MathAction SandBoxLorentzTransformation

Topics

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- SandBox
  - SandBox Categorical Relativity
    - SandBoxLorentzTransformation <-- You are here.

Lorentz transformations.

References

Space-time without reference frames T. Matolcsi
The Lorentz boost-link is not unique Zbigniew Oziewicz

Mathematical Preliminaries

A vector is represented as a $n\times 1$ matrix (column vector)

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(1) -> Scalar := Expression Integer

$\label{eq1}\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ })$

(1)

Type: Type

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vect(x:List Scalar):Matrix Scalar == matrix map(y+->[y],x)

   Function declaration vect : List(Expression(Integer)) -> Matrix(
      Expression(Integer)) has been added to workspace.

Type: Void

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vect [a0,a1,a2,a3]

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Compiling function vect with type List(Expression(Integer)) -> 
      Matrix(Expression(Integer))

$\label{eq2}\left[ \begin{array}{c} a 0 \ a 1 \ a 2 \ a 3$

(2)

Type: Matrix(Expression(Integer))

Identity

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ID:=diagonalMatrix([1,1,1,1])

$\label{eq3}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(3)

Type: Matrix(Integer)

Verification

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htrigs2exp == rule
  cosh(a) == (exp(a)+exp(-a))/2
  sinh(a) == (exp(a)-exp(-a))/2

Type: Void

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sinhcosh == rule
  ?c*exp(a)+?c*exp(-a) == 2*c*cosh(a)
  ?c*exp(a)-?c*exp(-a) == 2*c*sinh(a)
  ?c*exp(a-b)+?c*exp(b-a) == 2*c*cosh(a-b)
  ?c*exp(a-b)-?c*exp(b-a) == 2*c*sinh(a-b)

Type: Void

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expandhtrigs == rule
  cosh(:x+y) == sinh(x)*sinh(y)+cosh(x)*cosh(y)
  sinh(:x+y) == cosh(x)*sinh(y)+sinh(x)*cosh(y)
  cosh(2*x) == 2*cosh(x)^2-1
  sinh(2*x) == 2*sinh(x)*cosh(x)

Type: Void

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expandhtrigs2 == rule
  cosh(2*x+2*y) == 2*cosh(x+y)^2-1
  sinh(2*x+2*y) == 2*sinh(x+y)*cosh(x+y)
  cosh(2*x-2*y) == 2*cosh(x-y)^2-1
  sinh(2*x-2*y) == 2*sinh(x-y)*cosh(x-y)

Type: Void

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Simplify(x:Scalar):Scalar == htrigs sinhcosh simplify htrigs2exp x

   Function declaration Simplify : Expression(Integer) -> Expression(
      Integer) has been added to workspace.

Type: Void

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possible(x)==subst(x, map(y+->(y=(random(100) - random(100))),variables x) )

Type: Void

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is?(eq:Equation Scalar):Boolean == (Simplify(lhs(eq)-rhs(eq))=0)::Boolean

   Function declaration is? : Equation(Expression(Integer)) -> Boolean 
      has been added to workspace.

Type: Void

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Is?(eq:Equation(Matrix(Scalar))):Boolean == _
(map(Simplify,lhs(eq)-rhs(eq)) :: Matrix Expression AlgebraicNumber = _
zero(nrows(lhs(eq)),ncols(lhs(eq)))$Matrix Expression AlgebraicNumber )::Boolean

   Function declaration Is? : Equation(Matrix(Expression(Integer))) -> 
      Boolean has been added to workspace.

