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Edit detail for SandBoxLorentzTransformation revision 16 of 30

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Editor: Bill Page
Time: 2013/09/24 23:21:54 GMT+0
Note: tidy

changed:
-Lorentz Form (metric) produces a co-vector (represent as a 1xn matrix or row vector)
Lorentz Form (metric)

added:
\end{axiom}

applied to a vector produces a co-vector (represent as a 1xn matrix or row vector)
\begin{axiom}

changed:
-  An object (also referred to as an observer) is represented by a
  A material object (also referred to as an observer) is represented by a

Lorentz transformations.

Book by T. Matolcsi

Mathematical Preliminaries

A vector is represented as a nx1 matrix (column vector)

fricas
vect(x:List Expression Integer):Matrix Expression Integer == matrix map(y+->[y],x)
Function declaration vect : List(Expression(Integer)) -> Matrix( Expression(Integer)) has been added to workspace.
Type: Void
fricas
vect [a0,a1,a2,a3]
fricas
Compiling function vect with type List(Expression(Integer)) -> 
      Matrix(Expression(Integer))

\label{eq1}\left[ 
\begin{array}{c}
a 0 
\
a 1 
\
a 2 
\
a 3 
(1)
Type: Matrix(Expression(Integer))

Identity

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

\label{eq2}\left[ 
\begin{array}{cccc}
1 & 0 & 0 & 0 
\
0 & 1 & 0 & 0 
\
0 & 0 & 1 & 0 
\
0 & 0 & 0 & 1 
(2)
Type: Matrix(Integer)

Verification

fricas
possible(x)==subst(x, map(y+->(y=(random(100) - random(100))),variables x) )
Type: Void
fricas
is?(eq:Equation EXPR INT):Boolean == (lhs(eq)-rhs(eq)=0)::Boolean
Function declaration is? : Equation(Expression(Integer)) -> Boolean has been added to workspace.
Type: Void
fricas
Is?(eq:Equation(Matrix(EXPR(INT)))):Boolean == _
( (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)

fricas
G:=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)

applied to a vector produces a co-vector (represent as a 1xn matrix or row vector)

fricas
g(x)==transpose(x)*G
Type: Void
fricas
g(vect [a0,a1,a2,a3])
fricas
Compiling function g with type Matrix(Expression(Integer)) -> Matrix
      (Expression(Integer))

\label{eq4}\left[ 
\begin{array}{cccc}
- a 0 & a 1 & a 2 & a 3 
(4)
Type: Matrix(Expression(Integer))

Scalar product

fricas
dot(x,y)== (g(x)*y)::Expression Integer
Type: Void
fricas
dot(vect [a0,a1,a2,a3], vect [b0,b1,b2,b3])
fricas
Compiling function dot with type (Matrix(Expression(Integer)),Matrix
      (Expression(Integer))) -> Expression(Integer)

\label{eq5}{a 3 \  b 3}+{a 2 \  b 2}+{a 1 \  b 1}-{a 0 \  b 0}(5)
Type: Expression(Integer)

Tensor product

fricas
tensor(x,y) == x*g(y)
Type: Void
fricas
tensor(vect [a0,a1,a2,a3], vect [b0,b1,b2,b3])
fricas
Compiling function tensor with type (Matrix(Expression(Integer)),
      Matrix(Expression(Integer))) -> Matrix(Expression(Integer))

\label{eq6}\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}
(6)
Type: Matrix(Expression(Integer))

Massive Objects

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

fricas
P:=vect [sqrt(p1^2+p2^2+p3^2+1),-p1,-p2,-p3];
Type: Matrix(Expression(Integer))
fricas
dot(P,P)

\label{eq7}- 1(7)
Type: Expression(Integer)
fricas
Q:=vect [sqrt(q1^2+q2^2+q3^2+1),-q1,-q2,-q3];
Type: Matrix(Expression(Integer))
fricas
R:=vect [sqrt(r1^2+r2^2+r3^2+1),-r1,-r2,-r3];
Type: Matrix(Expression(Integer))
fricas
S:=1/sqrt(1-s1^2-s2^2-s3^2)*vect [1,-s1,-s2,-s3]

\label{eq8}\left[ 
\begin{array}{c}
{1 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
-{s 1 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
-{s 2 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
-{s 3 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
(8)
Type: Matrix(Expression(Integer))
fricas
dot(S,S)

