Suppose that you can start a the Minkowski vacuum $(H-E_{\Omega})|{\Omega}\rangle=0$. Then for any time-like killing vector (which I’ll think of as specifying a time-like curve or some accelerated observer) we can ask whether there is vacuum. Locally the region in space over which the killing field is defined can be put in the form of Rindler coordinates. In other words, at each instance of proper time we know what the acceleration is and general covariance tells you that locally physics is the same as Minkowski space. So the Minkowski vacuum for this observer should look like a thermal state, maybe with a varying temperature. In other words, an accelerated observer always sees an effective horizon to which one can assign a temperature, so your questions should be answered by the Unruh effect.

Suppose that you can start a the Minkowski vacuum $(H-E_{\Omega})|{\Omega}\rangle=0$. Then for any time-like Killing vector (which I’ll think of as specifying a time-like curve or some accelerated observer) we can ask whether there is vacuum. Locally the region in space over which the killing field is defined can be put in the form of Rindler coordinates. In other words, at each instance of proper time we know what the acceleration is and general covariance tells you that locally physics is the same as Minkowski space. So the Minkowski vacuum for this observer should look like a thermal state, maybe with a varying temperature. In other words, an accelerated observer always sees an effective horizon to which one can assign a temperature, so your questions should be answered by the Unruh effect.