Vectors. Three dimensions. (a) Cartesian coordinates ds is the distance from x to x + dx. ds 2 = dx 2 + dy 2 + dz 2 = g ij dx i dx j (1)

Transcription

1 Vectors (Dated: September017 I. TENSORS Three dimensions (a Cartesian coordinates ds is the distance from x to x + dx ds dx + dy + dz g ij dx i dx j (1 Here dx 1 dx, dx dy, dx 3 dz, and tensor g ij is diagonal: g ij δ ij ( The determinant of g ij, g det(g ij 1. (b Spherical coordinates ds dr + r 3 dθ + r sin θdφ g ij dx i dx j (3 Here dx 1 dr, dx dθ, dx 3 dφ, and the diagonal tensor g ij is given by g ij 0 r r sin θ (4 i.e. g rr 1, g θθ r and g φφ r sin θ. The determinant of g ij, g det(g ij r 4 sin θ. Three-dimensional volume element: gdx dv dx dx 3 dxdydz (r sin θdrdθdφ. (5

2 II. TRANSFORMATION OF VECTORS AND TENSORS For an arbitrary transformation x µ x µ, the following objects are defined: a Contravariant vector Example: b Covariant vector An example is the gradient: V µ (x x µ x ν V ν (x. (6 dx µ x µ x ν dxν (7 V µ(x xν x µ V ν(x. (8 c Mixed Tensor dφ xν dφ dx µ x µ dx ν (9 T µλ ν (x V µ (xw λ (xu ν (x (10 That means that this third rank tensor is a quantity that transforms like a product of two contravariant vectors and one covariant vector. Upon transformation of the coordinate system, the tensor T transforms as µλ (x x µ x λ x γ x α x β x µ T αβ γ (x (11 d One of the most important tensors is the metric tensor that defines a coordinate system we are in. g µν (x ξα x µ ξ β x ν η αβ(ξ. (1 Here ξ characterize Minkowski flat space, and x are any other coordinates.

3 3 To prove that g µν is a tensor, we have to demostrate that upon transformation of coordinates it transforms as a product of two covariant vectors. Indeed, using the rule for transformation of vectors, properties of partial derivatives and the definitoion of the metric tensor, we obtain g µν(x ξα ξ β x µ x ν η αβ ( ξ α x ρ ( ξ β x σ x ρ x µ x σ x ν ( ( ( x ρ x σ ξ α x µ x ( ( ν x ρ x σ x µ x ν η αβ (ξ x ρ ξ β x σ η αβ(ξ g ρσ (x, (13 that is, upon transformation of coordinates metric tensor transforms as a product of two covariant vectors. A. Equivalence of contravariant and covariant vectors in Cartesian coordinates In old notation: In new notation: Inverting Eq. (15, we obtain x i a ij x j a ij x i x j. (14 x i a i jx j a i j xi x j. (15 a m i x i a m i a i jx j δ m j x j x m a m i x i x m a m i xm. (16 xi Replacing m by j, we get Comparing this to Eq. (15 and changing notation i j, we have Finally comparing Eqs (17 and (18, we find a j i xj x i (17 a j i xj x i. (18 x j x i xj x i. (19 Thus, there is no difference whether the primed coordinate is in the numerator or denominator, or in other words, contravariant covariant.

4 4 III. MANIPULATING TENSORS a Contraction ( generalized trace T µρ T µρν ν. (0 On the left in this equation is a second rank tensor, on the right there is summation over repeated index ν. We need to demonstrate how tensor T µρ is transformed upon a change of cooordinate system. We set λ ν and sum over repeating indices Using the identity we find T µρν λ (x x µ x ρ x ν x ɛ αβγ x α x β x γ T x λ ɛ (x (1 µρν (x x µ x ρ x ν x ɛ αβγ x α x β x γ T x ν ɛ (x. ( x ν x γ x ɛ x ν δ ɛγ, (3 µρν (x x µ x ρ x α x β T γ αβγ (x. (4 This shows that T µρν ν transforms like a second rank contravariant tensor. b Raising and lowering indices We define the fifth rank tensor We then set ν µ and sum over repeating indices. S µρ νλσ g νλt µρ σ (5 S µρ µλσ Sρ λσ g µλt µρ σ (6 The effect of g µλ is to lower an index of the original tensor T σ µρ. Indices can also be raised using the inverse tensor to g µλ, g λν : g λν (xg µν (x δ λ ν, (7 which is a Kronecker symbol.

5 5 IV. TENSOR DENSITIES We already defined g(x detg µν (x. (8 Because g µν (x is a second rank covariant tensor it transforms as We can regard xρ x µ g µν(x xρ x µ x σ x ν g ρσ(x. (9 f ρ µ as the matrix relating x and x.eq. (9 can then be written in the form g µν f ρ µg ρσ f σ ν. (30 Taking the determinant of both sides of (30 we obtain g det ( g µν ( det f ρ µ g ρσ fν σ ( det f ρ µ det (gρσ det (fν σ (31 We therefore have for the determinants g (x (detf g(x x x g(x, (3 where x x is the Jacobian of the transformation from x to x. Since g(x is a scalar, and has no indices, the expected transformation law might be g (x g(x, without the additional factor (detf x x. The presence of additional factor makes g(x not a scalar but a scalar density. More generally, a quantity which transforms as a tensor except for additional factors of x x w is a tensor density of weight w. As easy to see x x x x 1 (33 Therefore we have g (x x x g(x (34 Hence g(x is a scalar density of weight (-. It is possible to show that any tensor of weight w can be expressed as a product of ordinary tensor multiplied by a factor g w/. To see this, let J µ ν (x be a tensor density of weight w so that