Imaginary time
| Statics (geometric = no time): | ||
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[xlength] | x coordinate |
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[X] | proportionality constant |
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[ylength] | y coordinate |
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[ylength / xlength] | slope |
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[X ylength^2 / xlength^2] | proportional to curvature |
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[X ylength^2 / xlength^2] | original shape |
 ![= \int \left[ \frac{m}{2} \left(\frac{dq(s)}{ds}\right)^2 + V(s) \right] ds](http://l.wordpress.com/latex.php?latex=%3D+%5Cint+%5Cleft%5B+%5Cfrac%7Bm%7D%7B2%7D+%5Cleft%28%5Cfrac%7Bdq%28s%29%7D%7Bds%7D%5Cright%29%5E2+%2B+V%28s%29+%5Cright%5D+ds&bg=fff&fg=222&s=0 "= \int \left[ \frac{m}{2} \left(\frac{dq(s)}{ds}\right)^2 + V(s) \right] ds") |
[X ylength^2 / xlength] | distortion |
| Statics (with energy): | ||
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[xlength] | x coordinate |
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[mass xlength / time^2] | force due to stretching spring by dx |
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[ylength] | y coordinate |
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[ylength / xlength] | slope at s |
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[mass ylength^2 / time^2 xlength] | stretching energy density |
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[mass ylength^2 / time^2 xlength] | gravitational energy density |
![= \int \left[ \frac{F}{2} \left(\frac{dq(s)}{ds}\right)^2 + V(s) \right] ds](http://l.wordpress.com/latex.php?latex=%3D+%5Cint+%5Cleft%5B+%5Cfrac%7BF%7D%7B2%7D+%5Cleft%28%5Cfrac%7Bdq%28s%29%7D%7Bds%7D%5Cright%29%5E2+%2B+V%28s%29+%5Cright%5D+ds&bg=fff&fg=222&s=0 "= \int \left[ \frac{F}{2} \left(\frac{dq(s)}{ds}\right)^2 + V(s) \right] ds") |
[mass ylength^2 / time^2] | energy (dS = 0 at equilibrium) |
Dynamics ( ): |
||
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[time] | time |
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[mass] | mass |
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[ylength] | y coordinate |
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[ylength / i time] | i * velocity |
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[mass ylength^2 / time^2] | kinetic energy |
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[mass ylength^2 / time^2] | potential energy |
 ![= \int \left[ \frac{m}{2} \left(\frac{dq(t)}{i dt}\right)^2 + V(t) \right] (i dt)](http://l.wordpress.com/latex.php?latex=%3D+%5Cint+%5Cleft%5B+%5Cfrac%7Bm%7D%7B2%7D+%5Cleft%28%5Cfrac%7Bdq%28t%29%7D%7Bi+dt%7D%5Cright%29%5E2+%2B+V%28t%29+%5Cright%5D+%28i+dt%29&bg=fff&fg=222&s=0 "= \int \left[ \frac{m}{2} \left(\frac{dq(t)}{i dt}\right)^2 + V(t) \right] (i dt)") ![= -i \int \left[ \frac{m}{2} \left(\frac{dq(t)}{dt}\right)^2 - V(t) \right] dt](http://l.wordpress.com/latex.php?latex=%3D+-i+%5Cint+%5Cleft%5B+%5Cfrac%7Bm%7D%7B2%7D+%5Cleft%28%5Cfrac%7Bdq%28t%29%7D%7Bdt%7D%5Cright%29%5E2+-+V%28t%29+%5Cright%5D+dt&bg=fff&fg=222&s=0 "= -i \int \left[ \frac{m}{2} \left(\frac{dq(t)}{dt}\right)^2 - V(t) \right] dt") |
[mass ylength^2 / i time] | i * action |
See also Toby Bartels‘ sci.physics post.
I am fascinated. And rueful over my failed phone call this weekend. And massively sleep-deprived after a full-weekend battle with children-borne night terrors and consequent lack of sleep. So: when I call you up (tonight?) can we please talk also about material manifestations, if such there may be, of imaginary time? Does the universe actually manifest this and its manifold derivatives?