MathAction SandBox Categorical Relativity

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  - SandBox Categorical Relativity <-- You are here.

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)

axiom

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

axiom

vect [a1,a2,a3]

axiom

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

axiom

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

axiom

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

axiom

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

axiom

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

Type: Void

or a scalar

axiom

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.

axiom

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

Type: Void

axiom

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

axiom

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 $\sqrt{n}$ .

axiom

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

axiom

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

axiom

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

Type: Matrix(Expression(Integer))

axiom

g(P,P)

axiom

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

$\label{eq5}- 1$

(5)

Type: Expression(Integer)

axiom

Q:=vect [sqrt(q1^2+q2^2+q3^2+1),q1,q2,q3];

Type: Matrix(Expression(Integer))

axiom

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

axiom

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

Type: Void

Lorentz factor

axiom

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

Type: Void

Binary Boost

axiom

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

Type: Void

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

axiom

u:=w(P,Q);

axiom

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

Type: Matrix(Expression(Integer))

axiom

g(P,u)

$\label{eq7}0$

(7)

Type: Expression(Integer)

axiom

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

axiom

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

$\label{eq8}0.99999991406714860312$

(8)

Type: Expression(Float)

axiom

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

axiom

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

axiom

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

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

(9)

Type: Boolean

u is velocity of object Q

axiom

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

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

(10)

Type: Boolean

Observer Q is u-boost of P

axiom

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

axiom

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

axiom

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

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

(11)

Type: Boolean

Inverse velocity is measured by Q

axiom

u' := w(Q,P);

Type: Matrix(Expression(Integer))

axiom

g(Q,u')

$\label{eq12}0$

(12)

Type: Expression(Integer)

Inverse velocity is not reciprocal

axiom

IsPossible2?(-u=u')

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

(13)

Type: Boolean

Object P is u'-boost of Q

axiom

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'

axiom

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

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

(15)

Type: Boolean

axiom

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

axiom

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':

$\digraph[scale=0.75]{CategoricalRelativity1}{rankdir=LR; P->Q [label="u"]; Q->L [label="v"]; Q->P [label="u'"];}$

axiom

v := alpha*u';

Type: Matrix(Expression(Integer))

axiom

L := b(Q,v);

Type: Matrix(Expression(Integer))

axiom

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

axiom

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)

axiom

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

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

(19)

Type: Boolean

General addition of relative velocities (Oziewicz)

axiom

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

Type: Void

axiom

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

axiom

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

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

(20)

Type: Boolean

axiom

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

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

(21)

Type: Boolean

Associativity

Unlike Einstein addition of velocities, addition of relative velocities is associative:

$\digraph[scale=0.75]{CategoricalRelativity2}{rankdir=LR; P->Q [label="u"]; Q->R [label="v"]; R->S [label="w"]; P->R [label="v + u"]; Q->S [label="w + v"]; P->S [label="(w+v)+u=w+(v+u)"]; }$

axiom

R:=vect [sqrt(r1^2+r2^2+r3^2+1),r1,r2,r3];

Type: Matrix(Expression(Integer))

axiom

g(R,R)

$\label{eq22}- 1$

(22)

Type: Expression(Integer)

axiom

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

Type: Matrix(Expression(Integer))

axiom

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).

axiom

--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)))

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

(24)

Type: Boolean

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)))
0

$\label{eq25}0$

(25)

Type: NonNegativeInteger

Subject: Be Bold !!

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