We note that massless fields within the future and past light cone may be quantized as independent systems. The vacuum is shown to be a nonseparable state of these systems, exactly mirroring the known entanglement between the spacelike separated Rindler wedges. This leads to a notion of timelike entanglement. We describe an inertial detector which exhibits a thermal response to the vacuum when switched on at t=0, due to this property. The feasibility of detecting this effect is discussed, with natural experimental parameters appearing at the scale of 100 GHz.