Posts tagged with Albert Einstein

As every sci-fi geek knows, matter may travel faster than the speed of light as long as its mass is imaginary (a multiple of √−1). A so-called tachyon would not overturn special relativity—and it would provide a handy way of resolving any conflicts in a given Star Trek plot.

  • 14th Law of How to Write Star Trek: Whenever you’ve written yourself into a hole, instead of re-writing the show so that it’s better, simply make characters issue the word “tachyon” several times toward the end. Everything is magically resolved, returning all aspects of life to the way the show started with no long-term consequences for the characters—which by the way is a great lesson to teach to young adults—and then Spock or Data has an “a-ha!” moment wherein he throws around jargon to further justify the deus ex machina.

The only problem with tachyons, as any sci-fi geek can attest, is that “imaginary” mass is pure fiction! How could anything weigh an imaginary amount?


Well, I’m not sure that tachyons do exist—although if someone wants to post some arXiv links to relevant papers that would be awesome—but, I will say that “imaginary mass” isn’t that ridiculous of a concept.

As Tristan Needham said in the best book about complex numbers ever, the “imaginary” descriptor only reflects the historical prejudice against √−1.

Do imaginary numbers exist? No. But neither do counting numbers. Numbers are linguistic entities that humans communicate with. Sort of like how trees, flowers, bushes, shrubs, brambles, and vines all exist in nature, but those classifications, concepts, words, groupings are human-language mental constructs. “Five” doesn’t “exist” per se, but mathematical models built with the-thing-that-satisfies-the-properties making five five, do wonderfully at prediction of physics experiments.

Anyway, imaginary numbers exist just as much as other numbers. Just like rational numbers, they’re generated by an operation that comes up as a matter of course in algebra. And algebra seems to have something to do with nature. God knows why. (ohh! which way did I mean it?!)

So I’m not saying imaginary mass exists, but here are some good ways to think about imaginary numbers.

  • imaginary numbers are twisted numbers
  • imaginary numbers are phase-shifted like a sine wave versus a cosine wave
  • an imaginary current heats up a wire but does no useful work

If the mass of a particle is an imaginary number, then … that might help you make sense of tachyons.


Nerdy side note: E=MC² is not the real equation to describe the conversion of energy into matter or vice-versa.

  • E=MC² tells you how to convert stationary matter into energy.
  • The real equation is E² = [mc²]² + [pc]².
  • (p is momentum.)
  • (Notice that the real equation is of the form A²+B²=C². i.e., Energy is the hypotenuse (C) to the triangle sides B=mc² and A=p•c)

You can casually start/interrupt conversations with this knowledge the next time you attend a kegger / black-tie affair. Doing so will win handsome glances from potential sex partners. Also, there is a 0% chance that anyone will think you’re an insufferable know-it-all.

Branes, D-branes, M-theory, K-theory … news articles about theoretical physics often mention “manifolds”.  Manifolds are also good tools for theoretical psychology and economics. Thinking about manifolds is guaranteed to make you sexy and interesting.

Fortunately, these fancy surfaces are already familiar to anyone who has played the original Star Fox—Super NES version.

In Star Fox, all of the interactive shapes are built up from polygons.  Manifolds are built up the same way!  You don’t have to use polygons per se, just stick flats together and you build up any surface you want, in the mathematical limit.

The point of doing it this way, is that you can use all the power of linear algebra and calculus on each of those flats, or “charts”.  Then as long as you’re clear on how to transition from chart to chart (from polygon to polygon), you know the whole surface—to precise mathematical detail.

Regarding curvature: the charts don’t need the Euclidean metric.  As long as distance is measured in a consistent way, the manifold is all good.  So you could use hyperbolic, elliptical, or quasimetric distance. Just a few options.


Manifolds are relevant because according to general relativity, spacetime itself is curved.  For example, a black hole or star or planet bends the “rigid rods" that Newton & Descartes supposed make up the fabric of space.

bent spacetime

black hole photo

In fact, the same “curved-space” idea describes racism. Psychological experiments demonstrate that people are able to distinguish fine detail among their own ethnic group, whereas those outside the group are quickly & coarsely categorized as “other”.

This means a hyperbolic or other “negatively curved" metric, where the distance from 0 to 1 is less than the distance from 100 to 101.  Imagine longitude & latitude lines tightly packed together around "0", one’s own perspective — and spread out where the “others” stand.  (I forget if this paradigm changes when kids are raised in multiracial environments.)

Experiments verify that people see “other races” like this. I think it applies also to any “othering” or “alienation” — in the postmodern / continental sense of those words.


The manifold concept extends rectilinear reasoning familiar from grade-school math into the more exciting, less restrictive world of the squibbulous, the bubbulous, and the flipflopflegabbulous.

ga zair bison and monkey

calabi-yau manifold

cat detective

"General relativity can be summed up in two statements:

  1. Spacetime is a curved pseudo-Riemannian manifold with a metric of signature (− + + + ).
  2. The relationship between matter and the curvature of spacetime is contained in the equation
    R_{\mu \nu} - {1 \over 2} \, R \; g_{\mu \nu} = 8 \pi \; G \, T_{\mu \nu}.”

Sean Carroll

There you have it, ladies and gentlemen! The universe summed up in two sentences.

Regarding the first statement: a metric of signature (− + + + ) just means that time goes the opposite direction as space. (Think light cones.)

No matter how fast you go — or if you stand still — time is flowing through you at the speed of light.

Another way to think about going somewhere, say Bermuda, is that you will meet up with Bermuda’s future. To hurry to Bermuda is to meet an earlier future — but no matter how fast you go, the Bermuda that is now will be gone by the time you get there.

And so we beat on, boats against the tide, borne back ceaselessly into the past.