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Lorentz transformations.

References

Mathematical Preliminaries

A vector is represented as a $n\times 1$ matrix (column vector) \begin{axiom} Scalar := Expression Integer vect(x:List Scalar):Matrix Scalar == matrix map(y+->[y],x) vect [a0,a1,a2,a3] \end{axiom}

Identity \begin{axiom} ID:=diagonalMatrix([1,1,1,1]) \end{axiom}

Verification \begin{axiom} htrigs2exp == rule cosh(a) == (exp(a)+exp(-a))/2 sinh(a) == (exp(a)-exp(-a))/2 sinhcosh == rule ?cexp(a)+?cexp(-a) == 2ccosh(a) ?cexp(a)-?cexp(-a) == 2csinh(a) ?cexp(a-b)+?cexp(b-a) == 2ccosh(a-b) ?cexp(a-b)-?cexp(b-a) == 2csinh(a-b) expandhtrigs == rule cosh(:x+y) == sinh(x)sinh(y)+cosh(x)cosh(y) sinh(:x+y) == cosh(x)sinh(y)+sinh(x)cosh(y) cosh(2x) == 2cosh(x)^2-1 sinh(2x) == 2sinh(x)cosh(x) expandhtrigs2 == rule cosh(2x+2y) == 2cosh(x+y)^2-1 sinh(2x+2y) == 2sinh(x+y)cosh(x+y) cosh(2x-2y) == 2cosh(x-y)^2-1 sinh(2x-2y) == 2sinh(x-y)*cosh(x-y) Simplify(x:Scalar):Scalar == htrigs sinhcosh simplify htrigs2exp x possible(x)==subst(x, map(y+->(y=(random(100) - random(100))),variables x) ) is?(eq:Equation Scalar):Boolean == (Simplify(lhs(eq)-rhs(eq))=0)::Boolean Is?(eq:Equation(Matrix(Scalar))):Boolean == (map(Simplify,lhs(eq)-rhs(eq)) :: Matrix Expression AlgebraicNumber = zero(nrows(lhs(eq)),ncols(lhs(eq)))$Matrix Expression AlgebraicNumber )::Boolean \end{axiom}

Lorentz Form (metric) \begin{axiom} G:=diagonalMatrix [-1,1,1,1] \end{axiom}

applied to a vector produces a co-vector (represent as a $1\times n$ matrix or row vector) \begin{axiom} g(x) == transpose(x)*G g(vect [a0,a1,a2,a3]) \end{axiom}

Scalar product \begin{axiom} dot(x,y) == (g(x)*y)::Scalar dot(vect [a0,a1,a2,a3], vect [b0,b1,b2,b3]) \end{axiom}

Tensor product \begin{axiom} tensor(x,y) == x*g(y) tensor(vect [a0,a1,a2,a3], vect [b0,b1,b2,b3]) \end{axiom}

Massive Objects

A material object (also referred to as an observer) is represented by a time-like 4-vector \begin{axiom} P:=vect [sqrt(p1^2+p2^2+p3^2+1),-p1,-p2,-p3]; dot(P,P) Q:=vect [sqrt(q1^2+q2^2+q3^2+1),-q1,-q2,-q3]; R:=vect [sqrt(r1^2+r2^2+r3^2+1),-r1,-r2,-r3]; S:=1/sqrt(1-s1^2-s2^2-s3^2)*vect [1,-s1,-s2,-s3] dot(S,S) T:=1/sqrt(1-t1^2-t2^2-t3^2)*vect [1,-t1,-t2,-t3] U:=vect [cosh(u),sinh(u),0,0] simplify dot(U,U) V:=vect [cosh(v),sinh(v),0,0] Simplify dot(U,V) W:=vect [cosh(w),0,sinh(w),0] Simplify dot(U,W) \end{axiom}

