Единая геометрическая теория классических полей (стр. 3 из 3)

’ # , "

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1. 0 - -% :

∆αµν = Γµαν + Kα⋅µν

Kαµν = −Kµαν

2. ." % :

^σ =∂^µg , # g = det g_µν

Γµσ

2g

3. $ # :

Ωαµν = ∆αµν − ∆ανµ = Kαµν − Kανµ

Kαµν =

1 (Ωαµν − Ωµαν − Ωναµ)

2

4. :

δuµ = −∆µαβuαdxβ, δuµ = ∆αµβuαdxβ

5. % # :

∇µuν = ∂µuν + ∆νσµuσ, ∇~µuν = ∂µuν + Γσνµuσ

∇µuν = ∂µuν − ∆σνµuσ, ∇~µuν = ∂µuν − Γνσµuσ

6. % # # ∆^α_µν = Γ_µ^α_ν + iA^α_⋅µν:

Aα⋅µα = Aα⋅(µν) = 0, ∆αµα = Γµαα , ∆α(µν) = Γµαν

∇µuµ = ∂µuµ+ ∆µσµuσ = ∂µuµ+ Γσµµuσ

∇_µT (µν⁾ = ∂_µT (µν⁾ +∆µ_σµT (σν⁾ + ∆ν_(σµ)T (µσ⁾ = ∂_µTµν + Γ_σµ_µT (σν⁾ + Γ_σν_µT (µσ⁾

7. 1 - :

(∇µ∇ν −∇ν∇µ)uλ = Rλ⋅σµνuσ + Ωσ⋅µν∇σuλ

Rα⋅βµν = ∂µ∆αβν − ∂ν∆αβµ+ ∆ατµ∆τβν − ∆ατν∆τβµ

Ω^α_⋅µν = ∆^α_µν − ∆^α_νµ

8. - - :

R

+∇~ α −∇~νKα⋅βµ+ Kα⋅τµKτ⋅βν− Kα⋅τνKτ⋅βµ

µK ⋅βν

9. 1 2 3 :

ε_αβγλ= _g [αβγλ], ε^αβγλ=− ¹ [αβγλ]

+1, αβγλ - " 0123

[αβγ^λ]= −1, αβγλ - " 0123

0, αβγλ #

10. * ’- :

δα⋅β⋅γ⋅ λ⋅µνστ ≡ −εαβγλεµνστ δα⋅β⋅γ⋅µνσ ≡ −εαβγτεµνστ

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1. Einstein A., The Meaning of Relativity, Princeton Univ. Press, Princeton, N.Y, 1950

(* #: (!& ! )., . , 2, ., 1955).

2. ). (!& ! , . & #, 1. 1-2, #- «) », ., 1966.

3. E. Schrodinger, Space-Time Structure, Cambridge University Press, 1960 (* #:

(. 6#, * - , , )7 ,

2000).

4. * *."., * & +.,., 1 , #- «) », ., 1973.

5. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, Freeman, San Francisco,

1973 (* #: -. , , . . , " . / , / , #- « », .,

1977).

6. 0.). " $ , .1. % , )... 2 , . : #

, #- «) », ., 1986.

7. E. Cartan, Lecons sur la Geometrie des Espaces de Riemann, Gauthier-Villars, Paris, 1928 and 1946 (* #: % (., -

, #- /, ., 1960).

8. +. Cartan, On Manifolds with an Affine Connection and the Theory of General Relativity, translated by A. Magnon and A. Ashtekar (Bibliopolis, Naples, 1986).

9. %.%. 1 , ) # -

, #- «+# -..», 2002 .

10. 3. . - $ , 0 & # , 7),

1 119. . 3, 1976.

11. Alberto Saa, Einstein-Cartan theory of gravity revisited, gr-qc/9309027 (1993).

12. Hong-jun Xie and Takeshi Shirafuji, Dynamical torsion and torsion potential, gr-qc/9603006 (1996).

13. V.C. de Andrade and J.G. Pereira, Torsion and the Electromagnetic Field, gr-qc/9708051 (1999).

14. Yuyiu Lam, Totally Asymmetric Torsion on Riemann-Cartan Manifold, gr-qc/0211009 (2002).