User:Tom Allen/ta/AddendumX04P09
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To generate spacetime subject to torsion from flat spacetime each spacetime coordinate in
flat spacetime is folded into the corresponding spacetime coordinate in spacetime subject
to torsion.
Values associated with the curvature of spacetime subject to torsion are calculated in flat
spacetime at each point in spacetime.
The content of this page explores the concept of spacetime subject to torsion using waves in
3-dimensional spacetime.
The plots shown on this page describe waves moving along circular paths.
blue -> red -> green -> purple -> pink
The 3 waves plotted in the 3-dimensional flat spacetime do not have the same wavelength.
The 3 waves plotted in the 3-dimensional spacetime subject to torsion have the same wavelength.
The 3 waves plotted in the 3-dimensional flat spacetime do not have the same velocity.
The 3 waves plotted in the 3-dimensional spacetime subject to torsion have the same velocity.
... waves at a radius of 0.50 ...
...
... 3-dimensional flat spacetime ...
...
load 'plot' pd 'reset' a=:1"_ k=:2"_ w=:1"_ r=:0.50"_ NB. ... wave is at this radius ... t=:0.00p1"_ pd (];a*sin@((k*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:0.25p1"_ pd (];a*sin@((k*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:0.50p1"_ pd (];a*sin@((k*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:0.75p1"_ pd (];a*sin@((k*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:1.00p1"_ pd (];a*sin@((k*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 pd 'aspect 1' pd 'pdf'
NB. ... execute (ijx) ... NB. ... number of wavelengths around circumference ... k'' 2 NB. ... time for wave to travel a distance of 1 wavelength ... (2p1%w)'' 6.28319 NB. ... time for wave to travel around circumference ... ((2p1%w)*k)'' 12.5664 NB. ... distance around circumference ... (2p1*r)'' 3.14159 NB. ... velocity of the wave ... ((2p1*r)%(2p1%w)*k)'' 0.25
... blue wave ...
...
... 3-dimensional spacetime subject to torsion ...
...
load 'plot' pd 'reset' a=:1"_ k=:2"_ w=:1"_ r=:0.50"_ t=:0.00p1"_ pd ((r*]);a*sin@((k*r*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:0.25p1"_ pd ((r*]);a*sin@((k*r*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:0.50p1"_ pd ((r*]);a*sin@((k*r*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:0.75p1"_ pd ((r*]);a*sin@((k*r*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:1.00p1"_ pd ((r*]);a*sin@((k*r*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 pd 'aspect 1' pd 'pdf'
NB. ... execute (ijx) ... NB. ... number of wavelengths around circumference ... (k*r)'' 1 NB. ... time for wave to travel a distance of 1 wavelength ... (2p1%w)'' 6.28319 NB. ... time for wave to travel around circumference ... ((2p1%w)*k*r)'' 6.28319 NB. ... distance around circumference ... (2p1*r)'' 3.14159 NB. ... velocity of the wave ... ((2p1*r)%(2p1%w)*k*r)'' 0.5
... red wave (constant velocity) ...
...
... curvature of spacetime is a result of the constant velocity of a wave of a particular type ...
...
load 'plot' pd 'reset' a=:1"_ k=:2"_ w=:1"_ r=:0.50"_ t=:2.00p1"_ NB. ... blue wave ... NB. ... NB. ... wave in 3-dimensional flat spacetime ... NB. ... at ... NB. ... radius 0.50 ... NB. ... and ... NB. ... time 2.00p1 ... NB. ... as ... NB. ... space varies from 0 to 0.50*2p1 ... pd (];a*sin@((k *])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 NB. ... red wave ... NB. ... NB. ... wave in 3-dimensional spacetime subject to torsion ... NB. ... at ... NB. ... radius 0.50 ... NB. ... and ... NB. ... time 2.00p1 ... NB. ... as ... NB. ... space varies from 0 to 0.50*2p1 ... pd (];a*sin@((k*r*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 NB. ... NB. ... NB. ... NB. ... the blue wave is folded into the red wave ... NB. ... at ... NB. ... each point in spacetime ... NB. ... NB. ... the resulting curvature of spacetime ... NB. ... is ... NB. ... calculated from the blue wave at each point in spacetime ... NB. ... NB. ... the red wave travels at a constant velocity ... NB. ... (analogous to electromagnetic waves) ... pd 'aspect 1' pd 'pdf'
... waves at a radius of 1.00 ...
...
... 3-dimensional flat spacetime ...
...
load 'plot' pd 'reset' a=:1"_ k=:2"_ w=:1"_ r=:1.00"_ NB. ... wave is at this radius ... t=:0.00p1"_ pd (];a*sin@((k*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:0.25p1"_ pd (];a*sin@((k*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:0.50p1"_ pd (];a*sin@((k*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:0.75p1"_ pd (];a*sin@((k*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:1.00p1"_ pd (];a*sin@((k*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 pd 'aspect 1' pd 'pdf'
NB. ... execute (ijx) ... NB. ... number of wavelengths around circumference ... k'' 2 NB. ... time for wave to travel a distance of 1 wavelength ... (2p1%w)'' 6.28319 NB. ... time for wave to travel around circumference ... ((2p1%w)*k)'' 12.5664 NB. ... distance around circumference ... (2p1*r)'' 6.28319 NB. ... velocity of the wave ... ((2p1*r)%(2p1%w)*k)'' 0.5
... blue wave ...
...
... 3-dimensional spacetime subject to torsion ...
