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The zero-temperature response of an interacting electron liquid to a time-dependent vector potential of wave vector q and frequency ω, such that q≪ q F, qv F≪ ω≪ E F/ħ (where q F, v F, and E F are the Fermi wave vector, velocity, and energy, respectively), is equivalent to that of a continuous elastic medium with nonvanishing shear modulus μ, bulk modulus K, and viscosity coefficients η and ζ. We establish the relationship between the viscoelastic coefficients and the long-wavelength limit of the “dynamical local-field factors” G L (T)(q, ω), which are widely used to describe exchange-correlation effects in electron liquids. We present several exact results for μ, including its expression in terms of Landau parameters, and practical approximate formulas for μ, η, and ζ as functions of density. These are used to discuss the possibility of a transverse collective mode in the electron liquid at sufficiently low density. Finally, we …