And as Wiki notes, modeling quantum dots showcases the interplay of quantum mechanical, semiclassical, and classical physics: “A variety of theoretical frameworks exist to model optical, electronic, and structural properties of quantum dots. These may be broadly divided into quantum mechanical, semiclassical, and classical.”

Phys.org > “Direct visualization of quantum dots reveals shape of quantum wave function” by University of California – Santa Cruz (Nov 24, 2020)

]]>Trapping and controlling electrons in bilayer graphene quantum dots yields a promising platform for quantum information technologies. Researchers at UC Santa Cruz have now achieved the first

direct visualization of quantum dots in bilayer graphene, revealingthe shape of the quantum wave function of the trapped electrons.Understanding the nature of the quantum dot wave function in bilayer graphene is important because this basic property determines several relevant features for quantum information processing, such as the electron energy spectrum, the interactions between electrons, and the coupling of electrons to their environment.

“The peaks represent sites of high amplitude in the wave function. Electrons have a dual wave-particle nature, and we are visualizing the wave properties of the electron in the quantum dot.”

• Overview of operations [PDF]

The ISS is

approximately 260 miles above the surface of the Earth, just a bit farther than the distance between Washington, D.C. and New York City.The space station is 357 feet wide, one yard shy of the length of an American football field including the end zones.

Up to eight spacecraft can connect to the space station at any time.

[Publications]

Biology and Biotechnology

Earth and Space Science

Human Research

Technology Development and Demonstration

Physical Science

Educational Activities and OutreachIt can take as few as 6 hours for a spacecraft to arrive to the station from Earth.

The ISS travels at a speed of 5 miles per second or 17,500 mph.In 24 hours, the space station makes 16 orbits of Earth, traveling through 16 sunrises and sunsets.

The Water Recovery System recycles 93% of space station waste water, reducing the dependence on water delivered by a cargo spacecraft.

• NASA > Johnson Space Center > “NASA Collaborates with Peanuts Worldwide in Celebration of International Space Station’s 20th Anniversary” by Heather Kinney (October 15, 2020)

NASA is weeks away [on November 2] from 20 years of continuous human presence on the International Space Station (ISS). In celebration, the NASA Space Flight Awareness Program (SFA) and the International Space Station (ISS) Program partnered with Peanuts Worldwide, LLC, to develop a unique product series featuring your favorite Beagle!

• YouTube > NASA > “A Bridge Above: 20 Years of the International Space Station” (Aug 10, 2020)

“What if we built a bridge, between and above all nations, to jointly discover the galaxy’s great unknowns?” Join us this fall as we prepare to celebrate the 20th anniversary of the International Space Station. As a global endeavor, 240 people from 19 countries have visited the unique microgravity laboratory, which has hosted more than 2,800 research investigations from scientists in over 100 nations.

• New NASA Space Flight Awareness posters

What might be expected if you asked, à la a Jay Leno “Jaywalking” segment, some random people this question: “Where’s the nearest black hole?” Well, tallying those that know what a black hole is [5] … those that know what the Milky Way is … those that know our Sun is the nearest star … [4]

Anyway, how did we determine that there’s a supermassive black hole in the center of our Milky Way? Black hole tropes have been around for decades. I’ve taken it for granted that almost every galaxy has a massive black hole in its center; so, why not ours? This post was inspired by recent articles about three scientists receiving the 2020 Nobel Prize in Physics for this very discovery.

• NPR > “3 Scientists Awarded Nobel Prize In Physics For Discoveries Related To Black Holes” by Geoff Brumfiel (October 6, 2020)

The prize was awarded to

Roger Penroseof the University of Oxford, for demonstrating that the general theory of relativity leads to the formation of black holes; and toReinhard Genzelof the Max Planck Institute for Extraterrestrial Physics andAndrea Ghezof the University of California, Los Angeles, for the discovery of a compact object at the center of the Milky Way galaxy that governs the orbits of stars, for which a black hole is the only known explanation.… for years after the prediction [shortly after Albert Einstein unveiled his general theory of relativity in 1915], researchers remained unsure whether black holes could form in the real universe, where conditions were often much more complicated than Einstein’s rarefied equations. It was

Penrose who found a more complex mathematical description of black hole formation that matched with the natural world. Published in 1965, his work “is still regarded as the most important contribution to the general theory of relativity since Einstein,” according to the Nobel Prize committee, which awarded him half of the prize for his work.

Genzel and Ghez won the other half for painstaking observations of the supermassive black hole at the center of our own galaxy.Known as Sagittarius A, it is more than 4 million times the mass of our sun. Sagittarius Ais shrouded behind a cloud of gas at the very core of the Milky Way, but undeterred, Genzel and Ghez used infrared telescopes to look through the gas. They painstakingly developed technologies to remove distortions caused by the gas and by Earth’s own atmosphere to track objects orbiting very close to the black hole.

• Scientific American > “How Andrea Ghez Won the Nobel for an Experiment Nobody Thought Would Work” by Hilton Lewis [W. M. Keck Observatory Director] (October 8, 2020) – She provided conclusive evidence for a supermassive black hole at the core of the Milky Way.

Standing in my office 25 years ago was an unknown, newly minted astronomer … She had come with an outrageous request … to carry out an experiment that was basically a waste of time and couldn’t be done – to prove that a massive black hole lurked at the center of our Milky Way.

