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SandBox Idempotent Observers

last edited 7 years ago by Bill Page

Edit detail for SandBox Idempotent Observers revision 6 of 29

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Editor: Bill Page Time: 2013/10/14 03:40:39 GMT+0
Note: Momentum

changed:
-C:=map(Simplify,PP*QQ+QQ*PP-PP-QQ);
-N:=map(Simplify,2/trace(C)*C);
n:=map(Simplify,PP*QQ+QQ*PP-PP-QQ);
N:=map(Simplify,2/trace(n)*n);

added:

Momentum
\begin{axiom}
m:=(PP*QQ+QQ*PP)/dot(P,Q)-PP-QQ;
M:=1/trace(m)*m;
trace M
Is?(M*M=M)
\end{axiom}

Mathematical Preliminaries

Space-time without reference frames

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

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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]

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

(9)

Type: Matrix(Expression(Integer))

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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]

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

(11)

Type: Matrix(Expression(Integer))

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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))

fricas

Simplify dot(U,V)

fricas

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

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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)

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

(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))

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

(20)

Type: Matrix(Expression(Integer))

fricas

ω(S,T)

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

(21)

Type: Matrix(Expression(Integer))

Idempotent Observers

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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)

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

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

(23)

Type: Boolean

fricas

trace(PP)

$\label{eq24}1$

(24)

Type: Expression(Integer)

fricas

QQ:=tensor(-Q,Q)

$\label{eq25}\left[ \begin{array}{cccc} {{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}&{q 1 \ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}&{q 2 \ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}&{q 3 \ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}} \ -{q 1 \ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}& -{{q 1}^{2}}& -{q 1 \ q 2}& -{q 1 \ q 3} \ -{q 2 \ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}& -{q 1 \ q 2}& -{{q 2}^{2}}& -{q 2 \ q 3} \ -{q 3 \ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}& -{q 1 \ q 3}& -{q 2 \ q 3}& -{{q 3}^{2}}$

(25)

Type: Matrix(Expression(Integer))

fricas

is?(trace(PP*QQ)=dot(P,Q)^2)

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

$\label{eq26} \mbox{\rm true}$

(26)

Type: Boolean

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Is?(PP*QQ*PP = trace(PP*QQ)@Scalar * PP)

$\label{eq27} \mbox{\rm true}$

(27)

Type: Boolean

Unit

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n:=map(Simplify,PP*QQ+QQ*PP-PP-QQ);

Type: Matrix(Expression(Integer))

fricas

N:=map(Simplify,2/trace(n)*n);

Type: Matrix(Expression(Integer))

fricas

Simplify trace N

$\label{eq28}2$

(28)

Type: Expression(Integer)

fricas

Is?(N*N=N)

$\label{eq29} \mbox{\rm true}$

(29)

Type: Boolean

fricas

Is?(PP*N=PP)

$\label{eq30} \mbox{\rm true}$

(30)

Type: Boolean

fricas

Is?(QQ*N=QQ)

$\label{eq31} \mbox{\rm true}$

(31)

Type: Boolean

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Is?(N*PP=PP)

$\label{eq32} \mbox{\rm true}$

(32)

Type: Boolean

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Is?(N*QQ=QQ)

$\label{eq33} \mbox{\rm true}$

(33)

Type: Boolean

Momentum

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m:=(PP*QQ+QQ*PP)/dot(P,Q)-PP-QQ;

Type: Matrix(Expression(Integer))

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M:=1/trace(m)*m;

Type: Matrix(Expression(Integer))

fricas

trace M

$\label{eq34}1$

(34)

Type: Expression(Integer)

fricas

Is?(M*M=M)

$\label{eq35} \mbox{\rm true}$

(35)

Type: Boolean