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Lorentz symmetry (LS), one of the most fundamental physical symmetries, hasbeen extensively studied in the context of quantum gravity and unificationtheories. Many of these theories predict a LS violation, which could arise fromthe discreteness of spacetime, or extra dimensions. Standard-model extension(SME) is an effective field theory to describe Lorentz violation whose effectscan be explored using precision instruments such as atomic clocks andgravitational-wave (GW) detectors. Considering the pure-gravity sector andmatter-gravity coupling sector in the SME, we studied the leadingLorentz-violating modifications to the time delay of light and the relativisticfrequency shift of the clock in the space-based GW detectors. We found that thesix data streams from the GW mission can construct various combinations ofmeasurement signals, such as single-arm round-trip path, interference path,triangular round-trip path, etc. These measurements are sensitive to thedifferent combinations of SME coefficients and provide novel linearcombinations of SME coefficients different from previous studies. Based on theorbits of TianQin, LISA, and Taiji missions, we calculated the response ofLorentz-violating effects on the combinations of the measurement signal datastreams. Our results allow us to estimate the sensitivities for SMEcoefficients: $10^{-6}$ for the gravity sector coefficient $\bar{s}^{TT}$,$10^{-6}$ for matter-gravity coupling coefficients$(\bar{a}^{(e+p)}_{\text{eff}})_{T}$ and $\bar{c}^{(e+p)}_{TT}$, and $10^{-5}$for $(\bar{a}^{n}_{\text{eff}})_{T}$ and $\bar{c}^{n}_{TT}$.

Testing Lorentz symmetry with space-based gravitational-wave detectors