Type: Void

Lorentz Form (metric)

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G:=diagonalMatrix [-1,1,1,1]

$\label{eq4}\left[ \begin{array}{cccc} - 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(4)

Type: Matrix(Integer)

applied to a vector produces a co-vector (represent as a $1\times n$ matrix or row vector)

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g(x) == transpose(x)*G

Type: Void

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g(vect [a0,a1,a2,a3])

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Compiling function g with type Matrix(Expression(Integer)) -> Matrix
      (Expression(Integer))

$\label{eq5}\left[ \begin{array}{cccc} - a 0 & a 1 & a 2 & a 3$

(5)

Type: Matrix(Expression(Integer))

Scalar product

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dot(x,y) == (g(x)*y)::Scalar

Type: Void

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dot(vect [a0,a1,a2,a3], vect [b0,b1,b2,b3])

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Compiling function dot with type (Matrix(Expression(Integer)), 
      Matrix(Expression(Integer))) -> Expression(Integer)

$\label{eq6}{a 3 \ b 3}+{a 2 \ b 2}+{a 1 \ b 1}-{a 0 \ b 0}$

(6)

Type: Expression(Integer)

Tensor product

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tensor(x,y) == x*g(y)

Type: Void

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tensor(vect [a0,a1,a2,a3], vect [b0,b1,b2,b3])

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Compiling function tensor with type (Matrix(Expression(Integer)), 
      Matrix(Expression(Integer))) -> Matrix(Expression(Integer))

$\label{eq7}\left[ \begin{array}{cccc} -{a 0 \ b 0}&{a 0 \ b 1}&{a 0 \ b 2}&{a 0 \ b 3} \ -{a 1 \ b 0}&{a 1 \ b 1}&{a 1 \ b 2}&{a 1 \ b 3} \ -{a 2 \ b 0}&{a 2 \ b 1}&{a 2 \ b 2}&{a 2 \ b 3} \ -{a 3 \ b 0}&{a 3 \ b 1}&{a 3 \ b 2}&{a 3 \ b 3}$

(7)

Type: Matrix(Expression(Integer))

Massive Objects

A material object (also referred to as an observer) is represented by a time-like 4-vector

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P:=vect [sqrt(p1^2+p2^2+p3^2+1),-p1,-p2,-p3];

Type: Matrix(Expression(Integer))

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dot(P,P)

$\label{eq8}- 1$

(8)

Type: Expression(Integer)

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Q:=vect [sqrt(q1^2+q2^2+q3^2+1),-q1,-q2,-q3];

Type: Matrix(Expression(Integer))

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R:=vect [sqrt(r1^2+r2^2+r3^2+1),-r1,-r2,-r3];

Type: Matrix(Expression(Integer))

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S:=1/sqrt(1-s1^2-s2^2-s3^2)*vect [1,-s1,-s2,-s3]

$\label{eq9}\left[ \begin{array}{c} {\frac{1}{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}} \ -{\frac{s 1}{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}} \ -{\frac{s 2}{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}} \ -{\frac{s 3}{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}$

(9)

Type: Matrix(Expression(Integer))

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dot(S,S)

$\label{eq10}- 1$

(10)

Type: Expression(Integer)

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T:=1/sqrt(1-t1^2-t2^2-t3^2)*vect [1,-t1,-t2,-t3]

$\label{eq11}\left[ \begin{array}{c} {\frac{1}{\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}} \ -{\frac{t 1}{\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}} \ -{\frac{t 2}{\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}} \ -{\frac{t 3}{\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}$

(11)

Type: Matrix(Expression(Integer))

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U:=vect [cosh(u),sinh(u),0,0]

$\label{eq12}\left[ \begin{array}{c} {\cosh \left({u}\right)} \ {\sinh \left({u}\right)} \ 0 \ 0$

(12)

Type: Matrix(Expression(Integer))

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simplify dot(U,U)

$\label{eq13}- 1$

(13)

Type: Expression(Integer)

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V:=vect [cosh(v),sinh(v),0,0]

$\label{eq14}\left[ \begin{array}{c} {\cosh \left({v}\right)} \ {\sinh \left({v}\right)} \ 0 \ 0$

(14)

Type: Matrix(Expression(Integer))

