From: mk_thisisit
A significant mathematical problem, open for 120 years, has been solved, contributing greatly to the understanding of black holes [00:00:02]. This breakthrough, achieved by a Polish mathematician at the University of Cambridge (with the key insight occurring in Austria) [00:00:02], aims to bring humanity closer to understanding the nature of the universe [00:00:21]. Comprehending black holes is considered a “quantum leap” in understanding [00:00:27].
The Metrizable Problem
The problem, posed by French mathematician Roger Liouville at the end of the 19th century, is called the “metrizable problem” [00:00:58]. It asks whether it’s possible to recreate a concept of distance from a given family of curves, such that these curves are the shortest paths [00:01:15]. An everyday example is straight lines on a plane, where distance results from the Pythagorean theorem, and straight lines are the shortest paths between two points [00:01:26]. The inverse question is: can a concept of distance be recreated from the existence of straight lines? [00:01:44]
The key idea for solving this problem occurred to the mathematician around 2007 or 2008 while on skis in Austria [00:02:01]. Further work was done in Cambridge with colleagues Mikee from Australia and Robert Bry from Berkeley [00:02:17].
Connection to Einstein’s Theory of Relativity
The solution to the metrizable problem has profound implications for Einstein’s theory of relativity and black holes [00:02:42]. According to Einstein’s theory of gravity, what is perceived as movement in a gravitational field (e.g., galaxies, planets, comets) is actually movement along “geodesic lines” [00:02:53]. These are the shortest lines in a specific concept of “distance of space-time metrics” [00:03:06]. The “metric” (how to find this distance) is the unknown in Einstein’s equations [00:03:15].
Einstein’s brilliant idea was to make this metric and the curvature of space-time dependent on matter [00:03:24]. The problem in the context of relativity theory is: if we have the trajectory of all celestial bodies, can we recreate the metric—the entire space-time—from only these paths? [00:03:38]
Black Holes and Singularities
The paths of objects falling into black holes are geodesics, but they disappear into regions of space-time from which they cannot escape [00:04:01]. Their paths are only known up to the moment of “singularity” [00:04:18]. The question then becomes: can the metric of the universe still be recreated from such incomplete geodesics? It turns out that usually, yes [00:04:26].
In Einstein’s theory, the Pythagorean theorem is modified for space-time because it includes both spatial and time directions [00:05:25]. The metric is a formula or algorithm that describes how to consistently calculate distance in different points of space-time where curvature varies [00:05:46]. Nature chooses paths where this distance is either minimal or maximal, the most “economical” way of moving [00:06:11].
Typically, in classical Einstein’s theory of gravity, one starts with the metric and calculates trajectories [00:06:29]. The solved problem reverses this: given all possible paths, can the metric be uncovered? [00:07:03] This is the physical application of a problem posed by a pure mathematician [00:07:18].
The Mysteries of Black Holes
Understanding black holes is critical because our current understanding of physics “collapses” within them at the point of singularity [00:09:16]. Einstein’s theory of gravity predicts its own collapse within black holes [00:09:47].
Sir Roger Penrose, along with Stephen Hawking, showed that the existence of “singularities” is inevitable under physically reasonable mathematical assumptions [01:11:19]. A singularity is a point or region in space-time where space-time curvature becomes infinitely large, or where geodesics (preferred curved lines) have a beginning but no end, terminating somewhere outside space-time [01:11:30]. This is where physics, as currently understood, ends [01:12:04].
Penrose received the Nobel Prize in 2020 for explaining how the existence of black holes results from the theory of relativity [01:16:16]. However, the work of Penrose and Hawking proves the existence of singularities (end points of space-time where physics ends), but not necessarily that these singularities must be hidden within black holes (regions from which nothing can escape) [01:28:38].
The Cosmic Censor Hypothesis
The “cosmic censor hypothesis” proposes that all singularities must be hidden inside an event horizon, preventing observation of their nature [01:41:20]. Whether this hypothesis is true—whether a horizon must always surround a singularity—remains an open problem in both physics and mathematics [01:44:42]. This involves solving complex nonlinear differential equations [01:50:06]. Many researchers have been working on this for over 50 years with moderate success [01:54:50].
Hawking Radiation and the Information Paradox
A significant recent breakthrough in black hole research is the idea that what falls into a black hole might eventually come out [01:59:53]. This challenges the classical view from Einstein’s theory of relativity that nothing, not even light or information, can escape a black hole [01:37:37].
In the 1970s, Stephen Hawking attempted to reconcile quantum mechanics with the theory of gravity, despite the lack of a full quantum theory of gravity [01:54:51]. He calculated what happens to quantum mechanical particles near a black hole’s horizon, using the concept of “pair creation” (particle-antiparticle pairs created at high energies) [01:57:05]. His idea suggests that an antiparticle might fall into the black hole while its paired particle escapes to infinity, an effect known as “Hawking radiation” [01:59:58].
