Spinor Info

Many textbooks and many popular science books tell you that the event horizon, the so-called “point of no return” in the vicinity of a black hole is nothing special. Apart from increasing (but finite!) tidal forces, an observer would not notice anything special when he crosses the horizon. But is this really true?

If this statement were true, it would mean, in essence, that there is no measurement an observer could perform in his immediately vicinity to determine if his vicinity is at or near the event horizon. But this may not be the case; there may, in fact, be a quantity that is measurable (at least in principle).

The curvature of spacetime is described by the Riemann tensor \(R^{\mu\nu\rho\sigma}\). The gradient (covariant derivative) of this quantity is \(R^{\mu\nu\rho\sigma;\kappa}\). Forming the scalar product of this quantity with itself, we obtain an invariant scalar quantity,

\[ K=R^{\mu\nu\rho\sigma;\kappa}R_{\mu\nu\rho\sigma;\kappa}. \]

If we calculate \(K\) for the Schwarzschild metric of a nonrotating, uncharged black hole, we get

\[ K=720m^2\frac{r-2m}{r^9}. \]

This quantity becomes zero and changes sign at the Schwarzschild horizon \(r=2m\).

So never mind what the books say. In principle, an observer can measure the curvature tensor and its gradient, and therefore, can construct an instrument that measures this invariant \(K\). (Note that although I used the letter \(K\), this is not to be confused with the better known Kretchmann invariant.) If this is true, what other effects might there be that make the event horizon a special place?

There is another thing to think about. Often you hear that the Rindler horizon seen by an accelerating observer, or the cosmological horizon in an expanding universe are “just like” the Schwarzschild horizon (perhaps even suggesting that we might be living inside a black hole.) But this cannot be so! Two observers who are not moving the same way do not see the same Rindler horizon or the same cosmological horizon. These are only apparent horizons, their presence, even their existence dependent on the observer’s motion. In contrast, the Schwarzschild horizon is real: two observers can agree on its location regardless of their own location and motion.

I am still evaluating WordPress, but I thought it’d be instructive to write something about physics. (For one thing, it’d allow me to check how well I can include HTML equations in a WordPress post.)

Notably, about neutrino masses. The Standard Model (SM) of particle physics says that neutrinos are massless. Fermions in general are organized into left-handed doublets (with neutrinos and leptons forming one pair, while up-type and down-type quarks forming the other) and right-handed singlets, but there is no right-handed neutrino. An important claim of fame of the SM is that it is a theory that is unitary and renormalizable.

Trouble is, there is now plenty of observational evidence telling us that neutrinos are not massless. Why is that a problem? You take a left-handed neutrino that is massless, and it moves at the speed of light. You take a left-handed neutrino that is massive, and it moves slower than the speed of light. So, you move fast enough to catch up with it, pass it, and look back, and what do you see? A right-handed neutrino, that’s what. But in the SM, no right-handed neutrino exists.

So what happens when you add a right-handed neutrino? There are two basic choices: the neutrino may be represented by either a Majorana spinor or a Dirac spinor. Without going into needless detail, the first choice basically means that the neutrino is its own antiparticle, ν = ν. So why is this a problem? Because neutrinos carry lepton numbers (very simplistically, an electron can be thought of as the sum of a W^– particle that carries its electric charge and an electron neutrino that carries its “electronness”. That “electronness” is the lepton number. Now if a neutrino is its own antiparticle, that means that two neutrinos (each carrying a unit of “electronness”) can annihilate, and the two units of “electronness” disappear: lepton number is not conserved. In addition to aesthetic concerns, there are also stringent experimental limits on the violation of lepton number conservation.

The other possibility is that neutrinos are Dirac neutrinos. Dirac neutrinos have genuine antiparticles, so lepton number violation is not a problem. One big problem, however, is the smallness of the neutrino mass; as I understand it, the main objection is that the dimensionless coupling constant that governs the neutrino mass in the Lagrangian is small (of order 10^–12) and this worries a lot of people.

Yet another alternative is to account for the observed neutrino oscillations by putting in new interaction terms. While this can be done without a right-handed neutrino, it breaks the renormalizability of the standard model. Loosely speaking, this means that one can no longer assume that things happening at a low energy scale are not affected by things that only happen at high energy. This is not how we experience nature. On the contrary: we can build mechanical devices (relying on low-energy interactions between surface atoms) without worrying about chemistry; we can perform chemical experiments without worrying about nuclear physics; we can do nuclear physics without having to worry about the internal structure of protons and neutrons; and so on. In other words, most physical phenomena can be “renormalized”, treated as if the upper limit of its applicability was infinite, and still yield meaningful results; you only need to worry about higher energy phenomena once you reach that higher energy scale. Why would neutrino physics be different?