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Simplify dot(U,V)

fricas

Compiling body of rule htrigs2exp to compute value of type Ruleset(
      Integer,Integer,Expression(Integer))

fricas

Compiling body of rule sinhcosh to compute value of type Ruleset(
      Integer,Integer,Expression(Integer))

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Compiling function Simplify with type Expression(Integer) -> 
      Expression(Integer)

$\label{eq15}-{\cosh \left({v - u}\right)}$

(15)

Type: Expression(Integer)

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W:=vect [cosh(w),0,sinh(w),0]

$\label{eq16}\left[ \begin{array}{c} {\cosh \left({w}\right)} \ 0 \ {\sinh \left({w}\right)} \ 0$

(16)

Type: Matrix(Expression(Integer))

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Simplify dot(U,W)

$\label{eq17}\frac{-{\cosh \left({w + u}\right)}-{\cosh \left({w - u}\right)}}{2}$

(17)

Type: Expression(Integer)

Observer "at rest"

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vect [1,0,0,0]

$\label{eq18}\left[ \begin{array}{c} 1 \ 0 \ 0 \ 0$

(18)

Type: Matrix(Expression(Integer))

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dot(%,%)

$\label{eq19}- 1$

(19)

Type: Expression(Integer)

Associated with each such vector is the orthogonal 3-d Euclidean subspace $E_P =\{x | P \cdot x = 0\}$

Relative Velocity

An object P has a unique relative velocity ω(P,Q) with respect to object Q given by

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ω(P,Q)==-P/dot(P,Q)-Q

Type: Void

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ω(P,Q)

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Compiling function ω with type (Matrix(Expression(Integer)), Matrix(
      Expression(Integer))) -> Matrix(Expression(Integer))

$\label{eq20}\left[ \begin{array}{c} {\frac{{{\left({p 3 \ q 3}+{p 2 \ q 2}+{p 1 \ q 1}\right)}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}+{{\left(-{{q 3}^{2}}-{{q 2}^{2}}-{{q 1}^{2}}\right)}\ {\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}}}{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ q 3}-{p 2 \ q 2}-{p 1 \ q 1}}} \ {\frac{{q 1 \ {\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ q 1 \ q 3}-{p 2 \ q 1 \ q 2}-{p 1 \ {{q 1}^{2}}}- p 1}{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ q 3}-{p 2 \ q 2}-{p 1 \ q 1}}} \ {\frac{{q 2 \ {\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ q 2 \ q 3}-{p 2 \ {{q 2}^{2}}}-{p 1 \ q 1 \ q 2}- p 2}{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ q 3}-{p 2 \ q 2}-{p 1 \ q 1}}} \ {\frac{{q 3 \ {\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ {{q 3}^{2}}}+{{\left(-{p 2 \ q 2}-{p 1 \ q 1}\right)}\ q 3}- p 3}{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ q 3}-{p 2 \ q 2}-{p 1 \ q 1}}}$

(20)

Type: Matrix(Expression(Integer))

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ω(S,T)

$\label{eq21}\left[ \begin{array}{c} {\frac{{{t 3}^{2}}-{s 3 \ t 3}+{{t 2}^{2}}-{s 2 \ t 2}+{{t 1}^{2}}-{s 1 \ t 1}}{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}} \ {\frac{-{s 1 \ {{t 3}^{2}}}+{s 3 \ t 1 \ t 3}-{s 1 \ {{t 2}^{2}}}+{s 2 \ t 1 \ t 2}- t 1 + s 1}{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}} \ {\frac{-{s 2 \ {{t 3}^{2}}}+{s 3 \ t 2 \ t 3}+{{\left({s 1 \ t 1}- 1 \right)}\ t 2}-{s 2 \ {{t 1}^{2}}}+ s 2}{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}} \ {\frac{{{\left({s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ t 3}-{s 3 \ {{t 2}^{2}}}-{s 3 \ {{t 1}^{2}}}+ s 3}{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}$

(21)