It was my first encounter with … Andrea Ghez, one of three winners of this year’s Nobel Prize in Physics, for her work on providing the conclusive experimental evidence of a supermassive black hole with the mass of four million suns residing at the center of the Milky Way galaxy.

For 25 years she has focused almost exclusively on Sagittarius A* – the name of our own local supermassive black hole. It is remarkable that an entire field of study has grown up in the intervening quarter century, of searching for and finding evidence of these monsters thought to lie at the heart of every large galaxy. And Andrea is without question one of the great pioneers in this search.

Andrea’s co-prizewinner

Reinhard Genzelhas been involved in the same research from the outset—and it is the work of these two teams, each led by a formidable intellect and usingtwo different observatories in two different hemispheresthat has brought astronomy to this remarkable result – the confirmation of another of the predictions of Einstein’s more than century-old theory of general relativity.Andrea is a great scientist … In addition to doing research, she has created the

UCLA Galactic Center Groupto coordinate research and technical developments.Today, Andrea sits at the pinnacle of scientific recognition for her achievements. But as she would be the very first to acknowledge, this triumph represents the combined efforts of so many. … the product of the work of thousands.

• The Caltech Weekly > “Two Caltech Alumni Win 2020 Nobel Prizes” (Oct 8, 2020)

Andrea Ghez (MS ’89, PhD ’92) wins Nobel Prize in Physics for research demonstrating the presence of a supermassive black hole at the heart of the Milky Way galaxy, and virologist Charles M. Rice (PhD ’81) receives the 2020 Nobel Prize in Physiology or Medicine for his work on curing hepatitis C. Both are Distinguished Caltech Alumni.

• Caltech > News > “Alumna Andrea Ghez Awarded 2020 Nobel Prize in Physics” (October 6, 2020)

Andrea Ghez (MS ’89, PhD ’92), the Lauren B. Leichtman and Arthur E. Levine Professor of

Astrophysics at UCLA, has won the 2020 Nobel Prize in Physics for pioneering research that helped reveal a supermassive black hole lurking at the center of the Milky Way galaxy. She shares half the Nobel Prize with Reinhard Genzel of UC Berkeley and the Max Planck Institute for Extraterrestrial Physics. Together, Ghez and Genzel are being honored “for the discovery of a supermassive compact object at the centre of our galaxy.”The other half of the Nobel Prize goes to Roger Penrose of the University of Oxford, “for the discovery that black hole formation is a robust prediction of the general theory of relativity.”

At Caltech, Ghez’s PhD advisor was the late Gerry Neugebauer (PhD ’60), formerly the Robert Andrews Millikan Professor of Physics, Emeritus, and a founder of the field of

infrared astronomy. Ghez’s PhD thesis looked at the frequency of multiple-star systems and stellar evolution usingCaltech’s Palomar Observatory. She was named a Caltech Distinguished Alumna in 2012.At UCLA, where Ghez joined the faculty in 1994, she and her team began mapping stars in a region at the center of our galaxy known as

Sagittarius A*, around which all the stars in the Milky Way orbit.

• UCLA > Newsroom > “Andrea Ghez wins 2020 Nobel Prize in physics” by Stuart Wolpert (October 6, 2020) [includes video below] – UCLA professor is honored for her pioneering research on the Milky Way’s supermassive black hole.

In July 2019, the journal Science published a study by Ghez and her research group that is the most comprehensive test of Albert Einstein’s iconic general theory of relativity near

the monstrous black hole at the center of our galaxy. Although she concluded that “Einstein’s right, at least for now,” the research group is continuing to testEinstein’s theory, which she sayscannot fully explain gravity inside a black hole.Ghez studies more than

3,000 starsthat orbit the supermassive black hole. Black holes have such high density that nothing can escape their gravitational pull, not even light.The center of the vast majority of galaxies appears to have a supermassive black hole, she said.The

National Science Foundationfunded Ghez’s research for the past 25 years. More recently, her research has also been funded by the W.M. Keck Foundation, the Gordon and Betty Moore Foundation and the Heising-Simons Foundation, Lauren Leichtman and Arthur Levine, and Howard and Astrid Preston.Ghez earned a bachelor’s degree in physics from

MITin 1987 and a doctorate fromCaltechin 1992, and she has been a member of theUCLAfaculty since 1994. When she was young, she wanted to be the first woman to walk on the moon.

• YouTube > UCLA > “Andrea Ghez reacts to winning the Nobel prize in physics” (Oct 6, 2020) – “A full view of how this dance really works …”

• YouTube > UCLA > “Testing Einstein’s theory of relativity near a black hole” (Jul 25, 2019)

[1] As in: “Yes, Virginia, there is a Santa Claus.“

[2] Does Sgr A* have a companion?

• UCLA College > “That Supermassive Black Hole in our Galaxy? It has a Friend” by Smadar Naoz, associate professor of physics and astronomy in the UCLA College (December 20, 2019)

Almost every galaxy, including our Milky Way, has a supermassive black hole at its heart, with masses of millions to billions of times the mass of the sun. Astronomers are still studying why the heart of galaxies often hosts a supermassive black hole. One popular idea connects to the possibility that supermassive holes have friends.The supermassive black hole that lurks at the center of our galaxy, called Sgr A

, has a mass of about 4 million times that of our sun. A black hole is a place in space where gravity is so strong that neither particles or light can escape from it. Surrounding Sgr Ais a dense cluster of stars. Precise measurements of the orbits of these stars allowed astronomers to confirm the existence of this supermassive black hole and to measure its mass.For more than 20 years, scientists have been monitoring the orbits of these stars around the supermassive black hole.Based on what we’ve seen, my colleagues and I show that if there is a friend there, it might be a second black hole nearby that is at least 100,000 times the mass of the sun.

