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	SandBox Categorical Relativity
SandBox Lorentz Transformations ←	↑	→ SandBoxRelativeVelocity

SandBoxLorentzTransformation

last edited 10 years ago by Bill Page

Edit detail for SandBoxLorentzTransformation revision 11 of 30

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Editor: Bill Page Time: 2013/09/24 21:03:17 GMT+0
Note: boost

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

changed:
-Q:=vect [sqrt(q1^2+q2^2+q3^2+1),q1,q2,q3];
-R:=vect [sqrt(r1^2+r2^2+r3^2+1),r1,r2,r3];
Q:=vect [sqrt(q1^2+q2^2+q3^2+1),-q1,-q2,-q3];
R:=vect [sqrt(r1^2+r2^2+r3^2+1),-r1,-r2,-r3];
S:=1/sqrt(1-s1^2-s2^2-s3^2)*vect [1,-s1,-s2,-s3]
dot(S,S)

changed:
-L(vect [1,0,0,0],vect [u0,u1,u2,u3])
L(vect [1,0,0,0],vect [u0,-u1,-u2,-u3])
map(x+->eval(x,[u0=_\kappa,u1=_\kappa*v',u2=0,u3=0]),%)

added:

Lorentz transformations.

Book by T. Matolcsi

Space-time without reference frames

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]

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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

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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)

Applying the Lorentz form produces a co-vector (represent as a 1xn matrix or row)

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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)

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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))

$\label{eq4}\left[ \begin{array}{cccc} - a 0 & a 1 & a 2 & a 3$

(4)

Type: Matrix(Expression(Integer))

And scalar product is

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dot(x,y)== (g(x)*y)::Expression Integer

Type: Void

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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 is

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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))

$\label{eq6}\left[ \begin{array}{cccc} -{a 0 \ b 0}&{a 0 \ b 1}&{a 0 \ b 2}&{a 0 \ b 3} \ -{a 1 \ b 0}&{a 1 \ b 1}&{a 1 \ b 2}&{a 1 \ b 3} \ -{a 2 \ b 0}&{a 2 \ b 1}&{a 2 \ b 2}&{a 2 \ b 3} \ -{a 3 \ b 0}&{a 3 \ b 1}&{a 3 \ b 2}&{a 3 \ b 3}$

(6)

Type: Matrix(Expression(Integer))

Verification

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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 EXPR INT):Boolean == (lhs(eq)-rhs(eq)=0)::Boolean

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

Type: Void

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Is2?(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 Is2? : Equation(Matrix(Expression(Integer)))
       -> Boolean has been added to workspace.

Type: Void

Massive Objects

An object (also referred to as an obserser) 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))

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dot(P,P)

$\label{eq7}- 1$

(7)

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]

$\label{eq8}\left[ \begin{array}{c} {1 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}} \ -{s 1 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}} \ -{s 2 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}} \ -{s 3 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}$

(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 .

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)

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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

Is2?(L(P,P) = ID)

fricas

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

fricas

Compiling function Is2? with type Equation(Matrix(Expression(Integer
      ))) -> Boolean

$\label{eq14} \mbox{\rm true}$

(14)

Type: Boolean

fricas

Is2?(L(P,Q)*L(Q,P) = ID)

$\label{eq15} \mbox{\rm true}$

(15)

Type: Boolean

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Is2?(L(P,Q)*Q=P)

$\label{eq16} \mbox{\rm true}$

(16)

Type: Boolean

fricas

Is2?(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

Is2?(L(R,P)*L(P,Q) = L(R,Q))

$\label{eq18} \mbox{\rm false}$

(18)

Type: Boolean

Lorentz boost with respect to observer "at rest"

fricas

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

$\label{eq19}\left[ \begin{array}{cccc} u 0 & u 1 & u 2 & u 3 \ u 1 &{{{{u 1}^{2}}+ u 0 + 1}\over{u 0 + 1}}&{{u 1 \ u 2}\over{u 0 + 1}}&{{u 1 \ u 3}\over{u 0 + 1}} \ u 2 &{{u 1 \ u 2}\over{u 0 + 1}}&{{{{u 2}^{2}}+ u 0 + 1}\over{u 0 + 1}}&{{u 2 \ u 3}\over{u 0 + 1}} \ u 3 &{{u 1 \ u 3}\over{u 0 + 1}}&{{u 2 \ u 3}\over{u 0 + 1}}&{{{{u 3}^{2}}+ u 0 + 1}\over{u 0 + 1}}$

(19)

Type: Matrix(Expression(Integer))

fricas

map(x+->eval(x,[u0=_\kappa,u1=_\kappa*v',u2=0,u3=0]),%)

$\label{eq20}\left[ \begin{array}{cccc} \kappa &{\kappa \ v'}& 0 & 0 \ {\kappa \ v'}&{{{{{\kappa}^{2}}\ {{v'}^{2}}}+ \kappa + 1}\over{\kappa + 1}}& 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(20)

Type: Matrix(Expression(Integer))