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SandBox Idempotent Observers

last edited 7 years ago by Bill Page

Edit detail for SandBox Idempotent Observers revision 13 of 29

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Editor: Bill Page Time: 2013/10/14 17:36:01 GMT+0
Note: tidy

changed:
-m:=map(x+->factor(numer x)/factor(denom x),(PP*QQ+QQ*PP)/dot(P,Q)-PP-QQ)
m:=map(x+->factor(numer x)/factor(denom x),-(PP*QQ+QQ*PP)/dot(P,Q)+PP+QQ)
Is?(m = tensor(-(P+Q),(P+Q)))

Preliminaries

Space-time without reference frames

Lorentz Form (metric) applied to a vector ( $n\times 1$ matrix) produces a co-vector ( $1\times n$ matrix). Scalar and tensor products use matrix multiplication.

fricas

ID:=diagonalMatrix [1,1,1,1];

Type: Matrix(Integer)

fricas

G:=diagonalMatrix [-1,1,1,1]

$\label{eq1}\left[ \begin{array}{cccc} - 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1$

(1)

Type: Matrix(Integer)

fricas

Scalar := Expression Integer

$\label{eq2}\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ })$

(2)

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

g(x:Matrix Scalar):Matrix Scalar == transpose(x)*G

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

Type: Void

fricas

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

   Function declaration dot : (Matrix(Expression(Integer)),Matrix(
      Expression(Integer))) -> Expression(Integer) has been added to 
      workspace.

Type: Void

fricas

tensor(x:Matrix Scalar,y:Matrix Scalar):Matrix Scalar == x*g(y)

   Function declaration tensor : (Matrix(Expression(Integer)),Matrix(
      Expression(Integer))) -> Matrix(Expression(Integer)) has been 
      added to workspace.

Type: Void

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

Massive Objects

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

fricas

P:=vect [p0,p1,p2,p3]

fricas

Compiling function vect with type List(Expression(Integer)) -> 
      Matrix(Expression(Integer))

$\label{eq3}\left[ \begin{array}{c} p 0 \ p 1 \ p 2 \ p 3$

(3)

Type: Matrix(Expression(Integer))

fricas

dot(P,P)

fricas

Compiling function g with type Matrix(Expression(Integer)) -> Matrix
      (Expression(Integer))

fricas

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

$\label{eq4}{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}$

(4)

Type: Expression(Integer)

fricas

solve(%=-1,p0)

$\label{eq5}\left[{p 0 ={\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}}, \:{p 0 = -{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}}\right]$

(5)

Type: List(Equation(Expression(Integer)))

fricas

Q:=vect [q0,q1,q2,q3];

Type: Matrix(Expression(Integer))

fricas

S:=1/sqrt(1-s1^2-s2^2-s3^2)*vect [1,-s1,-s2,-s3]

$\label{eq6}\left[ \begin{array}{c} {1 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}} \ -{s 1 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}} \ -{s 2 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}} \ -{s 3 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}$

(6)

Type: Matrix(Expression(Integer))

fricas

dot(S,S)

$\label{eq7}- 1$

(7)

Type: Expression(Integer)

fricas

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

Type: Matrix(Expression(Integer))

fricas

U:=vect [cosh(u),sinh(u),0,0]

$\label{eq8}\left[ \begin{array}{c} {\cosh \left({u}\right)} \ {\sinh \left({u}\right)} \ 0 \ 0$

(8)

Type: Matrix(Expression(Integer))

fricas

simplify dot(U,U)

$\label{eq9}- 1$

(9)

Type: Expression(Integer)

fricas

V:=vect [cosh(v),sinh(v),0,0];

Type: Matrix(Expression(Integer))

Observer "at rest"

fricas

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

$\label{eq10}\left[ \begin{array}{c} 1 \ 0 \ 0 \ 0$

(10)

Type: Matrix(Expression(Integer))

fricas

dot(R,R)

$\label{eq11}- 1$

(11)

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

map(Simplify, ω(P,Q))

fricas

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

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)

