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Edit detail for SandBox Categorical Relativity revision 4 of 11

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Editor: Bill Page
Time: 2009/08/04 16:05:42 GMT-7
Note: fix links


        

Special relativity without Lorentz transformations.

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

and the book by T. Matolcsi

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

Mathematical Preliminaries

A vector is represented as a nx1 matrix (column vector) LatexWiki Image Then a row vector is LatexWiki Image Inner product is LatexWiki Image and tensor product is LatexWiki Image

Applying the Lorentz form produces a row vector LatexWiki Image or a scalar LatexWiki Image For difficult verifications it is sometimes convenient to replace symbols by random numerical values. LatexWiki Image The AlgebraicNumber domain can test for numerical equality of complicated expressions involving LatexWiki Image. LatexWiki Image

Massive Objects

An object (also referred to as an obserser) is represented by a time-like 4-vector LatexWiki Image Associated with each such vector is the orthogonal 3-d Euclidean subspace LatexWiki Image

Relative Velocity

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

Lorentz factor LatexWiki Image

Binary Boost LatexWiki Image

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

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u is velocity of object Q LatexWiki Image

Observer Q is u-boost of P LatexWiki Image

Inverse velocity is measured by Q LatexWiki Image

Inverse velocity is not reciprocal LatexWiki Image

Object P is u'-boost of Q LatexWiki Image

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

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

Collinear Velocities

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

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Composition of collinear velocities

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

General addition of relative velocities (Oziewicz) LatexWiki Image

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Associativity

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

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

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Some or all expressions may not have rendered properly, because Axiom returned the following error:
Error: export AXIOM=/usr/local/lib/fricas/target/x86_64-unknown-linux; export ALDORROOT=/usr/local/aldor/linux/1.1.0; export PATH=LatexWiki ImagePATH; export HOME=/var/zope2/var/LatexWiki; ulimit -t 120; $AXIOM/bin/AXIOMsys < /var/zope2/var/LatexWiki/3933104534659023437-25px.axm

Checking for foreign routines AXIOM="/usr/local/lib/fricas/target/x86_64-unknown-linux" spad-lib="/usr/local/lib/fricas/target/x86_64-unknown-linux/lib/libspad.so" foreign routines found openServer result -2 FriCAS (AXIOM fork) Computer Algebra System Version: FriCAS 2009-10-10 Timestamp: Tuesday October 13, 2009 at 18:58:11 ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave FriCAS and return to shell. -----------------------------------------------------------------------------

(1) -> (1) -> (1) -> (1) -> (1) -> 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 vect [a1,a2,a3]

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

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Type: Matrix(Expression(Integer)) (3) -> transpose(vect [a1,a2,a3])

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Type: Matrix(Expression(Integer)) (4) -> transpose(vect [a1,a2,a3])*vect [b1,b2,b3]

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Type: Matrix(Expression(Integer)) (5) -> vect [a1,a2,a3]*transpose(vect [b1,b2,b3])

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Type: Matrix(Expression(Integer)) (6) -> g(x)==transpose(x)*diagonalMatrix [-1,1,1,1]

Type: Void (7) -> g(x,y)== (transpose(x)*diagonalMatrix([-1,1,1,1])*y)::EXPR INT

Type: Void (8) -> possible(x)==subst(x, map(y+->(y=(random(100) - random(100))),variables x) )

Type: Void 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 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 (11) -> 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 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 (13) -> P:=vect [sqrt(p1^2+p2^2+p3^2+1),p1,p2,p3];

Type: Matrix(Expression(Integer)) g(P,P)

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

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

Type: Matrix(Expression(Integer)) g(Q,Q)

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Type: Expression(Integer) (17) -> w(P,Q)==-Q/g(P,Q)-P

Type: Void (18) -> gamma(v)==1/sqrt(1-g(v,v))

Type: Void (19) -> b(P,v)==gamma(v)*(P+v)

Type: Void (20) -> u:=w(P,Q);

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

Type: Matrix(Expression(Integer)) g(P,u)

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Type: Expression(Integer) possible(g(u,u))::EXPR Float

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

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Type: Expression(Float) (23) -> IsPossible?(gamma(u)=-g(P,Q))

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

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Type: Boolean (24) -> IsPossible?(g(Q,u)=gamma(u)-1/gamma(u))

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Type: Boolean (25) -> IsPossible2?(Q=b(P,u))

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

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Type: Boolean (26) -> u' := w(Q,P);

Type: Matrix(Expression(Integer)) g(Q,u')

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Type: Expression(Integer) (28) -> IsPossible2?(-u=u')

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Type: Boolean (29) -> IsPossible2?(P=b(Q,u'))

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Type: Boolean (30) -> IsPossible2?(P = -1/g(u,u)*(u+u'/gamma(u)))

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

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Type: Boolean (32) -> IsPossible?(g(u,u)=g(u',u'))

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Type: Boolean (33) -> v := alpha*u';

Type: Matrix(Expression(Integer)) L := b(Q,v);

Type: Matrix(Expression(Integer)) Is2?(v=w(Q,L))

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

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Type: Boolean (36) -> Is2?(w(P,L)=(u-alphau)/(1-alphag(u,u)))

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Type: Boolean (37) -> addition(v,u,u') == ( u + v/gamma(u) - g(v,u')/g(u,u)*(u + u'/gamma(u)) ) / (1-g(v,u'))

Type: Void (38) -> Is2?(w(P,L)=addition(v,u,u'))

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

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Type: Boolean IsPossible2?(w(P,L)=addition(w(Q,L),w(P,Q),w(Q,P)))

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Type: Boolean (40) -> R:=vect [sqrt(r1^2+r2^2+r3^2+1),r1,r2,r3];

Type: Matrix(Expression(Integer)) g(R,R)

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

Type: Matrix(Expression(Integer)) g(S,S)

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Type: Expression(Integer) (44) -> --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)))