Posts tagged with spacetime

Tracks in the Snow

We went hiking and I noticed animal tracks in the snow.

• rabbit
• bighorn sheep
• fox
• dog
• something hard to identify

It occurred to me that these are curves γ(t) and that I’m looking into the past at them.

$\large \dpi{200} \bg_white \gamma(\lfloor t-t_0 \rfloor) \\ \\ \xi_{\rm{\; sheep}} \\ \gamma_{\rm{\; rabbit}} \\ \Gamma_{\rm{\; human\ shoe}}$

In Zelda: Twilight Princess, when Link turns into a wolf, his world is dominated by the sense of smell.

It’s an interesting way of interacting with the world, seeing the spatial projection of the past. A bit like the Tralfamadorians.

Death and Symplecticity

The universe is receding behind you every second. One of the lessons of special relativity is the −ct term:
$\large \dpi{200} \bg_white (- + + +)$

• you can stand still where you are,
• you can run away as fast as you can,
• you can stop and go and wander around,
• you can focus like a nail and pound deep into something,
• you can get bored or be excited,
• you can build something & raise the Lagrangian or veg & leave it low,

time is still flowing past you, that metric subtracting −ct ticks at a rate of one tick per tick.

"Your prison is walking through this world all alone"

In other words, freedom and independence, too, have a cost, perhaps exactly equal to the cost of

• or spending your “best years” raising children instead of “achieving” career-wise.

A tumbleweed sees more but also less than a tree.

If you want to think about lifetime as being a fixed length (ignoring that its length comes from a probability distribution, which itself is conditional on your choices) then you can derive my favourite equation:

$\large \dpi{200} \bg_white \text{consumption} = \text{wage} \cdot (\text{lifespan}-\text{leisure})$

the tradeoff between work, leisure, and wealth. That idea as well is symplectic. And many other such tradeoffs ∃. Symplecticity is the theoretical basis of all budget constraints. It’s another way of talking about all the tradeoffs that make choice meaningful and also unavoidable (even not-choosing is a choice). You can strain and strive as much as you want, all you will do is slide amongst alternatives and never do everything.

If you want to use a picture of the form of Christopher Alexander’s

and just substitute in names of various other things that you want—then the “metric signature”, due to time flowing over and beyond us like a river always, is − in so many of the pursuits one might like to do, such as

• making money
• learning algebraic topology
• spending time with kids
• learning to do a backflip
• travelling in Asia
• playing guitar
• writing an opera
• living so you get to Heaven after this life (ok, I said I wouldn’t bring in any probability distributions but I had to cheat on this one. It’s an interesting measure theory question, isn’t it? If there is even a finite chance of getting an infinite payoff, then unless the utility function becomes flat above a certain payoff, then the only logical thing to do is make 100% sure you get the infinite payoff. OK, /rant)
• making the sex, many times. Or, not:

Sure, sometimes one lucks out and there is a positive association between two things, like learning mathematics and being a quant—but the magnitude might be less than you expect. (Pure maths alone is insufficient and unnecessary to finance.)

In terms of the 10,000-hours-to-expertise paradigm—despite some complementarities (+)—there are only so many 10,000-hour blocks in your life. And the Type A personality who squeezes out the most 10,000-hour blocks, gets the most toys or becomes the world’s best cyclist or visits all the countries, learns the most languages, or whatever, still miss out on something.

Leaving aside that the human encyclopedia and Tony Hawk also will turn back to dust, just even evaluating only the finite path [0,1] → life , that busy body necessarily misses out on

• the down moments,
• the still time,
• the zoning out,
• the chilling,
• the doing nothing and being OK with it,
• the taking in instead of forcing out,
• and perhaps those have some value as well.

In English it sounds so obvious to be trivial: you can’t do everything, because nothing is also something and if you’re doing something you can’t be doing nothing.

But the mathematical language, in addition to sounding more exotic and smartypants, adds something real, at least for me—which is the sense of those − signs attaching me to everything. Every time I do something, I’ve lost some other opportunity. Every person I become, I drift further away from the possibilities of who else I might have been. Every commitment loses a freedom and every freedom wastes a commitment. Every nothing wastes a something and every something forgoes a nothing. Everything is receding, decaying, entropying, with or without me, until eventually the waters will cover my head and I never surface again.

Sufficiently convolved with the

$\large \dpi{200} \bg_white x+y+z + \varsigma + \xi = 100\%$

all the paths sum to a constant and that constant quantity eventually runs out.

Geotagging

Geotagged photos (e.g. flickr) and text (e.g. twitter) associate data to a particular point on the globe × time. In other words, a fibre bundle over S²×T.