$\label{eq12}\left[ \begin{array}{c} {{-{p 3 \ q 0 \ q 3}-{p 2 \ q 0 \ q 2}-{p 1 \ q 0 \ q 1}+{p 0 \ {{q 0}^{2}}}- p 0}\over{{p 3 \ q 3}+{p 2 \ q 2}+{p 1 \ q 1}-{p 0 \ q 0}}} \ {{-{p 3 \ q 1 \ q 3}-{p 2 \ q 1 \ q 2}-{p 1 \ {{q 1}^{2}}}+{p 0 \ q 0 \ q 1}- p 1}\over{{p 3 \ q 3}+{p 2 \ q 2}+{p 1 \ q 1}-{p 0 \ q 0}}} \ {{-{p 3 \ q 2 \ q 3}-{p 2 \ {{q 2}^{2}}}+{{\left(-{p 1 \ q 1}+{p 0 \ q 0}\right)}\ q 2}- p 2}\over{{p 3 \ q 3}+{p 2 \ q 2}+{p 1 \ q 1}-{p 0 \ q 0}}} \ {{-{p 3 \ {{q 3}^{2}}}+{{\left(-{p 2 \ q 2}-{p 1 \ q 1}+{p 0 \ q 0}\right)}\ q 3}- p 3}\over{{p 3 \ q 3}+{p 2 \ q 2}+{p 1 \ q 1}-{p 0 \ q 0}}}$

(12)

Type: Matrix(Expression(Integer))

fricas

map(Simplify, ω(P,R))

$\label{eq13}\left[ \begin{array}{c} 0 \ {p 1 \over p 0} \ {p 2 \over p 0} \ {p 3 \over p 0}$

(13)

Type: Matrix(Expression(Integer))

fricas

map(Simplify, ω(S,T))

$\label{eq14}\left[ \begin{array}{c} {{{{t 3}^{2}}-{s 3 \ t 3}+{{t 2}^{2}}-{s 2 \ t 2}+{{t 1}^{2}}-{s 1 \ t 1}}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}} \ {{-{s 1 \ {{t 3}^{2}}}+{s 3 \ t 1 \ t 3}-{s 1 \ {{t 2}^{2}}}+{s 2 \ t 1 \ t 2}- t 1 + s 1}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}} \ {{-{s 2 \ {{t 3}^{2}}}+{s 3 \ t 2 \ t 3}+{{\left({s 1 \ t 1}- 1 \right)}\ t 2}-{s 2 \ {{t 1}^{2}}}+ s 2}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}} \ {{{{\left({s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ t 3}-{s 3 \ {{t 2}^{2}}}-{s 3 \ {{t 1}^{2}}}+ s 3}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}$

(14)

Type: Matrix(Expression(Integer))

fricas

map(Simplify, ω(S,R))

$\label{eq15}\left[ \begin{array}{c} 0 \ - s 1 \ - s 2 \ - s 3$

(15)

Type: Matrix(Expression(Integer))

fricas

map(Simplify, ω(U,V))

$\label{eq16}\left[ \begin{array}{c} {{-{\cosh \left({{2 \ v}- u}\right)}+{\cosh \left({u}\right)}}\over{2 \ {\cosh \left({v - u}\right)}}} \ {{-{\sinh \left({{2 \ v}- u}\right)}+{\sinh \left({u}\right)}}\over{2 \ {\cosh \left({v - u}\right)}}} \ 0 \ 0$

(16)

Type: Matrix(Expression(Integer))

Idempotent Observers

fricas

PP:=tensor(-P,P)

fricas

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

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

(17)

Type: Matrix(Expression(Integer))

fricas

QQ:=tensor(-Q,Q);

Type: Matrix(Expression(Integer))

fricas

RR:=tensor(-R,R)

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

(18)

Type: Matrix(Expression(Integer))

fricas

SS:=map(Simplify,tensor(-S,S))

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

(19)

Type: Matrix(Expression(Integer))

fricas

Is?(SS*SS=SS)

fricas

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

$\label{eq20} \mbox{\rm true}$

(20)

Type: Boolean

fricas

trace(SS)

$\label{eq21}1$

(21)

Type: Expression(Integer)

fricas

TT:=map(Simplify,tensor(-T,T));

Type: Matrix(Expression(Integer))

fricas

is?(trace(SS*TT)=dot(S,T)^2)

fricas

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

$\label{eq22} \mbox{\rm true}$

(22)

Type: Boolean

fricas

Is?(SS*TT*SS = dot(S,T)^2 * SS)

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

(23)

Type: Boolean

fricas

UU:=map(Simplify,tensor(-U,U))

$\label{eq24}\left[ \begin{array}{cccc} {{{\cosh \left({2 \ u}\right)}+ 1}\over 2}& -{{\sinh \left({2 \ u}\right)}\over 2}& 0 & 0 \ {{\sinh \left({2 \ u}\right)}\over 2}&{{-{\cosh \left({2 \ u}\right)}+ 1}\over 2}& 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(24)

Type: Matrix(Expression(Integer))

fricas

VV:=map(Simplify,tensor(-V,V));

