| ∵Kμν＝1/1+1/μ+1/ν | ﾊｲｾﾞﾝﾍﾞﾙｸﾞ | p151 | |||||||||
| 竹内 | 『ｱｲﾝｼｭﾀｲﾝとﾌｧｲﾝﾏﾝ…』p170 | ||||||||||
| 球面の次元 | 1 2 3 4 5 | 6 7 8 9 10 | 11 | ||||||||
| 微分構造数 | 1 1 1 1 1 | 1 28 2 8 6 | 992 | ||||||||
| Kμν | (q/3+2) | ⓪ｵｲﾗｰ数と4πΓ(9/(2･ｵｲﾗｰ数)) | |||||||||
| 角形 | ｵｲﾗｰ数 | /(q/3+1) | ⋍9/(2･ｵｲﾗｰ) | Gammaln(⋆) | exp(⋆⋆) | 4π | ﾘｰﾏﾝ零点 | ||||
| ① | 3 | (3/1) | 1.5 | ⋆ | -0.120782238 | ⋆ | 0.886227 | ・ | 11.13666 | ⋆ | 文献値 |
| ② | 2 | (2/1) | 2.25 | ⋆ | 0.124871715 | ⋆ | 1.133003 | K | 14.23774 | ∼ | 14.13472 |
| ③ | 1.6666667 | (5/3) | 2.7 | ⋆ | 0.434820554 | ⋆ | 1.544686 | ・ | 19.41109 | ∼ | 21.02203 |
| ④ | 1.5 | (3/2) | 3 | ⋆ | 0.69314718 | ⋆ | 2 | ・ | 25.13274 | ∼ | 25.01085 |
| ⑤ | 1.4 | (7/5) | 3.21428571 | ⋆ | 0.899711269 | ⋆ | 2.458893 | ・ | 30.89936 | ∼ | 30.42487 |
| 5 | 1.3636363 | (30/22) | 3.30000015 | ⋆ | 0.987098737 | ⋆ | 2.683438 | ・ | 33.72107 | ∼ | 32.93506 |
| ⑥ | 1.3333333 | (4/3) | 3.375 | ⋆ | 1.065695898 | ⋆ | 2.902858 | ・ | 36.47839 | ∼ | 37.58617 |
| ⑦ | 1.2857143 | (9/7) | 3.5 | ⋆ | 1.200973602 | ⋆ | 3.323351 | ・ | 41.76246 | ∼ | 40.91871 |
| Σ7.5 | 1.2666667 | (9/7.1) | 3.55263158 | ⋆ | 1.25948943 | ⋆ | 3.523622 | ・ | 44.27914 | ∼ | 43.32707 |
| ⑧ | 1.25 | (5/4) | 3.6 | ⋆ | 1.312923308 | ⋆ | 3.717024 | ・ | 46.7095 | ∼ | 48.00515 |
| ⑨ | 1.2222222 | (11.1/9) | 3.68181818 | ⋆ | 1.406902695 | ⋆ | 4.083289 | ・ | 51.31212 | ∼ | 49.77383 |
| 9.5 | 1.2105263 | (12.1/10) | 3.7173913 | ⋆ | 1.448417452 | ⋆ | 4.256373 | ・ | 53.48716 | ∼ | 52.97 |
| ⑩ | 1.2 | (6/5) | 3.75 | ⋆ | 1.486815578 | ⋆ | 4.422988 | ・ | 55.58091 | ∼ | 56.446 |
| ⑪ | 1.1818182 | (31/27) | 3.80769231 | ⋆ | 1.5555441 | ⋆ | 4.737664 | ・ | 59.53524 | ∼ | 59.347 |
| ⑫ | 1.1666667 | (7/6) | 3.85714286 | ⋆ | 1.61524993 | ⋆ | 5.029145 | ・ | 63.1981 | ∼ | 60.831 |
| 13 | 1.16 | (20/17) | 3.87931034 | ⋆ | 1.642249998 | ⋆ | 5.166782 | ・ | 64.92769 | ∼ | 65.112 |
