1 There gives 4 dimensional sphere that contains space and time.
2 The sphere has unit radius 1. x2+y2+z2+t2=1 The sphere is called sphere4.
3 Sphere4 has projection to 3 dimensional sphere. Projection is from sphere4 to the center of the same center of the two spheres.
4 3 dimensional sphere has unit radius 1. x2+y2+z2=1 The sphere is called sphere3.
5 On sphere4, t=
6 On sphere4, time has symmetry
7 On sphere4, when 1- ( x2+y2+z2)
viz x2+y2+z2<1, time becomes real number
8 On sphere4, when 1- ( x2+y2+z2)
viz x2+y2+z2=1, time becomes 0.
9 On sphere4, when 1- ( x2+y2+z2)
viz x2+y2+z2>1, time becomes imaginary number.
10 On sphere4, peculiar points on time are below.
x2+y2+z2=0 t=
1
x2+y2+z2=1 t=0
x2+y2+z2=2 t=i
11 Now put f=x2+y2+z2. On sphere4, there gives orthogonal coordinates about f and t.
Coordinate f has the center that is f=1.
When f>1, t becomes imaginary number. Sphere3 is larger than sphere4. (11-1)
When f<1, t becomes real number. Sphere3 is smaller than sphere4. (11-2)
12 Now there gives a membrane that has f-t orthogonal coordinates.
13 The membrane has smooth bent at f=1. The membrane is called escalator-membrane, abbreviated to EMB or EB.
14 EB’s time is visible by (11-2).
15 EB’s time is invisible by (11-1).
Tokyo January 21, 2007
1 There gives 4 dimensional sphere that contains space and time.
2 The sphere has unit radius 1. x2+y2+z2+t2=1 The sphere is called sphere4.
3 Sphere4 has projection to 3 dimensional sphere. Projection is from sphere4 to the center of the same center of the two spheres.
4 3 dimensional sphere has unit radius 1. x2+y2+z2=1 The sphere is called sphere3.
5 On sphere4, t=
6 On sphere4, time has symmetry
7 On sphere4, when 1- ( x2+y2+z2) viz x2+y2+z2<1, time becomes real number
8 On sphere4, when 1- ( x2+y2+z2) viz x2+y2+z2=1, time becomes 0.
9 On sphere4, when 1- ( x2+y2+z2) viz x2+y2+z2>1, time becomes imaginary number.
10 On sphere4, peculiar points on time are below.
x2+y2+z2=0 t=1
x2+y2+z2=1 t=0
x2+y2+z2=2 t=i
11 Now put f=x2+y2+z2. On sphere4, there gives orthogonal coordinates about f and t.
Coordinate f has the center that is f=1.
When f>1, t becomes imaginary number. Sphere3 is larger than sphere4. (11-1)
When f<1, t becomes real number. Sphere3 is smaller than sphere4. (11-2)
12 Now there gives a membrane that has f-t orthogonal coordinates.
13 The membrane has smooth bent at f=1. The membrane is called escalator-membrane, abbreviated to EMB or EB.
14 EB’s time is visible by (11-2).
15 EB’s time is invisible by (11-1).
Tokyo January 21, 2007
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