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We consider perturbative gravity for small fluctuations of the metric and derive the quadratic Fierz–Pauli action for them. We derive its gauge invariance and fix it by the de Donder gauge, equivalent of the Lorenz gauge of electromagnetism, and leading to the KG equation. We also define synchronous and Newtonian gauges. Gravitational waves are defined, and their generation via retarded sources is shown. The formula for the gravitational wave perturbation in terms of the time variation of a quadrupole momentum is proven. Finally, the exact cylindrical wave of Einstein and Rosen is found.
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