Observer "at rest" \begin{axiom} vect [1,0,0,0] dot(%,%) \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 P has a unique relative velocity ω(P,Q) with respect to object Q given by \begin{axiom} ω(P,Q)==-P/dot(P,Q)-Q ω(P,Q) ω(S,T) \end{axiom}

In two dimensions \begin{axiom} map(x+->Simplify x,ω(U,V)) vect [cosh(u)/cosh(u-v)-cosh(v),sinh(u)/cosh(u-v)-sinh(v),0,0] Is?(% = ω(U,V)) map(x+->Simplify x,ω(U,W)) \end{axiom}

Observer P measures velocity ω(Q,P). ω(Q,P) is space-like \begin{axiom} dot(ω(P,vect [1,0,0,0]),ω(P,vect [1,0,0,0])) \end{axiom}

and in $E_P$ \begin{axiom} dot(P,ω(Q,P)) possible dot(ω(Q,P),ω(Q,P))::EXPR Float dot(Q,ω(P,Q)) possible dot(ω(P,Q),ω(P,Q))::EXPR Float \end{axiom}

Velocity with respect to observer "at rest" \begin{axiom} ω(vect [u0,u1,u2,u3],vect [1,0,0,0]) ω(R,vect [1,0,0,0]) ω(S,vect [1,0,0,0]) map(Simplify, ω(U,vect [1,0,0,0])) \end{axiom}

Non-reciprocal velocities \begin{axiom} ω(vect [1,0,0,0],S) ω(vect [1,0,0,0],R) is?(dot(ω(P,Q),ω(P,Q))=dot(ω(Q,P),ω(Q,P))) \end{axiom}

Lorentz Boost

is a linear bijection $E_Q \leftrightarrow E_P$ that preserves $E_Q \cap E_P$ and maps orthogonal compliments into each other. \begin{axiom} L(P,Q) == ID + tensor(P+Q,P+Q)/(1-dot(P,Q)) - 2tensor(P,Q) Is?(L(P,P) = ID) Is?(L(P,Q)L(Q,P) = ID) Is?(L(P,Q)Q=P) Is?(L(P,Q)ω(P,Q) = -ω(Q,P)) \end{axiom}

Most General Lorentz Boost (Oziewicz, 2006)

is given by three non-coplanar vectors \begin{axiom} B(P,Q,X) == ID - ( tensor( 2X, dot(P-Q,P-Q)X - 2dot(X,P)(P-Q) ) + tensor( _ P-Q, 2dot(X,X)(P-Q)+4dot(X,Q)X ) ) / ( _ dot(X,X)dot(P-Q,P-Q)+4dot(X,P)dot(X,Q) _ ) Is?(B(P,P,R) = ID) Is?(B(P,Q,R)B(Q,P,R) = ID) Is?(B(P,Q,R)P=Q) --Is?(B(P,Q,R)ω(P,Q) = -ω(Q,P)) Is?(L(P,Q)=B(Q,P,qQ+pP)) \end{axiom}

In two dimensions \begin{axiom} map(x+->simplify expandhtrigs2 Simplify x, L(U,V)) map(x+->simplify expandhtrigs expandhtrigs2 Simplify x, L(U,W)) \end{axiom}

Composition of two Lorentz boosts is not a Lorentz boost unless all three observers are in the same plane. \begin{axiom} Is?(L(R,P)L(P,Q) = L(R,Q)) RQ:=aR+bQ; rq:=solve(dot(RQ,RQ)=-1,b); #rq RQ1:=eval(RQ,rq.1); dot(RQ1,RQ1) Is?(L(R,RQ1)L(RQ1,Q) = L(R,Q)) RQ2:=eval(RQ,rq.2); Is?(RQ1=RQ2) dot(RQ2,RQ2) Is?(L(R,RQ2)*L(RQ2,Q) = L(R,Q)) \end{axiom}

but the composition does preserve observers and magnitudes \begin{axiom} LRPQ := L(R,P)L(P,Q); Is?(LRPQQ = L(R,Q)Q) is?(dot(LRPQω(S,Q),LRPQ*ω(S,Q))=dot(L(R,Q)*ω(S,Q),L(R,Q)*ω(S,Q))) \end{axiom}

