general relativity - Is there such thing as imaginary time dilation?

When I was doing research on General Relativity, I found Einstein's equation for Gravitational Time Dilation. I discovered that when you plugged in a large enough value for $M$ (around $10^{19}$ kilograms), and plugged in $1$ for $r$, then the equation would give an imaginary answer. What does this mean?

Answer

Nice discovery! The formula for time dilation outside a spherical body is

$$\tau = t\sqrt{1-\frac{2GM}{c^2r}}$$

where $\tau$ is the proper time as measured by your object at coordinate radius $r$, $t$ is the time as measured by an observer at infinity, $M$ the mass of the spherical body, and $G$ and $c$ the gravitational constant and the speed of light. You have noticed that when $r$ gets small enough, the square root can become imaginary. To get a real result you must have

$$r>\frac{2GM}{c^2}=r_S$$

where I have defined $r_S$, the Schwarzschild radius.

Well, there's a simple reason for this. If your body has a radius smaller than $r_S$, then it's a black hole, and the formula doesn't apply because objects inside the black hole (that is, with $r

Indeed, as $r$ approaches the Schwarzschild radius (from above) the redshift approaches infinity; this is why it is said that if you observe from far away a probe falling into a black hole, you will see it getting redder and moving slower as it falls; you'll never actually see it get into the black hole.

To answer the question in the title: no, there's no such thing as imaginary time dilation. Getting an imaginary result here is a sign that the formula doesn't always make sense.