Strongly lensed gravitational waves as the probes to test the cosmic distance duality relation

The cosmic distance relation (DDR) associates the angular diameters distance ($D_A$) and luminosity distance ($D_L$) by a simple formula, i.e., $D_L=(1+z)^2D_A$. The strongly lensed gravitational waves (GWs) provide a unique way to measure $D_A$ and $D_L$ simultaneously to the GW source, hence can be used as probes to test DDR. In this paper, we prospect the use of strongly lensed GW events from the future Einstein Telescope to test DDR. We write the possible deviation of DDR as $(1+z)^2D_A/D_L=\eta(z)$, and consider two different parametrizations of $\eta(z)$, namely, $\eta_1(z)=1+\eta_0 z$ and $\eta_2(z)=1+\eta_0 z/(1+z)$. Numerical simulations show that, with about 100 strongly lensed GW events observed by ET, the parameter $\eta_0$ can be constrained at $1.3\%$ and $3\%$ levels for the first and second parametrizations, respectively.