  Quantum geometry is a key quantity that distinguishes electrons in a crystal
from those in the vacuum. Its study continues to provide insights into quantum
materials, uncovering new design principles for their discovery. However,
unlike the Berry curvature, an intuitive understanding of the quantum metric is
lacking. Here, we show that the quantum metric of Bloch electrons leads to a
momentum-space gravity. In particular, by extending the semiclassical
formulation of electron dynamics to second order, we find that the resulting
velocity is modified by a geodesic term and becomes the momentum-space dual of
the Lorentz force in curved space. We calculate this geodesic response for
magic-angle twisted bilayer graphene and show that moir\'e systems with flat
bands are ideal candidates to observe this effect. Extending this analogy with
gravity further, we find that the momentum-space dual of the Einstein field
equations remains sourceless for pure states while for mixed states it acquires
a source term that depends on the von Neumann entropy, for small entropies. We
compare this stress-energy equation with the weak-field limit of general
relativity and conclude that the von Neumann entropy is the momentum-space dual
of the gravitational potential. Consequently, the momentum-space geodesic
equation for mixed states is modified by a term resembling an entropic force.
Our results highlight connections between quantum geometry, momentum-space
gravity and quantum information, prompting further exploration of this dual
gravity in quantum materials.
