Edit detail for SandBox Lorentz Transformations revision 3 of 3

Lorentz transformations relate one object or observer (represented by a time-like 4-vector) to another object or observer. A Lorentz transformation is not uniquely given by the relative velocity of these objects.

References

Z. Oziewicz, 2005

Special relativity without Lorentz group

Donald Fahnline, AJP, 1982

A covariant four-dimensional expression for Lorentz transformations

Daniel Gottlieb, 1996

Skew Symmetric Bundle Maps on Space-Time

Mathematical Preliminaries (from: SandBoxCategoricalRelativity)

This package implements a vector as a matrix (column vector), a co-vector as a matrix (row vector), inner and outter (tensor) products with Minkowski signature and fast routines for checking equations.

(1) -> <aldor>
#include "axiom.as"
#pile

-- Try RealClosure instead of AlgebraicNumber in order to
-- avoid error message #305 ... but it takes too long.
RAN    ==> AlgebraicNumber
--RAN  ==> RealClosure( Fraction(Integer) )
EQ     ==> Equation
EXPR   ==> Expression
INT    ==> Integer
NNI    ==> NonNegativeInteger
SYMBOL ==> Symbol
LIST   ==> List
MATRIX ==> Matrix

SCALAR ==> Expression Integer

import from
  INT, NNI
  EXPR INT
  EXPR RAN
  EQ EXPR INT
  LIST EXPR INT
  LIST LIST SYMBOL
  ListFunctions2(SCALAR, LIST SCALAR)
  ListFunctions2(SYMBOL, EQ SCALAR)
  ListFunctions2(SYMBOL, EQ EXPR RAN)
  ListFunctions2(SCALAR, LIST SYMBOL)
  ListFunctions2(EXPR RAN, LIST SYMBOL)
  MATRIX SCALAR
  MATRIX EXPR RAN

