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Edit detail for SandBox Idempotent Observers revision 11 of 29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
Editor: Bill Page
Time: 2013/10/14 16:38:21 GMT+0
Note: tidy

changed:
-Mathematical Preliminaries
Preliminaries

changed:
-A vector is represented as a $n\times 1$ matrix (column vector)
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.

added:
ID:=diagonalMatrix [1,1,1,1];
G:=diagonalMatrix [-1,1,1,1]

changed:
-vect [a0,a1,a2,a3]
g(x:Matrix Scalar):Matrix Scalar == transpose(x)*G
dot(x:Matrix Scalar,y:Matrix Scalar):Scalar == g(x)*y 
tensor(x:Matrix Scalar,y:Matrix Scalar):Matrix Scalar == x*g(y)

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

changed:
-Lorentz Form (metric)
Massive Objects

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

changed:
-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];
P:=vect [p0,p1,p2,p3]

changed:
-Q:=vect [sqrt(q1^2+q2^2+q3^2+1),-q1,-q2,-q3];
solve(%=-1,p0)
Q:=vect [q0,q1,q2,q3];

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

changed:
-V:=vect [cosh(v),sinh(v),0,0]
-Simplify dot(U,V)
-W:=vect [cosh(w),0,sinh(w),0]
-Simplify dot(U,W)
V:=vect [cosh(v),sinh(v),0,0];

changed:
-Is?(PP*PP=PP)
-trace(PP)
-QQ:=tensor(-Q,Q)
-is?(trace(PP*QQ)=dot(P,Q)^2)
-Is?(PP*QQ*PP = trace(PP*QQ)@Scalar * PP)
QQ:=tensor(-Q,Q);

changed:
-TT:=map(Simplify,tensor(-T,T))
Is?(SS*SS=SS)
trace(SS)
TT:=map(Simplify,tensor(-T,T));
is?(trace(SS*TT)=dot(S,T)^2)
Is?(SS*TT*SS = dot(S,T)^2 * SS)

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

changed:
-n:=map(Simplify,PP*QQ+QQ*PP-PP-QQ);
PP*QQ+QQ*PP-PP-QQ;
2/trace(%)*%;
trace %
n:=map(Simplify,SS*TT+TT*SS-SS-TT);

changed:
-Is?(PP*N=PP)
-Is?(QQ*N=QQ)
-Is?(N*PP=PP)
-Is?(N*QQ=QQ)
Is?(SS*N=SS)
Is?(TT*N=TT)
Is?(N*SS=SS)
Is?(N*TT=TT)

removed:
---SS*TT+TT*SS-SS-TT;
---2/trace(%)*%

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

changed:
-map(x+->(sqrt(p1^2+p2^2+p3^2+1)*numer(x))/(sqrt(p1^2+p2^2+p3^2+1)*denom(x)),1/trace(%)*%)
1/trace(%)*%

Preliminaries

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:=(PP*QQ+QQ*PP)/dot(P,Q)-PP-QQ;
Type: Matrix(Expression(Integer))
fricas
M:=1/trace(m)*m

