Contravariant and covariant vectors

Transcription

1 Faculty of Engineering and Physical Sciences Department of Physics Module PHY08 Special Relativity Tensors You should have acquired familiarity with the following ideas and formulae in attempting the questions on problem sheet 1. Contravariant and covariant vectors The n + 1 quantities V µ, µ = 0, 1,..., n, measured in a frame of reference S with respect to the coordinate axes x 0, x 1,..., x n, are said to form the components of a contravariant vector if, under a transformation of coordinates to a new frame of reference S with coordinate axes x 0, x 1,..., x n, the components V µ are related to those in S by V µ = n ν=0 x µ x ν V ν = x µ x ν V ν. (1) In the case of the coordinate transformation implied by the Lorentz transformation with velocity v along the x axis of frame S, i.e. x 0 = γ v (x 0 β v x 1 ), x 1 = γ v (x 1 β v x 0 ), x 2 = x 2, x = x, this transformation rule for contravariant vectors is, in matrix form V 0 V 1 V 2 V γ v β v γ v 0 0 β v γ v γ v V 0 V 1 V 2 V. (2) The matrix of coefficients in this transformation is denoted γ v β v γ v 0 0 L µ ν = x µ x β v γ v γ v 0 0 ν , () and in this notation the matrix equation, equation (2), reads V µ = L µ νv ν. (4) 1

2 We have seen that, in special relativity, many such contravariant vectors arise; that is quantites which transform under a Lorentz transformation according to equations (2) or (4). These are position four vector x µ = (x 0, x 1, x 2, x ) = (ct, r), (5) velocity four vector U µ = (U 0, U 1, U 2, U ) = (γ u c, γ u u), (6) momentum four vector p µ = (p 0, p 1, p 2, p ) = (ε/c, p), (7) force four vector F µ = (F 0, F 1, F 2, F ) = (( F u)/c, γ u f), (8) wave four vector k µ = (k 0, k 1, k 2, k ) = (ω/c, k), (9) current four vector j µ = (j 0, j 1, j 2, j ) = (cρ, ȷ), (10) potential four vector A µ = (A 0, A 1, A 2, A ) = (ϕ/c, A). (11) In all cases the property of the vector which is invariant under the Lorentz transformation is (V 0 ) 2 (V 1 ) 2 (V 2 ) 2 (V ) 2 = (V 0 ) 2 (V 1 ) 2 (V 2 ) 2 (V ) 2. (12) In similar fashion, n + 1 quantities W µ, µ = 0, 1,..., n measured in S are said to form the components of a covariant vector if, under the transformation of coordinates to S, the components W µ are related to those in S by W µ = n ν=0 x ν x µ W ν = xν x µ W ν. (1) In the case of the Lorentz transformation with velocity v along the x axis of frame S this transformation rule for covariant vectors is, in matrix form W 0 W 1 W 2 W γ v β v γ v 0 0 β v γ v γ v W 0 W 1 W 2 W. (14)

3 The matrix of coefficients in this transformation is denoted γ v β v γ v 0 0 Lµ ν = xν x β v γ v γ v 0 0 µ , (15) and in this notation the matrix equation, equation (14), reads W µ = L ν µ W ν. (16) We have seen that, in special relativity, the space time derivatives µ transform precisely according to equations (14) and (16), where µ = ( 0, 1, 2, ) = ( ct, x, y, z ). (17) It is easily proven that the scalar product of an arbitary contravariant vector A µ with a covariant vector B µ is invariant under a transformation, that is or, in the case of the Lorentz transformation A µ B µ = A µ B µ (18) A 0 B 0 + A 1 B 1 + A 2 B 2 + A B = A 0 B 0 + A 1 B 1 + A 2 B 2 + A B. (19) Tensors If (n + 1) 2 quantities T µν, µ, ν = 0, 1,..., n, measured in a frame of reference S with respect to the coordinate axes x 0, x 1,..., x n, and the corresponding (n + 1) 2 quantities T µν, measured in frame S with coordinate axes x 0, x 1,..., x n, are related by T µν = x µ x α x ν x β T αβ, (20) then T µν is defined as a twice contravariant tensor. That is it has two contravariant indices, and, under transformation, each index transforms by the contravariant rule of equation (1). It follows that the (n + 1) 2 quantities C µν obtained from the product of any pair of contravariant vectors A µ and B µ according to C µν = A µ B ν (21)

