Edit detail for SandBox Categorical Relativity revision 1 of 11

changed:
-
Special relativity without Lorentz transformations.

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

  - "How do you add relative velocities?":http://portal.axiom-developer.org/Members/billpage/catrel/group25.pdf/file_view

  - "What Is Categorical Relativity?":http://portal.axiom-developer.org/Members/billpage/catrel/ladek.pdf/file_view

    *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

  - "Spacetime without reference frames":http://portal.axiom-developer.org/Members/billpage/physics/matolcsi.pdf/file_view

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

Mathematical Preliminaries

  A vector is represented as a nx1 matrix (column vector)
\begin{axiom}
vect(x:List Expression Integer):Matrix Expression Integer == matrix map(y+->[y],x)
vect [a1,a2,a3]
\end{axiom}
Then a row vector is
\begin{axiom}
transpose(vect [a1,a2,a3])
\end{axiom}
Inner product is
\begin{axiom}
transpose(vect [a1,a2,a3])*vect [b1,b2,b3]
\end{axiom}
and tensor product is
\begin{axiom}
vect [a1,a2,a3]*transpose(vect [b1,b2,b3])
\end{axiom}

Applying the Lorentz form produces a row vector
\begin{axiom}
g(x)==transpose(x)*diagonalMatrix [-1,1,1,1]
\end{axiom}
or a scalar
\begin{axiom}
g(x,y)== (transpose(x)*diagonalMatrix([-1,1,1,1])*y)::EXPR INT
\end{axiom}
For difficult verifications it is sometimes convenient to replace
symbols by random numerical values.
\begin{axiom}
possible(x)==subst(x, map(y+->(y=(random(100) - random(100))),variables x) )
Is?(eq:Equation EXPR INT):Boolean == (lhs(eq)-rhs(eq)=0)::Boolean
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

\end{axiom}
The AlgebraicNumber domain can test for numerical equality of complicated
expressions involving $\sqrt{n}$.
\begin{axiom}
IsPossible?(eq:Equation EXPR INT):Boolean == _
  (possible(lhs(eq)-rhs(eq)) :: Expression AlgebraicNumber=0)::Boolean
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
\end{axiom}

Massive Objects

  An object (also referred to as an obserser) is represented by a
time-like 4-vector
\begin{axiom}
P:=vect [sqrt(p1^2+p2^2+p3^2+1),p1,p2,p3];
g(P,P)
Q:=vect [sqrt(q1^2+q2^2+q3^2+1),q1,q2,q3];
g(Q,Q)
\end{axiom}
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
\begin{axiom}
w(P,Q)==-Q/g(P,Q)-P
\end{axiom}

Lorentz factor
\begin{axiom}
gamma(v)==1/sqrt(1-g(v,v))
\end{axiom}

Binary Boost
\begin{axiom}
b(P,v)==gamma(v)*(P+v)
\end{axiom}

Observer P measures velocity u. u is space-like and in $E_P$.
\begin{axiom}
u:=w(P,Q);
g(P,u)
possible(g(u,u))::EXPR Float
\end{axiom}

\begin{axiom}
IsPossible?(gamma(u)=-g(P,Q))
\end{axiom}

u is velocity of object Q
\begin{axiom}
IsPossible?(g(Q,u)=gamma(u)-1/gamma(u))
\end{axiom}

Observer Q is u-boost of P
\begin{axiom}
IsPossible2?(Q=b(P,u))
\end{axiom}

Inverse velocity is measured by Q
\begin{axiom}
u' := w(Q,P);
g(Q,u')
\end{axiom}

Inverse velocity is not reciprocal
\begin{axiom}
IsPossible2?(-u=u')
\end{axiom}

Object P is u'-boost of Q
\begin{axiom}
IsPossible2?(P=b(Q,u'))
\end{axiom}

Objects P and Q are completely determined by velocities u and u'
\begin{axiom}
IsPossible2?(P = -1/g(u,u)*(u+u'/gamma(u)))
IsPossible2?(Q = -1/g(u,u)*(u'+u/gamma(u)))
\end{axiom}

The magnitude of the inverse velocity is the same as the velocity
\begin{axiom}
IsPossible?(g(u,u)=g(u',u'))
\end{axiom}

Collinear Velocities

  Suppose the velocity v of some object L is collinear with reciprocal velocity u':
\begin{latex}
\digraph[scale=0.75]{CategoricalRelativity1}{rankdir=LR; P->Q [label="u"];
  Q->L [label="v"]; Q->P [label="u'"];}
\end{latex}

\begin{axiom}
v := alpha*u';
L := b(Q,v);
Is2?(v=w(Q,L))
\end{axiom}

Composition of collinear velocities

  For velocity v collinear with reciprocal velocity u' we have Matolcsi (4.3.3)
\begin{axiom}
Is2?(w(P,L)=(u-alpha*u)/(1-alpha*g(u,u)))
\end{axiom}

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}

Associativity

  Unlike Einstein addition of velocities, addition of relative velocities
is associative:
\begin{latex}
\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)"]; }
\end{latex}

\begin{axiom}
R:=vect [sqrt(r1^2+r2^2+r3^2+1),r1,r2,r3];
g(R,R)
S:=vect [sqrt(s1^2+s2^2+s3^2+1),s1,s2,s3];
g(S,S)
\end{axiom}

Unfortunately Axiom is not able to evaluate all of these in a reasonable
amount of time (within the 1 minute wiki limit).
\begin{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)))
\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)))
0
\end{axiom}

Special relativity without Lorentz transformations.

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

• How do you add relative velocities?
• What Is Categorical Relativity?

  In categorical relativity the inverse relative velocity morphism is observer dependent, and not absolute as in the usual formulation where .

and the book by T. Matolcsi

• Spacetime 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

(1)

Type: Matrix Expression Integer

Then a row vector is

axiom

transpose(vect [a1,a2,a3]?)

(2)

Type: Matrix Expression Integer

Inner product is

axiom

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

(3)

Type: Matrix Expression Integer

and tensor product is

axiom

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

(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 .

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

(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)

(6)

Type: Expression Integer

Associated with each such vector is the orthogonal 3-d Euclidean subspace

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)

(7)

Type: Expression Integer

axiom

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

axiom

Compiling function possible with type Expression Integer -> 
      Expression Integer

(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

(9)

Type: Boolean

u is velocity of object Q

axiom

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

(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

(11)

Type: Boolean

Inverse velocity is measured by Q

axiom

u' := w(Q,P);

Type: Matrix Expression Integer

axiom

g(Q,u')

(12)

Type: Expression Integer

Inverse velocity is not reciprocal

axiom

IsPossible2??(-u=u')

(13)

Type: Boolean

Object P is u'-boost of Q

axiom

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

(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)))

(15)

Type: Boolean

axiom

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

(16)

Type: Boolean

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

axiom

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

(17)

Type: Boolean

Collinear Velocities

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

(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)))

(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

(20)

Type: Boolean

axiom

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

(21)

Type: Boolean

Associativity

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

axiom

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

Type: Matrix Expression Integer

axiom

g(R,R)

(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)

(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)))

(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

(25)

Type: NonNegativeInteger?