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  • richardmitnick 8:22 pm on March 18, 2018 Permalink | Reply
    Tags: , , Mathematics vs Physics, Minkowski space, , , Shake a Black Hole, , The black hole stability conjecture   

    From Quanta: “To Test Einstein’s Equations, Poke a Black Hole” 

    Quanta Magazine
    Quanta Magazine

    mathematical physics
    https://sciencesprings.wordpress.com/2018/03/17/from-ethan-siegel-where-is-the-line-between-mathematics-and-physics/

    March 8, 2018
    Kevin Hartnett

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    Fantastic animation. Olena Shmahalo/Quanta Magazine

    In November 1915, in a lecture before the Prussian Academy of Sciences, Albert Einstein described an idea that upended humanity’s view of the universe. Rather than accepting the geometry of space and time as fixed, Einstein explained that we actually inhabit a four-dimensional reality called space-time whose form fluctuates in response to matter and energy.

    Einstein elaborated this dramatic insight in several equations, referred to as his “field equations,” that form the core of his theory of general relativity. That theory has been vindicated by every experimental test thrown at it in the century since.

    Yet even as Einstein’s theory seems to describe the world we observe, the mathematics underpinning it remain largely mysterious. Mathematicians have been able to prove very little about the equations themselves. We know they work, but we can’t say exactly why. Even Einstein had to fall back on approximations, rather than exact solutions, to see the universe through the lens he’d created.

    Over the last year, however, mathematicians have brought the mathematics of general relativity into sharper focus. Two groups have come up with proofs related to an important problem in general relativity called the black hole stability conjecture. Their work proves that Einstein’s equations match a physical intuition for how space-time should behave: If you jolt it, it shakes like Jell-O, then settles down into a stable form like the one it began with.

    “If these solutions were unstable, that would imply they’re not physical. They’d be a mathematical ghost that exists mathematically and has no significance from a physical point of view,” said Sergiu Klainerman, a mathematician at Princeton University and co-author, with Jérémie Szeftel, of one of the two new results [https://arxiv.org/abs/1711.07597].

    To complete the proofs, the mathematicians had to resolve a central difficulty with Einstein’s equations. To describe how the shape of space-time evolves, you need a coordinate system — like lines of latitude and longitude — that tells you which points are where. And in space-time, as on Earth, it’s hard to find a coordinate system that works everywhere.

    Shake a Black Hole

    General relativity famously describes space-time as something like a rubber sheet. Absent any matter, the sheet is flat. But start dropping balls onto it — stars and planets — and the sheet deforms. The balls roll toward one another. And as the objects move around, the shape of the rubber sheet changes in response.

    Einstein’s field equations describe the evolution of the shape of space-time. You give the equations information about curvature and energy at each point, and the equations tell you the shape of space-time in the future. In this way, Einstein’s equations are like equations that model any physical phenomenon: This is where the ball is at time zero, this is where it is five seconds later.

    “They’re a mathematically precise quantitative version of the statement that space-time curves in the presence of matter,” said Peter Hintz, a Clay research fellow at the University of California, Berkeley, and co-author, with András Vasy, of the other recent result [https://arxiv.org/abs/1606.04014].

    In 1916, almost immediately after Einstein released his theory of general relativity, the German physicist Karl Schwarzschild found an exact solution to the equations that describes what we now know as a black hole (the term wouldn’t be invented for another five decades). Later, physicists found exact solutions that describe a rotating black hole and one with an electrical charge.

    These remain the only exact solutions that describe a black hole. If you add even a second black hole, the interplay of forces becomes too complicated for present-day mathematical techniques to handle in all but the most special situations.

    Yet you can still ask important questions about this limited group of solutions. One such question developed out of work in 1952 by the French mathematician Yvonne Choquet-Bruhat. It asks, in effect: What happens when you shake a black hole?

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    Lucy Reading-Ikkanda/Quanta Magazine

    This problem is now known as the black hole stability conjecture. The conjecture predicts that solutions to Einstein’s equations will be “stable under perturbation.” Informally, this means that if you wiggle a black hole, space-time will shake at first, before eventually settling down into a form that looks a lot like the form you started with. “Roughly, stability means if I take special solutions and perturb them a little bit, change data a little bit, then the resulting dynamics will be very close to the original solution,” Klainerman said.

