Foundations

Lorentz transformations preserve a combined space-time separation.

Free test motion follows the metric connection.

Spacetime curvature responds to stress and energy.

Clocks at different radii measure different light frequencies.

Homogeneity and isotropy reduce gravitational expansion to two balances.

For inertial frames in standard configuration with S′ moving at v along S's x axis, x′=γ_v(x−vt), t′=γ_v(t−vx/c²), y′=y and z′=z, where γ_v=(1−v²/c²)⁻¹ᐟ².

Along a timelike worldline in an inertial coordinate frame, dτ=dt√(1−u²/c²), and elapsed proper time is its integral.

With S′ moving at v along x, a particle's velocities obey u′_x=(u_x−v)/(1−vu_x/c²) and u′_y,z=u_y,z/[γ_v(1−vu_x/c²)].

A particle of invariant mass m has p=γ_u m u, E=γ_u mc² and K=(γ_u−1)mc²; E²−|p|²c²=m²c⁴.

For a light source and observer receding directly from each other at relative speed v, f_obs=f_source√[(1−v/c)/(1+v/c)].

The asymptotically flat, spherical vacuum exterior has ds²=−(1−r_s/r)c²dt²+(1−r_s/r)⁻¹dr²+r²dΩ², r_s=2GM/c².

Two static clocks outside a Schwarzschild source compare proper rates through √(1−r_s/r).

Weak far-zone gravitational waves change proper separations through transverse-traceless strain.

For a differentiable local field Lagrangian, a continuous global symmetry that changes it by a total divergence yields an on-shell conserved Noether current.

LIGO's 2016 report of the 14 September 2015 event combined a coherent transient in two detectors with noise-background tests and waveform comparison.