Type: Matrix(Expression(Integer))

fricas

map(Simplify, UU*VV)

$\label{eq25}\left[ \begin{array}{cccc} {{{\cosh \left({2 \ v}\right)}+{\cosh \left({{2 \ v}-{2 \ u}}\right)}+{\cosh \left({2 \ u}\right)}+ 1}\over 4}&{{-{\sinh \left({2 \ v}\right)}-{\sinh \left({{2 \ v}-{2 \ u}}\right)}-{\sinh \left({2 \ u}\right)}}\over 4}& 0 & 0 \ {{{\sinh \left({2 \ v}\right)}-{\sinh \left({{2 \ v}-{2 \ u}}\right)}+{\sinh \left({2 \ u}\right)}}\over 4}&{{-{\cosh \left({2 \ v}\right)}+{\cosh \left({{2 \ v}-{2 \ u}}\right)}-{\cosh \left({2 \ u}\right)}+ 1}\over 4}& 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(25)

Type: Matrix(Expression(Integer))

Unit

fricas

PP*QQ+QQ*PP-PP-QQ;

Type: Matrix(Expression(Integer))

fricas

2/trace(%)*%;

Type: Matrix(Expression(Integer))

fricas

trace %

$\label{eq26}2$

(26)

Type: Expression(Integer)

fricas

n:=map(Simplify,SS*TT+TT*SS-SS-TT);

Type: Matrix(Expression(Integer))

fricas

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

Type: Matrix(Expression(Integer))

fricas

Simplify trace N

$\label{eq27}2$

(27)

Type: Expression(Integer)

fricas

Is?(N*N=N)

$\label{eq28} \mbox{\rm true}$

(28)

Type: Boolean

fricas

Is?(SS*N=SS)

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

(29)

Type: Boolean

fricas

Is?(TT*N=TT)

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

(30)

Type: Boolean

fricas

Is?(N*SS=SS)

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

(31)

Type: Boolean

fricas

Is?(N*TT=TT)

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

(32)

Type: Boolean

fricas

PP*RR+RR*PP-PP-RR;

Type: Matrix(Expression(Integer))

fricas

2/trace(%)*%

$\label{eq33}\left[ \begin{array}{cccc} {{{2 \ {{p 0}^{2}}}- 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}& 0 & 0 & 0 \ 0 &{{2 \ {{p 1}^{2}}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ p 1 \ p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ p 1 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}} \ 0 &{{2 \ p 1 \ p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ {{p 2}^{2}}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ p 2 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}} \ 0 &{{2 \ p 1 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ p 2 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}&{{2 \ {{p 3}^{2}}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+{{p 0}^{2}}- 1}}$

(33)

Type: Matrix(Expression(Integer))

fricas

SS*RR+RR*SS-SS-RR;

Type: Matrix(Expression(Integer))

fricas

2/trace(%)*%

$\label{eq34}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 &{{{s 1}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{s 1 \ s 2}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{s 1 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}} \ 0 &{{s 1 \ s 2}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{{s 2}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{s 2 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}} \ 0 &{{s 1 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{s 2 \ s 3}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}&{{{s 3}^{2}}\over{{{s 3}^{2}}+{{s 2}^{2}}+{{s 1}^{2}}}}$

(34)

Type: Matrix(Expression(Integer))

fricas

UU*RR+RR*UU-UU-RR;

Type: Matrix(Expression(Integer))

fricas

map(Simplify,2/trace(%)*%)

$\label{eq35}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(35)

Type: Matrix(Expression(Integer))

fricas

UU*VV+VV*UU-UU-VV;

Type: Matrix(Expression(Integer))

fricas

map(Simplify,2/trace(%)*%)

$\label{eq36}\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(36)

Type: Matrix(Expression(Integer))

Momentum

fricas

m:=map(x+->factor(numer x)/factor(denom x),-(PP*QQ+QQ*PP)/dot(P,Q)+PP+QQ)