Type: Matrix(Expression(Integer))

In two dimensions

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map(x+->Simplify x,ω(U,V))

$\label{eq22}\left[ \begin{array}{c} {\frac{-{\cosh \left({{2 \ v}- u}\right)}+{\cosh \left({u}\right)}}{2 \ {\cosh \left({v - u}\right)}}} \ {\frac{-{\sinh \left({{2 \ v}- u}\right)}+{\sinh \left({u}\right)}}{2 \ {\cosh \left({v - u}\right)}}} \ 0 \ 0$

(22)

Type: Matrix(Expression(Integer))

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vect [cosh(u)/cosh(u-v)-cosh(v),sinh(u)/cosh(u-v)-sinh(v),0,0]

$\label{eq23}\left[ \begin{array}{c} {\frac{-{{\cosh \left({v - u}\right)}\ {\cosh \left({v}\right)}}+{\cosh \left({u}\right)}}{\cosh \left({v - u}\right)}} \ {\frac{-{{\cosh \left({v - u}\right)}\ {\sinh \left({v}\right)}}+{\sinh \left({u}\right)}}{\cosh \left({v - u}\right)}} \ 0 \ 0$

(23)

Type: Matrix(Expression(Integer))

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Is?(% = ω(U,V))

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Compiling function Is? with type Equation(Matrix(Expression(Integer)
      )) -> Boolean

$\label{eq24} \mbox{\rm true}$

(24)

Type: Boolean

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map(x+->Simplify x,ω(U,W))

$\label{eq25}\left[ \begin{array}{c} {\frac{-{\cosh \left({2 \ w}\right)}+ 1}{2 \ {\cosh \left({w}\right)}}} \ {\frac{2 \ {\sinh \left({u}\right)}}{{\cosh \left({w + u}\right)}+{\cosh \left({w - u}\right)}}} \ -{\sinh \left({w}\right)} \ 0$

(25)

Type: Matrix(Expression(Integer))

Observer P measures velocity ω(Q,P). ω(Q,P) is space-like

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dot(ω(P,vect [1,0,0,0]),ω(P,vect [1,0,0,0]))

$\label{eq26}\frac{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}}{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}$

(26)

Type: Expression(Integer)

and in

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dot(P,ω(Q,P))

$\label{eq27}0$

(27)

Type: Expression(Integer)

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possible dot(ω(Q,P),ω(Q,P))::EXPR Float

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Compiling function possible with type Expression(Integer) -> 
      Expression(Integer)

$\label{eq28}0.9999995659 \<u> 3941083651$

(28)

Type: Expression(Float)

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dot(Q,ω(P,Q))

$\label{eq29}0$

(29)

Type: Expression(Integer)

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possible dot(ω(P,Q),ω(P,Q))::EXPR Float

$\label{eq30}0.9999994628 \<u> 9493005142$

(30)

Type: Expression(Float)

Velocity with respect to observer "at rest"

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ω(vect [u0,u1,u2,u3],vect [1,0,0,0])

$\label{eq31}\left[ \begin{array}{c} 0 \ {\frac{u 1}{u 0}} \ {\frac{u 2}{u 0}} \ {\frac{u 3}{u 0}}$

(31)

Type: Matrix(Expression(Integer))

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ω(R,vect [1,0,0,0])

$\label{eq32}\left[ \begin{array}{c} 0 \ -{\frac{r 1}{\sqrt{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}+ 1}}} \ -{\frac{r 2}{\sqrt{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}+ 1}}} \ -{\frac{r 3}{\sqrt{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}+ 1}}}$

(32)

Type: Matrix(Expression(Integer))

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ω(S,vect [1,0,0,0])

$\label{eq33}\left[ \begin{array}{c} 0 \ - s 1 \ - s 2 \ - s 3$

(33)

Type: Matrix(Expression(Integer))

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map(Simplify, ω(U,vect [1,0,0,0]))