| ⑬ | 1.1538462 | (9/8) | 3.9 | ⋆ | 1.667580347 | ⋆ | 5.29933 | ・ | 66.59334 | ∼ | 75.8 |
| ⓪ｵｲﾗｰ数と4πΓ(9/(2･ｵｲﾗｰ数)) | (q/3+2) | ⓪ｵｲﾗｰ数と4πΓ(9/(2･ｵｲﾗｰ数)) | ||||||||
| ⒬ | 頂点 | 辺 | 面 | 角形 | ｵｲﾗｰ数 | /(q/3+1) | ⋍9/(2･ｵｲﾗｰ) | Gammaln(⋆) | ||
| ⋆角形 | 1 | 1 | 1 | 1 | ① | 3 | (3/1) | 1.5 | ⋆ | -0.120782238 |
| 2角形 | 2 | 2 | 2 | 1 | ② | 2 | (2/1) | 2.25 | ⋆ | 0.124871715 |
| 3角形 | 3 | 3 | 3 | 1 | ③ | 1.6666667 | (5/3) | 2.7 | ⋆ | 0.434820554 |
| 4角形 | 4 | 4 | 4 | 1 | ④ | 1.5 | (3/2) | 3 | ⋆ | 0.69314718 |
| 5角形 | 5 | 5 | 5 | 1 | ⑤ | 1.4 | (7/5) | 3.2142857 | ⋆ | 0.899711269 |
| 5.0125 | 5.01 | 5.01 | 1 | 5 | 1.3636363 | (30/22) | 3.3000002 | ⋆ | 0.987098737 | |
| 6角形 | 6 | 6 | 6 | 1 | ⑥ | 1.3333333 | (4/3) | 3.375 | ⋆ | 1.065695898 |
| 7角形 | 7 | 7 | 7 | 1 | ⑦ | 1.2857143 | (9/7) | 3.5 | ⋆ | 1.200973602 |
| 7.5 | ⋆ | 7.5 | 7.5 | 1 | 7.5 | 1.2666667 | (9/7.1) | 3.5526316 | ⋆ | 1.25948943 |
| 8角形 | 8 | 8 | 8 | 1 | ⑧ | 1.25 | (5/4) | 3.6 | ⋆ | 1.312923308 |
| 9角形 | 9 | 9 | 9 | 1 | ⑨ | 1.2222222 | (11.1/9) | 3.6818182 | ⋆ | 1.406902695 |
| 9.5 | ⋆ | 9.5 | 9.5 | 1 | 9.5 | 1.2105263 | (12.1/10) | 3.7173913 | ⋆ | 1.448417452 |
| 10角形 | 10 | 10 | 10 | 1 | ⑩ | 1.2 | (6/5) | 3.75 | ⋆ | 1.486815578 |
| 11角形 | 11 | 11 | 11 | 1 | ⑪ | 1.1818182 | (31/27) | 3.8076923 | ⋆ | 1.5555441 |
| 12角形 | 12 | 12 | 12 | 1 | ⑫ | 1.1666667 | (7/6) | 3.8571429 | ⋆ | 1.61524993 |
| 12.5 | 12.5 | 12.5 | 1 | 13 | 1.16 | (20/17) | 3.8793103 | ⋆ | 1.642249998 | |
| 13角形 | 13 | 13 | 13 | 1 | ⑬ | 1.1538462 | (9/8) | 3.9 | ⋆ | 1.667580347 |
| (q/3+2) | ⓪ｵｲﾗｰ数と4πΓ(9/(2･ｵｲﾗｰ数)) | ||||||||||
| 角形 | ｵｲﾗｰ数 | /(q/3+1) | ⋍9/(2･ｵｲﾗｰ) | Gammaln(⋆) | exp(⋆⋆) | 4π | ﾘｰﾏﾝ零点 | ||||
| ① | 3 | (3/1) | 1.5 | ⋆ | -0.120782238 | ⋆ | 0.886227 | ・ | 11.13666 | ⋆ | 文献値 |
| ② | 2 | (2/1) | 2.25 | ⋆ | 0.124871715 | ⋆ | 1.133003 | K | 14.23774 | ∼ | 14.13472 |
