Special Relativity

Special Relativity

(Lorentz transformations)

Dr. Luigi E. Masciovecchio email: [email protected]

(first published on http://mio.discoremoto.alice.it/luigimasciovecchio/, november 2009) available as notebook and PDF on http://sites.google.com/site/luigimasciovecchio/

2017.06.19

Print@"Document revision: ", IntegerPart@Date@DDD Document revision: 82017, 6, 19, 7, 57, 44<

D

This is my personal Mathematica notebook on special relativity focussing on the Lorentz transformations. This document wasn't originally intended for publication, but a few formulas are maybe of interest to you, so here they are. The code seems to work well and I added some comments to make it more understandable. This is not an introduction to this field, so use it at your own risk! I have stolen many ideas (with some corrections) from Ladislau Radu "Herleitung der Lorentz-Transformation" (2006, Internet).

In Einstein's theory of special relativity (first published in 1905) the Lorentz transformation converts between two different inertial observers' measurements of space and time, where one observer is in (constant) motion with respect to the other. Here I present a derivation of the Lorentz transformation without the assumption that the speed of light is a constant. For more details about this approach see Jean- Marc Lévy-Leblond, "One more derivation of the Lorentz transformation", American Journal of Physics 44, 271-277 (1976).

1. Spezielle Lorentztransformation (Lorentz transformation for frames in standard configuration)

Ÿ 1.1 Konstruktion der Standardkonfiguration. Construction of the standard configuration.

inertial frames O and O' with Cartesian system of coordinates (for space) in both frames inertial frame O: space-time coordinates (t, x, y, z), velocity of O' measured in O: v inertial frame O': space-time coordinates (t', x', y', z'), velocity of O measured in O': v'

Lorentz transformation: (t', x', y', z') = f (t, x, y, z) We have to find the exact form of the function f !

A) homogeneity of space-time Þ space-time transformations between inertial frames are linear. This implies:

t ' x ' y ' z '

=fHt, x, y, zL=

L00 L01 L02 L03

L10 L11 L12 L13

L20 L21 L22 L23

L30 L31 L32 L33

. t x y z

+ Α0

Α1

Α2

Α3

(Poincaré-Transformation)

Remove@"Global`*"D

L=Table@L@i,jD,8i, 0, 3<, 8j, 0, 3<D;Event= t x y z

;

MatrixFormž8L.Event+A<

:

Α@0D+tL@0, 0D+xL@0, 1D+yL@0, 2D+zL@0, 3D Α@1D+tL@1, 0D+xL@1, 1D+yL@1, 2D+zL@1, 3D Α@2D+tL@2, 0D+xL@2, 1D+yL@2, 2D+zL@2, 3D Α@3D+tL@3, 0D+xL@3, 1D+yL@3, 2D+zL@3, 3D

>

B) Assume the coincidence of space-time origins of the two frames. Þ Αi=0 t '

x ' y ' z '

=

L00 L01 L02 L03

L10 L11 L12 L13

L20 L21 L22 L23

L30 L31 L32 L33

. t x y z

(Lorentz-Transformation)

: 0 0 0 0

, "=",L.

0 0 0 0

+A>

8880<,80<,80<,80<<,=,88Α@0D<,8Α@1D<,8Α@2D<,8Α@3D<<<

C) Assume that at t = 0 the axis of O overlap the axis of O'. Þ (0, x, 0, 0) ® (t', x', 0, 0) and (0, 0, y, 0) ® (t', 0, y', 0) and (0, 0, 0, z) ® (t', 0, 0, z')

Þ transformation matrix must be like this:

Lsc=

L@0, 0D L@0, 1D L@0, 2D L@0, 3D L@1, 0D L@1, 1D 0 0 L@2, 0D 0 L@2, 2D 0 L@3, 0D 0 0 L@3, 3D

;

MatrixFormž:L.

0 x 0 0

,L.

0 0 y 0

,L.

0 0 0 z

>

MatrixFormž:Lsc.

0 x 0 0

,Lsc.

0 0 y 0

,Lsc.

