What is a singularity

Singularities are mathematical

Singularities are exceptional mathematical situations in which the equations go crazy. Different types of singularities can be distinguished, but not all are as alarming as the putative initial singularity of the Big Bang.

• Mathematical singularities occur in many functions. An example is the function f (x) = 1 / x, if one substitutes 0 for x.

• Numerical singularities are a hindrance in many areas of science, for example in hydrodynamics. They come about solely because of the limited calculation accuracy. For example, (1 + 10–12) - 1 = 0 if you calculate to the nearest eleven digits. Division by such an expression would have fatal consequences for the entire calculation, because you cannot divide by 0. A clever reformulation, however, gets such a numerical singularity under control: (1 - 1) + 10–12 results in eleven digits of computational accuracy still 10–1 2.

• Coordinate singularities are only the result of a certain system of description. The usual coordinate system of the earth has a singularity at the north and south pole, because the meridians overlap there. Now it is bitterly cold at the poles - but the laws of physics are not going crazy. Of course, coordinate singularities cannot simply be eliminated using computer tricks. “The only thing that helps is looking for a better coordinate system. There is still no suitable standard procedure for this, ”says Werner Berger, who at the Max Planck Institute for Gravitational Physics in Golm near Potsdam simulates the collision of black holes on the computer - a tricky matter because the metric is at its event horizon (the“ edge ”of the Black hole) becomes singular in the Schwarzschild solution. Put simply, this means: Radial distances to the imaginary “surface” of a black hole are infinitely large, although paradoxically, the circumference and surface can be calculated. Coordinate singularities are also artifacts without physical reality and can therefore in principle be avoided. “Choosing a good coordinate system is one of the secret recipes for a physically trustworthy simulation,” says Berger.

• “But even the best coordinate system is of no use if you come across the most delicate point in spacetime: the physical singularity,” continues Berger. “No other theory or methodology can give advice here. By definition, this point is not calculable. And here only one thing helps: to avoid this point computationally by all means. ”When simulating colliding black holes, this succeeds by simply“ cutting out ”the singularity - especially since the conditions inside black holes, beyond the event horizon, from the outside anyway are not visible. The big bang singularity cannot be avoided in this way, because our entire universe emerged from it. So the crucial question is whether such a physical singularity is real - a stop sign, as it were, for our knowledge and the end of all explanations - or whether it too only appears as an artifact of an inadequate theory and can be overcome with a better one. RV

November 2, 2004

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