| ③ | 1.6666667 | (5/3) | 2.7 | ⋆ | 0.434820554 | ⋆ | 1.544686 | ・ | 19.41109 | ∼ | 21.02203 |
| ④ | 1.5 | (3/2) | 3 | ⋆ | 0.69314718 | ⋆ | 2 | ・ | 25.13274 | ∼ | 25.01085 |
| ⑤ | 1.4 | (7/5) | 3.2142857 | ⋆ | 0.899711269 | ⋆ | 2.458893 | ・ | 30.89936 | ∼ | 30.42487 |
| 5 | 1.3636363 | (30/22) | 3.3000002 | ⋆ | 0.987098737 | ⋆ | 2.683438 | ・ | 33.72107 | ∼ | 32.93506 |
| ⑥ | 1.3333333 | (4/3) | 3.375 | ⋆ | 1.065695898 | ⋆ | 2.902858 | ・ | 36.47839 | ∼ | 37.58617 |
| ⑦ | 1.2857143 | (9/7) | 3.5 | ⋆ | 1.200973602 | ⋆ | 3.323351 | ・ | 41.76246 | ∼ | 40.91871 |
| 7.5 | 1.2666667 | (9/7.1) | 3.5526316 | ⋆ | 1.25948943 | ⋆ | 3.523622 | ・ | 44.27914 | ∼ | 43.32707 |
| ⑧ | 1.25 | (5/4) | 3.6 | ⋆ | 1.312923308 | ⋆ | 3.717024 | ・ | 46.7095 | ∼ | 48.00515 |
| ⑨ | 1.2222222 | (11.1/9) | 3.6818182 | ⋆ | 1.406902695 | ⋆ | 4.083289 | ・ | 51.31212 | ∼ | 49.77383 |
| 9.5 | 1.2105263 | (12.1/10) | 3.7173913 | ⋆ | 1.448417452 | ⋆ | 4.256373 | ・ | 53.48716 | ∼ | 52.97 |
| ⑩ | 1.2 | (6/5) | 3.75 | ⋆ | 1.486815578 | ⋆ | 4.422988 | ・ | 55.58091 | ∼ | 56.446 |
| ⑪ | 1.1818182 | (31/27) | 3.8076923 | ⋆ | 1.5555441 | ⋆ | 4.737664 | ・ | 59.53524 | ∼ | 59.347 |
| ⑫ | 1.1666667 | (7/6) | 3.8571429 | ⋆ | 1.61524993 | ⋆ | 5.029145 | ・ | 63.1981 | ∼ | 60.831 |
| 13 | 1.16 | (20/17) | 3.8793103 | ⋆ | 1.642249998 | ⋆ | 5.166782 | ・ | 64.92769 | ∼ | 65.112 |
| ⑬ | 1.1538462 | (9/8) | 3.9 | ⋆ | 1.667580347 | ⋆ | 5.29933 | ・ | 66.59334 | ∼ | 75.8 |
| ⓪ | ｵｲﾗ数 | ①Γ | (An) | 固有値問題 | ||||||
| 2角形 | 3角形 | 4角形 | 5角形 | 6角形 | 7角形 | 8角形 | 9角形 | 10角形 | ||
| n | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 頂点 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 辺 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
| 面 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| ⓪ | ｵｲﾗｰ数 | 2 | 1.667 | 1.5 | 1.4 | 1.33333 | 1.28571 | 1.25 | 1.22222 | 1.2 |