\label{eq37}\left[ 
\begin{array}{cccc}
{{-{{q 0}^{2}}-{2 \  p 0 \  q 0}-{{p 0}^{2}}}\over{{{q 3}^{2}}+{2 \  p 3 \  q 3}+{{q 2}^{2}}+{2 \  p 2 \  q 2}+{{q 1}^{2}}+{2 \  p 1 \  q 1}-{{q 0}^{2}}-{2 \  p 0 \  q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}}}&{{{{\left(q 0 + p 0 \right)}\  q 1}+{p 1 \  q 0}+{p 0 \  p 1}}\over{{{q 3}^{2}}+{2 \  p 3 \  q 3}+{{q 2}^{2}}+{2 \  p 2 \  q 2}+{{q 1}^{2}}+{2 \  p 1 \  q 1}-{{q 0}^{2}}-{2 \  p 0 \  q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}}}&{{{{\left(q 0 + p 0 \right)}\  q 2}+{p 2 \  q 0}+{p 0 \  p 2}}\over{{{q 3}^{2}}+{2 \  p 3 \  q 3}+{{q 2}^{2}}+{2 \  p 2 \  q 2}+{{q 1}^{2}}+{2 \  p 1 \  q 1}-{{q 0}^{2}}-{2 \  p 0 \  q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}}}&{{{{\left(q 0 + p 0 \right)}\  q 3}+{p 3 \  q 0}+{p 0 \  p 3}}\over{{{q 3}^{2}}+{2 \  p 3 \  q 3}+{{q 2}^{2}}+{2 \  p 2 \  q 2}+{{q 1}^{2}}+{2 \  p 1 \  q 1}-{{q 0}^{2}}-{2 \  p 0 \  q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}}}
\
{{{{\left(- q 0 - p 0 \right)}\  q 1}-{p 1 \  q 0}-{p 0 \  p 1}}\over{{{q 3}^{2}}+{2 \  p 3 \  q 3}+{{q 2}^{2}}+{2 \  p 2 \  q 2}+{{q 1}^{2}}+{2 \  p 1 \  q 1}-{{q 0}^{2}}-{2 \  p 0 \  q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}}}&{{{{q 1}^{2}}+{2 \  p 1 \  q 1}+{{p 1}^{2}}}\over{{{q 3}^{2}}+{2 \  p 3 \  q 3}+{{q 2}^{2}}+{2 \  p 2 \  q 2}+{{q 1}^{2}}+{2 \  p 1 \  q 1}-{{q 0}^{2}}-{2 \  p 0 \  q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}}}&{{{{\left(q 1 + p 1 \right)}\  q 2}+{p 2 \  q 1}+{p 1 \  p 2}}\over{{{q 3}^{2}}+{2 \  p 3 \  q 3}+{{q 2}^{2}}+{2 \  p 2 \  q 2}+{{q 1}^{2}}+{2 \  p 1 \  q 1}-{{q 0}^{2}}-{2 \  p 0 \  q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}}}&{{{{\left(q 1 + p 1 \right)}\  q 3}+{p 3 \  q 1}+{p 1 \  p 3}}\over{{{q 3}^{2}}+{2 \  p 3 \  q 3}+{{q 2}^{2}}+{2 \  p 2 \  q 2}+{{q 1}^{2}}+{2 \  p 1 \  q 1}-{{q 0}^{2}}-{2 \  p 0 \  q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}}}
\
{{{{\left(- q 0 - p 0 \right)}\  q 2}-{p 2 \  q 0}-{p 0 \  p 2}}\over{{{q 3}^{2}}+{2 \  p 3 \  q 3}+{{q 2}^{2}}+{2 \  p 2 \  q 2}+{{q 1}^{2}}+{2 \  p 1 \  q 1}-{{q 0}^{2}}-{2 \  p 0 \  q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}}}&{{{{\left(q 1 + p 1 \right)}\  q 2}+{p 2 \  q 1}+{p 1 \  p 2}}\over{{{q 3}^{2}}+{2 \  p 3 \  q 3}+{{q 2}^{2}}+{2 \  p 2 \  q 2}+{{q 1}^{2}}+{2 \  p 1 \  q 1}-{{q 0}^{2}}-{2 \  p 0 \  q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}}}&{{{{q 2}^{2}}+{2 \  p 2 \  q 2}+{{p 2}^{2}}}\over{{{q 3}^{2}}+{2 \  p 3 \  q 3}+{{q 2}^{2}}+{2 \  p 2 \  q 2}+{{q 1}^{2}}+{2 \  p 1 \  q 1}-{{q 0}^{2}}-{2 \  p 0 \  q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}}}&{{{{\left(q 2 + p 2 \right)}\  q 3}+{p 3 \  q 2}+{p 2 \  p 3}}\over{{{q 3}^{2}}+{2 \  p 3 \  q 3}+{{q 2}^{2}}+{2 \  p 2 \  q 2}+{{q 1}^{2}}+{2 \  p 1 \  q 1}-{{q 0}^{2}}-{2 \  p 0 \  q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}}}
\
{{{{\left(- q 0 - p 0 \right)}\  q 3}-{p 3 \  q 0}-{p 0 \  p 3}}\over{{{q 3}^{2}}+{2 \  p 3 \  q 3}+{{q 2}^{2}}+{2 \  p 2 \  q 2}+{{q 1}^{2}}+{2 \  p 1 \  q 1}-{{q 0}^{2}}-{2 \  p 0 \  q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}}}&{{{{\left(q 1 + p 1 \right)}\  q 3}+{p 3 \  q 1}+{p 1 \  p 3}}\over{{{q 3}^{2}}+{2 \  p 3 \  q 3}+{{q 2}^{2}}+{2 \  p 2 \  q 2}+{{q 1}^{2}}+{2 \  p 1 \  q 1}-{{q 0}^{2}}-{2 \  p 0 \  q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}}}&{{{{\left(q 2 + p 2 \right)}\  q 3}+{p 3 \  q 2}+{p 2 \  p 3}}\over{{{q 3}^{2}}+{2 \  p 3 \  q 3}+{{q 2}^{2}}+{2 \  p 2 \  q 2}+{{q 1}^{2}}+{2 \  p 1 \  q 1}-{{q 0}^{2}}-{2 \  p 0 \  q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}}}&{{{{q 3}^{2}}+{2 \  p 3 \  q 3}+{{p 3}^{2}}}\over{{{q 3}^{2}}+{2 \  p 3 \  q 3}+{{q 2}^{2}}+{2 \  p 2 \  q 2}+{{q 1}^{2}}+{2 \  p 1 \  q 1}-{{q 0}^{2}}-{2 \  p 0 \  q 0}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}-{{p 0}^{2}}}}
(37)
Type: Matrix(Expression(Integer))
fricas
trace M

\label{eq38}1(38)
Type: Expression(Integer)
fricas
Is?(M*M=M)

\label{eq39} \mbox{\rm true} (39)
Type: Boolean

fricas
(PP*RR+RR*PP)/dot(P,R)-PP-RR;
Type: Matrix(Expression(Integer))
fricas
1/trace(%)*%

\label{eq40}\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}}
(40)
Type: Matrix(Expression(Integer))
fricas
trace %

\label{eq41}1(41)
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{eq42}\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 
(42)
Type: Matrix(Expression(Integer))