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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?

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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 $r<r_S$) can't send signals to the outside, so the notion of time dilation of a signal (also called redshift in this context) doesn't make sense.

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.

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    $\begingroup$ What does "doesn't make sense" mean mathematically? $\endgroup$ – Era May 9 '16 at 14:06
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    $\begingroup$ @Era "Doesn't make sense" in this sense probably means that the mathematical model does not properly model real-world observations for all numerical values that could be plugged into the model. E.g., you can use a quadratic function to model the path of a thrown object under the influence of gravity, and that function can have values plugged into it that will produce "results" that make no sense in the real world, often because the input numbers would be modeling things that can't happen in the real world (like throwing a ball underground). That situation is analogous to this one. $\endgroup$ – Todd Wilcox May 9 '16 at 14:34
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I almost agree with Javier, except one point: Space and time flip positions inside a black hole, and so the imaginary time dilation is basically a space dilation. In short, imaginary time is space and imaginary space is time (because multiplication with $i$ changes the sign of the term in the metric)

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