Why would it be (or it be not) valid to take that neutrinos can follow the same path as null geodesics?

Question

Basically my main question is the one in the heading of my post, because apparently "Gravitation" by Misner, Thorne and Wheeler takes it directly as true and valid (a common place for null geodesics).

We know that the rest mass of neutrinos is very very low, although we are sure it is not entirely zero. According to Wikipedia (sorry, I don't know where to read more about this topic), their lowest rest mass can be around $\sim$ 0.1 - 1 eV/$c²$. So they don't have null mass, unlike photons which are regarded as particles of mass zero.

In General Relativity we can regard null geodesics as paths followed by photons, stuck onto the surface of null cones. So why is it valid (or maybe not) that we can take as an approximation that neutrinos follow those same null geodesics?

I can imagine that, because their rest mass is extremely small, it is more straightforward to consider that the energy of neutrinos can be simply $E\approx pc$, just like photons. And I can see that since their speed is nearly $c$, then their line element must be very close to that of a null event: $ds =c d\tau \approx 0$. So then, their proper time could be approximated too as null: $d\tau\approx 0$? And can we then assume they also lay in the null cone like photons, because an affine parameter for their geodesics will be always very close to zero?

Furthermore, I can imagine ultra relativistic electrons or muons going pretty close to $c$, however I know maybe it will not be true that their proper time is $d\tau \approx 0$.

Answer 1

As ProfRob mentions, most of the neutrinos (and antineutrinos) in the universe decoupled from the rest of the matter in the early moments of the Big Bang, when the universe was around 1 second old. These neutrinos now form the cosmic neutrino background (CNB). At the time of decoupling, the CNB neutrinos had a mean kinetic energy around 2.5 MeV, equivalent to ~17.4 billion K, but due to expansion their temperature is now ~1.95 K, and their mean kinetic energy is now estimated to be $10^{-6}$ to $10^{-4}$ eV.

Neutrinos are notoriously difficult to detect. Our best neutrino detectors, using the Ga → Ge → Ga reaction sequence (aka, the "Alsace-Lorraine" technique), have a threshold of 233 keV. And even then only a tiny fraction of the neutrinos passing through the detector actually trigger the reaction sequence. A well-known heuristic is that a light-year of lead would only absorb ~50% of the neutrinos passing through it (assuming the neutrinos have energies typical of those we can detect).

So neutrinos are effectively invisible unless their kinetic energy is at least on the order of a million times greater than their rest mass. That is, we can't detect neutrinos unless their Lorentz $\gamma$ is >1000000. The trajectories of such neutrinos are very close to the light-cone.

For example, the neutrinos emitted by the core-collapse supernova have energies in the range 10-30 MeV. We detected 19 neutrinos from the supernova SN1987A located in the Large Magellanic Cloud, at a distance of ~168,000 light-years. If we assume that neutrinos have a rest mass of 0.1 eV, a 20 MeV neutrino from SN1987A would arrive at Earth about 66 microseconds after a photon that left SN1987A at the same time.

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Wikipedia mentions that there are prospects of detecting CNB neutrinos, using tritium. I suppose we might be able to detect them by running the Alsace-Lorraine reaction in a particle accelerator like the LHC. But even if we can detect their presence, that won't give us any data on their trajectory.