Metric Tensor - Inverse and Raising and Lowering Indices

METRIC TENSOR - INVERSE AND RAISING AND LOWERING INDICES

Link to: physicspages home page. To leave a comment or report an error, please use the auxiliary blog and include the title or URL of this post in your comment. Post date: 30 Jan 2021. The inverse metric tensor gij is deﬁned so that

ij i g gjk = δ k (1) If the metric tensor is viewed as a matrix, then this is equivalent to saying ij −1 ij g = [gij] . The transformation property of g can be worked out by i direct calculation, using the transformation of gij and the fact that δ k is invariant.

0ij 0 i g gjk = δ k (2) ∂xl ∂xm = g0ij g (3) ∂x0j ∂x0k lm We can try the transformation

∂x0i ∂x0j g0ij = gab (4) ∂xa ∂xb Substituting, we get

∂x0i ∂x0j ∂xl ∂xm g0ijg0 = gab g (5) jk ∂xa ∂xb ∂x0j ∂x0k lm ∂x0i ∂xm = gabδl g (6) ∂xa b ∂x0k lm ∂x0i ∂xm = gal g (7) ∂xa ∂x0k lm ∂x0i ∂xm = δa (8) ∂xa ∂x0k m ∂x0i ∂xm = (9) ∂xm ∂x0k i = δ k (10) 1 METRIC TENSOR - INVERSE AND RAISING AND LOWERING INDICES 2

0j l l ∂x ∂x = ∂x = δl galg = δa On line 2 we used ∂xb ∂x0j ∂xb b and on line 4 we used lm m. ij Thus g is a rank-2 contravariant tensor, and is the inverse of gij which is a rank-2 covariant tensor. Since the matrix inverse is unique (basic fact from matrix algebra), we can use the standard techniques of matrix algebra to calculate the inverse. ij In rectangular coordinates, g = gij since the metric is diagonal with all diagonal elements equal to 1. In polar coordinates in 2-d,

1 0 g = (11) ij 0 r2 so the inverse is

1 0 gij = (12) 0 r−2 A contravariant vector vi can be lowered (converted to a covariant vector) by multiplying by gij:

i vi = gijv (13) The covariant vector can be converted back into a contravariant vector by raising its index:

ij ij k g vj = g gjkv (14) i k = δ kv (15) = vi (16) If we start with a vector vi in rectangular coordinates, we can convert it to polar coordinates:

vr = vx cosθ + vy sinθ (17) sinθ cosθ vθ = −vx + vy (18) r r We can lower these components by multiplying by gij

x y vr = v cosθ + v sinθ (19) sinθ cosθ v = r2 −vx + vy (20) θ r r = −rvx sinθ + rvy cosθ (21) METRIC TENSOR - INVERSE AND RAISING AND LOWERING INDICES 3

The magnitude is found by combining the covariant and contravariant vec- tors (which can also be viewed as a one-form operating on a vector, or vice versa):

i r θ v vi = v vr + v vθ (22) = (vx cosθ + vy sinθ)2 + (−vx sinθ + vy cosθ)2 (23) = (vx)2 + (vy)2 (24) (No implied sum on the RHS in line 1.) The same result can be obtained by using the appropriate form of the metric on either the vector or one-form version. Using the vector, we have:

2 i j v = gijv v (25) r r θ θ = grrv v + gθθv v (26) sinθ cosθ2 = (vx cosθ + vy sinθ)2 + r2 −vx + vy (27) r r = (vx)2 + (vy)2 (28) Using the one-form, we have

2 ij v = g vivj (29) rr θθ = g vrvr + g vθvθ (30) 1 = (vx cosθ + vy sinθ)2 + (−rvx sinθ + rvy cosθ)2 (31) r2 = (vx)2 + (vy)2 (32)

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