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54 lines
3.5 KiB
Markdown
54 lines
3.5 KiB
Markdown
---
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title: How Imaginary Universes Can Help us to Understand the Real One
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excerpt:
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layout: post
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image:
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---
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I've just finished reading the first book of Greg Eagan's excellent series Orthogonal. The book is set in a universe with a subtly different spacetime structure to ours that leads to radically different physics. What I particularly liked about it was that struggling to understand the unfamiliar physics of this imaginary universe helped me to understand the physics of our universe a little bit better too.
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## Three different kinds of physics
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At humanish scales and speeds, moving objects basically behave according to *Newtonian Physics* (except light, light is weird). Once you start going faster, a more realistic model of motion is Special Relativity (SR). In the Orthogonal universe, Newtonian Physics *also* holds at slow speeds (subject to similar caveats to do with light) and the equivalent of SR I'll call Orthogonal Relativity (OR).
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### Newtonian Physics
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In this model, time and space are entirely separate. All objects move through time at exactly one second per second and they can move through the three spatial dimensions at whatever speed they like. You have the normal equations for kinetic energy and momentum:
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Kinetic Energy: $E_k = \frac{1}{2} m v^2$
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Momentum: $p = mv$
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Also relevant will be how we measure length, in our normal 3d world you measure length with good ol' Pythagoras' theorem.
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$$ s^2 = x^2 + y^2 + z^2 $$
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This also works for other vectors, if your velocity is $\vec{v} = (v_x, v_y, v_z)$ then the same formula holds for your overall velocity $ v^2 = v_x^2 + v_y^2 + v_z^2 $.
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### Going faster
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In SR and OR, we have a four dimensional spacetime with some rules:
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- All objects move through space-time at a 4-velocity with a constant length.
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- We can make that length equal to 1 with the right units.
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- When an object is acted on by a force, it's 4-velocity can **rotate** but cannot change length.
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The way you measure the length is different between SR and OR and this turns out to make a huge difference in how the two universes behave!
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In OR, the Pythagorean theorem continues to hold in 4d and velocities must have length 1.
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$$1 = v_t^2 + v_x^2 + v_y^2 + v_z^2 $$
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In SR, it's slightly modified
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$$ v^2 = v_t^2 - v_x^2 + v_y^2 + v_z^2 $$
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There's some stuff to do with what units you measure velocity and time-velocity in but we'll get to that later.
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## Deriving Newtonian physics
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## The arrow of time and the past hypothesis
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In OR it is very clear that there is now special direction of time. OR doesn't really have a spacetime, it's just a 4D space. What I find amazing about this is that nevertheless the characters **experience** time in a very similar very to us. That is, as long as they stay put on their planet and don't move about at relativistic speeds motion seems relatively normal.
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How can this be? Well all we actually need is to find ourselves a big clump of gas that is mostly moving with a consistent 4-velocity. Whatever direction this velocity happens to point in becomes our arrow of time.
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This relates to the 'past hypothesis' in our universe. As far as I understand, SR does give us some sense of the direction of the arrow of time, not fully but a bit. What it doesn't tell us however is the sign of that arrow, it could be pointing ahead or it could be pointing behind. In OR it is clear that we need to make some assumptions about the origin of the universe in order to explain why time flows in the perceived direction that it does.
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Ok and this also relates to entropy and the second law! How? ....
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