Posts tagged with special relativity

## 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.

## Religious Infinity

When I was young, I used to — as an exercise — try to conceive of ∞. We would hear in Sunday School that God is Infinite, that you can’t comprehend God’s Infinite-ness.

I would imagine myself in a spaceship flying out to the edge of the universe. I would imagine all of the stuff we had left behind us, flying at the speed of imagination.

Then I would zoom out the camera, seeing that in fact we had only gotten to the edge of a tiny speck. I would recurse this and try to recurse the recursions until my brain got tired. “Infinity is so big,” I would think. “The Universe is so big. God is so big.”

All of this was on purpose. I wanted ∞ to fill up my mind. I think there are lots of religious people who do this — meditate, in a way, on ∞.

Imagining ∞ as a mathematician is easy in comparison. Using the M.O. of stereographic projection, I can conceive of infinity in an Augenblick.

Nowadays it’s up to me, whether I want to view ∞ as large or small.

## 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