Remarks Around Lorentz Transformation

Transcription

1 Remarks Around Lorentz Transformation Arm Boris Nima Abstrat After diagonalizing the Lorentz Matrix, we find the frame where the Dira equation is one derivation and we alulate the speed of the Shwarshild metri

2 Introdution The Lorentz transformation is the only transformation whih leaves Maxwell equations invariants. As a matter of fat, the matrix representation of the Lorentz belongs to what we all the Poinare group. However, the Lorentz matrix is often express in the Cartesian oordinates x, y, z, t. Furthermore, I wondered what are the eigenvalues and the eigenvetors of the Lorentz matrix. After diagonalizing the Lorentz matrix, we see that the first oordinates of its eigenvetors are x + t and x t and that the eigenvalues are inverse eah others. In that ase the idea ame to me to express some well-known physis equation in the oordinates x+t, x t. But the only physi equation I found where I ould do it was the Dira equation. Also, there was some derivation in the Dira equation and I searhed whih frame I have to derivate by to obtain the derivations in the Dira equation. The answer of this question is not x + t but Ux, t tl 4 + j α i x i 0. where the α i are related with the Pauli matries. So the derivation by the metri Ux, t gives the Dira equation ψx, t i m α 0 ψx, t 0.2 U with α 0 γ 0. Moreover, I remarked that, if we deompose the well known Shwarshild metri, we have a matrix with two eigenvalues inverse eah others, whih remind us the diagonalization of the Lorentz matrix. Then we deide to alulate the orresponidng speed of the Shwarshild metri whih gives us v 2GM 0.3 In the first part, we alulate the eigenvalues and the eigenvetors of the Lorentz matrix expressed with hyperboli funtions. We reall the definition of O, and the Poinare group and we give the diagonalized matrix in funtion of the speed v of the translation of the frame. In the seond part, we use the oordinates x + t found in the eigenvetors to express the Dira equation with a derivation by a frame funtion of the Pauli matries. In the third part, we find the orresponding speed of the Shwarzshild metri in deomposing it and omparing it with the diagonalization of the Lorentz matrix.

3 The Poinare Group We reall the definition of the Lorentz transformation in a diretion x taken on Wikipedia t x y z oshα sinhα sinhα oshα } {{ } L t x y z.4 with oshα γ, sinhα βγ and γ, β v β 2. We often say that L belongs to the Poinare group. Now we study the matrix g oshα sinhα sinhα oshα O,.5 where O, has been defined by [] as } O, {g GL2, R g J t g J { α R g exp α }.6 with J.7 You an hek that oshα sinhα sinhα oshα t oshα sinhα sinhα oshα.8 Now we alulate the eigenvalues of g detg λid 2 oshα λ sinhα sinhα oshα λ 0.9 So we have two eigenvalues λ ± given by and the orresponding eigenvetors sinhα sinhα ker sinhα sinhα sinhα sinhα ker sinhα sinhα λ ± e ±α.0. 2

4 So we an diagonalize the matrix L : oshα sinhα sinhα oshα 2 e α e α and the equation.4 t x t + x y z e α e α t x t + x y z.2 Beause oshα γ, sinhα βγ, we an write : t x +β β 2 t + x y z β β 2 } {{ } L 0 t x t + x y z.3 where β v. Thus wee have seen in.3 that the oordinates x + t and x t are very speial oordinates beause these are the oordinates where Lorentz transformation matries are diagonales. Now we try to write the Dira equation in these kind of oordinates. 2 The Dira Equation We know the Dira equation under the form f. Wikipedia i γ µ µ m ψx But the expliit form is given by i ψx, t t m 2 α 0 i j α j x j ψx, t 2.5 where and α 0 α j l 2 l 2 σ j σ j

5 for eah Pauli matries σ σ 2 i i σ The matries γ j in 2.4 are defined by γ 0 α 0 γ j α 0 α j 2.9 Now we an write 2.5 in the form l 4 i t + j α j x j ψx, t m 2 α 0 ψx, t 2.20 Beause l l and α j α, we an rewrite 2.20 as j l4 i t + j α j x j For a general funtion Ux, x 2, x 3, t Ux, t, we have Ux, t t Ux, t t + x Ux, t x + If we take the frame Ux, t tl j α i x i, we have ψx, t m α 0 ψx, t 2.2 x 2 Ux, t x 2 + x 3 Ux, t x Ux, t l 4 t + α x + α 2 x 2 + α 3 x whih is the derivation in 2.2. Finally we an express the derivate in 2.2 as i ψx, t U Then we an see Ux, t as a metri given by Ux, t m α 0 ψx, t 2.24 t 0 x 3 x ix 2 0 t x + ix 2 x 3 x 3 x ix 2 t 0 x + ix 2 x 3 0 t

6 3 Lorentz transformation in the Shwarshild metri? We onsider the Shwarshild metri 2GM g µν 2GM r 2 r 2 sin 2 θ 3.26 Now what I will do is really speulative. We imagine that the Shwarzshild metri is the produt of a Lorentz transformation and the Minskowskian polar metri g µν 2GM 2GM Now we ompare the first matrix of 3.27 with.3 and we identify + β β 2 + β + β GM 2GM + β 2GM v 2GM r 2 r 2 sin 2 θ The expression 3.28 has to be ompared with the liberation speed of a blak hole gravitation v lib 2GM

7 Référenes [] Daniel Bump, Lie Groups p34 6