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SandBox Lorentz Transformations ←	↑	→ SandBoxRelativeVelocity

SandBoxLorentzTransformation

last edited 10 years ago by Bill Page

Edit detail for SandBoxLorentzTransformation revision 7 of 30

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Editor: Bill Page Time: 2013/09/22 22:23:13 GMT+0
Note:

changed:
-vect [a1,a2,a3,a4]
vect [a0,a1,a2,a3]

removed:
-
-Applying the Lorentz form produces a row vector
-\begin{axiom}
-g(x)==transpose(x)*diagonalMatrix [-1,1,1,1]
-\end{axiom}
-
-Then the dual is a row vector is
-\begin{axiom}
-g(vect [a1,a2,a3,a4])
-\end{axiom}
-
-And scalar product is
-\begin{axiom}
-dot(x,y)== (g(x)*y)::Expression Integer 
-dot(vect [a1,a2,a3,a4], vect [b1,b2,b3,b4])
-\end{axiom}
-
-Tensor product is
-\begin{axiom}
-tensor(x,y) == x*g(y)
-tensor(vect [a1,a2,a3,a4], vect [b1,b2,b3,b4])
-\end{axiom}

added:

Applying the Lorentz form produces a co-vector (represent as a 1xn matrix or row)
\begin{axiom}
G:=diagonalMatrix [-1,1,1,1]
g(x)==transpose(x)*G
g(vect [a0,a1,a2,a3])
\end{axiom}

And scalar product is
\begin{axiom}
dot(x,y)== (g(x)*y)::Expression Integer 
dot(vect [a0,a1,a2,a3], vect [b0,b1,b2,b3])
\end{axiom}

Tensor product is
\begin{axiom}
tensor(x,y) == x*g(y)
tensor(vect [a0,a1,a2,a3], vect [b0,b1,b2,b3])
\end{axiom}

changed:
-  An object Q has a unique relative velocity w(P,Q) with respect
-to object P given by
  An object P has a unique relative velocity w(P,Q) with respect
to object Q given by

changed:
-w(P,Q)==-Q/dot(P,Q)-P
v(P,Q)==-P/dot(P,Q)-Q

removed:
-
-Lorentz factor
-\begin{axiom}
-gamma(v)==1/sqrt(1-g(v,v))
-\end{axiom}
-
-Binary Boost
-\begin{axiom}
-b(P,v)==gamma(v)*(P+v)
-\end{axiom}

changed:
-u:=w(P,Q);
-dot(P,u)
-possible dot(u,u)::EXPR Float
-v:=w(Q,P);
-dot(Q,v)
-possible dot(v,v)::EXPR Float
dot(Q,v(P,Q))
possible dot(v(P,Q),v(P,Q))::EXPR Float
dot(P,v(Q,P))
possible dot(v(Q,P),v(Q,P))::EXPR Float

changed:
-L(R,P)
-Is2?(L(P,Q)*w(P,Q) = -w(Q,P))
L(vect [1,0,0,0],vect [u0,u1,u2,u3])
Is2?(L(P,Q)*v(P,Q) = -v(Q,P))

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]

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)

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

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)

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))

And scalar product is

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 is

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))

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

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))

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

dot(Q,Q)

$\label{eq8}- 1$

(8)

Type: Expression(Integer)

fricas

R:=vect [1,0,0,0]

$\label{eq9}\left[ \begin{array}{c} 1 \ 0 \ 0 \ 0$

(9)

Type: Matrix(Expression(Integer))

fricas

dot(R,R)

$\label{eq10}- 1$

(10)

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 u. u is space-like and in .

fricas

dot(Q,v(P,Q))

fricas

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

$\label{eq11}0$

(11)

Type: Expression(Integer)

fricas

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

fricas

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

$\label{eq12}0.9999999805_5823520184$

(12)

Type: Expression(Float)

fricas

dot(P,v(Q,P))

$\label{eq13}0$

(13)

Type: Expression(Integer)

fricas

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

$\label{eq14}0.9999999746_9826498313$

(14)

Type: Expression(Float)

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

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

(15)

Type: Boolean

fricas

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

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

(16)

Type: Boolean

fricas

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

$\label{eq17}\left[ \begin{array}{cccc} u 0 & - u 1 & - u 2 & - u 3 \ - u 1 &{{{{u 1}^{2}}+ u 0 + 1}\over{u 0 + 1}}&{{u 1 \ u 2}\over{u 0 + 1}}&{{u 1 \ u 3}\over{u 0 + 1}} \ - u 2 &{{u 1 \ u 2}\over{u 0 + 1}}&{{{{u 2}^{2}}+ u 0 + 1}\over{u 0 + 1}}&{{u 2 \ u 3}\over{u 0 + 1}} \ - u 3 &{{u 1 \ u 3}\over{u 0 + 1}}&{{u 2 \ u 3}\over{u 0 + 1}}&{{{{u 3}^{2}}+ u 0 + 1}\over{u 0 + 1}}$

(17)

Type: Matrix(Expression(Integer))

fricas

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

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

(18)

Type: Boolean