    So-called “stability” results are an important test of any physical theory. To understand why, it’s useful to consider an example that’s more familiar than a black hole.

    Imagine a pond. Now imagine that you perturb the pond by tossing in a stone. The pond will slosh around for a bit and then become still again. Mathematically, the solutions to whatever equations you use to describe the pond (in this case, the Navier-Stokes equations) should describe that basic physical picture. If the initial and long-term solutions don’t match, you might question the validity of your equations.

    “This equation might have whatever properties, it might be perfectly fine mathematically, but if it goes against what you expect physically, it can’t be the right equation,” Vasy said.

    For mathematicians working on Einstein’s equations, stability proofs have been even harder to find than solutions to the equations themselves. Consider the case of flat, empty Minkowski space — the simplest of all space-time configurations. This solution to Einstein’s equations was found in 1908 in the context of Einstein’s earlier theory of special relativity. Yet it wasn’t until 1993 that mathematicians managed to prove that if you wiggle flat, empty space-time, you eventually get back flat, empty space-time. That result, by Klainerman and Demetrios Christodoulou, is a celebrated work in the field.

    One of the main difficulties with stability proofs has to do with keeping track of what is going on in four-dimensional space-time as the solution evolves. You need a coordinate system that allows you to measure distances and identify points in space-time, just as lines of latitude and longitude allow us to define locations on Earth. But it’s not easy to find a coordinate system that works at every point in space-time and then continues to work as the shape of space-time evolves.

    “We don’t know of a one-size-fits-all way to do this,” Hintz wrote in an email. “After all, the universe does not hand you a preferred coordinate system.”

    The Measurement Problem

    The first thing to recognize about coordinate systems is that they’re a human invention. The second is that not every coordinate system works to identify every point in a space.

    Take lines of latitude and longitude: They’re arbitrary. Cartographers could have anointed any number of imaginary lines to be 0 degrees longitude.

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    And while latitude and longitude work to identify just about every location on Earth, they stop making sense at the North and South poles. If you knew nothing about Earth itself, and only had access to latitude and longitude readings, you might wrongly conclude there’s something topologically strange going on at those points.

    This possibility — of drawing wrong conclusions about the properties of physical space because the coordinate system used to describe it is inadequate — is at the heart of why it’s hard to prove the stability of space-time.

    “It could be the case that stability is true, but you’re using coordinates that are not stable and thus you miss the fact that stability is true,” said Mihalis Dafermos, a mathematician at the University of Cambridge and a leading figure in the study of Einstein’s equations.

    In the context of the black hole stability conjecture, whatever coordinate system you’re using has to evolve as the shape of space-time evolves — like a snugly fitting glove adjusting as the hand it encloses changes shape. The fit between the coordinate system and space-time has to be good at the start and remain good throughout. If it doesn’t, there are two things that can happen that would defeat efforts to prove stability.

    First, your coordinate system might change shape in a way that makes it break down at certain points, just as latitude and longitude fail at the poles. Such points are called “coordinate singularities” (to distinguish them from physical singularities, like an actual black hole). They are undefined points in your coordinate system that make it impossible to follow an evolving solution all the way through.

    Second, a poorly fitting coordinate system might disguise the underlying physical phenomena it’s meant to measure. To prove that solutions to Einstein’s equations settle down into a stable state after being perturbed, mathematicians must keep careful track of the ripples in space-time that are set in motion by the perturbation. To see why, it’s worth considering the pond again. A rock thrown into a pond generates waves. The long-term stability of the pond results from the fact that those waves decay over time — they grow smaller and smaller until there’s no sign they were ever there.

    The situation is similar for space-time. A perturbation will set off a cascade of gravitational waves, and proving stability requires proving that those gravitational waves decay. And proving decay requires a coordinate system — referred to as a “gauge” — that allows you to measure the size of the waves. The right gauge allows mathematicians to see the waves flatten and eventually disappear altogether.

    “The decay has to be measured relative to something, and it’s here where the gauge issue shows up,” Klainerman said. “If I’m not in the right gauge, even though in principle I have stability, I can’t prove it because the gauge will just not allow me to see that decay. If I don’t have decay rates of waves, I can’t prove stability.”

