Physics and mathematics their interplay and implications

From: mk_thisisit

Professor Maciej Dunajski, a Polish mathematician, recently solved a problem that had remained open for 120 years [00:00:05]. This breakthrough, achieved in Austria [00:00:17], holds significant implications for the understanding of black holes and the nature of the universe [00:00:08]. His work highlights the profound interplay between mathematics and physics in advancing scientific knowledge [01:01:10].

The Metrizability Problem and its Physical Significance

Problem Definition

The problem solved by Professor Dunajski, known as the metrizable problem, was posed by French mathematician Roger Liouville at the end of the 19th century [01:00:58], [01:06:06]. It asks whether it’s possible to reconstruct a concept of “distance” from a given family of curves such that these curves represent the shortest paths [01:15:15]. An everyday example is straight lines on a plane, where distance is derived from the Pythagorean theorem, and these lines are known to be the shortest paths between two points [01:26:18]. The problem reverses this: given straight lines, can one recreate the concept of distance [01:44:03]?

Connection to Einstein’s Theory of Relativity

In 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” [02:53:07]. These are the shortest lines within a particular concept of distance in space-time, known as the “metric” [03:06:07]. Einstein’s equations typically seek to find this metric, which dictates how distance is measured and how the curvature of space-time depends on matter [03:15:06].

The metrizability problem, in the context of relativity, asks: if we know the trajectories of all celestial bodies in the universe, can we reconstruct the entire space-time metric from these paths alone [03:40:08]?

Recreating the Universe’s Metric

The problem becomes even more complex when considering black holes [04:08:08]. Geodesics falling into a black hole enter a region from which they cannot escape, meaning their entirety is unknown [04:11:08]. The question then arises whether the universe’s metric can be reconstructed from such incomplete geodesics [04:26:08]. It turns out that, usually, this is possible [04:32:02]. The metric is a mathematical term for what describes distance and how space-time evolves [04:43:01].

Understanding Black Holes

Limitations of Current Knowledge

Despite advancements, our understanding of black holes remains limited [08:56:04]. If we truly understood black holes, it would represent a “quantum leap” in scientific understanding [09:30:08]. Einstein’s theory itself predicts the existence of black holes and paradoxically “collapses” within them at the point of singularity [09:12:08], [09:47:04].

Singularities and Their Implications

Roger Penrose and Stephen Hawking demonstrated that, under physically reasonable mathematical assumptions, the existence of “singularities” is inevitable [11:24:08]. A singularity is a point or region in space-time where the curvature can become infinitely large [11:34:04], or where geodesic lines have a beginning but no end within space-time, effectively ending where physics as we know it ceases [11:49:03]. Penrose received the Nobel Prize in 2020 for his work explaining how the existence of black holes arises from the theory of relativity [12:20:00]. However, their work shows the existence of singularities, not necessarily that these must be hidden within a black hole’s event horizon [12:44:03], [14:06:01].

The Cosmic Censor Hypothesis

Black holes are regions from which nothing, not even light, can escape [13:00:03]. The “Cosmic Censor Hypothesis” proposes that all singularities must be hidden inside an event horizon, preventing observers from seeing what happens within [13:48:07], [14:20:01]. This is a major open problem in physics and mathematics, requiring solutions to complex non-linear differential equations [14:50:00].

The Information Paradox (Hawking Radiation)

A significant recent breakthrough suggests that what falls into a black hole might eventually come out [16:00:02]. Classically, Einstein’s theory states nothing can escape [16:37:05]. However, in the 1970s, Stephen Hawking attempted to reconcile quantum mechanics with gravity by calculating what happens to quantum mechanical particles near a black hole’s horizon [16:51:00]. Using the concept of pair creation (particle-antiparticle pairs), he theorized that one particle could fall into the black hole while the other escapes to infinity, a phenomenon now called Hawking radiation [17:24:06]. This implies black holes, though massive, eventually evaporate [18:03:07]. The “information paradox” then asks whether the information of matter falling into a black hole is destroyed or can be retrieved through Hawking radiation [18:54:02]. This remains an unsolved, highly interesting problem [19:35:01].

Observational Evidence

While black holes cannot be directly observed as they are “black” [20:49:09], advancements in gravitational wave telescopes and instruments like the James Webb Telescope provide direct and indirect evidence of their existence [20:34:00]. Scientists observe the evolution of matter near black holes, such as the spiral motion of nebulae or galaxies, which can only be explained by the immense gravitational attraction of a hidden black hole [20:59:02]. This allows them to deduce information about black holes by observing their influence on space-time and celestial bodies [21:31:02].

The Relationship Between Mathematics and Physics

Mathematics as a Tool for Understanding Nature

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