  It is well established that linear dispersive modes in a flowing quantum
fluid behave as though they are coupled to an Einstein-Hilbert metric and
exhibit a host of phenomena coming from quantum field theory in curved space,
including Hawking radiation. We extend this analogy to any nonrelativistic
Goldstone mode in a flowing spinor Bose-Einstein condensate. In addition to
showing the linear dispersive result for all such modes, we show that the
quadratically dispersive modes couple to a special nonrelativistic spacetime
called a Newton-Cartan geometry. The kind of spacetime (Einstein-Hilbert or
Newton-Cartan) is intimately linked to the mean-field phase of the condensate.
To illustrate the general result, we further provide the specific theory in the
context of a pseudo-spin-1/2 condensate where we can tune between relativistic
and nonrelativistic geometries. We uncover the fate of Hawking radiation upon
such a transition: it vanishes and remains absent in the Newton-Cartan geometry
despite the fact that any fluid flow creates a horizon for certain wave
numbers. Finally, we use the coupling to different spacetimes to compute and
relate various energy and momentum currents in these analog systems. While this
result is general, present day experiments can realize these different
spacetimes including the magnon modes for spin-1 condensates such as $^{87}$Rb,
$^{7}$Li, $^{41}$K (Newton-Cartan), and $^{23}$Na (Einstein-Hilbert).
