fields $X,Y$. The flat reference is the Minkowski metric $eta = diag(-1,1,1,1)$, with conversions $metric_to_matrix$ and $inverse_metric$ provided for coordinate computations.

background

In the Recognition Science approach to relativity, the metric arises from the J-cost functional on recognition events. The module builds on the Tensor module for index handling and the Derivatives module for basis vectors. It introduces MetricTensor as the core object and eta as the standard Minkowski form. Upstream, the Tensor module provides the scaffold for multilinear forms, while Derivatives supplies the coordinate basis.

proof idea

This is a definition module, no proofs. The single symmetry statement follows immediately from the definition of the metric tensor as a symmetric bilinear form.

why it matters in Recognition Science

This module supplies the metric foundation for downstream results in BlackHoleEntropy, where it enables derivation of Bekenstein-Hawking entropy from ledger capacity, and in Curvature for Christoffel symbols. It fills the geometry layer in the discrete-to-continuum bridge, connecting J-cost lattice to Einstein tensor. It touches the open question of how recognition flux induces the metric signature.

scope and limits

used by (11)

From the project-wide theorem graph. These declarations reference this one in their body.

depends on (2)

Lean names referenced from this declaration's body.

declarations in this module (11)