1 Language is expressed by word, phrase and sentence in Aurora Theory, abbreviated to AT.
Refer to the following paper.
2 Dictoron reaches Riemann sphere that has the radius r1 of 1 by the time t1 from the origin of coordinates.
Riemann sphere is expressed by x2 + y2 + z2 = 1.
Here c is light velocity, r1 = c t1 = 1.
Refer to the following paper.
3 Dictoron has distance that is abbreviated to ds.
In Aurora Theory, phrase is expressed on Riemann sphere by dictoron’s radiation from coordinate’s origin.
Phrase takes shape when ds = 1.
Phrase does not take shape when ds < 1 or ds > 1.
4 Now in Aurora Time Theory, abbreviated to ATT, 4 dimensional sphere that has the radius 1 is expressed by x2 + y2 + z2 + t2 = 1.
Here t is time dimension.
5 In ATT, phrase takes shape when ds = 1 and phrase does not take shape when ds < 1 or ds > 1.
6 In ATT, t2 = 1 – (x2 + y2 + z2).
When x2 + y2 + z2 > 1, t2< 0.
t becomes imaginary number.
7 In AT and ATT, word is expressed on complex plane and phrase is expressed on the sphere. ds has real number 1.
8 In ATT, t is expressed by 1/c. Here c is light velocity. So t be imaginary number, the expression is i/c.
9 4 dimensional coordinates in ATT, when t is expressed by i, the sphere is expressed by x2 + y2 + z2 = 2.
10 In ATT, when ds = 1 and the sphere is x2 + y2 + z2 = 2, dictoron is in the sphere. So phrase does not make shape.
11 Dictoron in ATT has magnitude by the definition that is expressed by ds.
When t is imaginary unit i, remained magnitude to the destination for phrase is 1.
12 If t be imaginary unit i, non-existential space that is measured by remained magnitude 1 is to be existent.
13 When t is imaginary number, now verified imaginary unit i, t is called imaginary time.
14 Imaginary time makes <should-not-be-existent-but-logically demanded> space in the ATT’s sphere.
The space is called imaginary space.
Tokyo November 11, 2006
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