| 逆数 | 0.5 | 0.6 | 0.67 | 0.714 | 0.75 | 0.77778 | 0.8 | 0.81818 | 0.83333 | |
| ⓪ | ｵｲﾗｰ数 | (2/1) | (10/6) | (3/2) | (7/5) | (4/3) | (5/4) | (6/5) | ||
| ①Γ | (An) | (2/1) | ⋆ | (3/2) | ⋆⋆ | (4/3) | ⋆⋆ | (5/4) | ⋆⋆ | (6/5) |
| ② | α | (2/1) | ⋆ | (3/2) | ⋆ | (4/2) | ⋆ | (5/2) | ⋆ | (6/2) |
| ②11(k＾2+α) | ⋍[k[2k]]^(α) | 連分数 | (n/3+2） | |||||||
| n | α | ①Γ(An) | /(n/3+1) | |||||||
| 1 | ⇒ | a=(n-1)(n/2+1) | 2 | (2/1) | ||||||
| 2 | (2/3) | a⋆=a/2b=n/4+1/2 | 1.63636 | ⋆ | ||||||
| 3 | (1/1) | a(分子) | b=n-1 | 母 | a/2b | 1.5 | (3/2) | |||
| 4 | (3/2) | 9 | ･ | 3 | ･ | 1.5 | 1.40625 | (20/14) | ||
| 5 | (3.5/2) | 14 | ･ | 4 | ･ | 1.75 | 1.36364 | |||
| 6 | (4/2) | 20 | ･ | 5 | ･ | 2 | 1.32353 | (4/3) | ||
| 7 | (4.5/2) | 27 | ･ | 6 | ･ | 2.25 | 1.28571 | |||
| 8 | (5/2) | 35 | ･ | 7 | ･ | 2.5 | 1.26761 | |||
| 9 | (5.5/2) | 44 | ･ | 8 | ･ | 2.75 | 1.25 | (5/4) | ||
| 10 | (6/2) | 54 | ･ | 9 | ･ | 3 | 1.23288 | |||
| 11 | (6.5/2) | 65 | ･ | 10 | ･ | 3.25 | 1.21622 | |||
| 12 | (7/2) | 77 | ･ | 11 | ･ | 3.5 | 1.2 | (6/5) | ||
| 13 | (7.5/2) | 90 | ･ | 12 | ･ | 3.75 | 1.18421 | |||
| 14 | (8/2) | 104 | ･ | 13 | ･ | 4 | 1.16883 | (20/17) | ||
| 15 | (8.5/2) | 119 | ･ | 14 | ･ | 4.25 | 1.15385 | (7/6) | ||
| 16 | (9/2) | 135 | ･ | 15 | ･ | 4.5 | 1.16883 | |||
| ﾃﾞｨﾗｯｸ(p45) | ﾆﾕｰﾄﾝ近似 | ～ｱｲﾝｼｭﾀｲﾝg00⋍1-2m/r | ||||
| 時間の単位を適当にとってg00が近似的に1に等しくなるようにすることができる。 | ||||||
| g00=1+2V | (16.6) | |||||
| のVは小さい。√g00⋍1+Vとなり､このVがﾎﾟﾃﾝｼｬﾙである｡これはﾗﾌﾟﾗｽの方程式 | ||||||
| をみたすのだから､ﾆｭｰﾄﾝ･ﾎﾟﾃﾝｼｬﾙと同定できる｡質量mが原点にある…なら | ||||||
| V=-m/rである。符号を確かめるには… | ||||||
| 加速度=-gradV | ||||||
| ∴g00⋍1-2m/r | (18.5)p82 | |||||
| [ｽｶﾗｰ場の曲率] | ||||||
| 2ﾓﾉﾎﾟｰﾙのｽｶﾗｰ場バは | φ⋍1ｰ2/r | 1/φ=1-(2/r) | ||||
| 逆数は | 1/φ=1/[1-(2/r)]⋍1+(2/r)+(2/r)^2+… | |||||
| r/φ=r+2+4/r… | ∴r/4φ⋍r/4+1/2+… | |||||
| 曲率Faは | ∴Fa | =n/4+1/2 | ||||
| 曲率方程式 | μ(Fa+)=… | |||||