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SandBox Idempotent Observers

last edited 7 years ago by Bill Page

Edit detail for SandBox Idempotent Observers revision 1 of 29

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Editor: Bill Page Time: 2013/10/09 23:35:00 GMT+0
Note: new

changed:
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Mathematical Preliminaries

  - "Space-time without reference frames":http://axiom-wiki.newsynthesis.org/uploads/matolcsi.pdf

A vector is represented as a $n\times 1$ matrix (column vector)
\begin{axiom}
Scalar := Expression Integer
vect(x:List Scalar):Matrix Scalar == matrix map(y+->[y],x)
vect [a0,a1,a2,a3]
\end{axiom}

Identity
\begin{axiom}
ID:=diagonalMatrix([1,1,1,1])
\end{axiom}

Verification
\begin{axiom}
htrigs2exp == rule
  cosh(a) == (exp(a)+exp(-a))/2
  sinh(a) == (exp(a)-exp(-a))/2
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)
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)
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)
Simplify(x:Scalar):Scalar == htrigs sinhcosh simplify htrigs2exp x
possible(x)==subst(x, map(y+->(y=(random(100) - random(100))),variables x) )
is?(eq:Equation Scalar):Boolean == (Simplify(lhs(eq)-rhs(eq))=0)::Boolean
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
\end{axiom}

Lorentz Form (metric)
\begin{axiom}
G:=diagonalMatrix [1,-1,-1,-1]
\end{axiom}

applied to a vector produces a co-vector (represent as a $1\times n$ matrix or row vector)
\begin{axiom}
g(x) == transpose(x)*G
g(vect [a0,a1,a2,a3])
\end{axiom}

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

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

Massive Objects

  A material object (also referred to as an observer) is represented by a
time-like 4-vector
\begin{axiom}
P:=vect [sqrt(p1^2+p2^2+p3^2+1),-p1,-p2,-p3];
dot(P,P)
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)
T:=1/sqrt(1-t1^2-t2^2-t3^2)*vect [1,-t1,-t2,-t3]
U:=vect [cosh(u),sinh(u),0,0]
simplify dot(U,U)
V:=vect [cosh(v),sinh(v),0,0]
Simplify dot(U,V)
W:=vect [cosh(w),0,sinh(w),0]
Simplify dot(U,W)
\end{axiom}

Observer "at rest"
\begin{axiom}
vect [1,0,0,0]
dot(%,%)
\end{axiom}

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
\begin{axiom}
ω(P,Q)==-P/dot(P,Q)-Q
ω(P,Q)
ω(S,T)
\end{axiom}

Idempotent Observers

\begin{axiom}
PP:=tensor(P,P)
Is?(PP*PP=PP)
\end{axiom}

Mathematical Preliminaries

Space-time without reference frames

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

fricas

Scalar := Expression Integer

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

(1)

Type: Type

fricas

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

fricas

vect [a0,a1,a2,a3]

fricas

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

fricas

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

fricas

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

Type: Void

fricas

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

fricas

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

fricas

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

fricas

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

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

Type: Void

fricas

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

Type: Void

fricas

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

fricas

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)

fricas

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)

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))

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

(5)

Type: Matrix(Expression(Integer))

Scalar product

fricas

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

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)

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

(6)

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))

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

(7)

Type: Matrix(Expression(Integer))

Massive Objects

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

fricas

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

Type: Matrix(Expression(Integer))

fricas

dot(P,P)

$\label{eq8}1$

(8)

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]

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

(9)

Type: Matrix(Expression(Integer))

fricas

dot(S,S)

$\label{eq10}1$

(10)

Type: Expression(Integer)

fricas

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

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

(11)

Type: Matrix(Expression(Integer))

fricas

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))

fricas

simplify dot(U,U)

$\label{eq13}1$

(13)

Type: Expression(Integer)

fricas

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))

fricas

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))

fricas

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

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

(15)

Type: Expression(Integer)

fricas

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))

fricas

Simplify dot(U,W)

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

(17)

Type: Expression(Integer)

Observer "at rest"

fricas

vect [1,0,0,0]

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

(18)

Type: Matrix(Expression(Integer))

fricas

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

fricas

ω(P,Q)==-P/dot(P,Q)-Q

Type: Void

fricas

ω(P,Q)

fricas

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

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

(20)

Type: Matrix(Expression(Integer))

fricas

ω(S,T)

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

(21)

Type: Matrix(Expression(Integer))

Idempotent Observers

fricas

PP:=tensor(P,P)

$\label{eq22}\left[ \begin{array}{cccc} {{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}&{p 1 \ {\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}}&{p 2 \ {\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}}&{p 3 \ {\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}} \ -{p 1 \ {\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}}& -{{p 1}^{2}}& -{p 1 \ p 2}& -{p 1 \ p 3} \ -{p 2 \ {\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}}& -{p 1 \ p 2}& -{{p 2}^{2}}& -{p 2 \ p 3} \ -{p 3 \ {\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}}& -{p 1 \ p 3}& -{p 2 \ p 3}& -{{p 3}^{2}}$

(22)

Type: Matrix(Expression(Integer))

fricas

Is?(PP*PP=PP)

fricas

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

$\label{eq23} \mbox{\rm true}$

(23)

Type: Boolean