is a Killing tensor field for Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime which models a homogeneous and isotropic universe. ( is the metric tensor, is the 4-velocity of observers comoving with the Hubble flow, and is the scale factor. The coordinate choice should not matter, as we can easily write the expression in a tensorial way. One can check that indeed , where the parentheses denote symmetrisation of the indices.) This is useful because Killing vectors and tensors correspond to symmetries of spacetime and hence conserved quantities. We can use to conveniently derive the cosmic redshift of photons, and decay of “peculiar velocity” for particles with mass. We start with the quantity:
which is conserved along a geodesic (more precisely, is the tangent vector to the geodesic, under some affine parameter e.g. proper time). It follows
For a photon, its tangent vector has norm-squared , and energy as measured by a Hubble observer (or proportional to this, depending on your choice of affine parameter). For a massive particle, its norm-squared is , and which is the Lorentz factor for the motion relative to the Hubble flow observer (assuming its worldline is parametrised by proper time). The usual results follow. In my view this is more elegant than most textbook approaches.
Note you can tweak some quantities, because multiplying a conserved quantity by a constant gives another conserved quantity, and the same applies for a Killing vector/tensor. So we may replace by the unnormalised scale factor , because is also a Killing tensor, with conserved quantity . Alternatively for a timelike particle with constant mass , we may define which is also conserved.
Curiously, the tensor contains the spatial projector for the Hubble observers. This is just the spatial part of the metric, along the usual homogeneous and isotropic slices. We can rearrange the conserved quantity to give: , then take the square root. In words, the spatial length of the tangent vector to a geodesic is inversely proportional to the scale factor. More precisely: this assumes an affine parameter, also the length is determined by Hubble flow observers. More simply, Hubble observers measure things to lessen? (I need to think about the spatial geodesic case more…)
The only textbook I know which identifies it as a Killing tensor is Carroll (2004, §8.5) . (However Wald (1984, §5.3a) deserves credit for using a Killing vector in the same fashion. See my forthcoming post on FLRW Killing vectors.) One source which gives the tensor is Maharaj & Maartens (1987a, §4) and (1987b, §3) . (While they identify the tensor, and the resulting conserved quantity which they interpret as linear momentum, the term “Killing tensor” is not mentioned.) Also there is similar content in much older sources, including Robertson & Noonan (1968) .