Physics / advanced
Geometric tools for spacetime physics
Build the geometric machinery of relativity through coordinate-invariant constructions, local frames, curvature, transport, and causal distinctions.
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Module 01 advanced
Geometric tools for spacetime physics
Build the geometric machinery of relativity through coordinate-invariant constructions, local frames, curvature, transport, and causal distinctions.
- 01
Coordinates label a patch rather than define spacetime
Use overlapping charts and nonsingular transitions to separate manifold structure from coordinate failure.
28 min - 02
A tensor stays fixed while its component arrays change
Transform tangent and cotangent components and verify invariant contractions under a shear.
28 min - 03
A Lorentzian metric distinguishes clocks, light cones, and volume
Declare the −+++ convention, compute proper time, and construct the coordinate-invariant integration measure.
28 min - 04
Two geometric conditions determine one connection
Derive the Levi-Civita coefficients and use flat polar coordinates to separate connection values from curvature.
28 min - 05
A geodesic transports its own tangent with an affine parameter
Separate parallel transport, coordinate acceleration, and reparameterization effects using a flat polar example.
28 min - 06
Curvature measures the failure of covariant derivatives to commute
Fix the Riemann sign convention and connect invariant curvature to oriented geodesic deviation.
28 min - 07
Curvature traces obey identities without containing every tidal component
Contract the Riemann tensor, derive the divergence-free Einstein tensor, and distinguish local identities from global flux statements.
28 min - 08
An orthonormal frame separates local measurements from coordinate labels
Construct a tetrad and its dual coframe, then distinguish a local Lorentz rotation from a coordinate transformation.
28 min - 09
Metric symmetries produce constants along affine geodesics
Derive Killing conservation and distinguish a symmetry charge from an observer’s local energy.
28 min - 10
Local light cones do not determine a spacetime’s global health
Separate causal character, extendibility, completeness, horizons, and curvature singularities through flat examples.
28 min