$\label{eq34}\left[ \begin{array}{c} 0 \ {\frac{\sinh \left({u}\right)}{\cosh \left({u}\right)}} \ 0 \ 0$

(34)

Type: Matrix(Expression(Integer))

Non-reciprocal velocities

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ω(vect [1,0,0,0],S)

$\label{eq35}\left[ \begin{array}{c} {\frac{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}}{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}} \ {\frac{s 1}{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}} \ {\frac{s 2}{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}} \ {\frac{s 3}{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}$

(35)

Type: Matrix(Expression(Integer))

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ω(vect [1,0,0,0],R)

$\label{eq36}\left[ \begin{array}{c} {\frac{-{{r 3}^{2}}-{{r 2}^{2}}-{{r 1}^{2}}}{\sqrt{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}+ 1}}} \ r 1 \ r 2 \ r 3$

(36)

Type: Matrix(Expression(Integer))

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is?(dot(ω(P,Q),ω(P,Q))=dot(ω(Q,P),ω(Q,P)))

fricas

Compiling function is? with type Equation(Expression(Integer)) -> 
      Boolean

$\label{eq37} \mbox{\rm true}$

(37)

Type: Boolean

Lorentz Boost

is a linear bijection $E_Q \leftrightarrow E_P$ that preserves $E_Q \cap E_P$ and maps orthogonal compliments into each other.

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L(P,Q) == ID + tensor(P+Q,P+Q)/(1-dot(P,Q)) - 2*tensor(P,Q)

Type: Void

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Is?(L(P,P) = ID)

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Compiling function L with type (Matrix(Expression(Integer)), Matrix(
      Expression(Integer))) -> Matrix(Expression(Integer))

$\label{eq38} \mbox{\rm true}$

(38)

Type: Boolean

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Is?(L(P,Q)*L(Q,P) = ID)

$\label{eq39} \mbox{\rm true}$

(39)

Type: Boolean

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Is?(L(P,Q)*Q=P)

$\label{eq40} \mbox{\rm true}$

(40)

Type: Boolean

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Is?(L(P,Q)*ω(P,Q) = -ω(Q,P))

$\label{eq41} \mbox{\rm true}$

(41)

Type: Boolean

Most General Lorentz Boost (Oziewicz, 2006)

is given by three non-coplanar vectors

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B(P,Q,X) == ID -                                _
( tensor(                                       _
    2*X, dot(P-Q,P-Q)*X - 2*dot(X,P)*(P-Q)      _
  ) +                                           _
  tensor(                                       _
    P-Q, 2*dot(X,X)*(P-Q)+4*dot(X,Q)*X          _
  )                                             _

fricas

) / (                                           _
      dot(X,X)*dot(P-Q,P-Q)+4*dot(X,P)*dot(X,Q) _
    )

Type: Void

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Is?(B(P,P,R) = ID)

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Compiling function B with type (Matrix(Expression(Integer)), Matrix(
      Expression(Integer)), Matrix(Expression(Integer))) -> Matrix(
      Expression(Integer))

$\label{eq42} \mbox{\rm true}$

(42)

Type: Boolean

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Is?(B(P,Q,R)*B(Q,P,R) = ID)

$\label{eq43} \mbox{\rm true}$

(43)

Type: Boolean

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Is?(B(P,Q,R)*P=Q)

$\label{eq44} \mbox{\rm true}$

(44)

Type: Boolean

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--Is?(B(P,Q,R)*ω(P,Q) = -ω(Q,P))
Is?(L(P,Q)=B(Q,P,q*Q+p*P))

$\label{eq45} \mbox{\rm true}$

(45)

Type: Boolean

In two dimensions

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map(x+->simplify expandhtrigs2 Simplify x, L(U,V))

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Compiling body of rule expandhtrigs2 to compute value of type 
      Ruleset(Integer,Integer,Expression(Integer))

