Mathematicians conceived of black holes long before astronomers actually found any?How?

Question

I read this statement in Lockhart's - "A Mathematician's Lament". But how could mathematicians figure out something like black holes even before astronomers noticing any of them?

Answer 1

Without going into the details (which I'm not intimately familiar with), it was possible because physicists and mathematicians (such as Lorentz, Einstein and Minkowski) had developed a good mathematical model of gravitation. Using this model - which is a system of partial differential equations - physicists and mathematicians were able to predict that gravitational singularities should form under certain conditions. As a very (very) simplified example of what I mean, suppose that we have modeled some quantity with the simple ODE

$$ \dot{x}=x^2,\quad x(0)=1 $$ The solution to this equation is the function

$$ x(t)=\frac{1}{1-t} $$

This function exhibits finite time blow-up, since when $t=1$, we have a singularity - a place where $x(t)$ becomes infinite. If the ODE models our system accurately, we should be able to observe this blow-up with measurements. The same is true of gravitational singularities - the model predicts a blow-up, so astronomers went looking for (and found) those singularities.