Imagine the position future historians will be in — if they can synthesise the petabytes of digital data we generate these days. I would love to have crowd-sourced pictures or live-tweeting bystanders’ microblogs of the Yan tie lun 鹽鐵論 (81 B.C.) — Rashomon effect be damned.

In the loop quantum gravity approach, space-time is quantized by a procedure that encodes it in a discretized structure, consisting of spin networks and spin foams.

A spin network consists of an oriented embedded graph in a 3-dimensional manifold with edges labelled by SU(2) representations and edges labelled by intertwiners between the representations attached to incoming and outgoing vertices. These representations relate to gravity in terms of holonomies of connections, and the formulation of Einstein’s equations in terms of vierbein, or tetrads, and dual co-tetrads.

Thus, to a spin networks, or the 1-skeleton of a triangulation by tetrahedra, one assigns operators of quantized area and volume, coming from counting intersection points of a surface, or 3-dimensional regions, with the edges or vertices of the spin network with a multiplicity given in terms of the spin representation attached to the edges and the intertwiners attached to the vertices.

SOURCE: Listening to Golem

Loop quantum gravity is one of a few general frameworks that may eventually form the basis of how physicists think of the smallest scales of time and distance.

(These frameworks are sometimes called Theories of Everything but they’re really just thoughts-on-the-way-to-theories of brief-and-tiny matter.)

GLOSSARY
• SU(2) is the group of 2×2 unitary matrices with determinant 1. They have the form
$\dpi{200} \bg_white U = \begin{pmatrix} \alpha & -\bar{\beta} \\ \beta & \bar{\alpha} \end{pmatrix}$
SU(2) is just like the unit quaternions, which represent rotations in 3-D. Here are the pieces that make up SU(2) .
$i\sigma_x = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$$i\sigma_y = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$$i\sigma_z = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}$
• 3-dimensional manifold — any shape that can be made with dough. Including that if you stretch the dough, the outside and the inside stretch.
• oriented graph — things like this:
• embedded oriented graph — oriented graphs sculpted in 3-D
• edge — arrows in the above pictures
• spin network — a graph like above and each circle has value  or −½
• representation — every group can be represented as a matrix
• holonomy — how to move things in parallel within the dough (curved 3-manifold)
• connection — parallel transport
• tetrad or vierbein — a 4-D spacetime frame of reference

I love randomly throwing out dense mathematical statements from theoretical physicists like this. Sometimes maths people sound like wizards canting magical runes.

Measure from Light

Time doesn’t pass for a light particle, I don’t think.  Just like Vonnegut’s Tralfamadorians, light “experiences” all its moments, everywhere it will ever go, at once.

I think this “inverted” way of thinking about time and space is the right one.  We aren’t moving around in time — we’re moving with respect to fixed light cones in both space and time.

This way of thinking makes unparadoxical the constancy of light’s speed in all reference frames. Rigid space-marking rods are not the guideposts by which the universe is measured — timeless, spatially extended photons are.

UPDATE: Nice video rendering of photons by NASA. http://www.nasa.gov/multimedia/videogallery/index.html?media_id=154874171

Time is Like Space

Here is the shortest way I can describe the difference between old-school (Newtonian) spacetime and modern, relativistic (Minkowskian) spacetime:  in the old view, time was something separate and distinct from space:

$\dpi{300} \bg_white \mathbb{S}^3 \times \mathbb{T}$

Now, in the modern view, time is another kind of thing that’s just like space.

$\dpi{300} \bg_white \mathbb{S}^4$

Read on if that doesn’t say it all.

You know how left-right and forward-back are only relative to some axis system?  Like if you turn to the left, then left-right are different.  Also if you lie down, then up-down are different.  Well, sooner-later is the same kind of thing as left-right.  If you go really fast (like 10% of the speed of light) then you notice this.

Going this fast relative to me, your time will be slower than my time — your watch will literally run slower.  Just like, if we were facing the same direction, and I turned 10 degrees to the left, then our “forwards” would be different.  Which means, walking at the same speed I would achieve less “forward” than you, according to your “forward”.

SOURCE: Sean Carroll

Manifolds, Metrics, Star Fox, and Self-versus-Other

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.

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.)

If you stitch together such non-Euclidean flats, you’ve again constructed a manifold.

Think about this: the pixel concept re-presents brush-stroke or natural images by a wall of sequential colored squares.  You could extend it to 3-D, for example representing humans by little blocks—white for the bone, burgundy for the blood, pink for the fingernails, etc.

In a similar fashion, 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.