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Using perturbation theory to describe the orbits of smaller bodies around large ones in space required Batygin to posit all objects in each specific orbit as a single entity and “smear” them into the form of a concentric ring, or wire. In the model, each such ring exhibited the same gravitational force as the combined individual objects, but uniformly distributed.
In such an approach, the solar system, for instance, would be represented by the sun, followed by a wire ring for each planet, plus others for the asteroid belt and Kuiper belt. Computer simulations representing millions of years showed that these rings behaved in ways that closely mirrored the behaviour of the real composite disc surrounding the sun.
Batygin then started refining the model, realising that he could portray any astrophysical system as a centre surrounded by ever more numerous, but ever thinner, wires until, inevitably, the wires blended into a single plane.
“Eventually, you can approximate the number of wires in the disk to be infinite, which allows you to mathematically blur them together into a continuum,” he says. “When I did this, astonishingly, the Schrödinger equation emerged in my calculations.”
This was a surprise, because the equation was thought to be only applicable to phenomena occurring on a quantum scale. It is used to describe one of the most bizarre aspects of quantum mechanics – the way in which subatomic particles behave simultaneously like particles and waves, a condition known as “wave-particle duality”.
“This discovery is surprising because the Schrödinger equation is an unlikely formula to arise when looking at distances on the order of light-years,” says Batygin.
"What I have presented in this paper is a framework," Batygin said. "I have attacked one particular problem with it, which is the problem of disk rigidity — the extent to which the disk can remain gravitationally rigid under external perturbations. There's a broad range of additional applications that I'm looking into at the moment."
One example is the evolution of the disk of debris that eventually formed our solar system, Batygin said. Another is the dynamics of rings around extrasolar planets. And a third is the disk of stars surrounding the black hole at the center of the Milky Way, which itself is highly bent.
Laughlin noted that the work should be particularly helpful in improving researchers' understanding of newborn star systems because they're harder to observe from afar, and researchers currently can't simulate their development from start to end.
I doubt it.
originally posted by: projectvxn
Quantum gravity research will likely benefit from this.
Also, no, this is not the fundamental link between general relativity and quantum mechanics that particle physicists are hunting for.
The General time-dependent Schrödinger equation
Ĥ | ψ (r,t)
In physical cosmology, cosmological perturbation theory is the theory by which the evolution of structure is understood in the big bang model. It uses general relativity to compute the gravitational forces causing small perturbations to grow and eventually seed the formation of stars, quasars, galaxies and clusters.
One method arose in the mathematical field of algebraic geometry. Here, mathematicians study polynomial equations — for example, x^2 + y^2 = 1 — by graphing their solutions (a circle, in this case). More-complicated equations can form elaborate geometric spaces. Mathematicians explore the properties of those spaces in order to better understand the original equations. Because mathematicians often use complex numbers, these spaces are commonly referred to as “complex” manifolds (or shapes).
The other type of geometric space was first constructed by thinking about physical systems such as orbiting planets. The coordinate values of each point in this kind of geometric space might specify, for example, a planet’s location and momentum. If you take all possible positions of a planet together with all possible momenta, you get the “phase space” of the planet — a geometric space whose points provide a complete description of the planet’s motion. This space has a “symplectic” structure that encodes the physical laws governing the planet’s motion.
Symplectic and complex geometries are as different from one another as beeswax and steel. They make very different kinds of spaces. Complex shapes have a very rigid structure. Think again of the circle. If you wiggle it even a little, it’s no longer a circle. It’s an entirely distinct shape that can’t be described by a polynomial equation. Symplectic geometry is much floppier. There, a circle and a circle with a little wiggle in it are almost the same.
“Algebraic geometry is a more rigid world, whereas symplectic geometry is more flexible,” said Nick Sheridan, a research fellow at Cambridge. “That’s one reason they’re such different worlds, and it’s so surprising they end up being equivalent in a deep sense.”
In the early 1990s, physicists were trying to figure out the details of string theory. They wanted to explain the physical world as a product of tiny, vibrating strings woven through an additional six dimensions of space. They tried to understand what the geometry of those six dimensions might be. The first option came from the mathematical field of algebraic geometry; a second one came from the mathematical field of symplectic geometry. To the trained mathematical eye, the two could hardly have seemed more different.
And yet, the physicists noticed some strange similarities between them. In particular, when they performed a calculation on one space, they generated numbers that matched the numbers they generated when they performed a very different type of calculation on the other side. “Two things that looked, in principle, unrelated, magically were equal,” said Denis Auroux, a mathematician at the University of California, Berkeley.
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In this new work, the four mathematicians use techniques from a field called tropical geometry. Using those techniques, they prove that these “special pieces” explain why numbers on opposite sides of the mirror differ by exactly a factor of zeta values. So far, their proof holds for many cases of mirror symmetry. The authors are waiting until they’ve been able to prove even more cases before they make the proof public.