1. notice
  3. English
  4. 1. feature
  6. logic-topic
  7. 2. logic
  8. 3. set-theory
  9. 4. map
  10. 5. order
  11. 6. combinatorics
  13. calculus
  14. 7. real-numbers
  15. 8. limit-sequence
  16. 9. โ„^n
  17. 10. Euclidean-space
  18. 11. Minkowski-space
  19. 12. polynomial
  20. 13. analytic-Euclidean
  21. 14. analytic-Minkowski
  22. 15. analytic-struct-operation
  23. 16. ordinary-differential-equation
  24. 17. volume
  25. 18. integral
  26. 19. divergence
  27. 20. limit-net
  28. 21. compact
  29. 22. connected
  30. 23. topology-struct-operation
  31. 24. exponential
  32. 25. angle
  34. geometry
  35. 26. manifold
  36. 27. metric
  37. 28. metric-connection
  38. 29. geodesic-derivative
  39. 30. curvature-of-metric
  40. 31. Einstein-metric
  41. 32. constant-sectional-curvature
  42. 33. simple-symmetric-space
  43. 34. principal-bundle
  44. 35. group-action
  45. 36. stereographic-projection
  46. 37. Hopf-bundle
  48. field-theory
  49. 38. point-particle-non-relativity
  50. 39. point-particle-relativity
  51. 40. scalar-field
  52. 41. scalar-field-current
  53. 42. scalar-field-non-relativity
  54. 43. projective-lightcone
  55. 44. spacetime-momentum-spinor-representation
  56. 45. Lorentz-group
  57. 46. spinor-field
  58. 47. spinor-field-current
  59. 48. electromagnetic-field
  60. 49. Laplacian-of-tensor-field
  61. 50. Einstein-metric
  62. 51. interaction
  63. 52. harmonic-oscillator-quantization
  64. 53. spinor-field-misc
  65. 54. reference
  67. ไธญๆ–‡
  68. 55. notice
  69. 56. feature
  71. ้€ป่พ‘
  72. 57. ้€ป่พ‘
  73. 58. ้›†ๅˆ่ฎบ
  74. 59. ๆ˜ ๅฐ„
  75. 60. ๅบ
  76. 61. ็ป„ๅˆ
  78. ๅพฎ็งฏๅˆ†
  79. 62. ๅฎžๆ•ฐ
  80. 63. ๆ•ฐๅˆ—ๆž้™
  81. 64. โ„^n
  82. 65. Euclidean ็ฉบ้—ด
  83. 66. Minkowski ็ฉบ้—ด
  84. 67. ๅคš้กนๅผ
  85. 68. ่งฃๆž (Euclidean)
  86. 69. ่งฃๆž (Minkowski)
  87. 70. ่งฃๆž struct ็š„ๆ“ไฝœ
  88. 71. ๅธธๅพฎๅˆ†ๆ–น็จ‹
  89. 72. ไฝ“็งฏ
  90. 73. ็งฏๅˆ†
  91. 74. ๆ•ฃๅบฆ
  92. 75. ็ฝ‘ๆž้™
  93. 76. ็ดง่‡ด
  94. 77. ่ฟž้€š
  95. 78. ๆ‹“ๆ‰‘ struct ็š„ๆ“ไฝœ
  96. 79. ๆŒ‡ๆ•ฐๅ‡ฝๆ•ฐ
  97. 80. ่ง’ๅบฆ
  99. ๅ‡ ไฝ•
  100. 81. ๆตๅฝข
  101. 82. ๅบฆ่ง„
  102. 83. ๅบฆ่ง„็š„่”็ปœ
  103. 84. Levi-Civita ๅฏผๆ•ฐ
  104. 85. ๅบฆ่ง„็š„ๆ›ฒ็Ž‡
  105. 86. Einstein ๅบฆ่ง„
  106. 87. ๅธธๆˆช้ขๆ›ฒ็Ž‡
  107. 88. simple-symmetric-space
  108. 89. ไธปไธ›
  109. 90. ็พคไฝœ็”จ
  110. 91. ็ƒๆžๆŠ•ๅฝฑ
  111. 92. Hopf ไธ›
  113. ๅœบ่ฎบ
  114. 93. ้ž็›ธๅฏน่ฎบ็‚น็ฒ’ๅญ
  115. 94. ็›ธๅฏน่ฎบ็‚น็ฒ’ๅญ
  116. 95. ็บฏ้‡ๅœบ
  117. 96. ็บฏ้‡ๅœบ็š„ๅฎˆๆ’ๆต
  118. 97. ้ž็›ธๅฏน่ฎบ็บฏ้‡ๅœบ
  119. 98. ๅ…‰้”ฅๅฐ„ๅฝฑ
  120. 99. ๆ—ถ็ฉบๅŠจ้‡็š„่‡ชๆ—‹่กจ็คบ
  121. 100. Lorentz ็พค
  122. 101. ๆ—‹้‡ๅœบ
  123. 102. ๆ—‹้‡ๅœบ็š„ๅฎˆๆ’ๆต
  124. 103. ็”ต็ฃๅœบ
  125. 104. ๅผ ้‡ๅœบ็š„ Laplacian
  126. 105. Einstein ๅบฆ่ง„
  127. 106. ็›ธไบ’ไฝœ็”จ
  128. 107. ่ฐๆŒฏๅญ้‡ๅญๅŒ–
  129. 108. ๆ—‹้‡ๅœบๆ‚้กน
  130. 109. ๅ‚่€ƒ

note-math

May migrate to principal-bundle-connection in the future?

cf. motivation-of-gauge-field

There are many possible connections

There are connections that cannot be local flat. For any local region, the connection field cannot be eliminated by changing the gauge.

Similar to flat-metric-iff-curvature-0

There exist local bundle coordinates or phases, where the connection is zero <==> the curvature is zero where , in coordinates

When not exist flat connection coordinate, choose minimal curvature, based on metric-volume-form

Example

case. is valued, and it is commutative . In this case , in coordinates

Transforming from local flat-connection coordinates to general coordinates gives a PDE

PDE solvable condition

From the solution of the PDE , integration gives and phase transformation

[electromagnetic-field]

In

There are many possible connections; select by minimizing curvature.

Note that means the definition of this action requires spacetime metric

eq

In coordinates

In spacetime decomposition coordinates

Of course, this decomposition method is not invariant

Question How to make obviously imply the form of the electromagnetic field equations? [Maxwell-equation]

Also

Where

Note that and exterior derivative are related to Hodge star. can also be related to Hodge star. It may be related to the behavior of Hodge star in the spatial coordinates of spacetime decomposition.

Note that Hodge star requires a metric

Using the special specification [Lorentz-gauge] , the equation becomes