0 0 0 z

,Lsc.Event>

:

xL@0, 1D xL@1, 1D xL@2, 1D xL@3, 1D

,

yL@0, 2D yL@1, 2D yL@2, 2D yL@3, 2D

,

zL@0, 3D zL@1, 3D zL@2, 3D zL@3, 3D

>

:

xL@0, 1D xL@1, 1D 0

0

,

yL@0, 2D 0

yL@2, 2D 0

,

zL@0, 3D 0

0

zL@3, 3D ,

tL@0, 0D+xL@0, 1D+yL@0, 2D+zL@0, 3D tL@1, 0D+xL@1, 1D

tL@2, 0D+yL@2, 2D tL@3, 0D+zL@3, 3D

>

Note: sc = standard configuration

Ÿ 1.2 Erstes Relativitätsargument. First relativity argument.

We choose the x-axis of O along v.

D) Two frames obtained by the substitutions y ® - z and z ® y in O and y' ® - z' and z' ® y' in O' are connected by the same transforma- tion matrix. This implies:

yzRotation=

1 0 0 0 0 1 0 0 0 0 0 -1 0 0 1 0

;

Lsc=Lsc.HReduce@yzRotation.LscŠLsc.yzRotation, Flatten@LDD .8Equal®Rule, And®List<L; MatrixFormž8%,Lsc.Event<

:

L@0, 0D L@0, 1D 0 0 L@1, 0D L@1, 1D 0 0

0 0 L@2, 2D 0

0 0 0 L@2, 2D

,

tL@0, 0D+xL@0, 1D tL@1, 0D+xL@1, 1D yL@2, 2D

zL@2, 2D

>

E) block structure of the transformation matrix for standard configuration Þ 1D-space case (t, x) can be treated separately from full 3D- space case (t, x, y, z).

The 1D-space case is best done à la Lévy-Leblond, see the next subsection.

Clear@LD

Ÿ 1.3 Jean-Marc Lévy-Leblond, "One more derivation of the Lorentz transformation", American Journal of Physics 44, 271-277 (1976).

Ÿ

1.3 Jean-Marc Lévy-Leblond, "One more derivation of the Lorentz transformation", American Journal of Physics 44, 271-277 (1976).

Ÿ PRELIMINARIES

"The principle of relativity is first stated in general terms, leading to the idea of equivalent frames of reference connected through "inertial"

transformations obeying a group law. The theory of relativity then is constructed by constraining the transformations through four succes- sive hypotheses: homogeneity of space-time, isotropy of space-time, group structure, causality condition. Only the Lorentz transformations and their degenerate Galilean limit obey these constraints."

"Relativity theory, in fact, is but the statement that all laws of physics are invariant under the Poincaré group (inhomogeneous Lorentz group)."

"I will take as a starting point the statement of the principle of relativity in a very general form: there exists an infinite continuous class of reference frames in space-time which are physically equivalent."

"Because of well-known physical considerations, I find it convenient to call "inertial frames" and "inertial transformations" the equivalent reference frames and the transformations connecting them. Indeed, the very existence of such equivalent reference frames corresponds to the validity of the principle of inertia, namely, that a physical object has no absolute state of motion or rest; for instance, a free body (with no "forces" acting on it) is characterized by an "inertial motion" which is not entirely determined, since it depends on "initial conditions", that is also to say, on the reference frame considered."

Let us call "inertial motions" those motions which are obtained from rest by an inertial transformation; an object has an inertial motion in some reference frame if it is at rest in another equivalent frame.

Number of independent parameters in the transformation formulas for the spatiotemporal coordinates (x, t) of an arbitrary event viewed from inertial frames:

Transformation parameters: a1... aN

8x '=f@x, t, a₁... a_ND, t '=g@x, t, a₁... a_ND<, N parameters in general 8x '=F@x, t, a₁... anD, t^'=G@x, t, a₁... anD<,

n = N -2 parametersHconsidering common space-time originsL

8x '=F@x, t, aD, t '=G@x, t, aD<, dependence on only one parameterHphysical considerationsL

Ÿ HYPOTHESIS 1: HOMOGENEITY OF SPACE-TIME

<<Utilities`Notation`

SymbolizeAx^'E;SymbolizeAt^'E;SymbolizeAxn^'E;SymbolizeAtn^'E; The direct approach doesn't work yet...