...
load 'plot' pd 'reset' a=:1"_ k=:2"_ w=:1"_ r=:1.00"_ t=:0.00p1"_ pd ((r*]);a*sin@((k*r*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:0.25p1"_ pd ((r*]);a*sin@((k*r*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:0.50p1"_ pd ((r*]);a*sin@((k*r*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:0.75p1"_ pd ((r*]);a*sin@((k*r*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:1.00p1"_ pd ((r*]);a*sin@((k*r*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 pd 'aspect 1' pd 'pdf'
NB. ... execute (ijx) ... NB. ... number of wavelengths around circumference ... (k*r)'' 2 NB. ... time for wave to travel a distance of 1 wavelength ... (2p1%w)'' 6.28319 NB. ... time for wave to travel around circumference ... ((2p1%w)*k*r)'' 12.5664 NB. ... distance around circumference ... (2p1*r)'' 6.28319 NB. ... velocity of the wave ... ((2p1*r)%(2p1%w)*k*r)'' 0.5
... red wave (constant velocity) ...
...
... at a radius of 1.00 the blue wave has the same velocity as the red wave ...
...
load 'plot' pd 'reset' a=:1"_ k=:2"_ w=:1"_ r=:1.00"_ t=:2.00p1"_ NB. ... blue wave ... NB. ... NB. ... wave in 3-dimensional flat spacetime ... NB. ... at ... NB. ... radius 1.00 ... NB. ... and ... NB. ... time 2.00p1 ... NB. ... as ... NB. ... space varies from 0 to 1.00*2p1 ... pd (];a*sin@((k *])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 NB. ... red wave ... NB. ... NB. ... the red wave overwrites the blue wave ... pd (];a*sin@((k*r*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 NB. ... NB. ... NB. ... NB. ... at a radius of 1.00 the red wave is identical to the blue wave ... NB. ... at ... NB. ... each point in spacetime ... NB. ... NB. ... the red wave travels at a constant velocity ... NB. ... (analogous to electromagnetic waves) ... pd 'aspect 1' pd 'pdf'
... waves at a radius of 1.50 ...
...
... 3-dimensional flat spacetime ...
...
load 'plot' pd 'reset' a=:1"_ k=:2"_ w=:1"_ r=:1.50"_ NB. ... wave is at this radius ... t=:0.00p1"_ pd (];a*sin@((k*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:0.25p1"_ pd (];a*sin@((k*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:0.50p1"_ pd (];a*sin@((k*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:0.75p1"_ pd (];a*sin@((k*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:1.00p1"_ pd (];a*sin@((k*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 pd 'aspect 1' pd 'pdf'
NB. ... execute (ijx) ... NB. ... number of wavelengths around circumference ... k'' 2 NB. ... time for wave to travel a distance of 1 wavelength ... (2p1%w)'' 6.28319 NB. ... time for wave to travel around circumference ... ((2p1%w)*k)'' 12.5664 NB. ... distance around circumference ... (2p1*r)'' 9.42478 NB. ... velocity of the wave ... ((2p1*r)%(2p1%w)*k)'' 0.75
... blue wave ...
...
... 3-dimensional spacetime subject to torsion ...
...
load 'plot' pd 'reset' a=:1"_ k=:2"_ w=:1"_ r=:1.50"_ t=:0.00p1"_ pd ((r*]);a*sin@((k*r*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:0.25p1"_ pd ((r*]);a*sin@((k*r*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:0.50p1"_ pd ((r*]);a*sin@((k*r*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:0.75p1"_ pd ((r*]);a*sin@((k*r*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 t=:1.00p1"_ pd ((r*]);a*sin@((k*r*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 pd 'aspect 1' pd 'pdf'
NB. ... execute (ijx) ... NB. ... number of wavelengths around circumference ... (k*r)'' 3 NB. ... time for wave to travel a distance of 1 wavelength ... (2p1%w)'' 6.28319 NB. ... time for wave to travel around circumference ... ((2p1%w)*k*r)'' 18.8496 NB. ... distance around circumference ... (2p1*r)'' 9.42478 NB. ... velocity of the wave ... ((2p1*r)%(2p1%w)*k*r)'' 0.5
... red wave (constant velocity) ...
...
... curvature of spacetime is a result of the constant velocity of a wave of a particular type ...
...
load 'plot' pd 'reset' a=:1"_ k=:2"_ w=:1"_ r=:1.50"_ t=:2.00p1"_ NB. ... blue wave ... NB. ... NB. ... wave in 3-dimensional flat spacetime ... NB. ... at ... NB. ... radius 1.50 ... NB. ... and ... NB. ... time 2.00p1 ... NB. ... as ... NB. ... space varies from 0 to 1.50*2p1 ... pd (];a*sin@((k *])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 NB. ... red wave ... NB. ... NB. ... wave in 3-dimensional spacetime subject to torsion ... NB. ... at ... NB. ... radius 1.50 ... NB. ... and ... NB. ... time 2.00p1 ... NB. ... as ... NB. ... space varies from 0 to 1.50*2p1 ... pd (];a*sin@((k*r*])-w*t)"0) n=: pTsteps _ 0p1 2p1 160 NB. ... NB. ... NB. ... NB. ... the blue wave is folded into the red wave ... NB. ... at ... NB. ... each point in spacetime ... NB. ... NB. ... the resulting curvature of spacetime ... NB. ... is ... NB. ... calculated from the blue wave at each point in spacetime ... NB. ... NB. ... the red wave travels at a constant velocity ... NB. ... (analogous to electromagnetic waves) ... pd 'aspect 1' pd 'pdf'
Download: File:LX04P0900G.txt