The nature of our universe, specifically the structure of space-time, raises fascinating questions about whether its complex features stem from physical laws or are purely products of human thought. It has exercised philosophers from time immemorial, but recent research delves mathematically into this intriguing issue, exploring the mathematical foundations that describe our reality. By examining the applicability of differential geometry to physical space-time, this study challenges traditional views on whether these mathematical tools are inherent to the universe or crafted by mathematicians.
Dr. Rathindra Nath Sen from Ben-Gurion University of the Negev, Beer Sheva, investigates whether the differential structure of space-time, a fundamental assumption of modern physics, derives from physical principles or is merely a mathematical construct. Published in the journal Entropy, this work delves into the longstanding debate on the nature of mathematics and its application to the physical world.
The investigation begins with a critical look at contrasting views on mathematics. Wigner’s famous assertion about the ”unreasonable effectiveness of mathematics in the natural sciences” contrasts sharply with Cantor’s claim that mathematics is a ”free creation of the human mind.” While Cantor’s power set construction has been pivotal in set theory, Dr. Sen argues it is not directly applicable to physics as it disrupts the essential notion of neighborhoods in space-time.
The concept of relativistic causality is central to this work. Basing himself of earlier work with the late Professor H.-J. Borchers of the University of Göttingen, Dr. Sen uses causality to explore whether space-time’s differential structure could inherently follow from physical principles. He points out that Einstein causality can be defined on a countably infinite set of points with no predefined mathematical structure. This endows the set with a Tychonoff topology, allowing it, after some mathematical gymnastics, to be embedded as a closed subspace of a product of real lines RJ . Dr. Sen emphasizes that ”this suggests that the differentiable structure of RJ may follow from the principle of causality,” although he acknowledges the empirical untestability of the completion processes required for this transition.
Throughout his work, Dr. Sen revisits historical milestones in the intersection of mathematics and physics. He discusses how Zeeman demonstrated that causality implies the Lorentz group, indicating that certain mathematical structures might stem from physical laws rather than human invention. The discovery of multiple differentiable structures on RJ by Donaldson and Gompf further complicates the picture, suggesting that the mathematical framework used in physics may not be unique or entirely grounded in empirical reality.
Dr. Sen’s research ultimately raises profound questions about the nature of mathematical constructs in physical theories. He argues that the traditional separation of a geometrical point from its neighborhood, fundamental to Cantor’s power set construction, is not physically meaningful. Instead, he suggests that the mathematics relevant to physics must maintain the intrinsic connection between points and their neighborhoods, as this reflects the empirical realities of space-time.
In conclusion, while Dr. Sen’s study does not definitively answer whether the differential structure of space-time is a discovery or an invention, it significantly advances the discourse. He proposes that “the continuum hypothesis and other set-theoretical constructs may not be necessary for the mathematical formulation of physical laws,” urging a reconsideration of how mathematics and physics intersect. As research in quantum gravity and other frontier fields continues, the insights from this research will likely inform future explorations into the fundamental nature of space-time.
Journal Reference
Sen, Rathindra Nath. “Does the Differential Structure of Space-Time Follow from Physical Principles?” Entropy 26, no. 3 (2024): 179. DOI: https://doi.org/10.3390/e26030179
About The Author
Dr. Rathindra Nath Sen was born in Calcutta (now Kolkata) and studied in Colvin Taluqdars’ School, Lucknow and St. Stephen’s College, Delhi. He did his Ph. D. with the late professor Giulio Racah at the Hebrew University of Jerusalem. He then spent four years in Naples and two in Milwaukee, and has lived in Israel since then. He has been at the Ben-Gurion University of the Negev since 1975, retiring in 2001. His scientific works include developing the theory of symmetry of infinite quantum-mechanical systems using fibre bundles; investigating the mathematical implications of Einstein-Weyl causality (with H.-J. Borchers) and, more recently, extending the work of G. L. Sewell to a complete resolution of the measurement problem in quantum mechanics.