minkowski1(): with
    +++   Colum vector
    vect:        LIST SCALAR             -> MATRIX SCALAR
      ++ is represented as a nx1 matrix (column vector)
    +++   The Lorentz form 
    g: ()                                -> MATRIX SCALAR
    g:           MATRIX SCALAR           -> MATRIX SCALAR
      ++   applied to vector produces a row vector
    g: (MATRIX SCALAR,MATRIX SCALAR)     -> SCALAR
      ++   as inner product of two vectors produces a scalar
    +++ replace symbols by random numerical values.
    possible:    SCALAR                  -> SCALAR
    possible:    EXPR RAN                -> EXPR RAN
    possible:    MATRIX SCALAR           -> MATRIX SCALAR
    possible:    EQ SCALAR               -> EQ SCALAR
    possible:    EQ MATRIX SCALAR        -> EQ MATRIX SCALAR
    +++ verify equality
    Is?:         EQ SCALAR               -> Boolean
    Is?:         EQ EXPR RAN             -> Boolean
    Is?:         EQ MATRIX EXPR RAN      -> Boolean
    +++ Massive Objects
    obs:         ()                      -> MATRIX SCALAR
    obs?:        MATRIX SCALAR           -> Boolean
    +++   Relative Velocity
    w: (MATRIX SCALAR,MATRIX SCALAR)     -> MATRIX SCALAR
    +++ outter (tensor) product
    /\: (MATRIX SCALAR,MATRIX SCALAR)    ->MATRIX SCALAR
    +++ Lorentz factor
    gamma:       MATRIX SCALAR           -> SCALAR
    +++ binary boost
    b: (MATRIX SCALAR,MATRIX SCALAR)     ->MATRIX SCALAR
    +++ addition of relative velocities
    addition: (MATRIX SCALAR,MATRIX SCALAR,MATRIX SCALAR) -> MATRIX SCALAR
  == add
    --
    -- Local functions
    randumb(y:SYMBOL):EQ SCALAR ==
      coerce(y)$SCALAR = coerce(random(100) - random(100))$SCALAR
    randumb(y:SYMBOL):EQ EXPR RAN ==
      coerce(y)$EXPR(RAN) =  coerce(random(100) - random(100))$EXPR(RAN)
    --
    -- get list of unique variables
    varList(x:MATRIX SCALAR):LIST Symbol ==
      removeDuplicates(reduce(append,map(variables,members(x))))
    varList(x:EQ MATRIX SCALAR):LIST Symbol ==
      removeDuplicates(reduce(append,
         append(map(variables,members(lhs x)),map(variables,members(rhs x)))
      ))
    varList(x:EQ SCALAR):LIST Symbol ==
      removeDuplicates(append(variables lhs x,variables rhs x))
    --
    -- Exported functions
    vect(x:LIST SCALAR):MATRIX SCALAR == matrix( map(list,x) )
    g():MATRIX SCALAR ==
      diagonalMatrix([-1,1,1,1])$MATRIX(SCALAR)
    g(x:Matrix SCALAR):MATRIX SCALAR ==
      transpose(x)*g()
    g(x:MATRIX SCALAR,y:MATRIX SCALAR):SCALAR ==
     (g(x)*y).(1,1)
    --
    --   For difficult verifications it is sometimes convenient to replace
    --   symbols by random numerical values.
    possible(x:SCALAR):SCALAR ==
      eval(x, map(randumb,variables x))
    possible(x:EXPR RAN):EXPR RAN ==
      eval(x, map(randumb,variables x))
    possible(x:MATRIX SCALAR):MATRIX SCALAR ==
      eval(x, map(randumb,varList x))
    --
    -- We must be careful to give each variable the same value
    possible(x:EQ MATRIX SCALAR):EQ MATRIX SCALAR ==
      rlist:LIST EQ SCALAR := map(randumb,varList x)
      eval(lhs x, rlist) = eval(rhs x, rlist)
    possible(x:EQ SCALAR):EQ SCALAR ==
      rlist:List EQ SCALAR := map(randumb,varList x)
      eval(lhs x, rlist) = eval(rhs x, rlist)
    --
    --   To verify equality, the AlgebraicNumber domain can test for
    --   equality of complicated expressions involving $\sqrt{n}$.
    Is?(eq:EQ SCALAR):Boolean ==
      zero?(lhs(eq)-rhs(eq))
    Is?(eq:EQ EXPR RAN):Boolean ==
      zero?(lhs(eq)-rhs(eq))
    Is?(eq:EQ Matrix EXPR RAN):Boolean ==
      every?(zero?,lhs(eq)-rhs(eq))
    --
    -- Fast check: If the result of Is?(possible(...)) is false then the
    --   equality does not hold on the other hand if it returns true, then
    --   it is only probably true.
    obs():MATRIX SCALAR ==
      p1:=new()$SYMBOL::SCALAR;
      p2:=new()$SYMBOL::SCALAR;
      p3:=new()$SYMBOL::SCALAR;
      vect [sqrt(p1^2::NNI+p2^2::NNI+p3^2::NNI+1),p1,p2,p3]
    obs?(P:MATRIX SCALAR):Boolean ==
      Is?( (g(P,P) = -1$SCALAR)$EQ(SCALAR) )
    w(P:MATRIX SCALAR,Q:MATRIX SCALAR):MATRIX SCALAR ==
      -Q/g(P,Q)-P
    (/\)(P:MATRIX SCALAR,K:MATRIX SCALAR):MATRIX SCALAR ==
       P*transpose(K) - K*transpose(P)
    gamma(v:MATRIX SCALAR):SCALAR ==
       1/sqrt(1-g(v,v))
    b(P:MATRIX SCALAR,v:MATRIX SCALAR):MATRIX SCALAR ==
      gamma(v)*(P+v)
    addition(v:MATRIX SCALAR,u:MATRIX SCALAR,uinv:MATRIX SCALAR):MATRIX SCALAR ==
      ( u + v/gamma(u) - g(v,uinv)/g(u,u)*(u + uinv/gamma(u)) ) / (1-g(v,uinv))
        ++ for u=w(A,B), uinv=w(B,A) and v=w(B,C) returns w(A,C)</aldor>

Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/mink1.as 
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      Use the system command )set compiler args to change these 
      options.
"/var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/mink1.as", line 1: 
#include "axiom.as"
^
[L1 C1] #1 (Error) Could not open file `axiom.as'.

