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SandBox Categorical Relativity

Edit detail for SandBox Categorical Relativity revision 8 of 11

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Editor: Bill Page Time: 2013/09/18 18:12:57 GMT+0
Note: fix graph formatting

changed:
-\begin{latex}
-\digraph[scale=0.9]{CategoricalRelativity1}{rankdir=LR; P->Q [label="u"];
\begin{latex}.
\digraph[scale=0.5]{CategoricalRelativity1}{rankdir=LR; P->Q [label="u"];

changed:
-General addition of relative velocities (Oziewicz)
-\begin{axiom}
-addition(v,u,u') == ( u + v/gamma(u) - g(v,u')/g(u,u)*(u + u'/gamma(u)) ) / (1-g(v,u'))
-\end{axiom}
-
-\begin{axiom}
-Is2?(w(P,L)=addition(v,u,u'))
-IsPossible2?(w(P,L)=addition(w(Q,L),w(P,Q),w(Q,P)))
-\end{axiom}
General composition of relative velocities (Oziewicz)
\begin{axiom}
compose(v,u,u') == ( u + v/gamma(u) - g(v,u')/g(u,u)*(u + u'/gamma(u)) ) / (1-g(v,u'))
\end{axiom}

\begin{axiom}
Is2?(w(P,L)=compose(v,u,u'))
IsPossible2?(w(P,L)=compose(w(Q,L),w(P,Q),w(Q,P)))
\end{axiom}

changed:
-  Unlike Einstein addition of velocities, addition of relative velocities
  Unlike Einstein addition of velocities, composition (denoted $\oplus$) of relative velocities

changed:
-\begin{latex}
\begin{latex}.
\psfrag{alpha}[cc][cc]{$v \oplus u$}
\psfrag{beta}[cc][cc]{$w \oplus v$}
\psfrag{gamma}[cc][cc]{$(w\oplus v)+u=w\oplus (v\oplus u)$}

changed:
-  Q->R [label="v"]; R->S [label="w"]; P->R [label="v + u"];  Q->S [label="w + v"];
-  P->S [label="(w+v)+u=w+(v+u)"]; }
  Q->R [label="v"]; R->S [label="w"]; P->R [label="alpha"];  Q->S [label="beta"];
  P->S [label="gamma"]; }

changed:
---IsPossible2?(w(P,R)=addition(w(Q,R),w(P,Q),w(Q,P)))
---IsPossible2?(w(R,P)=addition(w(Q,P),w(R,Q),w(Q,R)))
-IsPossible2?(w(Q,S)=addition(w(R,S),w(Q,R),w(R,Q)))
-\end{axiom}
-
-\begin{axiom}
---IsPossible2?(w(P,S)=addition(w(Q,S),w(P,Q),w(Q,P)))
---IsPossible2?(w(P,S)=addition(w(R,S),w(P,R),w(R,P)))
--IsPossible2?(w(P,R)=compose(w(Q,R),w(P,Q),w(Q,P)))
--IsPossible2?(w(R,P)=compose(w(Q,P),w(R,Q),w(Q,R)))
IsPossible2?(w(Q,S)=compose(w(R,S),w(Q,R),w(R,Q)))
\end{axiom}

\begin{axiom}
--IsPossible2?(w(P,S)=compose(w(Q,S),w(P,Q),w(Q,P)))
--IsPossible2?(w(P,S)=compose(w(R,S),w(P,R),w(R,P)))

Special relativity without Lorentz transformations.

Here are some sample computations based on papers by Z. Oziewicz

Relativity grupoid instead of relativity group
How do you add relative velocities?
What Is Categorical Relativity?
In categorical relativity the inverse relative velocity morphism $v^{-1}$ is observer dependent, and not absolute as in the usual formulation where $v^{-1} = -v$ .

and the book by T. Matolcsi

Space-time without reference frames

See also the slides: SandBoxRelativeVelocity? (presented at IARD 2006).

