Extrinsic Curvature as Geodesics Deviation

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i.e., Leibniz rule. And of course the operation d δ should be excluded from the wedge’s. We use a bracket on δ as d (δ) to express this fact explicitly. Before we continue, let’s introduce some notations C∑ N µν = g µν − h µν = (I) ɛ (I) n (I) µ n ν . (22) I=1 (I) N µν = N µν − (I) ɛ (I) n µ (I) n ν = C∑ (J) ɛ (J) n µ (J) n ν . (23) J=1,J≠I We notice that (I) n ν (I) n ν = (I) ɛ = const along Σ, thus we have ( d δ (I) n ν ) (I) n ν = 0, (24) which implies d (I) δ n ν = hδ α ∇ α (I) n ν = = ( ) hδ α ∇ α (I) n µ ( ) hδ α ∇ α (I) n µ (h ν µ + N ν µ ) = g ν µ ( ) ( hδ α ∇ α (I) n µ h ν µ + (I) N ν µ ) , (25) where we have used equation (24) in deriving the last equal sign. The above equation is of course just where we have defined d δ (I) n ν = (I) K δν + (IJ) B δ ≡ C∑ J=1,J≠I (J) ɛ (J) n ν (IJ) B δ , (26) ( (J) n µ d δ (I) n µ ) . (27) Overall, the above procedure is nothing but projecting d δ (I) n ν to the quantity lying on Σ and the quantity normalizing to Σ. Now we can use the fact and equation (26) to rewrite equation (21) as K δµβ...ν = (1) K (δ)µ ∧ (2) n β ∧ ... ∧ (C) n ν ... ∧ (J) n ν ∧ ... ∧ (J) n µ ∧ ... = 0, (28) + (1) n µ ∧ (2) K (δ)β ∧ ... ∧ (C) n ν + (1) n µ ∧ (2) n β ∧ ... ∧ (C) K (δ)ν . (29) Again, when we do wedge, the δ is not affected. It is apparent that the terms above are linearly independent, thus K δµβ...ν is a tensor whose coefficients of linearly independent tensors are (I) K αβ . In another word, there is a one-to-one correspondence between K δµβ...ν and a set of (I) K αβ 4 . Therefore, NVP extrinsic curvature K δµβ...ν is equivalent to C quantities (I) K µν , which characterize extrinsic curvature under GEP. Now we can conclude that GEP and NVP are equivalent for any codimension. 4 The above discuss can be clearer by the notation of [2], Actually this is how I found the way to solve it. In [2] tangent space and cotangent space are not distinguished (this is OK when metric tensor is defined on the manifold since metric defines an one-to-one map between tangent and cotangent space.) and vectors are 4
References [1] R. M. Wald, “General Relativity,” (The University of Chicago Press, Chicago and London, 1984), Chapter 10. [2] C. Misner, K. Thorne and J. A. Wheeler, “Gravitation,” (Freeman, San Francisco, 1973), Chapter 21. expressed explicitly by its coefficients and basis. Thus equation (26) is written as ∇ a (I) n = (I) K b a e b + C∑ J=1,J≠I where e a are basis vectors of Σ. And equation (29) can be rewritten as ( ) K = h ab e b ⊗ ∇ (1) a n ∧ (2) n ∧ ... ∧ (C) n = h ab e b ⊗ Now combine above two equations, we have K =h ab e b ⊗ =e a ⊗ C∑ I=1 C∑ I=1 C∑ I=1 (J) ɛ (J) n (IJ) B δ , (30) ( ( ) ) (1) n ∧ (2) n ∧ ... ∧ ∇ (I) a n ∧ ... ∧ (C) n . (31) ( ) (I) Ka c (1) n ∧ (2) n ∧ ... ∧ e c ∧ ... ∧ (C) n ( ) (I) K (1) ab n ∧ (2) n ∧ ... ∧ e b ∧ ... ∧ (C) n , (32) where we have defined e a ≡ h ab e b and in the above sum, e c (e b ) occupies the Ith position and (I) n is removed. Equation (32) shows clearly there is an one-to-one correspondence between K and (I) K ab . 5
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Page 3: Since K αβ only appear as contrac

Extrinsic Curvature as Geodesics Deviation

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