[3] Is there any relationship between the growth of black holes and their host galaxies? In particular, supermassive black holes (SMBHs). Here’s another example of research using sophisticated sets of (cosmological) simulations.

• Daily Galaxy > “‘Written in the Stars’ – Galaxies Supermassive Black Holes Linked to Stellar Growth” (Oct 2, 2019) – Astrophysicists continue to theorize about the origins of black holes, how they grow and glow, and how they interact with host galaxies in different astronomical environments.

“There has been a lot of uncertainty regarding the SMBH-galaxy connection, in particular whether SMBH growth was more tightly connected to the star formation rate or the mass of the host galaxy,” said Yale astrophysicist Priyamvada Natarajan, senior investigator of the new study, which appears in the journal Monthly Notices of the Royal Astronomical Society. “These results represent the most thorough theoretical evidence for the former – the growth rate of black holes appears to be tightly coupled to the rate at which stars form in the host.”

[4] So, what’s the answer? How far away is the nearest black hole? Regardless of mass – stellar class, massive, supermassive, …

These references (below) discuss the answer. Note that there is some debate about the status of HR 6819.

• Wiki > List of nearest black holes (within our Milky Way galaxy)

• Forbes > “How Close To Earth Is The Closest Black Hole?” by Ethan Siegel (May 11, 2020)

First predicted in 1916 in General Relativity, the first one wasn’t discovered in space until 1964:

Cygnus X-1.The second-largest black hole as seen from Earth, the one at the center of the galaxy

M87, …Sagittarius A*, at the center of the

Milky Way, is the closest supermassive black hole, some 25,000 light-years distant.A smaller one — just 6.6 solar masses — orbits a Sun-like star just 3,500 light-years away:

V616 Monocerotis.That distance record was shattered last week, by trinary system

HR 6819: two stars and a black hole 1,000 light-years distant.As our methods and surveys continue to improve, closer black holes will inevitably be discovered.

• BBC > “‘Nearest black hole to Earth discovered’” by Jonathan Amos (May 6, 2020)

Astronomers have a new

candidatein their search for the nearest black hole to Earth. It’s about 1,000 light-years away, or roughly 9.5 thousand, million, million km, in the Constellation Telescopium. [HR 6819]Astronomers have spotted only a couple of dozen black holes in our Milky Way Galaxy to date, nearly all of which strongly interact with their accretion discs.

“In the Milky Way, the idea is that there should be about 100 million black holes. So there should be perhaps a couple more that are closer by still,” Marianne Heida, a postdoctoral fellow at ESO, told BBC News.

[5] So, what is a black hole? Here’s an answer, in 5 levels of understanding.

WIRED has challenged NASA’s Varoujan Gorjian (Research Astronomer, NASA Jet Propulsion Lab) to explain black holes to 5 different people; a child, teen, a college student, a grad student and an expert.

• Wired > “Astronomer Explains One Concept in 5 Levels of Difficulty” (Season 1 Episode 6, Released on 07/05/2018).

(from transcript)

[Gorjian]

… in the AGN [**active galactic nuclei**] community, because we don’t know how the millions to billions solar mass black holes came to be. But it’s, at least we’re building up, or hopefully that at some point, and by understanding these lower mass, how these lower mass black holes came to be, then we can see where there are a large scale number of mergers can potentially give us this, or you really need something, other, another corridor to fundamentally get us something that’s a million solar masses, you know, on the minimum side, but definitely, you know, we’ve gotten those which are billion.

So we know we can merge the million solar mass black holes to get the bigger ones, but how do you get to those in the beginning, particularly so early in the universe, when you get quasars at really high red shifts, so they’re really early on.

[Madsen]

Yeah, it is odd, it is very odd. I mean, the other thing that’s a little odd, now we’re going back to stellar mass black holes is, so we look at a lot of supernova remnants, and we, so we see them, we can only really see them in our own galaxy, and so we have a lot of supernova remnants, and so how we see them is, you see the expelled mass from the star as it died, so that creates an extended source, and then you look for the compact object that was left behind.

**And what’s interesting is that you see, you quite often see the neutron stars because they pulse, so they’re easy to see, but so far, we’ve not found a single black hole at the center of a supernova remnant**. And so, which is interesting, so you say, you should see them, you know, you ought to see some fallback, you know, you need some matter, you need something, but no, never, not yet been detected.

Beyond the Milky Way – a game-changing discovery

]]>A recent YouTube video by Sabine Hossenfelder (below) is a useful start in answering the question: How does our “big and warm” everyday experience evince quantum effects?

(from transcript) … the weird stuff that’s typical for quantum mechanics – entanglement and quantum uncertainty and the ability of particles to act like waves – are under normal circumstances really really tiny for big and warm objects. I am here using the words “

big” and “warm” the way physicists do, so “warm” means anything more than a few degrees above absolute zero and “big” means anything exceeding the size of a molecule.

She presents a helpful summary of why quantum physics is not just about the “small” stuff. Such as your laser pointer.