This implies that black holes, despite their immense mass, will eventually evaporate [02:08:03]. While the exact quantum theory of gravity is unknown, if it reduces to classical gravity at low energy levels and quantum mechanics laws are correct, then Hawking radiation should exist [02:08:23].
A major unresolved question is the “information paradox”: what happens to the information (energy, matter) that falls into a black hole during its evaporation? [02:18:54] One view is that this information is destroyed forever; another is that some mechanism allows it to be obtained after evaporation via Hawking radiation [02:19:09].
Empirical Evidence for Black Holes
Scientific research on black holes is possible because the theory’s predictions align with observations [02:22:25]. This is why Penrose had to wait almost 60 years for his Nobel Prize; while his theory was indisputable, direct evidence was needed [02:25:25]. In recent years, gravitational wave telescopes and the James Webb Space Telescope have provided direct and indirect evidence for the existence of black holes [02:34:06]. Although black holes themselves cannot be observed directly because they are black [02:39:51], their gravitational influence on surrounding matter (e.g., spiral motion of nebulae, galaxies, intergalactic gas) can be observed [02:49:59]. This allows scientists to infer information about black holes by observing how they curve space-time and alter celestial body movements [02:59:39].
The Nature of Mathematics and its Role in Physics
Mathematics can define infinity as the limit of a sequence or series, or as the size of a set (e.g., infinitely many natural numbers) [03:00:00]. While mathematics can cope with infinity, physics and nature generally cannot; everything describable in physics uses finite (though possibly huge) numbers [03:00:00].
Mathematics: Discovered or Created?
A core philosophical question is whether mathematics describes or creates reality [02:48:06]. The prevailing view among mathematicians is that they discover mathematics; it exists independently [02:55:00]. Mathematics exists in its own world of abstract Platonic ideas [02:59:20]. Mathematical truths (e.g., the Pythagorean theorem, the infinitude of prime numbers) would remain true even if humanity or the universe ceased to exist [02:59:36]. Mathematics is built upon axioms, fundamental rules that are not proven but assumed [03:32:00]. Euclid, for instance, laid out the axioms of Euclidean geometry, and mathematicians then deduce truths from them [03:32:00]. These deduced theorems exist regardless of human knowledge or desire [03:37:39].
While mathematics is an exceptionally effective tool for physics, chemistry, and biology, mathematicians do not “create” it [02:59:03]. It’s about discovering the inherent mathematical world [03:00:51].
The Relationship Between Physics and Mathematics
Physics cannot exist without mathematics, but mathematics could exist without physics [03:07:07]. The laws of physics are described by mathematics, but mathematical laws exist irrespective of the universe [03:01:29]. Since the time of Newton, it became clear that mathematics is the cognitive method for physics; to understand celestial body movement, one must solve equations [03:02:05]. Understanding physics often involves rewriting it as mathematical equations, solving them, and then interpreting the results back into physics [03:02:55]. There is a synergy between the two fields [03:10:08].
The Challenge of Unification
For almost 50 years, elementary particle physics has not advanced significantly [03:51:00]. While physicists have many ideas and breakthroughs, the connection between these discoveries and experiments is minimal [03:57:07]. The fundamental interactions (electromagnetic, strong, weak nuclear forces) are understood, but a quantum theory of gravity or “grand unification theory” that connects them all doesn’t yet exist [03:59:29].
One perspective is that the lack of progress stems from too much faith in quantum mechanics [04:26:00]. Many physicists believe gravity theory should be changed to a quantum theory of gravity [04:35:00]. However, the professor and Roger Penrose suggest that gravity theory should remain, and quantum mechanics needs to be re-examined and modified [04:47:00].
Their idea, developed over decades by Penrose, is that quantum mechanics is not a complete theory, and the effects of gravity should be taken into account [04:54:00]. Specifically, gravity should modify the Schrödinger equation in quantum theory such that the “wave function collapse” is a physical process that takes place over time [04:58:00]. This collapse time would be inversely proportional to mass, explaining why massive objects seem to reduce immediately, while microscopic objects can remain in superposition for longer [04:58:00].
This approach is controversial among scientists [04:59:00], highlighting the difference between physics (which provokes speculation) and mathematics (where theorems, once proven, cannot be disputed) [04:59:00]. Their current work focuses on exploring this idea mathematically, potentially using “twistor theory” [03:52:00].
Professor Maciej Dunajski will be an invited guest at the Megabit Bomb Festival of Science, Culture and Technology in Łódź [01:11:36].