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• An analytical relationship between the generator terminal bus voltage and current phasors and the internal rotor speed is derived via the Thevenin equivalent. • A reformulation of inertia estimation is proposed based on a first-principles equation derived from Newton's law applied to rotating masses. Specifically, the measured voltage phasor is the model input, whereas the estimated rotor speed and the measured real and reactive power from PMUs are treated as model outputs.. • The rotor speed estimation in the proposed method is online and model-free, a big difference from existing dynamic state estimation methods using detailed generator models. This paper proposes an online decentralized and data-driven method for synchronous generator rotor speed and inertia estimation. First, an analytical relationship between the generator terminal voltage/current phasors and its internal rotor speed is derived using the Thevenin equivalent. The parameters of the latter are obtained via a recursive estimator, which allows us to estimate the generator's internal rotor speed from voltage/current phasors without the need for any generator information. Second, we reformulate the equivalent swing equation into the Kalman filter form, which enables us to use a priori-estimated internal rotor speed along with measured real and reactive power injections to obtain an estimate of the inertia for each online synchronous generator. This is achieved through an adaptive unscented Kalman filter. We demonstrate that the use of terminal bus frequency to approximate the rotor speed for inertia estimation by existing methods leads to large estimation biases/errors. Numerical simulations carried out on the IEEE 39-bus and Texas 2000-bus systems reveal that our proposed method is consistently more accurate than existing methods. Moreover, our proposed method is insensitive to both the type and location of system disturbances, and it is robust to different levels of measurement noise. [ABSTRACT FROM AUTHOR]