Hossenfelder also mentions demonstrations of “double-slit” interference of “macro” objects like atoms and molecules.

• YouTube > Sabine Hossenfelder > “Understanding Quantum Mechanics #6: It’s not just a theory for small things” (Sep 26, 2020) – ” *… the relevant point is that there is no limit in size or weight or distance where quantum effects suddenly stop. In principle, everything has quantum effects, even you. It’s just that those effects are so small you don’t notice.*“

Another science communicator, physicist Chad Orzel, discussed the topic in a 2015 video lecture: “Exotic Physics of an Ordinary Morning,” starting with a toaster oven.

Getting a general down-to-earth connection with quantum realities beyond popular hype, fictional devices, and heralded weirdness may take generations. A long game. More than just talking about “something hidden” and Big Science research.

A compelling connection might be biophysics and quantum biology. And medical tech.

Leon M. Lederman; Christopher T. Hill. **Quantum Physics for Poets**. Prometheus Books 2011. Kindle Edition.

Or does space curve in some way in higher dimensions? (“By gluing up a suitable chunk” of a 3-sphere, for example.)

What’s the evidence against a flat universe? The paths of “straight” lines and angles in triangles might tell.

• Quanta Magazine > “What Is the Geometry of the Universe?” by Erica Klarreich (March 16, 2020)

]]>There was a time, after all, when everyone thought the Earth was flat, because our planet’s curvature was too subtle to detect and a spherical Earth was unfathomable.

We can ask two separate but interrelated questions about the shape of the universe. One is about its

geometry: the fine-grained local measurements of things like angles and areas. The other is about itstopology: how these local pieces are stitched together into an overarching shape.

**Galilean**(Newtonian) static, separate space and time.**Relativistic**“flat” space-time – Minkowski / Poincaré / Einstein.**de Sitter**spherical space-time: “*… in the same way that the finite speed of light simplifies things, the finite radius makes the de Sitter group simpler and more unified than the Poincaré group.*“

Implicit in this article is a distinction between two perspectives. In the first perspective, there’s the **mathematical model** of the universe which allows us to predict things regardless of where & when we do so. This model is based on symmetries.

In the second perspective, there’s **the (real) landscape** and what “makes one place different from another.” Homogeneity in substance and form is equated with “perfect” symmetry. The non-uniform (variable) distribution of stuff “broke” symmetry: “*Broken symmetries are necessary for existence.*” Broke something real or a model? Such reification gives me pause.[1]

• Quanta Magazine > Abstractions Blog > “How (Relatively) Simple Symmetries Underlie Our Expanding Universe” by Natalie Wolchover (July 15, 2019) – Although Einstein’s theory of space-time seems more complicated than Newtonian physics, it greatly simplified the mathematical description of the universe.

… the symmetries of the universe, or

all the ways you can shift, rotate and move through it and still measure the same separation between objects or events as before. It is in the language of these symmetries that relativity simplified our mathematical description of the universe.In fact, the math becomes even nicer when the expansion of space-time is taken into account.

In

Newtonian physics, the distance between the start and finish lines and the time a sprinter takes to traverse that distance don’t depend on your point of view. You can carry your clock to a different place or hold the race at a different time, turn the clock upside down, or hop in a car and drive alongside the sprinter, and still you’ll record the same time as before, according to the equations. In other words, there are 10 “symmetries” of absolute space and time: rotations in any of three spatial directions (x, y and z), motion in those directions, and shifts to new positions in x, y, z and time. They’re known as theGalilean transformations.But those are not the true symmetries of nature.

Instead, as Einstein discovered, space and time are inextricably bound. … And yet, … the “

space-time interval” between two events – each person’s combined measurements of the length of the racetrack and the sprinter’s time – always stays the same regardless of one’s point of view.… the

Poincaré symmetries[the 10 ways of changing perspectives on space-time vs. the Galilean symmetries] still assume an infinity by specifying ways of transformingflat space-time, which extends uniformly forever in all directions.When the radius of the universe is finite — that is, when the space-time fabric looks like

the surface of an enormous sphererather than an infinite sheet of paper — the 10 Poincaré symmetries are replaced by a new group of 10 transformations known asthe de Sitter group. … in the same way that the finite speed of light simplifies things,the finite radius makes the de Sitter group simpler and more unified than the Poincaré group.In a de Sitter universe,

the space-time fabric is infused with energy, which not only causes it to curve like a sphere, but also makes it expand over time. … And space-time is indeed infused with energy — the “dark energy” discovered by astronomers in 1998. So do we live in a de Sitter universe, described by the simple de Sitter group of symmetries? The strange answer is: We will eventually.

[1] There’s a transition in the article from (1) **invariances** in applying the laws of physics – “all the ways you can shift, rotate and move through it [the universe] and still measure the same separation between objects or events” to (2) **variabilities** in how the universe looks from separate (perhaps cosmic) vantage points in space-time. Symmetries in the first case are exploited for **knowledge**. Variabilities in the second case are telltales of something deeper, a **metaphysics**.

The article concludes with a riff on the fate of the universe due to accelerating expansion, returning eventually to a pure unwrinkled (vacuum) state. (See post “A shelf life for the universe?“)

• Poincaré group of symmetries

• Symmetry Magazine > “Nature through the looking glass” by Oscar Miyamoto Gomez (09/22/20) – **Handedness** – and the related concept of **chirality** – are double-sided ways of understanding **how matter breaks symmetries**.

