Construction of Spacetime
Especially on Transformation with Boundary for Dimensions
1 Spacetime is putted to manifold.
2 When spacetime is expressed by sphere, spacetime becomes sphere symmetry.
Refer to the following paper.
3 The symmetry is special orthogonal transformational group.
4 Generally n-dimensional SO(n) is compact Rie group.
5 On symmetry of spacetime, geometric invariant is presented.
Now on invariant, metric tensor g is presented.
g = Σgij dxidxj
g’s invariant transformation is isometry group.
When gij → gji is established, infinitesimal motion (Killing vector field) is 0.
6 Here Ricci flow is expressed by the following.
g(t) = gij(t)
∂gij / ∂t = -2Rij
Rij is Ricci curvature tensor.
7 M defined by a time interval [0, T] has a unique g(0) = go on Ricci flow.
8 (S4,g0) is 4 dimensional sphere surface with radius 1.
g(t) = r(t)2g0
g(t) has finite time T = 1/2(n-1).
Ricci flow becomes a point after finite time.
Phenomenon what manifold becomes smaller dimensional space is called collapse.
9 Now word has meaning and time.
10 Here word is expressed by manifold.
Refer to the following paper.
Also refer to the following paper.
On time of language, refer to the following paper.
On time principle of language, refer to the following paper.
10 Here time of word is defined by manifold’s time of Ricci flow’s collapse.
11 By Thurston conjecture, when compact 3 dimensional manifold is cut and separated by 2 dimensional sphere and torus, while generated compact manifold has sphere boundary, sphere is put together there. As the result new generated manifold has locally uniform structure.
Refer to the following paper.
Also refer to the following paper.
12 Peculiar solution of Thurston conjecture is Poincaré conjecture that is what simply connected 3 dimensional compact manifold is homomorphic to 3 dimensional sphere surface S3.
13 According by Thurston conjecture, what put together newly generated 3 dimensional manifold is called connected sum.
Connected sum is expressed by M1#M2. M1 and M2 are 3 dimensional manifolds.
14 Here sentence is defined by manifold’s connected sum.
Tokyo April 24, 2007
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