Î DSolve@8

¶xF@x,t,aDŠH@aD,¶tF@x,t,aDŠ -K@aD,

¶xG@x,t,aDŠ -M@aD,¶tG@x,t,aDŠL@aD, F@0, 0,aDŠ0,G@0, 0,aDŠ0<,

8F@x,t,aD,G@x,t,aD<,8x,t<D

DSolveA9F^H1,0,0L@x, t, aDŠH@aD, F^H0,1,0L@x, t, aDŠ -K@aD, G^H1,0,0L@x, t, aDŠ -M@aD,

G^H0,1,0L@x, t, aDŠL@aD, F@0, 0, aDŠ0, G@0, 0, aDŠ0=,8F@x, t, aD, G@x, t, aD<,8x, t<E

...but the indirect approach leads to the wanted result:

DSolve@¶xF@x,t,aDŠH@aD,F@x,t,aD,8x,t<D;

DSolve@¶tH%@@1, 1, 2DDLŠ -K@aD, C@1D@tD,t, GeneratedParameters®HModule@8C<,CD&LD; x'= %%@@1, 1, 2DD .%@@1DD;

DSolve@¶xG@x,t,aDŠ -M@aD,G@x,t,aD,8x,t<D;

DSolve@¶tH%@@1, 1, 2DDLŠL@aD, C@1D@tD,t, GeneratedParameters®HModule@8C<,CD&LD; 8x',t'= %%@@1, 1, 2DD .%@@1DD<

Solve@8x'Š0,t'Š0,x==0,t==0<,8x'@@1DD,t'@@1DD<D; 8x',t'<=8x',t'< .%@@1DD

SolveAlways@8x'ŠΓHx-ΝtL,t'==ΓHΛt-ΜxL,v==K@aD H@aD<,8x,t<D

%@@2DD .8Rule®Equal,Γ®Γ@vD,Λ®Λ@vD,Μ®Μ@vD,Ν®Ν@vD< Flatten;

Solve@%,8H@aD, K@aD,L@aD,M@aD<D;

Print@"8x',t'< = ",A=8x',t'<=8x',t'< .%@@1DD SimplifyD 8C$104+x H@aD-t K@aD, C$128+t L@aD-x M@aD<

8x H@aD-t K@aD, t L@aD-x M@aD<

88L@aD®0, M@aD®0, K@aD®0, H@aD®0,Γ ®0<,8L@aD® Γ Λ, M@aD® Γ Μ, K@aD®vΓ,Ν ®v, H@aD® Γ<<

8x',t'< = 8H-t v+xLΓ@vD,Γ@vD HtΛ@vD-xΜ@vDL<

Ÿ HYPOTHESIS 2: ISOTROPY OF SPACE

B=8xn',tn'<=8-x',t'< .8x® -x,v®u< Simplify;

AŠBExpandAll

H8Coefficient@A,xD, Coefficient@A,tD< FlattenL==

H8Coefficient@B,xD, Coefficient@B,tD< FlattenL; eqn=Table@%@@1,iDDŠ %@@2,iDD,8i, 1, Length@%@@1DDD<D;

%TableForm Solve@eqn,uD

eqn=eqn.%Flatten r=eqn@@1DD . Equal®Rule;

8eqn@@1DD,eqn@@4DD .r,eqn@@2DD .r<;

