Schwarzschild coordinates beyond the event horizon

We can write down the metric of the Schwarzschild black hole in Schwarzschild coordinates.

On page 6 of the notes by Leonard Susskind of a course given at the Perimeter Institute titled 'Black Holes and Holography.' we find the following:

However, note that the Schwarzschild coordinates are only formally valid for $r > R_{s}$, and must be analytically continued within the event horizon.

Which aspect of the metric in Schwarzschild coordinates indicates that the coordinates are only valid outside the event horizon?

Coordinates are not sacred objects in GR. Any coordinate system is just as good as any other coordinate system. So to ask whether the Schwarzschild coordinates are valid or not is a meaningless question ${}^1$.

However it is reasonable to ask if coordinates have an intuitive meaning for some specified observer. So for example if we take an observer far from the massive object then the Schwarzschild time coordinate is the time as measured by that observer's clock, and the Schwarzschild radial coordinate is the circumference of a circle centred on the object divided by $2\pi$. Both these are intuitively meaningful measurements for our observer.

The problem with the interior of the black hole is that for our external observer anything falling into the black hole takes an infinite time even to reach the event horizon, let alone pass through it, so that makes us sceptical about any physical meaning for the Schwarzschild time inside the event horizon. And indeed if we write the Schwarzschild metric:

$$ds^2 = -\left(1-\frac{r_s}{r}\right)dt^2 + \frac{dr^2}{1-\frac{r_s}{r}}+r^2d\theta^2 + r^2\sin^2\theta d\phi^2$$

We find that the sign of the $dt^2$ and $dr^2$ terms changes as we move through the horizon. Since the sign tells us whether a term in the metric is spacelike or timelike this means inside the horizon $t$ behaves like a spatial coordinate and $r$ like a time coordinate.

This doesn't mean anything freaky, like time turning into space and vice versa as the more lurid popular science articles would have you believe, it just means the coordinates don't have the intuitive meaning that we associate with them outside the event horizon. However as I said right at the outset, they remain perfectly good coordinates and we just have to be careful about interpreting them.

${}^1$ the coordinates are singular at the horizon, i.e. at $r=r_s$, so they are not useful exactly at the horizon.

• John, if the sign of the metric changes inside the event horizon, what does it mean that time-like separated events become spacelike separated? If I had a radioactive particle, it's safe to say I wouldn't be able to "look around and see different decays arrayed out in space" correct? Is this just a mathematical blip without any meaning? So often these mathematical blips carry significance in the history of physics. Dec 7 '21 at 12:29
• @CasualScience that isn't a meaningful question because you cannot observe anything beyond the event horizon so it is meaningless to ask about observations of radioactive decays inside the horizon. Dec 7 '21 at 12:32

I think the coordinates are valid inside the horizon as well, albeit inside the horizon $r$ is timelike and $t$ is spacelike. They are not valid at the horizon since both $r$ and $t$ have zero coefficients in the metric there. That means you need some other coordinate system there to connect the exterior and interior solutions.