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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?

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    $\begingroup$ General relativity! $\endgroup$ – Qiaochu Yuan May 19 '13 at 19:33
  • $\begingroup$ Any more details would be wonderful!! $\endgroup$ – eminemence May 19 '13 at 19:39
  • $\begingroup$ One issue is that the escape velocity can be greater than the speed of light. What happens then? $\endgroup$ – Mark Bennet May 19 '13 at 19:42
  • $\begingroup$ Google the keywords "Schwarzschild solution". And it is in fact in the very FIRST place black hole exists in math (some object has a smaller radius than its Schwarzschild radius), then astronomers went on to design methods to detect them (like exploiting gravity lens effect). $\endgroup$ – Shuhao Cao May 19 '13 at 20:22
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    $\begingroup$ A word of caution about the looseness of that claim: Most everyone involved in developing the theory of black holes would be described primarily as a theoretical physicist/astrophysicist, not a mathematician. You could ask how does any theorist make predictions prior to having observations, but that's just what a prediction is, and it's exactly what theoretical scientists do all day long. $\endgroup$ – user43318 May 19 '13 at 22:31
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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.

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  • $\begingroup$ Are mathematical rules followed in case of a "blow-up"? Can mathematicians look beyond? $\endgroup$ – eminemence May 19 '13 at 19:50
  • $\begingroup$ @eminemence There are ways of dealing with singularities, yes. It's delicate, of course, but possible. $\endgroup$ – icurays1 May 19 '13 at 19:52
  • $\begingroup$ So mathematicians predict based on models. Chances are that things may not turn out as predicted. Black hole was the case where things were found as predicted. $\endgroup$ – eminemence May 19 '13 at 19:54
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    $\begingroup$ @eminemence Correct - if we weren't able to find black holes, we would probably try to fix the model. A model is only good if the data supports it! $\endgroup$ – icurays1 May 19 '13 at 19:59

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