    The trouble is, while the coordinate system is crucial, it’s not obvious which one to choose. “You have a lot of freedom about what this gauge condition can be,” Hintz said. “Most of these choices are going to be bad.”

    Partway There

    A full proof of the black hole stability conjecture requires proving that all known black hole solutions to Einstein’s equations (with the spin of the black hole below a certain threshold) are stable after being perturbed. These known solutions include the Schwarzschild solution, which describes space-time with a nonrotating black hole, and the Kerr family of solutions, which describe configurations of space-time empty of everything save a single rotating black hole (where the properties of that rotating black hole — its mass and angular momentum — vary within the family of solutions).

    Both of the new results make partial progress toward a proof of the full conjecture.

    Hintz and Vasy, in a paper posted to the scientific preprint site arxiv.org in 2016 [see above 1606.04014], proved that slowly rotating black holes are stable. But their work did not cover black holes rotating above a certain threshold.

    Their proof also makes some assumptions about the nature of space-time. The original conjecture is in Minkowski space, which is not just flat and empty but also fixed in size. Hintz and Vasy’s proof takes place in what’s called de Sitter space, where space-time is accelerating outward, just like in the actual universe. This change of setting makes the problem simpler from a technical point of view, which is easy enough to appreciate at a conceptual level: If you drop a rock into an expanding pond, the expansion is going to stretch the waves and cause them to decay faster than they would have if the pond were not expanding.

    “You’re looking at a universe undergoing an accelerated expansion,” Hintz said. “This makes the problem a little easier as it appears to dilute the gravitational waves.”

    Klainerman and Szeftel’s work has a slightly different flavor. Their proof, the first part of which was posted online last November [see above 1711.07597], takes place in Schwarzschild space-time — closer to the original, more difficult setting for the problem. They prove the stability of a nonrotating black hole, but they do not address solutions in which the black hole is spinning. Moreover, they only prove the stability of black hole solutions for a narrow class of perturbations — where the gravitational waves generated by those perturbations are symmetric in a certain way.

    Both results involve new techniques for finding the right coordinate system for the problem. Hintz and Vasy start with an approximate solution to the equations, based on an approximate coordinate system, and gradually increase the precision of their answer until they arrive at exact solutions and well-behaved coordinates. Klainerman and Szeftel take a more geometric approach to the challenge.

    The two teams are now trying to build on their respective methods to find a proof of the full conjecture. Some expert observers think the day might not be far off.

    “I really think things are now at the stage that the remaining difficulties are just technical,” Dafermos said. “Somehow one doesn’t need new ideas to solve this problem.” He emphasized that a final proof could come from any one of the large number of mathematicians currently working on the problem.

    For 100 years Einstein’s equations have served as a reliable experimental guide to the universe. Now mathematicians may be getting closer to demonstrating exactly why they work so well.

    See the full article here .

    Please help promote STEM in your local schools.

    STEM Icon

    Stem Education Coalition

    Formerly known as Simons Science News, Quanta Magazine is an editorially independent online publication launched by the Simons Foundation to enhance public understanding of science. Why Quanta? Albert Einstein called photons “quanta of light.” Our goal is to “illuminate science.” At Quanta Magazine, scientific accuracy is every bit as important as telling a good story. All of our articles are meticulously researched, reported, edited, copy-edited and fact-checked.

     
  • richardmitnick 12:35 pm on January 29, 2017 Permalink | Reply
    Tags: Einstein's theory of general relativity and the Metric Sensor or Reiman Metric, , , Minkowski space, Minowski developed the formalism of spacetime, , What Is Spacetime?   

    From Ethan Siegel: “What Is Spacetime?” 

    Ethan Siegel
    Jan 28, 2017

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    The fabric of the Universe, spacetime, is a tricky concept to understand. But we’re up to the challenge. Image credit: Pixabay user JohnsonMartin.