$\label{eq37}\left[ \begin{array}{cccc} {{\left(q 0 + p 0 \right)}^{2}}& -{{\left(q 0 + p 0 \right)}\ {\left(q 1 + p 1 \right)}}& -{{\left(q 0 + p 0 \right)}\ {\left(q 2 + p 2 \right)}}& -{{\left(q 0 + p 0 \right)}\ {\left(q 3 + p 3 \right)}} \ {{\left(q 0 + p 0 \right)}\ {\left(q 1 + p 1 \right)}}& -{{\left(q 1 + p 1 \right)}^{2}}& -{{\left(q 1 + p 1 \right)}\ {\left(q 2 + p 2 \right)}}& -{{\left(q 1 + p 1 \right)}\ {\left(q 3 + p 3 \right)}} \ {{\left(q 0 + p 0 \right)}\ {\left(q 2 + p 2 \right)}}& -{{\left(q 1 + p 1 \right)}\ {\left(q 2 + p 2 \right)}}& -{{\left(q 2 + p 2 \right)}^{2}}& -{{\left(q 2 + p 2 \right)}\ {\left(q 3 + p 3 \right)}} \ {{\left(q 0 + p 0 \right)}\ {\left(q 3 + p 3 \right)}}& -{{\left(q 1 + p 1 \right)}\ {\left(q 3 + p 3 \right)}}& -{{\left(q 2 + p 2 \right)}\ {\left(q 3 + p 3 \right)}}& -{{\left(q 3 + p 3 \right)}^{2}}$

(37)

Type: Matrix(Fraction(Factored(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer))))))

fricas

Is?(m = tensor(-(P+Q),(P+Q)))

$\label{eq38} \mbox{\rm true}$

(38)

Type: Boolean

fricas

factor trace m

$\label{eq39}\begin{array}{@{}l} \displaystyle -{\left({ \begin{array}{@{}l} \displaystyle {{q 3}^{2}}+{2 \ p 3 \ q 3}+{{q 2}^{2}}+{2 \ p 2 \ q 2}+{{q 1}^{2}}+{2 \ p 1 \ q 1}- \ \ \displaystyle {{q 0}^{2}}-{2 \ p 0 \ q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}$

(39)

Type: Factored(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer))))

fricas

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

Type: Matrix(Expression(Integer))

fricas

trace M

$\label{eq40}1$

(40)

Type: Expression(Integer)

fricas

Is?(M*M=M)

$\label{eq41} \mbox{\rm true}$

(41)

Type: Boolean

fricas

(PP*RR+RR*PP)/dot(P,R)-PP-RR;

Type: Matrix(Expression(Integer))

fricas

1/trace(%)*%

$\label{eq42}\left[ \begin{array}{cccc} {{-{{p 0}^{2}}-{2 \ p 0}- 1}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{{\left(p 0 + 1 \right)}\ p 1}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{{\left(p 0 + 1 \right)}\ p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{{\left(p 0 + 1 \right)}\ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}} \ {{{\left(- p 0 - 1 \right)}\ p 1}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{{p 1}^{2}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{p 1 \ p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{p 1 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}} \ {{{\left(- p 0 - 1 \right)}\ p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{p 1 \ p 2}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{{p 2}^{2}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{p 2 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}} \ {{{\left(- p 0 - 1 \right)}\ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{p 1 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{p 2 \ p 3}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}&{{{p 3}^{2}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}-{2 \ p 0}- 1}}$

(42)

Type: Matrix(Expression(Integer))

fricas

trace %

$\label{eq43}1$

(43)

Type: Expression(Integer)

fricas

--(SS*RR+RR*SS)/dot(S,R)-SS-RR;
--map(Simplify,1/trace(%)*%)
--(UU*VV+VV*UU)/dot(U,V)-UU-VV;
--1/trace(%)*%
(UU*RR+RR*UU)/dot(U,R)-UU-RR;

Type: Matrix(Expression(Integer))

fricas

map(Simplify,1/trace(%)*%)

$\label{eq44}\left[ \begin{array}{cccc} {{{\cosh \left({3 \ u}\right)}+{4 \ {\cosh \left({2 \ u}\right)}}+{7 \ {\cosh \left({u}\right)}}+ 4}\over{{4 \ {\cosh \left({2 \ u}\right)}}+{8 \ {\cosh \left({u}\right)}}+ 4}}&{{-{\sinh \left({3 \ u}\right)}-{2 \ {\sinh \left({2 \ u}\right)}}-{\sinh \left({u}\right)}}\over{{4 \ {\cosh \left({2 \ u}\right)}}+{8 \ {\cosh \left({u}\right)}}+ 4}}& 0 & 0 \ {{{\sinh \left({3 \ u}\right)}+{2 \ {\sinh \left({2 \ u}\right)}}+{\sinh \left({u}\right)}}\over{{4 \ {\cosh \left({2 \ u}\right)}}+{8 \ {\cosh \left({u}\right)}}+ 4}}&{{-{\cosh \left({3 \ u}\right)}+{\cosh \left({u}\right)}}\over{{4 \ {\cosh \left({2 \ u}\right)}}+{8 \ {\cosh \left({u}\right)}}+ 4}}& 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(44)

Type: Matrix(Expression(Integer))