Dealing with interference is one of the primary challenges to solve in the design of protocols for wireless ad-hoc networks. Most of the work in the literature assumes localized or hop-based interference models in which the effect of interference is neglected beyond a certain range from the transmitter. However, interference is a more complex phenomenon that cannot, in general, be captured by localized models, implying that protocols based on such models are not guaranteed to work in practice. This paper is the first to present and rigorously analyze a distributed dominating set protocol for wireless ad-hoc networks with O(l) approximation bound based on the physical interference model, which accounts for interference generated by all nodes in the network. The proposed protocol is fully distributed, randomized, and extensively uses physical carrier sensing to reduce message overhead. It does not need node identifiers or any kind of prior information about the system, and all messages are of constant size (in bits). We prove that, by appropriately choosing the threshold for physical carrier sensing, the protocol stabilizes within a logarithmic number of communication rounds, w.h.p., which is faster than the runtime of any known distributed protocol without prior knowledge about the system under any wireless model that does not abstract away collisions.