By Bertschinger E.
There arc 3 crucial principles underlying common relativity (OR). the 1st is that area time should be defined as a curved, 4-dimensional mathematical constitution referred to as a pscudo Ricmannian manifold. in short, time and house jointly include a curved 4 dimensional non-Euclidean geometry. therefore, the practitioner of OR has to be accustomed to the basic geometrical homes of curved spacctimc. particularly, the legislation of physics needs to be expressed in a sort that's legitimate independently of any coordinate procedure used to label issues in spacetimc.
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Example text
MTW eq. 14). Using equations (5), (12), and (13), one may show � � � µ α ˆ α α ν α ˆ α ˆ ω αˆµˆ ˆ E νˆ − ∇νˆ E µ ˆν = E α ∇µ ˆ = E µ ˆ E νˆ ∂µ E ν − ∂ν E µ � . (14) In general the commutator basis coefficients do not vanish. Despite the appearance of a second (coordinate) basis, the commutator basis coefficients are independent of any other basis besides the orthonormal one. The coordinate basis is introduced solely for the convenience of partial differentiation with respect to the coordinates. The commutator basis coefficients carry information about how the tetrad rotates as one moves to nearby points in the manifold.
We use equation (9) by expressing the tetrad components in a coordinate basis using equation (5). The result is (13) [eµˆ , eνˆ ] = ∂µˆ eνˆ − ∂νˆ eµˆ ≡ ω αˆµˆ ˆ , ˆν eα where ∂µˆ ≡ E µµˆ ∂µ . Equation (13) defines the commutator basis coefficients ω αˆµˆ ˆν (cf. MTW eq. 14). Using equations (5), (12), and (13), one may show � � � µ α ˆ α α ν α ˆ α ˆ ω αˆµˆ ˆ E νˆ − ∇νˆ E µ ˆν = E α ∇µ ˆ = E µ ˆ E νˆ ∂µ E ν − ∂ν E µ � . (14) In general the commutator basis coefficients do not vanish. Despite the appearance of a second (coordinate) basis, the commutator basis coefficients are independent of any other basis besides the orthonormal one.
14). Using equations (5), (12), and (13), one may show � � � µ α ˆ α α ν α ˆ α ˆ ω αˆµˆ ˆ E νˆ − ∇νˆ E µ ˆν = E α ∇µ ˆ = E µ ˆ E νˆ ∂µ E ν − ∂ν E µ � . (14) In general the commutator basis coefficients do not vanish. Despite the appearance of a second (coordinate) basis, the commutator basis coefficients are independent of any other basis besides the orthonormal one. The coordinate basis is introduced solely for the convenience of partial differentiation with respect to the coordinates. The commutator basis coefficients carry information about how the tetrad rotates as one moves to nearby points in the manifold.