TENSOR FORM OF SPECIAL RELATIVITY

Transcription

1 TENSOR FORM OF SPECIAL RELATIVITY We begin by realling that the fundamental priniple of Speial Relativity is that all physial laws must look the same to all inertial observers. This is easiest done by requiring them to be omposed of entities whih look the same to all inertial observers. We all suh entities tensors. They are four vetors et. beause spae-time is thought of as is a four dimensional geometry in whih the speed of light is for all observers. The simplest suh entity (tensor) is a onstant. We all suh quantities salars. Note that this is a hange from the normal usage of the term. Previously a salar was any quantity whih needed only a number to desribe it. Here we add the requirement that it must be the same for all observers. For example, the speed of light is a salar, but time is not. The next simplest tensor is a vetor. We now onsider them. FOUR VECTORS We use oordinates (t, x,y,z) so that a vetor has the form: We now adopt the notation with tˆ xˆ yˆ z ˆ 1 3 x t,x,y,z 1 3 x t,x x,x y,x z We also adopt the summation onvention: indiies repeated up and down are summed from to 3: Then a general 4-vetor an be written: x ˆ x ˆ x x ˆ x ˆ x ˆ Requiring that the speed of light be for all observers means that a light ray emitted at the origin will obey:

2 t x y z t x y z We therefore define the (length) of a vetor to be: To see that this is indeed invariant we hek 1 3 x x x x 1 3 vx ' 1 x x x x t' x' vt y z' v 4 t' v x' x't' v v y' z' t' x' y' z' We would like to be able to write this as the dot produt of the vetor with itself as with normal 3-vetors. Clearly this doesn t work, so instead we define two versions of the same physial entity (the vetor): with Then We then define: 1 3 A A ˆ A ˆ1 A ˆ A ˆ3 A A A A A 1 3 ˆ 1ˆ ˆ 3ˆ ˆ ˆ 1 A A A A A A A A A A A A, A1 A, A A, A3 A Similarly the dot produt of any two vetors is: ABA B A B The two sets of omponents are onneted by a matrix, η μν suh that:

3 x x u In matrix form this is Similarly, x x With and x x x x x 1 1 x x x 1 x x x x x However, if we want to use matrix multipliation to find the dot produt of two vetors, they annot both be olumn vetors one must be a row and one a olumn in the appropriate order: or b 1 AB a,a 1,a,a3 a b a1b ab a3b b b b 1 3

4 a 1 a b,b 1,b,b3 a a The vetor is a geometri objet whih is the same for all observers. However, its omponents will not be. What all will agree on is its length. Thus the transformation whih relates the omponents of the two systems must yield the same length in both. Suppose the transformation between omponents is given by: Then we must have But Thus Then In matrix form Then Then 7 ' x ' x x x' x x x' ' x b x b x x x b b v 1 b b b b

5 T T I We an hek this for the Lorentz Transformation in x Sine ηη=i we have A A A T I T I T 1 T as expeted sine the inverse transformation just hanges β to -β.

6 TENSORS IN GENERAL We imagine a tensor as a mahine into whih we plae vetors and whih then produes numbers. The mahine is a geometri entity whih is the same for all observers. Different mahines will aept different numbers of vetors. The number aepted is the rank of the tensor. For example a vetor is a tensor of rank one. To see how this works, all the vetor A. Then it has one entrane slot. If we insert ˆ (a ontravariant vetor) it produes the number A, where in our previous notation: A ˆ A Thus the mahine returns the omponents of the vetor. In general the mahine has slots whih aept ontravariant vetors: ˆ and slots whih aept ovariant vetors: Thus inserting ˆ gives the number A μ in ˆ A A ˆ TENSOR PRODUCTS We now define a speial mahine denoted by: u as follows. The mahine has two input slots into whih we insert opposite type vetors (if u is ontravariant we insert a ovariant vetor, σ, in its slot, and vie versa). It then produes the number: Writing this out in omponents we get: v u v, uv

