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In this paper, we discuss the possibility of using strongly lensedgravitational waves (GWs) and their electromagnetic (EM) counterparts aspowerful cosmic rulers. In the EM domain, it has been suggested that jointobservations of the time delay ($\Delta\tau$) between lensed quasar images andthe velocity dispersion ($\sigma$) of the lensing galaxy (i.e., the combination$\Delta\tau/\sigma^{2}$) are able to constrain the cosmological parameters morestrongly than $\Delta\tau$ or $\sigma^{2}$ separately. Here, for the firsttime, we propose that this $\Delta\tau/\sigma^{2}$ method can be applied to thestrongly lensed systems observed in both GW and EM windows. Combining theredshifts, images and $\sigma$ observed in the EM domain with the very precise$\Delta\tau$ derived from lensed GW signals, we expect that accuratemultimessenger cosmology can be achieved in the era of third-generation GWdetectors. Comparing with the constraints from the $\Delta\tau$ method, weprove that using $\Delta\tau/\sigma^{2}$ can improve the discrimination betweencosmological models. Furthermore, we demonstrate that with $\sim50$ stronglylensed GW-EM systems, we can reach a constraint on the dark energy equation ofstate $w$ comparable to the 580 Union2.1 Type Ia supernovae data. Much morestringent constraints on $w$ can be obtained when combining the $\Delta\tau$and $\Delta\tau/\sigma^{2}$ methods.

Strongly Lensed Gravitational Waves and Electromagnetic Signals as Powerful Cosmic Rulers