]]>Because our hands are

chiral, they do not interact with other objects and space in the exact same way. In nature, you will find this property in things like proteins, spiral galaxies and most elementary particles.These different-handed object pairs reveal some puzzling

asymmetries in the way our universe works.For example, the weak force – the force responsible for nuclear decay – has an effect only on particles that are left-handed. Also, life itself – every plant and creature we know – is built almost exclusively with right-handed sugars and left-handed amino acids.

A chiral twin has been found for every matter and antimatter particle in the Standard Model – with the exception of neutrinos.The difference between left-handed and right-handed could have influenced another

broken symmetry: the current predominance of matter over antimatter in our universe.According to de Gouvêa [a professor at Northwestern University], the main lesson that chirality teaches scientists is that we should always be prepared to be surprised.

“The big question is whether asymmetry is a property of our universe, or a property of the laws of nature,” he says. “We should always be willing to admit that our best ideas are wrong; nature does not do what we think is best.”

In his Q&A (where he responds to questions from prior videos), he notes a caveat about **the law of conservation of energy**. Energy may not be conserved … because space-time can change.

He offers some links for more thorough and technical explanations (see below).

In what special cases is this important? Models of the early universe and the cosmological constant … black holes … quantum gravity … Anything less cosmic?

• YouTube > Fermilab > Don Lincoln > “20 Subatomic Stories: Is the Planck length really the smallest?” (Aug 19, 2020)

Conservation of energy isn’t always real. Now, me saying that conservation of energy isn’t really true is pretty staggering and requires some explanation. Bear with me, because this is somewhat technical.

According to the theory, a conserved quantity implies that the equations don’t care where you set your zero. For instance, for

conservation of energy, the equations can’t care what moment you set as time zero. … If the laws of physics don’t care about that choice, energy will be conserved. Butan additional requirement that is never mentioned is that space and time must be static and unchanging. But, of course, in general relativity, that last requirement is not satisfied.Space and time can change and warp and distort. Accordingly, the law of conservation of energy doesn’t necessarily apply.So that’s the reason that it appears that energy is lost. It’s because non-conservation is not only allowed, it’s expected. Now, the description I’ve given here is the gist, but, because of the limited time we have, it’s very brief. So I’ve put links to two more thorough and technical explanations in the video description below.

• Sean Carroll’s blog > “Energy conservation in general relativity” (2010)

In his 2010 post, Carroll discusses something that’s long been understood about General Relativity (GR), namely, that **energy is not conserved because spacetime is not fixed (static)**.

The dynamic nature of spacetime is key to the theory of Big Bang Nucleosynthesis – the expansion and energy density of the universe.

He notes that some experts in cosmology and GR frame the physics in terms which preserve conservation by including gravitational field energy in the tally. He finds such an approach counterproductive due to:

- Ambiguous mapping at points of space: “
*Unlike with ordinary matter fields, there is no such thing as the density of gravitational energy. The thing you would like to define as the energy associated with the curvature of spacetime is not uniquely defined at every point in space.*“ - Dubious pedagogical benefit – introducing negative energy rather than saying “spacetime can give energy to matter, or absorb it from matter.”

The point is pretty simple: back when you thought energy was conserved, there was a reason why you thought that, namely time-translation invariance. A fancy way of saying “the background on which particles and forces evolve, as well as the dynamical rules governing their motions, are fixed, not changing with time.” But in general relativity that’s simply no longer true. Einstein tells us that space and time are dynamical, and in particular that they can evolve with time.

When the space through which particles move is changing, the total energy of those particles is not conserved.It’s not that all hell has broken loose; it’s just that we’re considering a more general context than was necessary under Newtonian rules. There is still a single important equation, which is indeed often called “energy-momentum conservation.”

… energy and momentum evolve in a precisely specified way in response to the behavior of spacetime around them. If that spacetime is standing completely still, the total energy is constant; if it’s evolving, the energy changes in a completely unambiguous way.

In the case of dark energy, that evolution is pretty simple:

the density of vacuum energy in empty space is absolute constant, even as the volume of a region of space (comoving along with galaxies and other particles) grows as the universe expands. So the total energy, density times volume, goes up.This bothers some people, but it’s nothing newfangled that has been pushed in our face by the idea of dark energy. It’s just as true for “radiation” — particles like photons that move at or near the speed of light. The thing about photons is that they redshift, losing energy as space expands. If we keep track of a certain fixed number of photons, the number stays constant while the energy per photon decreases, so the total energy decreases.

A decrease in energy is just as much a “violation of energy conservation” as an increase in energy, …

• University of California, Riverside > Math > “Is Energy Conserved in General Relativity?” (2017)

This article by Michael Weiss and John Baez addresses the question at a more technical level. In particular, the effort (indirectly referenced by Carroll) using “pseudo-tensors” to accomodate conservation.

In special cases, yes. In general, it depends on what you mean by “energy”, and what you mean by “conserved”.

In flat spacetime (the backdrop for special relativity), you can phrase energy conservation in two ways: as a differential equation, or as an equation involving integrals (gory details below). The two formulations are mathematically equivalent. But when you try to generalize this to curved spacetimes (the arena for general relativity), this equivalence breaks down. The differential form extends with nary a hiccup; not so the integral form.

