User:Tom Allen/ta/AddendumX04P06E
... geodesics follow paths resulting from the curvature of 4-dimensional spacetime ...
... and ...
... geodesics pass through points in 4-dimensional spacetime ...
... given ...
... the curvature of 4-dimensional spacetime ...
... then ...
... geodesics are never cyclic and never follow the same path ...
... (compare: the motion of the perihelion of Mercury) ...
... FOR THIS DIAGRAM ...
... the blue and pink lines travel in a wave in flat spacetime ...
... the blue and green lines travel in a wave in curved spacetime ...
... the blue pink and green lines indicate the distances traveled by the waves in the same time ...
... the initial configuration of the waves is at time=0 ...
... and ...
... the blue pink and green lines have no length ...
... the curvature tensor at the spacetime coordinate C is calculated in flat spacetime ...
... and ...
... the spacetime coordinate C moves clockwise as the radius increases for a fixed time>0 ...
... plots in flat spacetime along a fixed radius at a fixed time>0 ...
... are not ...
... an accurate display of the configuration of curved spacetime ...
... plots in flat spacetime along a fixed radius at time=0 ...
... are ...
... an accurate display of the configuration of curved spacetime ...
... the following 6 plots are plots in Cartesian coordinates around the poles (0.01p1<x1<1.99p1) at a radius of 1.05 ...
...
... where ...
...
... value in y0 direction is blue ...
... value in y1 direction is red ...
... value in y2 direction is green ...
... the electric field ...
... electric field-lines never form closed loops ...
load 'plot'
pd 'reset'
uQ0=:,:@(1 1 2 8 1 1 1 _1 0"_)
xCpts=:_ 1.05 1.05 0,_ 0.01p1 1.99p1 100,_ 0.0p1 0.0p1 0,:_ 0.00p1 0.00p1 0"_
pd (uQ0(cx1"_ 1;[(0{])"1@(-@EEMcn)yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts''
pd (uQ0(cx1"_ 1;[(1{])"1@(-@EEMcn)yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts''
pd (uQ0(cx1"_ 1;[(2{])"1@(-@EEMcn)yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts''
pd 'pdf'
... plot 0 ...
... the electric field with x2=0p1 ...
... direction of the electric field at time=0 (initial configuration) ...
load 'plot'
pd 'reset'
uQ0=:,:@(1 1 2 8 1 1 1 _1 0"_)
xCpts=:_ 1.05 1.05 0,_ 0.01p1 1.99p1 100,_ 0.5p1 0.5p1 0,:_ 0.00p1 0.00p1 0"_
pd (uQ0(cx1"_ 1;[(0{])"1@(-@EEMcn)yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts''
pd (uQ0(cx1"_ 1;[(1{])"1@(-@EEMcn)yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts''
pd (uQ0(cx1"_ 1;[(2{])"1@(-@EEMcn)yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts''
pd 'pdf'
... plot 1 ...
... the electric field with x2=0.5p1 ...
... direction of the electric field at time=0 (initial configuration) ...
... the magnetic field ...
load 'plot'
pd 'reset'
uQ0=:,:@(1 1 2 8 1 1 1 _1 0"_)
xCpts=:_ 1.05 1.05 0,_ 0.01p1 1.99p1 100,_ 0.0p1 0.0p1 0,:_ 0.00p1 0.00p1 0"_
pd (uQ0(cx1"_ 1;[(0{])"1@HEMcn yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts''
pd (uQ0(cx1"_ 1;[(1{])"1@HEMcn yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts''
pd (uQ0(cx1"_ 1;[(2{])"1@HEMcn yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts''
pd 'pdf'
... plot 2 ...
... the magnetic field with x2=0p1 ...
... direction of the magnetic field at time=0 (initial configuration) ...
load 'plot'
pd 'reset'
uQ0=:,:@(1 1 2 8 1 1 1 _1 0"_)
xCpts=:_ 1.05 1.05 0,_ 0.01p1 1.99p1 100,_ 0.5p1 0.5p1 0,:_ 0.00p1 0.00p1 0"_
pd (uQ0(cx1"_ 1;[(0{])"1@HEMcn yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts''
pd (uQ0(cx1"_ 1;[(1{])"1@HEMcn yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts''
pd (uQ0(cx1"_ 1;[(2{])"1@HEMcn yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts''
pd 'pdf'
... plot 3 ...