    When it comes to understanding the Universe, there are a few things everyone’s heard of: Schrödinger’s cat, the Twin Paradox and E = mc^2. But despite being around for over 100 years now, General Relativity — Einstein’s greatest achievement — is largely mysterious to everyone from the general public to undergraduate and graduate students in physics. For this week’s Ask Ethan, Katia Moskovitch wants that cleared up:

    Could you one day write a story explaining to a lay person what the metric is in GR?

    Before we get to “the metric,” let’s start at the beginning, and talk about how we conceptualize the Universe in the first place.

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    Quanta, whether waves, particles or anything in between, have properties that define what they are. But they require a stage on which to interact and play out the Universe’s story. Image credit: Wikimedia Commons user Maschen.

    At a fundamental level, the Universe is made up of quanta — entities with physical properties like mass, charge, momentum, etc. — that can interact with each other. A quantum can be a particle, a wave, or anything in some weird in-between state, depending on how you look at it. Two or more quanta can bind together, building up complex structures like protons, atoms, molecules or human beings, and all of that is fine. Quantum physics might be relatively new, having been founded in mostly the 20th century, but the idea that the Universe was made of indivisible entities that interacted with each other goes back more than 2000 years, to at least Democritus of Abdera.

    But no matter what the Universe is made of, the things it’s composed of need a stage to move on if they’re going to interact.

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    Newton’s law of Universal Gravitation has been superseded by Einstein’s general relativity, but relied on the concept of an instantaneous action (force) at a distance. Image credit: Wikimedia commons user Dennis Nilsson.

    In Newton’s Universe, that stage was flat, empty, absolute space. Space itself was a fixed entity, sort of like a Cartesian grid: a 3D structure with an x, y and z axis. Time always passed at the same rate, and was absolute as well. To any observer, particle, wave or quantum anywhere, they should experience space and time exactly the same as one another. But by the end of the 19th century, it was clear that Newton’s conception was flawed. Particles that moved close to the speed of light experienced time differently (it dilates) and space differently (it contracts) compared to a particle that was either slow-moving or at rest. A particle’s energy or momentum was suddenly frame-dependent, meaning that space and time weren’t absolute quantities; the way you experienced the Universe was dependent on your motion through it.

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    A “light clock” will appear to run different for observers moving at different relative speeds, but this is due to the constancy of the speed of light. Einstein’s law of special relativity governs how these time and distance transformations take place. Image credit: John D. Norton, via http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Special_relativity_clocks_rods/.

    That was where the notion of Einstein’s theory of special relativity came from: some things were invariant, like a particle’s rest mass or the speed of light, but others transformed depending on how you moved through space and time. In 1907, Einstein’s former professor, Hermann Minkowski, made a brilliant breakthrough: he showed that you could conceive of space and time in a single formulation. In one fell swoop, he had developed the formalism of spacetime. This provided a stage for particles to move through the Universe (relative to one another) and interact with one another, but it didn’t include gravity. The spacetime he had developed — still today known as Minkowski space — describes all of special relativity, and also provides the backdrop for the vast majority of the quantum field theory calculations we do.

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    Quantum field theory calculations are normally done in flat space, but general relativity goes beyond that to include curved space. QFT calculations are far more complex there. Image credit: SLAC National Accelerator Laboratory.

    If there were no such thing as the gravitational force, Minkowski spacetime would do everything we needed. Spacetime would be simple, uncurved, and would simply provide a stage for matter to move through and interact. The only way you’d ever accelerate would be through an interaction with another particle. But in our Universe, we do have the gravitational force, and it was Einstein’s principle of equivalence that told us that so long as you can’t see what’s accelerating you, gravitation treats you the same as any other acceleration.

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    The identical behavior of a ball falling to the floor in an accelerated rocket (left) and on Earth (right) is a demonstration of Einstein’s equivalence principle. Image credit: Wikimedia Commons user Markus Poessel, retouched by Pbroks13.

    It was this revelation, and the development to link this, mathematically, to the Minkowski-an concept of spacetime, that led to general relativity. The major difference between special relativity’s Minkowski space and the curved space that appears in general relativity is the mathematical formalism known as the Metric Tensor, sometimes called Einstein’s Metric Tensor or the Riemann Metric. Riemann was a pure mathematician in the 19th century (and a former student of Gauss, perhaps the greatest mathematician of them all), and he gave a formalism for how any fields, lines, arcs, distances, etc., can exist and be well-defined in an arbitrarily curved space of any number of dimensions. It took Einstein (and a number of collaborators) nearly a decade to cope with the complexities of the math, but all was said and done, we had general relativity: a theory that described our three-space-and-one-time dimensional Universe, where gravitation existed.