The differential form says, loosely speaking, that no energy is created in any infinitesimal piece of spacetime. The integral form says the same for a non-infinitesimal piece. (This may remind you of the “divergence” and “flux” forms of Gauss’s law in electrostatics, or the equation of continuity in fluid dynamics. Hold on to that thought!)

As often in physics, Weiss’ discussion of energy flux across infinitesimal (spacetime) volumes involves artful modeling. [1] Particularly when you encounter non-linear equations with synergetic effects and decide what level of detail is good enough. [2] In this case, as noted, conservation requires wrangling something to have an invariant meaning.

The article concludes:

We will not delve into definitions of energy in general relativity such as the hamiltonian (amusingly, the energy of a closed universe always works out to be zero according to this definition), various kinds of energy one hopes to obtain by “deparametrizing” Einstein’s equations, or “quasilocal energy”. There’s quite a bit to say about this sort of thing! Indeed, the issue of energy in general relativity has a lot to do with the notorious “problem of time” in quantum gravity… but that’s another can of worms.

[1] As well as skillfully applying corresponding equations, as discussed by Chad Orzel. When is a cow like a sphere, when is a cow treated as a point?

• “The Hardest Thing To Grasp In Physics? Thinking Like A Physicist” by Chad Orzel (Aug 29, 2016)

This kind of simplified model-building isn’t completely unique to physics, but we seem to rely on it more heavily than other sciences.

You need to know not just how to do calculations that treat cows as spheres, but when it’s appropriate to do that. And that helps make thinking like a physicist the hardest part of the discipline to learn.

[2] For example, perhaps gravitational waves interact and contribute to the energy tally (and so an additional source of gravity)? As well as the scale at which effective theories apply (as in scale analysis).

]]>In physical terms, the divergence of a vector field is the extent to which the vector field flux behaves like a source at a given point. It is a local measure of its “outgoingness” – the extent to which there is more of the field vectors exiting an infinitesimal region of space than entering it. A point at which the flux is outgoing has positive divergence, and is often called a “source” of the field. A point at which the flux is directed inward has negative divergence, and is often called a “sink” of the field. The greater the flux of field through a small surface enclosing a given point, the greater the value of divergence at that point. A point at which there is zero flux through an enclosing surface has zero divergence.

The Holy Grail of modern physics is a so-called theory of everything, a unified field theory, a theory which unifies all known “forces.” That is, unifies all the fundamental interactions of nature. The three “quantum” interactions (electromagnetism, weak, strong) and gravitation.

A conventional sequence of theories depicts final unification as occurring at the Planck energy (density) level.

…

electroweak unificationoccurs at around 100 GeV,grand unificationis predicted to occur at 10^16 GeV, andunification of the GUT force with gravityis expected at thePlanck energy, roughly 10^19 GeV.Electroweak unification is a

broken symmetry: the electromagnetic and weak forces appear distinct at low energies because the particles carrying the weak force, the W and Z bosons, have non-zero masses of 80.4 GeV/c2 and 91.2 GeV/c2, whereas the photon, which carries the electromagnetic force, is massless.At higher energies Ws and Zs can be created easily and the unified nature [symmetry?] of the force becomes apparent.

A recent Symmetry Magazine article (below) was inspiration for this post. I’ve encountered the topic – unification of forces at high enough energies – many times before. But this latest article, while an experimental milestone, struck me as quite particle oriented, not helping advance visualization based on quantum field theory (QFT).[1]

The premise of Grand Unified Theory models is merging of gauge interactions at extreme energies, energy densities possibly in the quite early universe.

A Grand Unified Theory (GUT) is a model in particle physics in which, at high energies, the three gauge interactions of the Standard Model comprising the electromagnetic, weak, and strong forces are merged into a single force. Although this unified force has not been directly observed, the many GUT models theorize its existence.

If unification of these three interactions is possible, it raises the possibility that there was a grand unification epoch in the very early universe in which these three fundamental interactions were not yet distinct.

I like to imagine that at high enough energies, all localized vibrations – excitations in the various fields associated with “particles” – become a mash-up of energy transformations. Where direct vs. conventional mediated interactions alter hallmark designations. Unfettered field interactions.[2]

• Symmetry Magazine (A joint Fermilab/SLAC publication) > “LHC creates matter from light” by Sarah Charley (August 24, 2020) – Scientists on an experiment at the Large Hadron Collider see **massive W particles emerging from collisions with electromagnetic fields**. How can this happen?

The

Large Hadron Colliderplays with Albert Einstein’s famous equation, E = mc², to transform matter into energy and then back into different forms of matter. But on rare occasions, it can skip the first step andcollide pure energy—in the form of electromagnetic waves.Last year, the ATLAS experiment at the LHC observed two photons, particles of light, ricocheting off one another and producing two new photons. This year, they’ve taken that research a step further and discovered

photons merging and transforming into something even more interesting: W bosons, particles that carry the weak force, which governs nuclear decay.This research doesn’t just illustrate the central concept governing processes inside the LHC: that energy and matter are two sides of the same coin. It also confirms that

at high enough energies, forces that seem separate in our everyday lives – electromagnetism and the weak force – are united.

