## Three levels of locality in quantum physics

Locality seems to be a deep principle of physics. It’s the idea that what happens at any given place in the universe doesn’t (directly) depend on things happening in other places far away, but only on what’s happening nearby. For instance, the lightspeed limit in relativity guarantees that if you want to know will happen here in the next nanosecond, all you have to care about is stuff happening within a light-nanosecond of here. Events farther than that are irrelevant. So cause and effect operate locally—events can’t have an effect too far away too quickly.

Quantum physics has thrown a bit of a wrench in conventional notions of locality. Experiments on entanglement have confirmed violations of Bell inequalities, establishing that correlations exist between events happening well outside each other’s lightspeed horizons—correlations that cannot be explained on the basis of “local realism”, the idea that the outcome of a measurement depends only on what’s going on in or near the measuring device, and not on what’s going on in a widely separated location. Continue reading

## Greg Egan’s Orthogonal: Alternate Physics, Familiar Politics

Science fiction works are commonly classified on a spectrum ranging from “soft” to “hard”. The distinction lies in the fiction’s attitude toward the physics (or other science) involved in the story. In soft sci-fi, the physics is ad-hoc, created as needed to serve the story, with no consistent theory behind it; only its outlines are apparent, with the details glossed-over. Most television sci-fi, such as Star Trek, falls into this category; so do many venerable sci-fi novels, such as Dune, Ender’s Game, and Foundation.

Conversely, hard sci-fi pursues a more committed relationship with the laws of physics. While the author may still add new physics for story purposes—cheap faster-than-light travel being a popular choice—it is constructed in a way that makes it fit in with real-world physics; it is plausible that there could be a deeper theory behind it. The details have been well-thought-out and may be discussed at length; likewise, the technologies encountered in the story have been conceived with real-world engineering principles in mind. Works that lean toward the “hard” end of the spectrum include most anything by Arthur C. Clarke, Robert A. Heinlein or Larry Niven; Peter F. Hamilton’s Night’s Dawn trilogy; Neal Stephenson’s Anathem. Continue reading

Posted in Physics, Technology | 6 Comments

## Nested Complex Numbers

As many of us learn in school, we can invent the complex numbers by starting with the real numbers, then simply “making up” a brand-new number, called $i$, which is not equal to any real number, and whose only defined property is $i^2 = -1$. If we then allow real numbers and $i$ to be freely intermixed in formulas involving any of the standard arithmetic operations, we are forced to create additional numbers to represent the results of these operations. When the dust settles, we have the entire complex plane, each of whose points can be expressed as $a + bi$, where $a$ and $b$ are real.

Historically, this process of creating extra numbers by fiat made many mathematicians uncomfortable, and it took time for complex numbers to be accepted as “legitimate” mathematical entities on an equal footing with the real numbers. However, the modern view of mathematics holds that creating new numbers or other new entities at will is perfectly fine—so long as the rules for working with them are well-defined and self-consistent. And indeed it does not appear that defining $i^2 = -1$ creates any inconsistencies. Moreover, complex numbers have been quite a fruitful invention, with applications in engineering and physics, and a whole new branch of mathematics to their credit.

Well, if making up just one new number is so great, why stop there? Let’s make up some more! Continue reading

Posted in Mathematics | 2 Comments

## E = mc² is only half the story

I’m sure you’ve seen the equation $E = mc^2$ many times. Probably the best-known equation in physics, it represents both a straightforward mathematical relationship and a deep physical principle. But did you know that this equation is incomplete? The full version of Einstein’s equation is:

$\displaystyle E = \sqrt{m^2 c^4 + p^2 c^2}$

This states that the relativistic energy, $E$, of a moving object is a function of its mass $m$ and its momentum, $p$ (as well as the speed of light, $c$). If you set $p = 0$ and simplify, you’ll get back to the usual $E = mc^2$. To be sure, the above equation doesn’t roll off the tongue quite as easily as $E = mc^2$…but with the momentum term included, it tells a fuller story about how relativity works. In fact, it looks very much like the Pythagorean theorem! Relativistic energy scales the same way as the hypotenuse of a right triangle whose legs are mass and momentum. This fact is not a coincidence, as we’ll see. Continue reading

## Dark energy and the cosmic horizon

In a previous post, I wrote about using numerical integration to calculate how the universe expands over time. It turned out that dark energy is making the expansion of the universe speed up, and (we presume) it will continue to do so forever. The universe’s destiny is a runaway accelerating expansion! Moreover, the details of this destiny depend sensitively on the detailed behavior of dark energy—specifically, how its density changes as space expands.

Matter and radiation naturally thin out as the universe expands: there’s a finite amount of each, spread over an increasing volume of space, and no more is being created (although a little matter converts into radiation from time to time). But dark energy works differently. Observations have established that its density remains very close to constant—as the volume of space grows, more and more dark energy comes into existence! But what observation can’t yet answer is: does dark energy’s density remain exactly constant? Or does the expansion of space dilute it just a little? Or might dark energy perversely become slightly more concentrated as the universe grows? Continue reading

Posted in Mathematics, Physics | 5 Comments

## Rotations and Infinitesimal Generators

In this post I’d like to talk about rotations in three-dimensional space. As you can imagine, rotations are pretty important transformations! They commonly show up in physics as an example of a symmetry transformation, and they have practical applications in fields like computer graphics.

Like any linear transformation, rotations can be represented by matrices, and this provides the best representation for actually computing with vectors, transformations, and suchlike. Various formulas for rotation matrices are well-known and can be found in untold books, papers, and websites, but how do you actually derive these formulas? Continue reading

Posted in Mathematics | 5 Comments

## Numerically modeling the expansion of the universe

Most people are familiar with the standard model of cosmology: the universe (or at least the part of it we can see) began in a Big Bang and has been expanding and cooling off ever since. Mathematical models of the expansion of the universe have been around since the 1920s, but only in the last decade or so have measurements become precise enough to plug specific numbers into those models. For instance, just a few years ago we knew only enough to put the age of the universe between 10 and 20 billion years old, but today we know it is 13.75 billion years old, and that’s accurate to within 1%!

Models of the expanding universe are all based on general relativity. GR is conceptually not too complicated—“spacetime tells matter how to move, and matter tells spacetime how to curve”, as the saying goes—but the mathematical machinery required to make it all precise is famously arcane. Fortunately, models of the expanding universe turn out to actually be quite simple. Continue reading

Posted in Coding, Mathematics, Physics | 9 Comments