7 Now take Thus ˆ ˆ ˆ ˆ ˆ ˆ u v u v u v, u v ˆ, ˆ. Then ˆ ˆ uv, u v u vu v ˆ (x) ˆ In general we an write any tensor in the form: SS ˆ ˆ ˆ ˆ et. The S are the omponents of the tensor in the system where the unit vetors are the ˆ and ˆ. TRANSFORMATION OF BASIS VECTORS AND COVARIANT COMPONENTS We know that the vetor is the same in any inertial oordinate system. Hene: B' B B 'ˆ ' B ˆ A B ˆ' B ˆ A ˆ' ˆ Then We have already found 1. 1 ˆ' ˆ

8 We next find the transformation for ovariant vetors. We find this as follows. x' x x' x a a a x' x B x a with B a a a In terms of matries this is: A A 1 or in the speial ase of motion along x We hek this as follows A RSR S R S a a But we have already found a a Hene RSR' S' R S R S a a As expeted. Realling our result above we an write the transformation equations for the basis vetors. 1 ˆ' ˆ

9 But 1 b b Hene ˆ' ˆ R' ˆ ' R ˆ ' R ˆ 1 R ˆ' R ˆ 1 ˆ ' ˆ But 1 Hene ˆ ' ˆ Note that in all ases we get the right result by simply lining up the indies orretly. We an now see how to transform the omponents of a tensor. Eah ontravariant index transforms as a ontravariant vetor, et: ab S x ' a b S et. ' ' S' S 'ˆ ' ˆ' ˆ ' S ˆ ˆ ˆ S All of the equations of physis an be written in terms of tensors defined as above. In fat, in order to do relativity either speial or general they must be. We are now ready to do relativisti eletriity and magnetism.

10 RELATIVISTIC ELECTRICITY AND MAGNETISM We start with Maxwell s equations in a vauum E B B E E BJ t t Let Then Thus B A A A E E t t A A E E t t The Maxwell equations are now A t A AJ t t But A A A Hene A AA J t t Let

11 1 Then A A J A x x Now let A A x x where A,A This is an invariant equation sine it is the produt of two 4-vetors x and A. Then A x But A x Thus x Putting these results together gives A J Where

12 x x x A A J,J This is learly also an invariant equation. Finally, let v A A F x x Then But F A x t 1 x E x A E A x t x t x x x Hene 1 E F x Similarly Ey 3 E F F z Next onsider F 1

13 F F F 1 x 13 x z 3 A Ay B y x A A B z x Ay Az B z y z y x Thus F Ex Ey Ez E E x y Ez Bz By Bz Bx By Bx This is the EM field tensor. As an example of the use of F μν onsider how EM fields transform. We have F ' ' ' F We would like to treat this as matrix multipliation. To do so we write it as But in our ase Λ T = Λ. Hene F ' ' F ' ' F ' T

14 E E x y Ez Ex Bz By ' F FA 1 E y 1 Bz Bx 1 1 Ez By Bx E E x Ex y Ez Ex Ex z By 1 Ey Ey 1 Bz Bz Bx Ez E B z y By Bx E E x y E B z z By x E Ey E z z By Ey Ey Bz Bz Bx Ez E B z y By Bx Ex Ex Ey Ey vbz Ez Ez vby Bx Bx By By Ez Bz Bz Ey Noting that is in the ˆx diretion we an rewrite these as E E B B 1 1 E E v B B B 1 E

15 RELATIVISTIC MECHANICS We an find some useful relations by starting with the oordinate 4-vetor x x,x x ˆ x i ˆi Take the derivative of both sides with proper time, τ dx dx dx, u d d d We know that dt = γdτ. Hene d dx dt u,, v d dt d Note that v uu uu v 1 Define p m u m, m v Sine p μ is a onstant times a 4-vetor, it is a 4-vetor. Similarly, p μ is a 4-vetor In the limit γ 1 we get Hene p m, mv m m m 1 m mv 1 This is the famous Einstein relation 1/ p m KE Energy

16 E m m with m m. Next onsider pp m mp Sine this is invariant we have p p m u u m Hene m m p 4 4 m m p E p 4 E p m