\label{eq9}- 1(9)
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 w(P,Q) with respect to object Q given by

fricas
v(P,Q)==-P/dot(P,Q)-Q
Type: Void

Observer P measures velocity v(Q,P). v(Q,P) is space-like and in E_P.

fricas
dot(P,v(Q,P))
fricas
Compiling function v with type (Matrix(Expression(Integer)),Matrix(
      Expression(Integer))) -> Matrix(Expression(Integer))

\label{eq10}0(10)
Type: Expression(Integer)
fricas
possible dot(v(Q,P),v(Q,P))::EXPR Float
fricas
Compiling function possible with type Expression(Integer) -> 
      Expression(Integer)

\label{eq11}0.9999999805_5823520184(11)
Type: Expression(Float)
fricas
dot(Q,v(P,Q))

\label{eq12}0(12)
Type: Expression(Integer)
fricas
possible dot(v(P,Q),v(P,Q))::EXPR Float

\label{eq13}0.9999999375_8760717897(13)
Type: Expression(Float)

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.

fricas
L(P,Q) == ID + tensor(P+Q,P+Q)/(1-dot(P,Q)) - 2*tensor(P,Q)
Type: Void
fricas
Is?(L(P,P) = ID)
fricas
Compiling function L with type (Matrix(Expression(Integer)),Matrix(
      Expression(Integer))) -> Matrix(Expression(Integer))
fricas
Compiling function Is? with type Equation(Matrix(Expression(Integer)
      )) -> Boolean

\label{eq14} \mbox{\rm true} (14)
Type: Boolean
fricas
Is?(L(P,Q)*L(Q,P) = ID)

\label{eq15} \mbox{\rm true} (15)
Type: Boolean
fricas
Is?(L(P,Q)*Q=P)

\label{eq16} \mbox{\rm true} (16)
Type: Boolean
fricas
Is?(L(P,Q)*v(P,Q) = -v(Q,P))

\label{eq17} \mbox{\rm true} (17)
Type: Boolean

Composition of two Lorentz boosts is not a Lorentz boost

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

\label{eq18} \mbox{\rm false} (18)
Type: Boolean

Velocity with respect to observer "at rest"

fricas
v(vect [u0,u1,u2,u3],vect [1,0,0,0])

\label{eq19}\left[ 
\begin{array}{c}
0 
\
{u 1 \over u 0}
\
{u 2 \over u 0}
\
{u 3 \over u 0}
(19)
Type: Matrix(Expression(Integer))
fricas
v(S,vect [1,0,0,0])

\label{eq20}\left[ 
\begin{array}{c}
0 
\
- s 1 
\
- s 2 
\
- s 3 
(20)
Type: Matrix(Expression(Integer))

Lorentz boost with respect to observer "at rest"

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

\label{eq21}\left[ 
\begin{array}{cccc}
u 0 & u 1 & u 2 & u 3 
\
u 1 &{{{{u 1}^{2}}+ u 0 + 1}\over{u 0 + 1}}&{{u 1 \  u 2}\over{u 0 + 1}}&{{u 1 \  u 3}\over{u 0 + 1}}
\
u 2 &{{u 1 \  u 2}\over{u 0 + 1}}&{{{{u 2}^{2}}+ u 0 + 1}\over{u 0 + 1}}&{{u 2 \  u 3}\over{u 0 + 1}}
\
u 3 &{{u 1 \  u 3}\over{u 0 + 1}}&{{u 2 \  u 3}\over{u 0 + 1}}&{{{{u 3}^{2}}+ u 0 + 1}\over{u 0 + 1}}
(21)
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{eq22}\left[ 
\begin{array}{cccc}
{1 \over{\sqrt{-{{v'}^{2}}+ 1}}}&{v' \over{\sqrt{-{{v'}^{2}}+ 1}}}& 0 & 0 
\
{v' \over{\sqrt{-{{v'}^{2}}+ 1}}}&{1 \over{\sqrt{-{{v'}^{2}}+ 1}}}& 0 & 0 
\
0 & 0 & 1 & 0 
\
0 & 0 & 0 & 1 
(22)
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{eq23} \mbox{\rm true} (23)
Type: Boolean