If you try to replicate this photon-colliding experiment at home by crossing the beams of two laser pointers, you won’t be able to create new, massive particles. Instead, you’ll see the two beams combine to form an even brighter beam of light.“If you go back and look at Maxwell’s equations for classical electromagnetism, you’ll see that two colliding waves sum up to a bigger wave,” says Simone Pagan Griso, a researcher at the US Department of Energy’s Lawrence Berkeley National Laboratory. “We only see these two phenomena recently observed by ATLAS when we

put together Maxwell’s equations with special relativity and quantum mechanicsin the so-called theory of quantum electrodynamics.”Inside CERN’s accelerator complex, protons are accelerated close to the speed of light. Their normally rounded forms squish along the direction of motion as special relativity supersedes the classical laws of motion for processes taking place at the LHC. The two incoming protons see each other as compressed pancakes accompanied by an

equally squeezed electromagnetic field(protons are charged, and all charged particles have an electromagnetic field). The energy of the LHC combined with the length contraction boosts the strength of the protons’ electromagnetic fields by a factor of 7500.

When two protons graze each other, their squished electromagnetic fields intersect. These fields skip the classical “amplify” etiquette that applies at low energies and instead follow the rules outlined by quantum electrodynamics. Through these new laws, the two fields can merge and become the “E” in E=mc².The LHC is one of the few places on Earth that can produce and collide energetic photons, and it’s the only place where scientists have seen two energetic photons merging and transforming into massive W bosons.

Just as photons carry the electromagnetic force, the W and Z bosons carry the weak force.

The reason photons can collide and produce W bosons in the LHC is that at the highest energies, those forces combine to make the electroweak force.

[1] And as to whether current QFT is incomplete – only a mathematical model of effective field theories. And the question as to whether unification requires additional dimensions and/or fields and interactions. Wiki notes:

Yet GUTs [Grand Unified Theories] are clearly not the final answer; both the current standard model and all proposed GUTs are quantum field theories which require

the problematic technique of renormalizationto yield sensible answers. This is usually regarded as a sign that these are onlyeffective field theories, omitting crucial phenomena relevant only at very high energies.

Is there a limit to probing those energies?

• YouTube > Sabine Hossenfelder > “Does nature have a minimal length?” (Feb 2, 2020)

The Planck length seems to be setting a limit to how small a structure can be so that we can still measure it. That’s because to measure small structures, we need to compress more energy into small volumes of space. That’s basically what we do with particle accelerators.

Higher energy allows us to find out what happens on shorter distances. But if you stuff too much energy into a small volume, you will make a black hole.

[2] As Wiki notes, extreme energy GUT models characterize such a state by a single unified coupling constant with yet several (so-called) force carriers. That is, in such a state, “particles” (excited fields) experience a single interaction strength while still mediated separately?

GUT models predict that at even higher energy, the strong interaction and the electroweak interaction will unify into

a single electronuclear interaction. This interaction is characterized by one larger gauge symmetry and thus several force carriers, but one unified coupling constant.

[3] Sometimes the tone of theories of everything strikes me as casting perfection onto the universe akin to that of ancient celestial spheres. A mathematical perfection or symmetry. Yet the universe need not play by any such vision. Perfection a quixotic quest.

]]>*Just a note before you go,A vision to be learnedTraveling near the speed of lightIt’s easy to get …*

Imagining how things would look when traveling near the speed of light is an interesting exercise. Using a freeware video game developed by MIT Game Lab (2012), this visualization (below) by The Action Lab is particularly interesting. An exploration of the various effects:

- Doppler effect
- Searchlight effect
- Time dilation
- Lorentz transformation
- Runtime effect (reference?)

“A Slower Speed of Light” hopefully corrects common misunderstandings – something that benefits contemporary physics classes.

OpenRelativity is a toolkit designed for use with the proprietary Unity game engine. It was developed by MIT Game Lab during the development of

A Slower Speed of Light. The toolkit allows for the accurate simulation of a 3D environment when light is slowed down.

A Slower Speed of Light was developed in hopes of being used as an educational tool to explain special relativity in an easy-to-understand fashion.The game is meant to be used as an interactive learning tool for those interested in physics.[1]

I get confused by the perspective blend of inertial and quasi-accelerating frames of reference: walking at the same speed but the speed of light getting slower and slower. As noted [1], relativistic effects are indeed why the game becomes increasingly difficult, more challenging.

• YouTube > The Action Lab > “Slowing the Speed of Light Down to 2 m/s [walking speed] – What Special Relativity Feels Like” (August 13, 2020). “When we start walking, we start to see relativistic effects that include time dilation and length contraction, and also doppler shifting.”

Description*: “In this video I show you what it would look like to slow the speed of light down to around walking speed. So with just walking around town you would experience relativistic effects. I talk about time dilation and length contraction and what it would look like to have it happen to you. Get the simulation created by MIT here.”*

The Action Lab • Pinned by The Action Lab

The Action Lab (August 13, 2020)

Interesting note:Even Einstein was mistaken on length contraction.He had said that a sphere would look like an ellipsoid. However, Penrose later proved that a sphere would still be spherical, although rotated.Notice in the simulation how the spheres are the only objects that don’t look distorted when moving at near light speeds!

