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.