From: mk_thisisit
The study of distance and paths, particularly geodesic lines, forms a fundamental bridge between mathematics and physics, offering insights into the nature of the universe, from the geometry of space-time to the behavior of black holes [00:00:21].
The Metrizable Problem: Recreating Distance from Curves
A long-standing mathematical problem, open for 120 years, centered on the concept of distance [00:00:50]. Posed by French mathematician Roger Liouville at the end of the 19th century, it is known as the metrizable problem [00:01:01]. The core question is whether it is possible to recreate a concept of distance from a given family of curves, such that these curves represent the shortest paths [00:01:15].
An everyday example is straight lines on a plane, where the concept of distance derives from the Pythagorean theorem, and these lines are known to be the shortest path between two points [00:01:26]. The problem reverses this, asking if a concept of distance can be recreated solely from the existence of such “straight lines” (or shortest curves) [00:01:44].
Professor Maciej Dunajski solved this problem, finding the key idea in Austria around 2007 or 2008, with further work done in Cambridge with colleagues Mike from Australia and Robert Bryan from Berkeley [00:02:01].
Geodesic Lines and Einstein’s Theory of Gravity
In Einstein’s theory of gravity, what we perceive as movement in a gravitational field—such as the motion of galaxies, planets, or comets—is actually movement along “geodesic lines” [00:02:53]. These are the shortest lines within a specific concept of space-time metrics (distance) [00:03:06].
The “metric” is the unknown in Einstein’s equations, representing how to find this distance [00:03:15]. Einstein’s brilliant insight was to link this metric and the curvature of space-time to matter [00:03:24].
The physical application of the metrizable problem asks: If we observe the trajectories of all celestial bodies in the universe, can we recreate the metric—and thus the entire space-time—solely from these paths? [00:03:38] It turns out, usually, the answer is yes [00:04:32].
The Metric as a Modified Pythagorean Theorem
In Euclidean geometry (plane geometry), distance is measured using the Pythagorean theorem [00:05:01]. However, in the context of space-time and Einstein’s theory, this theorem undergoes a change [00:05:25]. Space-time includes not only spatial directions but also the time direction, which is accounted for differently [00:05:33].
The “metric” (as Einstein defined it) is a formula or algorithm that describes how the Pythagorean theorem is modified at different points in space-time, where the curvature of space-time varies, to consistently calculate distance [00:05:46]. Nature, for reasons unknown, chooses paths where this distance is either minimal or maximal, essentially the most economical way of moving from point to point [00:06:11].
In classical Einstein’s theory of gravity, the starting point is the metric (concept of distance), from which trajectories are calculated by solving differential equations [00:06:29]. For example, knowing the metric of a black hole allows calculation of how a galaxy would fall into it [00:06:51]. The problem solved by Professor Dunajski reverses this: given all possible paths, can we infer the metric of the universe? [00:07:03]
Black Holes and Singularities: Where Physics Collapses
Understanding black holes is crucial for a “quantum leap” in scientific understanding [00:00:27]. In Einstein’s theory, the existence of black holes paradoxically suggests the theory’s own collapse [00:09:40]. Physics and mathematics reach their limits when describing black holes, particularly at the point of singularity [00:09:09].
Roger Penrose’s work in the mid-1960s, later joined by Stephen Hawking in the 1970s, showed that, under physically reasonable mathematical assumptions, the existence of a “singularity” is inevitable [00:11:01].
A singularity is a point or region in space-time where the curvature of space-time becomes infinitely large, or where geodesics (preferred curved lines in the universe) have a beginning but no end within space-time itself [00:11:34]. Physics, as we know it, effectively ends at such singularities [00:12:04].
Penrose received the Nobel Prize in 2020 for explaining how the existence of black holes results from general relativity [00:12:16]. However, his and Hawking’s work implies the existence of singularities, but not necessarily black holes themselves [00:12:44]. Black holes are defined as regions in space-time from which nothing, not even light or information, can escape once past the event horizon [00:13:00].
The Cosmic Censor Hypothesis
A crucial open problem is the cosmic censor hypothesis [00:09:59]. While singularities are predicted, it is not clear if every singularity must be hidden behind an event horizon, preventing observers from gaining insight into its interior [00:14:06]. This hypothesis suggests a “cosmic censor” that forbids us from looking into the nature of singularities [00:14:32]. This remains a significant open problem in both physics and mathematics, specifically in the field of nonlinear differential equations [00:14:50].
The Information Paradox and Hawking Radiation
A major breakthrough in black hole research, controversial to classical theory, suggests that what falls into a black hole might eventually come out [00:15:53]. Classically, Einstein’s theory of relativity states that nothing, not even information, can escape a black hole [00:16:37].
In the 1970s, Stephen Hawking attempted to reconcile quantum mechanics with gravity, despite the lack of a complete quantum theory of gravity [00:16:51]. His calculations, using the concept of pair creation (where a particle and its antiparticle can be created from high energies), showed that near the black hole’s horizon, an antiparticle might fall in while its partner particle escapes to infinity [00:17:22]. This phenomenon is known as Hawking radiation [00:17:58].
Hawking radiation implies that black holes, despite their immense mass, will eventually evaporate [00:18:03]. This leads to the “information paradox”: is the information about matter that falls into a black hole destroyed forever, or can it be recovered via Hawking radiation? This remains an extremely interesting and unsolved problem [00:19:12].
Observation and Validation
Despite never sending an object close to a black hole, our knowledge comes from observations and the predictive power of theories [00:19:45]. A theory’s validity rests on how well its predictions align with observations [00:20:07]. This is why Penrose waited nearly 60 years for his Nobel Prize, as direct evidence of black holes was needed [00:20:16].
Recent developments in gravitational wave telescopes and the James Webb Space Telescope have provided direct and indirect evidence for the existence of black holes [00:20:34]. While black holes themselves are unobservable (being “black”), scientists observe the evolution of matter near them [00:20:49]. For example, the spiraling motion of nebulae, galaxies, or intergalactic gas, explainable only by the gravitational attraction of a huge black hole, provides crucial information about their nature and how they curve space-time [00:21:05].
The Pursuit of a Grand Unification Theory
The current understanding of fundamental physics is largely based on separate theories for elementary particle interactions (electromagnetic, strong, and weak nuclear forces) and gravity [00:39:16]. A “grand unification theory” aims to connect these [00:39:39]. However, a complete quantum theory of gravity does not yet exist [00:39:29].
One perspective, championed by Roger Penrose and Professor Dunajski, suggests that the lack of progress in unification might stem from an over-reliance on standard quantum mechanics [00:40:23]. They propose that instead of altering gravity theory, it is quantum mechanics that needs modification, specifically by incorporating the effects of gravity into its mathematical structure, such as the Schrödinger equation [00:37:07]. This could potentially explain the “wave function collapse” or reduction as a physical process that occurs in time, with more massive objects experiencing immediate reduction [00:37:39].
This approach is highly controversial within the physics community [00:42:48]. The goal is to return to the foundational mathematical structure of quantum mechanics and resolve the problem of wave function reduction through gravitational interactions [00:40:57]. This path might lead closer to a comprehensive unification theory [00:41:41].