From transcript:

[Re

doppler shift] So what that means is that normally light that we can’t see, like infrared light, as we walk towards it, it gets shifted more towards the red end of the spectrum instead of the infrared part of the spectrum; so, it moves up in frequency, and since it moves up in frequency, it becomes visible light to us.… and also the effect is stronger – the brightness increases. that’s because as we’re walking towards it we’re actually hitting more photons along the way than we normally would, so basically it increases

the intensity as we’re walking towards it.Space and time are always connected. if you’re completely at rest – not moving, all of your movement is going through time and not through space. so you’re at rest but you’re still moving forward through time. but

if you start to move forward through space, then that means your movement through time has to decrease. so the faster you move through space, the slower your movement through time is going to be.Now of course this time dilation is only relative to somebody watching you do that movement. but

you yourself always experience time at the same rate. but how it portrays itself to you as thefirst person view– as the person who’s walking at close to speed of light speeds – is that it seems like you’re going faster, so you can get more movement through space in a given time.In this simulation, as i start to walk so as i approach the speed of light, things get stretched out. now the easiest way to see this is i’m going to turn off the doppler effect. collect my last orb, so now the speed of light is really close to my walking speed, so the relativistic effects are really strong here. so notice as i start to walk, now notice how far away the cliffs seem like they get, so the

length gets stretched out. but i just told you that when you go closer to the speeds of light length actually contracts.now this is a really confusing point and a lot of people have gotten this wrong.For the most part length contraction does not appear as something being squished, but it actually appears as something being stretched out and lengthened, and also

rotated a little bit. even though length contraction is occurring, this is not what you would see.what you would see is completely different than what you would measure as length, and that’s because the photons in your line of sight in the direction of your travel are leaving the thing thatyou’re seeing at different points in the past.So, for example, if you’re looking at something, you see it being stretched out because the [slower?] photons from the back of it take longer to get there than the front [faster photons?] of it, and so it actually appears as though the thing is being stretched out.

[1] Wiki:

In

A Slower Speed of Light, the player controls the ghost of a young child who was killed in an unspecified accident. The child wants to “become one with light”, but the speed of light is too fast for the child. This is solved through the use ofmagic orbswhich, as each are collected,slow down the speed of light, until by the end it is at walking speed. These orbs are spread throughout the level. At the beginning of the game, walking around and collecting these orbs is easy; however, as the game progresses,the effects of special relativitybecome apparent. This gradually increases the difficulty of the game.As the game progresses and light becomes slower, the effects of special relativity start to become more apparent. These effects include the

Doppler effect(red/blue-shifting of visible light and the shifting of ultraviolet and infrared into the visible spectrum), thesearchlight effect(increased brightness in the direction of travel),time dilation(difference between the passage of time perceived by the player and the outside world), theLorentz transformation(the perceived warping of the environment at near-light speeds), and theruntime effect(seeing objects in the past because of the speed of light). These effects combine as the game progresses to increase the difficulty and challenge the player.

Some commentary on the realism of this simulation – the optics of moving close to the speed of light:

• Stack Exchange > Physics > How realistic is the game “A slower speed of light”?

And the inherent limitations of OpenRelativity, as noted:

• Visualizing relativity: The OpenRelativity project (2015).

[2] A NASA cartoon about near-light-speed travel:

• YouTube > NASA Goddard > “NASA’s Guide to Near-light-speed Travel” (August 14, 2020)

• Biggest ideas in the universe – Sean Carroll chats concepts > The Biggest Ideas in the Universe | 6. Spacetime (Apr 28, 2020)

]]>A recent Nature article (below) was inspiration for this post. I’ve been encountering the use of topology in physics for some time. Typically the mathematics is elusive, but the notions are compelling.

Wiki > Topology

A continuous deformation (a type of homeomorphism) of a mug into a doughnut (torus) and a cow into a sphere.

In mathematics,

topology… is concerned withthe properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing.

A traditional joke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.

Homeomorphismcan be considered the most basic topological equivalence. Another is homotopy equivalence. This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from “squishing” some larger object.

Topology is relevant to physics in areas such as condensed matter physics, quantum field theory and physical cosmology.A

topological quantum field theory(or topological field theory or TQFT) is a quantum field theory that computes topological invariants.

Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things,knot theory, the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry.The topological classification of Calabi-Yau manifolds has important implications in

string theory, as different manifolds can sustain different kinds of strings.In cosmology, topology can be used to describe

the overall shape of the universe. This area of research is commonly known asspacetime topology.

Nature > News Q&A > “The mathematician who helped to reshape physics” by Davide Castelvecchi (August 4, 2020) – Barry Simon [who is at the California Institute of Technology in Pasadena] linked a phenomenon that had shocked physicists to topology, the branch of mathematics that studies shapes.

]]>In recent years, physics has been swept by ideas from a branch of mathematics called topology. Topology is the study of objects that deform continuously without tearing, for example through stretching or twisting. But it is now proving crucial to understanding

the shapes of quantum wavesformed by the electrons inside matter. These waves can form shapes such as vortices, knots and braids that give materials a variety of exotic properties. In 1983, Barry Simon was the first person to make the link between strange phenomena in materials and topology.German physicist Klaus von Klitzing won a Nobel prize for the [Hall] effect’s discovery in 1985. But it took several breakthroughs by theoretical physicists to begin to understand the phenomenon. And it took Simon — a mathematical physicist who uses mathematical tools to solve theoretical problems that emerge from nature — alongside collaborators, to recognize that equations created to describe the quantum Hall effect were a manifestation of topology. It was topology that was making the material’s resistance robust to small changes, allowing it to change in only discrete jumps [due to the topological effect, called a winding number].