archive old Section F_1

Archive old version of Section F.1 PhL 4.29.15 F.1 Definition and Interpretation of Γ : Γcab = ec (∂aeb) = Rci(∂aRbi) Context can be very confusing in a discussion of the affine connection Γ. We start with a modified Picture C in which the quasi-Cartesian space on the right is called ξ-space instead of x(0)-space as in Picture C. The notation ξi for the coordinates of ξ-space seems traditional in general relativity work where the Γ object appears frequently. (F.1.1) Recall from the discussion near (1.10) that the metric tensor G is a diagonal matrix whose elements are independently +1 or -1. Since the metric tensor transforms as a rank-2 tensor, we know from the last line of (7.5.8), adapted from Picture A to Picture C1, that gab = RaiRbjGij. Since G is diagonal, we write this as gab = RaiRbiGii and gab = RaiRbiGii (F.1.2) with a single implied sum on i. If G = 1, then the xi coordinates of x-space are "the curvilinear coordinates" and the ξi are "the Cartesian coordinates". The Role of Pictures in Understanding the Affine Connection Here we attempt to head off a confusion about the meaning of the symbol en, and confusion about the form taken by the definition of the affine connection. In Picture A the tangent base vectors exist in x-space and are called en. From (7.18.1) we know both that (en)i = Rni and the dot product en em = g'nm where g' is the metric tensor in x'-space on the left. In general en = en(x) as shown for example in Fig (3.4.3) for polar coordinates. Since x = F-1(x'), we could choose to express the dependence as en(x') instead of en(x). In Picture C1 the tangent base vectors exist in ξ-space and should be called en[ξ]. From (7.18.1) we know both that that (en[ξ])i = Rni and that the dot product en[ξ] em[ξ] = gnm where g is the metric tensor in x-space on the left. In general en[ξ] = en[ξ](ξ). Since ξ = F-1(x) , we could express the dependence as en[ξ](x) instead of en[ξ](ξ). For Picture C1, we shall make the dependency choice en[ξ](x) instead of en[ξ](ξ). Furthermore, to simplify notation, we shall refer to en[ξ] simply as en. This is then a new meaning for the symbol en. These new en are vectors in ξ-space of Picture C1, whereas the old en were vectors in x-space of Picture A. In both cases, the en are the tangent base vectors for "the space on the right", and in both pictures (en)i = Rni. In Picture C1, and with these new en, we have en em = gnm, where g is the metric tensor for the left space. If it happens that x-space of Picture C1 is Cartesian (g = 1) then en(x) = = constant vectors. In this case, we would certainly conclude that ∂ien(x) = ∂en(x)/∂xi = ∂i = 0. Below we shall define Γcin ≡ ec (∂ien) so if x-space is Cartesian, then certainly Γcab = 0, so Γ = 0 for a Cartesian x-space in the context of Picture C1. (F.1.3) Although Γcin is the dot product of two vectors in ξ-space, it is determined by the nature of x-space. In particular, Γ is determined by the metric tensor gnm of x-space, a fact that is explicitly shown in (F.4.1) below: Γdab = (1/2) gdc [ ∂agbc + ∂bgca – ∂cgab]. If we were to make a new version of Picture C1 with x'-space on the left, and x' = F'(ξ) with R' and S', we would then use the notation Γ'cin ≡ e'c (∂'ie'n) where this Γ' is determined by the g'nm of x'-space and e'n are the corresponding tangent base vectors in ξ-space given by (e'n)i = R'ni. Here is a comparison: Γcin = ec (∂ien) (en)i = Rni Γ'cin = e'c (∂'ie'n) (e'n)i = R'ni (F.1.4) Later on, we will want to talk about the affine connection in the context of Picture B. (F.1.4B) Γ'cin = ec (∂'ien) (en)i = Rni where en are the usual tangent base vectors of Chapter 3. Notice the primes on Γ' and ∂'i which correspond to the fact that the curvilinear coordinates in Picture B are x'. Since x-space is Cartesian in this Picture, we have Γ = 0 as in (F.1.3). The moral of the story here is that one must be very conscious of the Picture one is currently using. Definition of the Affine Connection in Picture C1 Each en(x) (by our new definition above) varies with x, and one wonders just how en varies with x. For a small variation dx in the x-space coordinates of Picture C1, one has, d(en)i = ∂j(en)i dxj // ∂j ≡ ∂/∂xi or (den)i = (∂jen)i dxj . // ∂x(fi) = = = (∂xfi) Since en and (∂jen) are both vectors in ξ-space, and since the ek are known to form a complete basis in ξ-space, it must be possible to expand (∂jen) on the ek with some appropriate coefficients, call them Γkjn : (∂jen) = Γkjn ek (den)i = Γkjn (ek)i dxj (F.1.5) Dotting the left equation into em , using em ek = δm,k as in (7.18.1), then doing m→ k gives Γkjn = ek (∂jen) = (ek)i(∂jen)i = Rki(∂jRni) (F.1.6) where we recall again from (7.18.1) that (ek)i = Rki and (en)i = Rni . (7.18.1) These coefficients Γkjn(x) comprise a tensor-like field called the affine connection, so we shall regard (F.1.5) as the definition of Γkjn in the context of Picture C1 above. With more standard index names, the above becomes Γcab = ec (∂aeb) = Rci(∂aRbi) . (F.1.7) From (7.5.2) we know that, adjusting to Picture C1, Rci = (∂xc/∂ξi) Rbi = (∂ξi/∂xb) (∂aRbi) = (∂2ξi/∂xa∂xb) = (∂bRai) . (F.1.8) Therefore one can write Γcab = Rci(∂aRbi) = (∂xc/∂ξi) (∂2ξi/∂xa∂xb) = , (F.1.9) which form appears in Weinberg p 100 (4.5.1). Notice in (F.1.8) that (∂bRcn) = (∂cRbn) so Γcbc is symmetric on the lower two indices. In the next section an alternate form for Γ is derived, so both forms will be stated here: Γcab = Rci(∂aRbi) // ∂a = ∂/∂xa Γcab = – Rbi (∂aRci) . (F.1.10)