By James Foster, J. D. Nightingale
This textbook offers a great creation to an issue that's tremendous effortless to get slowed down in. I took a one semester direction that used this article as an undergraduate, in which i assumed the e-book was once only respectable, yet then whilst I took a gradute path that used Carroll's Spacetime and Geometry is whilst i actually got here to understand the education this e-book gave me (not that Carroll's e-book is undesirable, I simply would not suggest it for a primary reading). let alone the booklet is lovely affordable so far as physics texts pass.
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Extra info for A Short Course in General Relativity
39) we have µ αµ = 1 µν g 2 ∂gνµ ∂gαµ ∂gνα + − µ α ∂x ∂x ∂xν . 74) αµ = g 2 ∂xα which can be simplified. , by the corresponding minor: dg = dgµν M µν where M µν is the minor of the component gµν . Now, g µν = Thus, M µν , g M µν = g µν g dg = gg µν dg µν = −g gµν dg µν The expression on the far right of the above equation follows from d(gµν g µν ) = d(δµµ ) = d(4) = 0. We then have ∂gµν ∂g ∂g µν µν = g g = −g g . 74) in the form µ αµ = √ 1 µν ∂gµν 1 ∂g ∂ ln −g g = = . 76) This expression is very useful.
The total number of free indexes of a tensor is called its rank (or order). We may set a subscript equal to a superscript and sum over all values of this index, which results in a tensor having two fewer free indexes than the original one. This process is called contraction. For example, if we start with a fourth-order tensor T µ νρ σ , one way of contracting it is to put σ = ρ, which gives the secondρ rank tensor T µ νρ , having only 16 components, arising from the four values of µ ρ and ν. We could contract again to get the scalar T µ µρ with just one component.
By virtue of the linearity of this transformation we also have δ Aµν = − Aµβ ν βα + Aβν µ βα dxα. Substituting this in D Aµν = d Aµν − δ Aµν = Aµν ;α d x α we get the covariant derivative of the tensor T µν in the form T µν ;α = ∂ T µν + ∂xα µ βα T βν ν + βα T µβ . 27) In similar fashion we obtain the covariant derivative of the mixed tensor T µ ν and the covariant tensor Tνµ in the form ∂ T µν − ∂xα ∂ Tµν = − ∂xα T µ ν;α = Tµν;α β β µ να T β µα Tβν µ + − β β βα T ν , να Tµβ . 29) One can similarly determine the covariant derivative of a tensor of arbitrary rank.
A Short Course in General Relativity by James Foster, J. D. Nightingale