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 [a1,a2,a3]

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

$\label{eq1}\left[ \begin{array}{c} a 1 \ a 2 \ a 3$

(1)

Type: Matrix(Expression(Integer))

Then a row vector is

fricas

transpose(vect [a1,a2,a3])

$\label{eq2}\left[ \begin{array}{ccc} a 1 & a 2 & a 3$

(2)

Type: Matrix(Expression(Integer))

Inner product is

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transpose(vect [a1,a2,a3])*vect [b1,b2,b3]

$\label{eq3}\left[ \begin{array}{c} {{a 3 \ b 3}+{a 2 \ b 2}+{a 1 \ b 1}}$

(3)

Type: Matrix(Expression(Integer))

and tensor product is

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vect [a1,a2,a3]*transpose(vect [b1,b2,b3])

$\label{eq4}\left[ \begin{array}{ccc} {a 1 \ b 1}&{a 1 \ b 2}&{a 1 \ b 3} \ {a 2 \ b 1}&{a 2 \ b 2}&{a 2 \ b 3} \ {a 3 \ b 1}&{a 3 \ b 2}&{a 3 \ b 3}$

(4)

Type: Matrix(Expression(Integer))

Applying the Lorentz form produces a row vector

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g(x)==transpose(x)*diagonalMatrix [-1,1,1,1]

Type: Void

or a scalar

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g(x,y)== (transpose(x)*diagonalMatrix([-1,1,1,1])*y)::EXPR INT

Type: Void

For difficult verifications it is sometimes convenient to replace symbols by random numerical values.

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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 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

The AlgebraicNumber? domain can test for numerical equality of complicated expressions involving $\sqrt{n}$ .

fricas

IsPossible?(eq:Equation EXPR INT):Boolean == _
  (possible(lhs(eq)-rhs(eq)) :: Expression AlgebraicNumber=0)::Boolean

   Function declaration IsPossible? : Equation(Expression(Integer)) -> 
      Boolean has been added to workspace.

Type: Void

fricas

IsPossible2?(eq:Equation(Matrix(EXPR(INT)))):Boolean == _
  ( map(possible,(lhs(eq)-rhs(eq))) :: Matrix Expression AlgebraicNumber = _
zero(nrows(lhs(eq)),ncols(lhs(eq)))$Matrix Expression AlgebraicNumber )::Boolean

   Function declaration IsPossible2? : 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

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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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g(P,P)

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

$\label{eq5}- 1$

(5)

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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g(Q,Q)

$\label{eq6}- 1$

(6)

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

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w(P,Q)==-Q/g(P,Q)-P

Type: Void

Lorentz factor

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gamma(v)==1/sqrt(1-g(v,v))

Type: Void

Binary Boost

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b(P,v)==gamma(v)*(P+v)

Type: Void

Observer P measures velocity u. u is space-like and in .

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u:=w(P,Q);

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

Type: Matrix(Expression(Integer))

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g(P,u)

$\label{eq7}0$

(7)

Type: Expression(Integer)

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possible(g(u,u))::EXPR Float

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

$\label{eq8}0.9999857413_817665124$

(8)

Type: Expression(Float)

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IsPossible?(gamma(u)=-g(P,Q))

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

fricas

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

$\label{eq9} \mbox{\rm true}$

(9)

Type: Boolean

u is velocity of object Q

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IsPossible?(g(Q,u)=gamma(u)-1/gamma(u))

$\label{eq10} \mbox{\rm true}$

(10)

Type: Boolean

Observer Q is u-boost of P

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IsPossible2?(Q=b(P,u))

fricas

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

fricas

Compiling function IsPossible2? with type Equation(Matrix(Expression
      (Integer))) -> Boolean

$\label{eq11} \mbox{\rm true}$

(11)

Type: Boolean

Inverse velocity is measured by Q

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u' := w(Q,P);

Type: Matrix(Expression(Integer))

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g(Q,u')

$\label{eq12}0$

(12)

Type: Expression(Integer)

Inverse velocity is not reciprocal

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IsPossible2?(-u=u')