%TableFormFullSimplify

8-t vΓ@vD+xΓ@vD, tΓ@vDΛ@vD-xΓ@vDΜ@vD<Š8t uΓ@uD+xΓ@uD, tΓ@uDΛ@uD+xΓ@uDΜ@uD<

Γ@vDŠ Γ@uD

- Γ@vDΜ@vDŠ Γ@uDΜ@uD -vΓ@vDŠuΓ@uD Γ@vDΛ@vDŠ Γ@uDΛ@uD 88u® -v<<

8Γ@vDŠ Γ@-vD,- Γ@vDΜ@vDŠ Γ@-vDΜ@-vD,-vΓ@vDŠ -vΓ@-vD,Γ@vDΛ@vDŠ Γ@-vDΛ@-vD<

Γ@-vDŠ Γ@vD

Γ@-vD HΛ@-vD- Λ@vDLŠ0 Γ@-vD HΜ@-vD+ Μ@vDLŠ0

Ÿ HYPOTHESIS 3: THE GROUP LAW (a) Identity transformation

SolveAlways@8x,t<==8H-tw+xLΓ@vD,Γ@vD HtΛ@vD-xΜ@vDL<,8x,t<D .v®w 88w®0,Μ@wD®0,Λ@wD®1,Γ@wD®1<<

(b) Inverse transformation

A=9xŠI-t^'w+x^'MΓ@wwD,tŠΓ@wwD It^'Λ@wwD-x^'Μ@wwDM=;

B=SolveA9x^',t^'=Š8H-tv+xLΓ@vvD,Γ@vvD HtΛ@vvD-xΜ@vvDL<,8x,t<E Flatten;

SolveAlwaysA8A@@1, 2DDŠB@@1, 2DD,A@@2, 2DD==B@@2, 2DD<,9x^',t^'=E . Rule®Equal;

Flatten@Table@Solve@%@@iDD,8w,Λ@wwD,Μ@wwD,Γ@wwD<D,8i, 1, 3<D, 1D .8vv®v,ww®w< MatrixForm

%@@2DD .8Rule®Equal,Λ@vD®1<

%@@4DD .w® -v.Γ@-vD®Γ@vD Μ@wD®0 w® - ^v

Λ@vD Λ@wD® ¹

Λ@vD Γ@wD® ¹

Γ@vD

w® - ^v

Λ@vD Μ@wD® -^Μ@vD

Λ@vD Λ@wD® ¹

Λ@vD Γ@wD® ^Λ@vD

Γ@vD HΛ@vD-vΜ@vDL

w®0 Λ@wD® _Λ@vD¹ Μ@wD® -^Μ@vD_Λ@vD Γ@wD® _Γ@vD¹ :wŠ -v,Μ@wDŠ - Μ@vD,Λ@wDŠ1,Γ@wDŠ 1

Γ@vD H1-vΜ@vDL>

Γ@vDŠ 1

Γ@vD H1-vΜ@vDL

Λ[v] ® 1: See Luigi Masciovecchio "Lévy-Leblond ab omni naevo vindicatus" for a proof that Λ(v) = 1.

(c) Composition law Solve@8

x1==Γ@v1D Hx-v1 tL,t1==Γ@v1D Ht-Μ@v1DxL, x2==Γ@v2D Hx1-v2 t1L,t2==Γ@v2D Ht1 -Μ@v2Dx1L<, 8x2,t2<,8x1,t1<D Flatten

V= -Coefficient@%@@1, 2DD,tD Coefficient@%@@1, 2DD,xD Simplify Solve@Coefficient@%%@@1, 2DD,xD==Coefficient@%%@@2, 2DD,tD,Μ@v1DD 8x2® -t v1Γ@v1DΓ@v2D-t v2Γ@v1DΓ@v2D+xΓ@v1DΓ@v2D+v2 xΓ@v1DΓ@v2DΜ@v1D,

t2®tΓ@v1DΓ@v2D-xΓ@v1DΓ@v2DΜ@v1D+t v1Γ@v1DΓ@v2DΜ@v2D-xΓ@v1DΓ@v2DΜ@v2D<

v1+v2 1+v2Μ@v1D

::Μ@v1D® v1Μ@v2D v2 >>

Î SolveAlways@Μ@v1D v1==Μ@v2D v2,8v1,v2<, InverseFunctions®FalseD SolveAlwaysBΜ@v1D

v1

Š Μ@v2D v2

,8v1, v2<, InverseFunctions®FalseF

:V=V.Μ@v_D® Αv, eq=Γv Š 1

ΓvH1-vΜ@vDL .Μ@vD® Αv>

Reduce@8eq,Γv³ 0<,Γv, RealsD

9"1L Non-causal:", ReduceA9%.Α® -Κ^-2,Κ>0=,Γv, RealsE,V.Α® -Κ^-2,

"2L Galilean:", Reduce@8%.Α®0<,Γv, RealsD,V.Α®0,

"3L Lorentzian:", ReduceA9%.Α®c^-2,c>0=,Γv, RealsE,V.Α®c^-2= TableForm : v1+v2

1+v1 v2Α,Γv Š 1 I1-v²ΑMΓv>

Α £0 &&ΓvŠ 1

1-v²Α ÈÈ Α >0 &&- 1

Α <v< 1

Α &&Γv Š 1 1-v²Α 1L Non-causal:

