I've read anything getting close to a regular black hole would experience spaghettification but not when you get close to super-massive black hole.

Is there a point of "peak spaghettification" where the mass of the black hole exerts the greatest tidal force? Or, have I misunderstood something along the way?

  • $\begingroup$ AFAIK, the tidal forces will always get larger as the black hole gets smaller since they are proportional to (powers of?) $r_\text{head}/r_\text{feet}$ (distances measured from the singularity). $\endgroup$ – Danu Nov 12 '15 at 12:27
  • $\begingroup$ @Danu the tidal forces at the event horizon are inversely proportional to the mass of the black hole, but the tidal forces at a particular distance from the singularity are directly proportional to the black hole mass. $\endgroup$ – Asher Nov 12 '15 at 14:31

A tidal force happens because different parts of the infalling object are trying to move along different geodesics. Suppose we take the 3-vector $\eta$ to be the distance between two points, then we can calculate how $\eta$ changes as the object falls inwards. If $\eta$ is constant then the distance between the points isn't changing and there is no tidal force. If $\eta$ is increasing then the points are being stretched apart while if $\eta$ is decreasing the points are being compressed together.

Anyhow, after some frenzied pen scratching (the details can be found in any GR textbook) we get:

$$ \frac{D^2\eta^r}{d\tau^2} = \frac{r_s}{r^3}\eta^r $$

$$ \frac{D^2\eta^\theta}{d\tau^2} = -\frac{r_s}{2r^3}\eta^\theta $$

$$ \frac{D^2\eta^\phi}{d\tau^2} = -\frac{r_s}{2r^3}\eta^\phi $$

where $r$, $\theta$ and $\phi$ are the spatial Schwarzschild coordinates and $r_s$ is the Schwarzschild radius $r_s = 2GM/c^2$. $D$ is the covariant derivative. The quantities on the left hand side are effectively an acceleration, so they are a force per unit mass that we can interpret as a tidal force.

The point of all this is that the tidal forces are proportional to $1/r^3$. This means they are greatest at $r \rightarrow 0$ i.e. they reach a maximum as the infalling object reaches the singularity.

Note that the tidal forces are finite at any value of $r \gt 0$ so they are finite at the event horizon. In fact at the horizon, i.e. when $r = r_s$, we get (I'll show just the radial equation):

$$ \frac{D^2\eta^r}{d\tau^2} = \frac{1}{r_s^2}\eta^r = \frac{c^4}{4G^2M^2}\eta^r $$

So the tidal force at the horizon decreases as the inverse square of the black hole mass.


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