... the magnetic field with x2=0.5p1 ...
... direction of the magnetic field at time=0 (initial configuration) ...
... the Poynting vector ...
... the Poynting vector cross product P=(E x H) ...
load 'plot'
pd 'reset'
uQ0=:,:@(1 1 2 8 1 1 1 _1 0"_)
xCpts=:_ 1.05 1.05 0,_ 0.01p1 1.99p1 100,_ 0.0p1 0.0p1 0,:_ 0.00p1 0.00p1 0"_
pd (uQ0(cx1"_ 1;[(0{])"1@(-@PEMcn)yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts''
pd (uQ0(cx1"_ 1;[(1{])"1@(-@PEMcn)yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts''
pd (uQ0(cx1"_ 1;[(2{])"1@(-@PEMcn)yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts''
pd 'pdf'
... plot 4 ...
... the energy flow with x2=0p1 ...
... direction of the energy flow at time=0 (initial configuration) ...
load 'plot'
pd 'reset'
uQ0=:,:@(1 1 2 8 1 1 1 _1 0"_)
xCpts=:_ 1.05 1.05 0,_ 0.01p1 1.99p1 100,_ 0.5p1 0.5p1 0,:_ 0.00p1 0.00p1 0"_
pd (uQ0(cx1"_ 1;[(0{])"1@(-@PEMcn)yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts''
pd (uQ0(cx1"_ 1;[(1{])"1@(-@PEMcn)yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts''
pd (uQ0(cx1"_ 1;[(2{])"1@(-@PEMcn)yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts''
pd 'pdf'
... plot 5 ...
... the energy flow with x2=0.5p1 ...
... direction of the energy flow at time=0 (initial configuration) ...
... magnetic field-lines form closed loops ...
load 'plot'
pd 'reset'
uQ0=:,:@(1 1 2 8 1 1 1 _1 0"_)
NB. ... wave in magnetic field (component in x2 direction) ...
NB. ... (x0 arbitrary / x1 arbitrary) ...
xCpts=:_ 1.05 1.05 0,_ 0.40p1 0.40p1 0,_ 0p1 2p1 100,:_ 0.00p1 0.00p1 0"_ NB. ... time=0 (initial configuration) ...
pd (uQ0(cx2"_ 1;[(2{])"1@(}:"1@}:"2@xdy smx"2 1 HEMcn)yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts'' NB. ... blue ...
xCpts=:_ 1.05 1.05 0,_ 0.40p1 0.40p1 0,_ 0p1 2p1 100,:_ 0.10p1 0.10p1 0"_ NB. ... time=0.10p1 (4-dimensional spacetime) ...
pd (uQ0(cx2"_ 1;[(2{])"1@(}:"1@}:"2@xdy smx"2 1 HEMcn)yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts'' NB. ... red ...
NB. ... wave in energy flow (component in x2 direction) ...
NB. ... (x0 arbitrary / x1 arbitrary) ...
xCpts=:_ 1.05 1.05 0,_ 0.40p1 0.40p1 0,_ 0p1 2p1 100,:_ 0.00p1 0.00p1 0"_ NB. ... time=0 (initial configuration) ...
pd (uQ0(cx2"_ 1;[(2{])"1@(}:"1@}:"2@xdy smx"2 1 -@PEMcn)yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts'' NB. ... green ...
xCpts=:_ 1.05 1.05 0,_ 0.40p1 0.40p1 0,_ 0p1 2p1 100,:_ 0.10p1 0.10p1 0"_ NB. ... time=0.10p1 (4-dimensional spacetime) ...
pd (uQ0(cx2"_ 1;[(2{])"1@(}:"1@}:"2@xdy smx"2 1 -@PEMcn)yRef@([hkyX04A zfyc"_ 1))]) pTtotal xCpts'' NB. ... purple ...
pd 'pdf'
