Module V·Article III·~1 min read
Geometry and Physics: From Euclid to Einstein
Affine and Projective Transformations
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Geometry and Physics
Euclidean Geometry and Classical Mechanics
Newtonian mechanics assumes Euclidean space: an absolute three-dimensional space whose metric does not depend on the observer. Time is absolute and independent.
Distances and angles are physically meaningful invariants. Newton’s laws are invariant under Galilean transformations: r' = r − vt, t' = t.
Special Theory of Relativity
Minkowski space-time: $\mathbb{R}^4$ with the metric $ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$.
Invariants: the interval $ds^2 = \text{const}$ under Lorentz transformations. Not a sum of squares, but a difference—a hyperbolic metric.
Light cone: $ds^2 = 0$ (for photons). Causal structure.
Euclidean distance is replaced by Lorentzian "distance." The geometry of space-time is pseudo-Euclidean (signature $-+++$).
General Theory of Relativity
Riemannian geometry: space-time is a four-dimensional Riemannian manifold with a variable metric $g_{\mu\nu}(x)$.
The metric is determined by the distribution of matter (Einstein’s equations). Geodesics (shortest paths) are the trajectories of freely falling particles.
Test of predictions: deflection of light by the Sun, precession of Mercury’s perihelion, gravitational redshift.
Analytical geometry became the language of physics—from Newtonian mechanics to quantum gravity.
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