$\label{eq46}\left[ \begin{array}{cccc} {\cosh \left({v - u}\right)}& -{\sinh \left({v - u}\right)}& 0 & 0 \ -{\sinh \left({v - u}\right)}&{\cosh \left({v - u}\right)}& 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(46)

Type: Matrix(Expression(Integer))

fricas

map(x+->simplify expandhtrigs expandhtrigs2 Simplify x, L(U,W))

fricas

Compiling body of rule expandhtrigs to compute value of type Ruleset
      (Integer,Integer,Expression(Integer))

$\label{eq47}\left[ \begin{array}{cccc} {\frac{{{\left({2 \ {{\cosh \left({u}\right)}^{2}}}- 1 \right)}\ {{\cosh \left({w}\right)}^{2}}}+{{\cosh \left({u}\right)}\ {\cosh \left({w}\right)}}-{{\cosh \left({u}\right)}^{2}}+ 1}{{{\cosh \left({u}\right)}\ {\cosh \left({w}\right)}}+ 1}}&{\frac{{\left({\cosh \left({w}\right)}+{\cosh \left({u}\right)}\right)}\ {\sinh \left({u}\right)}}{{{\cosh \left({u}\right)}\ {\cosh \left({w}\right)}}+ 1}}&{\frac{{\left({{\left(-{2 \ {{\cosh \left({u}\right)}^{2}}}+ 1 \right)}\ {\cosh \left({w}\right)}}-{\cosh \left({u}\right)}\right)}\ {\sinh \left({w}\right)}}{{{\cosh \left({u}\right)}\ {\cosh \left({w}\right)}}+ 1}}& 0 \ {\frac{{\left({2 \ {\cosh \left({u}\right)}\ {{\cosh \left({w}\right)}^{2}}}+{\cosh \left({w}\right)}-{\cosh \left({u}\right)}\right)}\ {\sinh \left({u}\right)}}{{{\cosh \left({u}\right)}\ {\cosh \left({w}\right)}}+ 1}}&{\frac{{{\cosh \left({u}\right)}\ {\cosh \left({w}\right)}}+{{\cosh \left({u}\right)}^{2}}}{{{\cosh \left({u}\right)}\ {\cosh \left({w}\right)}}+ 1}}&{\frac{{\left(-{2 \ {\cosh \left({u}\right)}\ {\cosh \left({w}\right)}}- 1 \right)}\ {\sinh \left({u}\right)}\ {\sinh \left({w}\right)}}{{{\cosh \left({u}\right)}\ {\cosh \left({w}\right)}}+ 1}}& 0 \ {\frac{{\left(-{\cosh \left({w}\right)}-{\cosh \left({u}\right)}\right)}\ {\sinh \left({w}\right)}}{{{\cosh \left({u}\right)}\ {\cosh \left({w}\right)}}+ 1}}&{\frac{{\sinh \left({u}\right)}\ {\sinh \left({w}\right)}}{{{\cosh \left({u}\right)}\ {\cosh \left({w}\right)}}+ 1}}&{\frac{{{\cosh \left({w}\right)}^{2}}+{{\cosh \left({u}\right)}\ {\cosh \left({w}\right)}}}{{{\cosh \left({u}\right)}\ {\cosh \left({w}\right)}}+ 1}}& 0 \ 0 & 0 & 0 & 1$

(47)

Type: Matrix(Expression(Integer))

Composition of two Lorentz boosts is not a Lorentz boost unless all three observers are in the same plane.

fricas

Is?(L(R,P)*L(P,Q) = L(R,Q))

$\label{eq48} \mbox{\rm false}$

(48)

Type: Boolean

fricas

RQ:=a*R+b*Q;

Type: Matrix(Expression(Integer))

fricas

rq:=solve(dot(RQ,RQ)=-1,b); #rq

$\label{eq49}2$

(49)

Type: PositiveInteger?

fricas

RQ1:=eval(RQ,rq.1);

Type: Matrix(Expression(Integer))

fricas

dot(RQ1,RQ1)