4 form the elements of a twice contravariant tensor. rule In similar fashion, a twice covariant tensor is defined by the transformation T µν = xα x µ x β x ν T αβ, (22) in which each covariant index transforms by the covariant rule of equation (1). It follows that the (n + 1) 2 quantities C µν obtained from the product of any pair of covariant vectors A µ and B µ according to C µν = A µ B ν (2) necessarily form the elements of a twice covariant tensor. Thus each index on a tensor transforms according to the covariant or contravariant rules depending on the nature of the index. One can thus define also a mixed tensor by the rule for which a protoype is the product of a contravariant with a covariant vector. T ν µ = xα x µ x ν x β T β α, (24) C ν µ = A µ B ν (25) In special relativity, using the L µ ν notation for the Lorentz transformation matrix, equation (), it follows that the rule for transforming tensors between two reference frames is, for example for a twice contravariant tensor, T µν = L µ αl ν βt αβ, (26) that is a Lorentz transformation is applied for each contravariant index. For a twice covariant tensor we will have, similarly with L ν µ given by equation (14). The metric tensor T µν = L α µ L β ν T αβ, (27) The metric tensor in a space of coordinates x 0, x 1,..., x n is defined in terms of the invariant distance element ds 2 = g µν dx µ dx ν, (28) 4

5 It follows, from the invariant property of the Lorentz transformation, that in the case of the four dimensional space time system of special relativity in which ds 2 = (dx 0 ) 2 (dx 1 ) 2 (dx 2 ) 2 (dx ) 2 = (dx 0 ) 2 (dx 1 ) 2 (dx 2 ) 2 (dx ) 2 (29) that the metric tensor is g µν The associated tensor to g µν, g µν, is defined by the relation. (0) g µα g αν = δ ν µ (1) where δ ν µ is the Kroneker delta symbol. It follows that, viewed as a matrix, g µν is simply the inverse matrix to g µν, and hence in the case of the four dimensional space time system of special relativity g µν a matrix of the same form as g µν , (2) These metric tensors have the property that, when acting on a contravariant index, g µν converts the index to a covariant index. That is g µα A α = A µ () where A µ has the transformation properties of a covariant vector. Similarly acting on a twice contravariant tensor g µα T αβ = T β µ. (4) That is, if T αβ T 00 T 01 T 02 T 0 T 10 T 11 T 12 T 1 T 20 T 21 T 22 T 2 T 0 T 1 T 2 T, (5) 5

6 then and so T β µ Similarly, and thus T µν g µα T αβ = T β µ T0 0 T0 1 T0 2 T0 T1 0 T1 1 T1 2 T1 T2 0 T2 1 T2 2 T2 T 0 T 1 T 2 T g µα g νβ T αβ = T µν T 00 T 01 T 02 T 0 T 10 T 11 T 12 T 1 T 20 T 21 T 22 T 2 T 0 T 1 T 2 T T 00 T 01 T 02 T 0 T 10 T 11 T 12 T 1 T 20 T 21 T 22 T 2 T 0 T 1 T 2 T T 00 T 01 T 02 T 0 T 10 T 11 T 12 T 1 T 20 T 21 T 22 T 2 T 0 T 1 T 2 T T 00 T 01 T 02 T 0 T 10 T 11 T 12 T 1 T 20 T 21 T 22 T 2 T 0 T 1 T 2 T T 00 T 01 T 02 T 0 T 10 T 11 T 12 T 1 T 20 T 21 T 22 T 2 T 0 T 1 T 2 T (6). (7). (8). (9) The tensor g µν thus performs an index lowering operation. The associated tensor g µν, by contrast, has the effect of raising indices so that and similarly for tensors. g µα A α = A µ, (40) In the context of special relativity it follows that the Lorentz matrices L ν µ and L µ ν can be related by the index raising and lowering operations g µα g νβ L α β = g µα L αν = L ν µ. (41) It also follows that from any contravariant vector A ν we can form an associated covariant vector A µ by the application of g µν. So, for each of the physical contravariant four vectors listed earlier there is a covariant partner, e.g. p µ = (p 0, p 1, p 2, p ) = g µν p ν = (p 0, p 1, p 2, p ) = (ε/c, p). (42) We can also define a contravariant derivative µ by µ = ( 0, 1, 2, ) = g µν ν = ( 0, 1, 2, ). (4) 6

7 Invariants Once one recognizes that the four quantities (A 0, A 1, A 2, A ) form the components of a covariant vector A µ we can attach new significance to the invariant of the Lorentz transformation, since equation (12) is simply rewritten, using the fact that (V 0 ) 2 (V 1 ) 2 (V 2 ) 2 (V ) 2 = V µ V µ = g µν V µ V ν. (44) It follows that in each case the invariant of the Lorentz transformation is the scalar product of the four vector with its covariant partner, which by equation (18) is always an invariant. Thus, for example, in the case of the four momentum transformation p µ p µ p µ = ε 2 /c 2 p 2 = ε 2 /c 2 p 2 = p µ p µ. (45) 7