3.1 Summary: Tensor derivatives Absolute derivative of a contravariant tensor over some path

a a c Dλ dλ b a dx = + λ Γbc ds ds ds gives a tensor field of same type (contravariant first order) in this case. The first bit takes into account both real physical changes in λa AND the way the curvature of the space can swing the vector while the second one takes out the swinging the vector from the curvature of space alone. So if there is no physical change then Dλa/ds = 0 and the vector is simply parallelly transported. Absolute derivative obeys all the normal rules for derivatives so D(λa + kµa)/ds = D(λa)/ds + kD(µa)/ds (linear) and D(λaµa)/ds = µaD(λa)/ds + λaD(µa)/ds (Leibniz’ rule). Absolute derivative of a scalar There is no space swing with a simple number, φ. so Dφ/ds = dφ/ds Absolute derivative of covariant tensors a Dφ/ds = dφ/ds so let φ = λ µa then

a a Dφ dφ D(λ µa) Dλ a Dµa = = = µa + λ ds ds ds ds ds a a c d(λ µa) dλ b a dx a Dµa = µa + λ Γbc + λ ds  ds ds  ds a a c a dµa dλ dλ b a dx a Dµa λ + µa = µa + µaλ Γbc + λ ds ds ds ds ds c a Dµa a dµa b a dx λ = λ − µaλ Γbc ds ds ds do some index manipulation and get c Dµa dµa b dx = − Γacµb ds ds ds Absolute derivative of higher order tensors a For example, to get the absolute derivative of a mixed tensor τb then look a a at the special case where τb = λ µb a a c a c Dτb a Dµb Dλ a dµb d dx dλ a d dx = λ + µb = λ ( − Γbcµd )+ µb( +Γdcλ ) ds ds ds ds ds ds ds

1 a c c a c c d(λ µb) a d dx a d dx d(τb ) a d dx d a dx = + µbΓdcλ − λ Γbcµd = +Γdcτb − Γbcτd ds ds ds ds ds ds Covariant derivative

a a ∂λ a b λ c = +Γbcλ ; ∂xc rules are as above, so for a second order contravariant tensor we have

ab a b a a τ ;c =(λ µ );c = λ ;cµb + λ µb;c

a b ∂λ b a d b a ∂µ a b d ab a db b ad = µ +Γdcλ µ + λ + λ Γdcµ = ∂cτ +Γdcτ +Γdcτ ∂xc ∂xc to find covarient derivative, again we use the fact that a scalar has no space swing so the covariant derivative is the same as partial derivative φ;c = a ∂cφ/, while we can write φ = λ µa so a a a φ;c = ∂cφ/ =(λ µa);c =(λ ;c)µa + λ (µa;c) a a a a d a ∂cφ =(∂cλ )µa + λ (∂cµa)=(∂cλ +Γ dcλ )µa + λ µa;c a a d a λ ∂cµa =Γ dcλ µa + λ µa;c a a b a a relabel indices to get all λ components as λ so λ ∂cµa =Γ acλ µb + λ µa;c b µa;c = ∂cµa − Γ acµb we can can do higher order covariant derivatives similarly

∂λab d d λab c = − Γacλdb − Γbcλad ; ∂xc Covariant differention forms a tensor field of 1 higher covariant order than the original tensor field. Covariant derivative of the metric In getting the Christoffel symbols (section 3.4) in terms of the metric we had ∂gab ∂ ~ea.~eb ∂ ~ea ∂~eb = = .~eb + ~ea. ∂xc ∂xc ∂xc ∂xc d d d d =Γac ~ed.~eb + ~ea.~edΓbc =Γacgdb +Γbcgad rearrange this to get

∂gab d d − Γacgdb − Γbcgad =0 ∂xc

2 but this is the definition of the covariant derivative of a second order covarient ab tensor. so gab;c = 0, and we can similarly get that g;c = 0. This means that index lowering can be swapped in and out of covariant differention. b b b b b Ra = gabR . Then Ra;c =(gabR );c = gab;cR + gabR ;c = gabR ;c ab likewise g ;c = 0 so index raising can also be swapped in and out of covariant differentiation

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