$\label{eq50}- 1$

(50)

Type: Expression(Integer)

fricas

Is?(L(R,RQ1)*L(RQ1,Q) = L(R,Q))

$\label{eq51} \mbox{\rm true}$

(51)

Type: Boolean

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RQ2:=eval(RQ,rq.2);

Type: Matrix(Expression(Integer))

fricas

Is?(RQ1=RQ2)

$\label{eq52} \mbox{\rm false}$

(52)

Type: Boolean

fricas

dot(RQ2,RQ2)

$\label{eq53}- 1$

(53)

Type: Expression(Integer)

fricas

Is?(L(R,RQ2)*L(RQ2,Q) = L(R,Q))

$\label{eq54} \mbox{\rm true}$

(54)

Type: Boolean

but the composition does preserve observers and magnitudes

fricas

LRPQ := L(R,P)*L(P,Q);

Type: Matrix(Expression(Integer))

fricas

Is?(LRPQ*Q = L(R,Q)*Q)

$\label{eq55} \mbox{\rm true}$

(55)

Type: Boolean

fricas

is?(dot(LRPQ*ω(S,Q),LRPQ*ω(S,Q))=dot(L(R,Q)*ω(S,Q),L(R,Q)*ω(S,Q)))

$\label{eq56} \mbox{\rm true}$

(56)

Type: Boolean

Lorentz boost with respect to observer "at rest"

fricas

LT:=L(vect [1,0,0,0],vect [u0,-u1,-u2,-u3])

$\label{eq57}\left[ \begin{array}{cccc} u 0 & u 1 & u 2 & u 3 \ u 1 &{\frac{{{u 1}^{2}}+ u 0 + 1}{u 0 + 1}}&{\frac{u 1 \ u 2}{u 0 + 1}}&{\frac{u 1 \ u 3}{u 0 + 1}} \ u 2 &{\frac{u 1 \ u 2}{u 0 + 1}}&{\frac{{{u 2}^{2}}+ u 0 + 1}{u 0 + 1}}&{\frac{u 2 \ u 3}{u 0 + 1}} \ u 3 &{\frac{u 1 \ u 3}{u 0 + 1}}&{\frac{u 2 \ u 3}{u 0 + 1}}&{\frac{{{u 3}^{2}}+ u 0 + 1}{u 0 + 1}}$

(57)

Type: Matrix(Expression(Integer))

fricas

map(simplify, L(vect [1,0,0,0], map(Simplify,U)))

$\label{eq58}\left[ \begin{array}{cccc} {\cosh \left({u}\right)}& -{\sinh \left({u}\right)}& 0 & 0 \ -{\sinh \left({u}\right)}&{\cosh \left({u}\right)}& 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(58)

Type: Matrix(Expression(Integer))

Two dimensional Lorentz Transformation

fricas

matrix [[1/sqrt(1-v'^2),v'/sqrt(1-v'^2),0,0],[v'/sqrt(1-v'^2),1/sqrt(1-v'^2),0,0],[0,0,1,0],[0,0,0,1]]

$\label{eq59}\left[ \begin{array}{cccc} {\frac{1}{\sqrt{-{{v'}^{2}}+ 1}}}&{\frac{v'}{\sqrt{-{{v'}^{2}}+ 1}}}& 0 & 0 \ {\frac{v'}{\sqrt{-{{v'}^{2}}+ 1}}}&{\frac{1}{\sqrt{-{{v'}^{2}}+ 1}}}& 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(59)

Type: Matrix(Expression(Integer))

fricas

Is?(%=map(x+->eval(x,[u0=1/sqrt(1-v'^2),u1=v'/sqrt(1-v'^2),u2=0,u3=0]),LT))

$\label{eq60} \mbox{\rm true}$

(60)

Type: Boolean

... --Bill Page, Wed, 09 Oct 2013 23:17:45 +0000 reply

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