$\label{eq13} \mbox{\rm false}$

(13)

Type: Boolean

Object P is u'-boost of Q

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IsPossible2?(P=b(Q,u'))

$\label{eq14} \mbox{\rm true}$

(14)

Type: Boolean

Objects P and Q are completely determined by velocities u and u'

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IsPossible2?(P = -1/g(u,u)*(u+u'/gamma(u)))

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

(15)

Type: Boolean

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IsPossible2?(Q = -1/g(u,u)*(u'+u/gamma(u)))

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

(16)

Type: Boolean

The magnitude of the inverse velocity is the same as the velocity

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IsPossible?(g(u,u)=g(u',u'))

$\label{eq17} \mbox{\rm true}$

(17)

Type: Boolean

Collinear Velocities

Suppose the velocity v of some object L is collinear with reciprocal velocity u':

$. \digraph[scale=0.5]{CategoricalRelativity1}{rankdir=LR; P->Q [label="u"]; Q->L [label="v"]; Q->P [label="u'"];}$

fricas

v := alpha*u';

Type: Matrix(Expression(Integer))

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L := b(Q,v);

Type: Matrix(Expression(Integer))

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Is2?(v=w(Q,L))

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Compiling function Is2? with type Equation(Matrix(Expression(Integer
      ))) -> Boolean

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

(18)

Type: Boolean

Composition of collinear velocities

For velocity v collinear with reciprocal velocity u' we have Matolcsi (4.3.3)

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Is2?(w(P,L)=(u-alpha*u)/(1-alpha*g(u,u)))

$\label{eq19} \mbox{\rm true}$

(19)

Type: Boolean

General composition of relative velocities (Oziewicz)

fricas

compose(v,u,u') == ( u + v/gamma(u) - g(v,u')/g(u,u)*(u + u'/gamma(u)) ) / (1-g(v,u'))

Type: Void

fricas

Is2?(w(P,L)=compose(v,u,u'))

fricas

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

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

(20)

Type: Boolean

fricas

IsPossible2?(w(P,L)=compose(w(Q,L),w(P,Q),w(Q,P)))

$\label{eq21} \mbox{\rm true}$

(21)

Type: Boolean

Associativity

Unlike Einstein addition of velocities, composition (denoted $\oplus$ ) of relative velocities is associative:

$. \psfrag{alpha}[cc][cc]{$v \oplus u$} \psfrag{beta}[cc][cc]{$w \oplus v$} \psfrag{gamma}[cc][cc]{$(w\oplus v)+u=w\oplus (v\oplus u)$} \digraph[scale=0.9]{CategoricalRelativity2}{rankdir=LR; P->Q [label="u"]; Q->R [label="v"]; R->S [label="w"]; P->R [label="alpha"]; Q->S [label="beta"]; P->S [label="gamma"]; }$

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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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g(R,R)

$\label{eq22}- 1$

(22)

Type: Expression(Integer)

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S:=vect [sqrt(s1^2+s2^2+s3^2+1),s1,s2,s3];

Type: Matrix(Expression(Integer))

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g(S,S)

$\label{eq23}- 1$

(23)

Type: Expression(Integer)

Unfortunately Axiom is not able to evaluate all of these in a reasonable amount of time (within the 1 minute wiki limit).

fricas

--IsPossible2?(w(P,R)=compose(w(Q,R),w(P,Q),w(Q,P)))
--IsPossible2?(w(R,P)=compose(w(Q,P),w(R,Q),w(Q,R)))
IsPossible2?(w(Q,S)=compose(w(R,S),w(Q,R),w(R,Q)))

$\label{eq24} \mbox{\rm true}$

(24)

Type: Boolean

fricas

--IsPossible2?(w(P,S)=compose(w(Q,S),w(P,Q),w(Q,P)))
--IsPossible2?(w(P,S)=compose(w(R,S),w(P,R),w(R,P)))
0

$\label{eq25}0$

(25)

Type: NonNegativeInteger?