# Mathematics, computer science and …black holes?

# BLOG: Heidelberg Laureate Forum

By now, the Lindau Lecture at the Heidelberg Laureate Forum is a treasured tradition: each HLF is joined by a Nobel Laureate talking about his or her work, and in turn, one of the mathematics or computer science laureates gives a lecture at each Lindau Nobel Laureate Meeting. But it’s rare for the connection with the HLF’s key topics of computer science and mathematics to mesh as tightly with a Nobel laureate talk than they did with this year’s talk by Reinhard Genzel, of the Max Planck Institute for Extraterrestrial Physics in Garching, who received the 2020 Nobel Prize in Physics (together with Andrea Ghez and Roger Penrose) for his work on the detection and characterisation of the black hole in the center of our very own Milky Way galaxy.

### Using mathematics to hunt for black holes – since 1784

The link with mathematics goes back a few hundred years: to Isaac Newton, who, had those two institutions existed back then, would surely have received both a physics Nobel Prize and a Fields Medal. After all, he invented in parallel a theory of mechanics and gravitational interaction, and the necessary formalism in the shape of the differential and integral calculus – which happens to be the basis of much of today’s mathematics. Genzel began the historical background portion of his talk with the two scientists who used Newton’s formalism to first derive something analogous to a black hole: a body so compact that its (Newtonian) escape velocity is greater than the speed of light; no light can escape from the surface of such a body to infinity.

One of them was Pierre-Simon Laplace, who is one of the mathematicians that brought Newton’s theory into the modern form we learn today. The other was John Michell, who in a 1784 article described the plan for what Genzel and his colleagues would achieve roughly 200 years later. Regarding the sort-of-black-holes he has just described, which are by definition themselves not visible, Michell speculates that “if other luminous bodies should happen to revolve about them we might still perhaps from the motions of these revolving bodies infer the existence of the central ones with some degree of probability” – which, of course, is just what Genzel’s group – and independently Andrea Ghez’s – did: they tracked the motion of stars around our Milky Way’s central black hole, and from the stellar motion deduced the black hole’s presence as well as the black hole’s mass.

### Journey with hard- and software

But the journey (here illustrated with a video from ESO, the European Southern Observatory, whose telescopes the Genzel group used) takes us from mathematics to computer science, as well.

The actual observations, as Genzel made clear, would not have been possible without cutting-edge hardware and software: control systems in the “adaptive optics” system that deform a mirror so as to counteract the image distortions due to turbulence in the Earth’s atmosphere, as well as camera chips sensitive to infrared radiation for the observations themselves. Last but not least, the computing power to put everything together: reconstruct the stellar motions around the black hole, and from there deduce the black hole’s mass. Genzel’s reply to the question by a participant about how computer scientists can help with this kind of research boiled down to the statement that astronomers have always been among the first customers for whatever new and powerful computers are available – from data analysis to comprehensive simulations of the history of the universe as a whole. When it comes to black holes, Lindau and Heidelberg can be much closer than you might think.

.. “

But the journey (here illustrated with a video from ESO, the European Southern Observatory, whose telescopes the Genzel group used) takes us from mathematics to computer science, as well. The actual observations, as Genzel made clear, would not have been possible without cutting-edge hardware and software.”….Is a trite statement, like: …”the paint was applied to the canvas with a brush, the artist’s hands, a palette knife or otherwise”….

With the difference:

Mathematics is free of real objects (see as a striking example the Banach-Tarski paradox).

Physical Models are in the ideal case axiomatically founded and consistent.

What is not the case here: Concretely: The General Theory of Relativity (GRT) was born among other things from the demand to be able to use arbitrary coordinate systems for the description of the laws of nature. According to the covariance principle, the form of the laws of nature should not depend decisively on the choice of the special coordinate system. This requirement leads to a variety of possible coordinate systems [metrics].

The equation systems (Einstein, Friedmann) of the general relativity theory, which are the basis of the statements of the standard model of cosmology, do not deliver analytical solutions. Only idealizations and approximations lead limitedly to computable solutions. The unavoidable (“covariant”) contradictions come with idealizations and approximations of the system of nonlinear, chained differential equations. Mathematically, the covariance principle cannot be “violated”, since it is axiomatically established. Only this axiomatic condition disappears with the idealization and approximation of the actual equations. In other words: The mathematically correct equations have no analytical solutions. The reduced equations (approximations, idealization) possess solutions, but these are not covariant. Thus, no solution possesses a real-physically justified meaning.

Exemplary contradiction to the existence of black holes

The physicist Walter Greiner (1935 – 2016, eight times honorary doctor, several times honorary professor, among others Max Born Laureate, Otto Hahn Laureate) should be known to all physicists, since he published an extensive textbook collection on theoretical physics, which has been the scientific basis for physics students since the mid-1970s. He fell out of favor in 2010 because he propagates a pulsating universe according to his current, independent calculations.

Greiner’s striking conclusion:there are no black holes. Greiner is a veteran and “heavyweight” of theoretical physics, thus of the mathematics on which it is based. On arxiv.org exist five papers by Walter Greiner on this topic and among others the article on scilogs:The Black Hole: To be or not to be!

What a weird hodgepodge of objections. At least those readers who are participating in the HLF should know better. Because no, the symmetries of underlying physical laws need not be satisfied by specific solutions of the differential equations – nobody demands that, neither in general relativity nor in other areas of physics. And as for “heavyweights” not being automatically correct, Atiyah’s purported proof of the Riemann Hypothesis, presented at the 2018 HLF is a cautionary tale regarding that kind of argument from authority. And of course the fact that the papers linked are all older than the data that has come in by now regarding the various tests where Greiner sees a difference between the predictions of general relativity and his own modification – relativistic effects for test body motion as seen by the GRAVITY instrument, the first direct detections of gravitational waves, the Event Horizon Telescope results – weakens the argument further. To the point where, going by past experience, my personal alarm bells are ringing and telling me I’m it’s much more likely I am not dealing with good-faith critical comments on Einstein’s theory (which are a valid part of scientific discussion!), but with a kind of fundamentalist anti-Einstein dogmatism (of which there are quite some examples in various corners of the internet).

Theory requirement and empirical findingAn experiment needs a concrete question for its conception. If the question is the result of a mathematical formalism, the experimental result is correspondingly theory-laden. If then, as usual in the context of standard models, the measurable results are preselected and only indirectly “connected” with the postulated theory objects, there is nothing left against the arbitrariness of interpretation.