Κ >0 &&Γv Š ^Κ2

v²+Κ² v1+v2

1-^{v1 v2} Κ2

2L Galilean:

Γv Š1 v1+v2

3L Lorentzian:

HHv£0 && c> -vL ÈÈ Hv>0 && c>vLL&&Γv Š ^c2

c²-v² v1+v2

1+^{v1 v2} c2

Ÿ HYPOTHESIS 4: CAUSALITY

Xnc@x_,t_D:=I1+v²‘Κ²M^-1‘2Hx- vtL

Tnc@x_,t_D:=I1+v²‘Κ²M^-1‘2It+vx ‘Κ²M Tnc@x₂,t₂D-Tnc@x₁,t₁D FullSimplify;

8"Non-causal: Dt ' = ",%.H-t₁+t₂L®Dt .H-x₁+x₂L®Dx<

:Non-causal: Dt ' = ,

vDx+ DtΚ²

1+ ^v2

Κ² Κ²

>

The sign of the time interval is arbitrary in the non-causal case.

Print@"Galilean: Dt ' = Dt , Dx ' = Dx"D Galilean: Dt ' = Dt , Dx ' = Dx

The sign of the time interval never changes in the Galilean case.

Xl@x_,t_D:=I1-v²‘c²M^-1‘2Hx- vtL

Tl@x_,t_D:=I1-v²‘c²M^-1‘2It-vx ‘c²M Tl@x₂,t₂D-Tl@x₁,t₁D FullSimplify;

Dt'= %.H-t₁+t₂L®Dt .Hx₁-x₂L® -Dx; Xl@x₂,t₂D-Xl@x₁,t₁D Simplify;

Dx'= %.v t₁-v t₂® -vDt .-Hx₁-x₂L®Dx;

8"Lorentzian: Dt ' = ",Dt', " Dx ' = ",Dx'<

:Lorentzian: Dt ' = ,

c²Dt-vDx

c² 1- ^v2

c²

, Dx ' = ,

-vDt+ Dx

1-^v2

c²

>

For | Dx/Dt | < c (time-like intervals) the sign of the time interval doesn't change in the Lorentzian case.

Ÿ 1.4 Zweites Relativitätsargument. Second relativity argument.

From the preceding section we have:

KL@0, 0D L@0, 1D

L@1, 0D L@1, 1D O=Γ 1 -v‘c²

-v 1

;

LscMatrixForm Γ -^vΓ

c² 0 0

-vΓ Γ 0 0

0 0 L@2, 2D 0

0 0 0 L@2, 2D

F) Two frames obtained by the substitutions x®- x and z®- z in O and x'®- x' and z'®- z' in O' are connected by the same transforma- tion matrix L, we have proved that v = - v' (reciprocity), we have also the continuity of L(v) and L(v=0) = 1 (identity). This implies:

xzInvertion=DiagonalMatrix@81,-1, 1,-1<D;

xzInvertionŠLsc.xzInvertion.Lsc.Γ®I1-v²‘c²M^-1‘2Simplify Solve@%D

Lsc=Lsc.%@@2DD; L@2, 2D²Š1

88L@2, 2D® -1<,8L@2, 2D®1<<

The Lorentz transformation for frames in standard configuration are finally:

MatrixFormž8Lsc,Lsc.Event< Simplify

:

Γ -^vΓ

c² 0 0 -vΓ Γ 0 0

0 0 1 0

0 0 0 1

,

Jt- ^{v x}

c²NΓ H-t v+xLΓ y

z

>