Einstein Velocity and Non-associative Addition ## Reference-
**Matolcsi 4.2.3 op. cit.** - In contradistinction to the non-relativistic case and to our habitual "evidence", the relative velocity of P with respect to Q is not the opposite of the relative velocity of Q with respect to P ... It is worth emphasizing this fact because in most textbooks one takes it for granted that w(P,Q) and -w(Q,P) are equal: "If an observer moves with velocity v relative to another then the second observer moves with velocity -v relative to the first one.". Nevertheless, no harm comes because vectors are given there by their components with respect to some convenient bases and then the components of w(P,Q) and w(Q,P) become opposite to each other.
## Minkowski PackageThis code extends the Minkowski package defined on the previous page SandBox Lorentz Transformations aldor #include "axiom.as" #pile aldor Compiling FriCAS source code from file /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/mink2.as using AXIOM-XL compiler and options -O -Fasy -Fao -Flsp -laxiom -Mno-ALDOR_W_WillObsolete -DAxiom -Y $AXIOM/algebra -I $AXIOM/algebra Use the system command )set compiler args to change these options. #1 (Warning) Could not use archive file `libaxiom.al'. #2 (Warning) Could not use archive file `libaxiom.al'. "/usr/local/aldor/linux/1.1.0/include/axiom.as", fricas )lib mink1 fricas 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 fricas )lib mink2 fricas Reading /var/aw/var/LatexWiki/mink2.asy minkowski2 is now explicitly exposed in frame initial minkowski2 will be automatically loaded when needed from /var/aw/var/LatexWiki/mink2 Objects and velocities fricas P:=obs(); Choose three arbitrary basis vectors in (i.e such that ) fricas e:=matrix [[e1,
Type: Matrix(Polynomial(Integer))fricas e:=eval(e, Express the coefficients of relative velocity u in this basis fricas u!:=matrix [[u1],
Type: Matrix(Polynomial(Integer))fricas u!:=eval(u!, The Lorentz transformation transforms the basis to in such a way that the components of the inverse velocity with respect to basis are just minus (-) the components of the velocity with respect to the basis . fricas e' := L(M(P, ## Composition of collinear velocities (again)For velocity v collinear with reciprocal velocity u' we can use Einstein's original formula. Reference: Matolcsi 4.3.4. op. cit. See SandBoxCategoricalRelativity. Find object L such that v=w(Q,L) where fricas v := alpha*u'; Type: Polynomial(Integer)fricas L := b(Q, First express the coefficients of the composite relative velocity w(P,L) and the relative velocity v=w(Q,L) in the bases and fricas w!:=matrix [[w1], Type: Matrix(Polynomial(Integer))fricas w!:=eval(w!, Type: Matrix(Polynomial(Integer))fricas v!:=eval(v!, Then in Einstein's equation we add u! and v! even though they have different bases: fricas abs(u)==sqrt((transpose(u)*u).(1, Type: Voidfricas Is?( w!=(u!+v!)/(1+abs(u!)*abs(v!)) ) fricas Compiling function abs with type Matrix(Polynomial(Integer)) -> Expression(Integer) |