Motivated by the recent developments in terahertz spectroscopy using pump-probe setups, we develop the theory of finite frequency nonlinear electro-optical responses in centrosymmetric metals starting from basic time dependent perturbation theory. We express the nonlinear current kernel as a sum of several causal response functions. These functions cannot be evaluated using perturbative field theory methods. Consequently, we associate each response function with an imaginary time ordered current correlation function, which can be factorized using Wicks theorem. The mapping between the response functions and the correlation functions, suitably analytically continued to real frequencies, is proven exactly. We derive constraints satisfied by the nonlinear current kernel and we prove a generalized $f$-sum rule for the nonlinear conductivity, all of which are consequences of particle number conservation. We apply the theory to compute the gauge invariant nonlinear conductivity of a system of noninteracting electrons in the presence of weak disorder. As special cases of this generalized nonlinear response, we discuss its third harmonic and its instantaneous terahertz Kerr signals. Our formalism can be used to compute the nonlinear conductivity in symmetry broken phases such as density waves and nematic states.
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