Cosmologic philosophy: new math supports flat universe (Introduction)

by David Turell , Thursday, November 30, 2023, 21:24 (359 days ago) @ David Turell

Using Einstein's math plus new techniques:

https://www.quantamagazine.org/a-century-later-new-

"While it might seem obvious that smaller mass would lead to smaller curvature, things are not so cut and dry when it comes to general relativity. According to the theory, dense concentrations of matter can “warp” a portion of space, making it highly curved. In some cases, this curvature can be extreme, possibly leading to the formation of black holes. This could occur even in a space with small amounts of matter, if it’s concentrated strongly enough.

"In a recent paper, Conghan Dong, a graduate student at Stony Brook University, and Antoine Song, an assistant professor at the California Institute of Technology, proved that a sequence of curved spaces with smaller and smaller amounts of mass will eventually converge to a flat space with zero curvature.

"This result is a noteworthy advance in the mathematical exploration of general relativity — a pursuit that continues to pay dividends more than a century after Einstein devised his theory. Dan Lee, a mathematician at Queens College who studies the mathematics of general relativity but was not involved in this research, said that Dong and Song’s proof reflects a deep understanding of how curvature and mass interact.

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"At the heart of the conjecture is a measurement of curvature. Space can curve in different ways, different amounts, and different directions — like a saddle (in two dimensions) that curves up going forward and back, but down going to the left and right. Dong and Song ignore those details. They use a concept called scalar curvature, which represents the curvature as a single number that summarizes the full curvature in all directions.

***

"Dong and Song’s new work, said Daniel Stern of Cornell University, is “one of the strongest results we have so far that shows us how scalar curvature controls [the] geometry” of the space as a whole. Their paper illustrates that “if we have nonnegative scalar curvature and small mass, we understand the structure of space very well.”

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"To simplify their task, Dong and Song adopted another mathematical trick from Stern and his co-authors, which showed that a three-dimensional space can be divided into infinitely many two-dimensional slices called level sets, much as a hard-boiled egg can be segmented into narrow sheets by the taut wires of an egg slicer.

"The level sets inherit the curvature of the three-dimensional space they comprise. By focusing their attention on level sets rather than on the bigger three-dimensional space, Dong and Song were able to reduce the dimensionality of the problem from three to two. That’s very beneficial, Song said, because “we know a lot about two-dimensional objects … and we have a lot of tools to study them.”

"If they could successfully show that each level set is “kind of flat,” Song said, this would allow them to attain their overall goal of showing that a three-dimensional space with little mass is close to flat. Fortunately, this strategy panned out."

Comment: further proof the space of the Universe is flat.