   The )library system command was not called after compilation.

)lib mink1

Reading /var/aw/var/LatexWiki/mink1.asy
   minkowski1 is now explicitly exposed in frame initial 
   minkowski1 will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/mink1

)show minkowski1

   >> System error:
   #<SB-SYS:FD-STREAM for "file /var/aw/var/LatexWiki/mink1.fasl" {1001D7DF43}> is
a fasl file compiled with SBCL 1.0.31, and can't be loaded into SBCL
2.2.9.debian.

Objects

P:=obs(); Q:=obs();

   >> System error:
   #<SB-SYS:FD-STREAM for "file /var/aw/var/LatexWiki/mink1.fasl" {1001D97923}> is
a fasl file compiled with SBCL 1.0.31, and can't be loaded into SBCL
2.2.9.debian.

Lorentz boost (Oziewicz)

M is a g-skew-symmetric endomorphism on the vector space

M(P,Q) == P*g(Q) - Q*g(P)

Is?(M(vect [a1,a2,a3,a4], vect [b1,b2,b3,b4]) = _
   (vect [a1,a2,a3,a4] /\ vect [b1,b2,b3,b4])*g())

   >> System error:
   #<SB-SYS:FD-STREAM for "file /var/aw/var/LatexWiki/mink1.fasl" {100219C813}> is
a fasl file compiled with SBCL 1.0.31, and can't be loaded into SBCL
2.2.9.debian.

The Lorentz transformation is an isometry (Oziewicz):

L(b) == 1 + b + b^2/(1+sqrt(1+1/2*trace(b^2)))

It maps P into Q,

Is?(possible( L(M(P,Q)) * P = Q ))

   There are 1 exposed and 0 unexposed library operations named g 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op g
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.
   Cannot find a definition or applicable library operation named g 
      with argument type(s) 
                                 Variable(Q)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are 1 exposed and 0 unexposed library operations named g 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op g
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named g 
      with argument type(s) 
                                 Variable(Q)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

maps observers into observers

S := L(M(P,Q)) * R;

   There are 1 exposed and 0 unexposed library operations named g 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op g
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.
   Cannot find a definition or applicable library operation named g 
      with argument type(s) 
                                 Variable(Q)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are 1 exposed and 0 unexposed library operations named g 
      having 1 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op g
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named g 
      with argument type(s) 
                                 Variable(Q)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

and the relative velocity of w(P,Q) into minus the inverse velocity w(Q,P)

u:=w(P,Q);

   There are 1 exposed and 0 unexposed library operations named w 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op w
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named w 
      with argument type(s) 
                                 Variable(P)
                                 Variable(Q)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

This Lorentz boost can be applied to other objects, for example we can apply the boost L(M(P,Q)) to the object R to obtain S. But in general the relative velocity w(P,Q) is not the same as the relative velocity w(R,S)! Further since u and v belong to separate sub-spaces, and respectively, we can not even directly compare their directions.

v:=w(R,S);

   There are 1 exposed and 0 unexposed library operations named w 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op w
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named w 
      with argument type(s) 
                                 Variable(R)
                                 Variable(S)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

The next command fails in some version of Axiom due to bug #305

Is?(possible( g(u,u) = g(v,v) ))

   There are 1 exposed and 0 unexposed library operations named g 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op g
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named g 
      with argument type(s) 
                                 Variable(u)
                                 Variable(u)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

The results on this page were computed with FriCAS

)version

"FriCAS 1.3.10 compiled at Wed 10 Jan 02:19:45 CET 2024"