Separating a particle's mass from its momentum

The Quantum Cheshire Cat experiment showed that when weak measurements are performed on pre- and post-selected systems, the counterintuitive result obtains that a neutron appears to be measured in one place without its spin, and its spin appears to be measured in another place without the neutron. A generalization of this effect is presented with a massive particle whose mass appears to be measured in one place with no momentum, while the momentum appears to be measured in another place without the mass. The new result applies to any massive particle, independent of its spin or charge. A gedanken experiment which illustrates this effect is presented using a nested pair of Mach-Zehnder interferometers, but with some of the mirrors and beam splitters moving relative to the laboratory frame. Ideally, the mass is measured using a gravitational detector, while the beams splitters, mirrors, and momentum detector couple to some other property of the particle, which may vary from one implementation to another.