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    The warping of spacetime by gravitational masses, as illustrated to represent General Relativity. Image credit: LIGO/T. Pyle.

    Conceptually, the metric tensor defines how spacetime itself is curved. Its curvature is dependent on the matter, energy and stresses present within it; the contents of your Universe define its spacetime curvature. By the same token, how your Universe is curved tells you how the matter and energy is going to move through it. We like to think that an object in motion will continue in motion: Newton’s first law. We conceptualize that as a straight line, but what curved space tells us is that instead an object in motion continuing in motion follows a geodesic, which is a particularly-curved line that corresponds to unaccelerated motion. Ironically, it’s a geodesic, not necessarily a straight line, that is the shortest distance between two points. This shows up even on cosmic scales, where the curved spacetime due to the presence of extraordinary masses can curve the background light from behind it, sometimes into multiple images.

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    An example/illustration of gravitational lensing, and the bending of starlight due to mass. Image credit: NASA / STScI, via http://hubblesite.org/newscenter/archive/releases/2000/07/image/c/.

    Physically, there are a number of different pieces that contribute to the Metric Tensor in general relativity. We think of gravity as due to masses: the locations and magnitudes of different masses determine the gravitational force. In general relativity, this corresponds to the mass density and does contribute, but it’s one of only 16 components of the Metric Tensor! There are also pressure components (such as radiation pressure, vacuum pressure or pressures created by fast-moving particles) that contribute, which are three additional contributors (one for each of the three spatial directions) to the Metric Tensor. And finally, there are six other components that tell us how volumes change and deform in the presence of masses and tidal forces, along with how the shape of a moving body is distorted by those forces. This applies to everything from a planet like Earth to a neutron star to a massless wave moving through space: gravitational radiation.

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    As masses move through spacetime relative to one another, they cause the emission of gravitational waves: ripples through the fabric of space itself. These ripples are mathematically encoded in the Metric Tensor. Image credit: ESO/L. Calçada.

    You might have noticed that 1 + 3 + 6 ≠ 16, but 10, and if you did, good eye! The Metric Tensor may be a 4 × 4 entity, but it’s symmetric, meaning that there are four “diagonal” components (the density and the pressure components), and six off-diagonal components (the volume/deformation components) that are independent; the other six off-diagonal components are then uniquely determined by symmetry. The metric tells us the relationship between all the matter/energy in the Universe and the curvature of spacetime itself. In fact, the unique power of general relativity tells us that if you knew where all the matter/energy in the Universe was and what it was doing at any instant, you could determine the entire evolutionary history of the Universe — past, present and future — for all eternity.

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    The four possible fates of the Universe, with the bottom example fitting the data best: a Universe with dark energy. Image credit: E. Siegel.

    This is how my sub-field of theoretical physics, cosmology, got its start! The discovery of the expanding Universe, its emergence from the Big Bang and the dark energy-domination that will lead to a cold, empty fate are all only understandable in the context of general relativity, and that means understanding this key relationship: between matter/energy and spacetime. The Universe is a play, unfolding every time a particle interacts with another, and spacetime is the stage on which it all takes place. The one key counterintuitive thing you’ve got to keep in mind? The stage isn’t a constant backdrop for everyone, but it, too, evolves along with the Universe itself.

    See the full article here .

    Please help promote STEM in your local schools.

    STEM Icon

    Stem Education Coalition

    “Starts With A Bang! is a blog/video blog about cosmology, physics, astronomy, and anything else I find interesting enough to write about. I am a firm believer that the highest good in life is learning, and the greatest evil is willful ignorance. The goal of everything on this site is to help inform you about our world, how we came to be here, and to understand how it all works. As I write these pages for you, I hope to not only explain to you what we know, think, and believe, but how we know it, and why we draw the conclusions we do. It is my hope that you find this interesting, informative, and accessible,” says Ethan

     
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