Einsteins Theory of Relativity 8 Curvature of Space
- Slides: 33
Einstein‘s Theory of Relativity 8. Curvature of Space, Curvature Tensor, Ricci Tensor, Bianchi Identity Ulrich R. M. E. Geppert 2/25/2021 U. R. M. E. Zielona Gora 1
parallel transport in 2 -D Euclidean space: 1. cartesian coordinates 2. polar coordinates: (calculation of inverse matrix ⇒ annex) 2/25/2021 U. R. M. E. Zielona Gora 2
parallel transport, covariant derivative: need Christoffel symbol 2/25/2021 U. R. M. E. Zielona Gora 3
for polar coordinates normalized basisvectors: the latter in more detail for polar coordinates: 2/25/2021 U. R. M. E. Zielona Gora 4
consider again parallel transport of the constant vector Q P 2/25/2021 U. R. M. E. Zielona Gora 5
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3. spherical coordinates, surface of a unit sphere 2/25/2021 U. R. M. E. Zielona Gora 7
coordinate singularity R Q P 2/25/2021 over Q to R back to P U. R. M. E. Zielona Gora 8
parallel transport of F from p to r either along C or along C‘ s p 2/25/2021 U. R. M. E. Zielona Gora r q 9
reminder: (Taylor expansion) 2/25/2021 U. R. M. E. Zielona Gora 10
(little exercise? ) 2/25/2021 U. R. M. E. Zielona Gora 11
~ geodesic deviation 2/25/2021 U. R. M. E. Zielona Gora 12
this was the more graphic derivation of the curvature tensor via the parallel transport here comes a more formal one ⇒ 2/25/2021 U. R. M. E. Zielona Gora 13
covariant derivatives of a vector are (i. g. ) not commutable: simply by counting and considering the indices: l. h. s. : tensor defines tensor too 2/25/2021 U. R. M. E. Zielona Gora vector 14
justifies the name „curvature tensor“ in flat space: proof: in flat space are cartesian coordinates allowed 2/25/2021 U. R. M. E. Zielona Gora 15
proof: reminder: covariant derivatives of vector and 2 nd rank tensor: 2/25/2021 U. R. M. E. Zielona Gora 16
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by use of the metric: fourfold covariant curvature tensor: one can read off the following symmetries: 2/25/2021 U. R. M. E. Zielona Gora 18
contractions of the curvature tensor: Einstein equations 2/25/2021 U. R. M. E. Zielona Gora 19
inverse metric: 2/25/2021 U. R. M. E. Zielona Gora 20
calculate here Christoffel symbols rather from LII than by its explicite formula: 2/25/2021 U. R. M. E. Zielona Gora 21
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calculate from curvature tensor Ricci tensor: calculate from Ricci tensor curvature scalar: 2/25/2021 U. R. M. E. Zielona Gora 24
synonymous: on a spherical surface there exist NO cartesian coordinates 2/25/2021 U. R. M. E. Zielona Gora 25
curvature tensor for polar coordinates: curvilinear coordinates but flat space 2/25/2021 U. R. M. E. Zielona Gora 26
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Bianchi identities: needed for derivation of Einstein equation (Riemann tensor equation) 2/25/2021 U. R. M. E. Zielona Gora 28
- 6 terms, 2 of them cancel, the same happens for 2. and 3. term covariance principle 2/25/2021 U. R. M. E. Zielona Gora 29
Annex 2/25/2021 U. R. M. E. Zielona Gora 30
transformation of cartesian into polar coordinates y x 2/25/2021 U. R. M. E. Zielona Gora 31
this returns for the time derivatives: 2/25/2021 U. R. M. E. Zielona Gora 32
calculation of the inverse matrix: 2/25/2021 U. R. M. E. Zielona Gora 33