The shear stress relaxation modulus G(t) may be determined from the shear stressτ (t) after switching on a tiny step strain γ or by inverse Fourier transformation of the storage modulus G (ω) or the loss modulus G (ω) obtained in a standard oscillatory shear experiment at angular frequency ω. It is widely assumed that G(t) is equivalent in general to the equilibrium stress autocorrelation function C(t) = βV δτ (t)δτ (0) which may be readily computed in computer simulations (β being the inverse temperature and V the volume). Focusing on isotropic solids formed by permanent spring networks we show theoretically by means of the fluctuation-dissipation theorem and computationally by molecular dynamics simulation that in general G(t) = Geq + C(t) for t > 0 with Geq being the static equilibrium shear modulus. A similar relation holds for G (ω). G(t) and C(t) must thus become different for a solid body and it is impossible to obtain Geq directly from C(t).

We investigate by means of molecular dynamics simulation a coarse-grained polymer glass model focusing on (quasistatic and dynamical) shear-stress fluctuations as a function of temperature T and sampling time Δt. The linear response is characterized using (ensemble-averaged) expectation values of the contributions (time averaged for each shear plane) to the stress-fluctuation relation μ_{sf} for the shear modulus and the shear-stress relaxation modulus G(t). Using 100 independent configurations, we pay attention to the respective standard deviations. While the ensemble-averaged modulus μ_{sf}(T) decreases continuously with increasing T for all Δt sampled, its standard deviation δμ_{sf}(T) is nonmonotonic with a striking peak at the glass transition. The question of whether the shear modulus is continuous or has a jump singularity at the glass transition is thus ill posed. Confirming the effective time-translational invariance of our systems, the Δt dependence of μ_{sf} and related quantities can be understood using a weighted integral over G(t).