Abstract: We present exact solutions for the quantum time evolution of two spatially separated, local inductor-capacitor (LC) oscillators that are coupled optomechanically to a long elastic strip that functions as a quantum thermal acoustic field bath. We show that the optomechanical coupling to the acoustic bath gives rise to entanglement dynamics between the two oscillators in the absence of photon exchange between them, and that significant entanglement develops periodically regardless of the bath temperature. Furthermore, the entanglement only occurs after the oscillators become causally connected. As a photon-phonon optomechanical analog of the quantum matter-graviton coupling, the considered model may shed light on the nature of  weak, quantum gravitational fields as mediators of quantum entanglement.