Lorentz boost with respect to observer "at rest" \begin{axiom} LT:=L(vect [1,0,0,0],vect [u0,-u1,-u2,-u3]) map(simplify, L(vect [1,0,0,0], map(Simplify,U))) \end{axiom}

Two dimensional Lorentz Transformation \begin{axiom} matrix [[1/sqrt(1-v'^2),v'/sqrt(1-v'^2),0,0],[v'/sqrt(1-v'^2),1/sqrt(1-v'^2),0,0],[0,0,1,0],[0,0,0,1]] Is?(%=map(x+->eval(x,[u0=1/sqrt(1-v'^2),u1=v'/sqrt(1-v'^2),u2=0,u3=0]),LT)) \end{axiom}

[SandBox Idempotent Observers]

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Checking for foreign routines FRICAS="/usr/local/lib/fricas/target/x86_64-linux-gnu" spad-lib="/usr/local/lib/fricas/target/x86_64-linux-gnu/lib/libspad.so" foreign routines found openServer result -2 FriCAS Computer Algebra System Version: FriCAS 1.3.10 built with sbcl 2.2.9.debian Timestamp: Wed 10 Jan 02:19:45 CET 2024 ----------------------------------------------------------------------------- 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) -> (1) -> Scalar := Expression Integer

$$ Expression \left( {Integer} \right) \leqno(1) $$

Type: Type vect(x:List Scalar):Matrix Scalar == matrix map(y+->[y],x)

Function declaration vect : List(Expression(Integer)) -> Matrix( Expression(Integer)) has been added to workspace. Type: Void vect [a0,a1,a2,a3]

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

$$ \left[ \begin{array}{c} a0 \ a1 \ a2 \ a3 \end{array} \right] \leqno(3) $$

Type: Matrix(Expression(Integer)) (4) -> ID:=diagonalMatrix([1,1,1,1])

$$ \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{array} \right] \leqno(4) $$

Type: Matrix(Integer) (5) -> htrigs2exp == rule cosh(a) == (exp(a)+exp(-a))/2 sinh(a) == (exp(a)-exp(-a))/2

Type: Void sinhcosh == rule ?cexp(a)+?cexp(-a) == 2ccosh(a) ?cexp(a)-?cexp(-a) == 2csinh(a) ?cexp(a-b)+?cexp(b-a) == 2ccosh(a-b) ?cexp(a-b)-?cexp(b-a) == 2csinh(a-b)

Type: Void expandhtrigs == rule cosh(:x+y) == sinh(x)sinh(y)+cosh(x)cosh(y) sinh(:x+y) == cosh(x)sinh(y)+sinh(x)cosh(y) cosh(2x) == 2cosh(x)^2-1 sinh(2x) == 2sinh(x)*cosh(x)

Type: Void expandhtrigs2 == rule cosh(2x+2y) == 2cosh(x+y)^2-1 sinh(2x+2y) == 2sinh(x+y)cosh(x+y) cosh(2x-2y) == 2cosh(x-y)^2-1 sinh(2x-2y) == 2sinh(x-y)cosh(x-y)

Type: Void Simplify(x:Scalar):Scalar == htrigs sinhcosh simplify htrigs2exp x

Function declaration Simplify : Expression(Integer) -> Expression( Integer) has been added to workspace. Type: Void possible(x)==subst(x, map(y+->(y=(random(100) - random(100))),variables x) )

Type: Void is?(eq:Equation Scalar):Boolean == (Simplify(lhs(eq)-rhs(eq))=0)::Boolean

Function declaration is? : Equation(Expression(Integer)) -> Boolean has been added to workspace. Type: Void Is?(eq:Equation(Matrix(Scalar))):Boolean == (map(Simplify,lhs(eq)-rhs(eq)) :: Matrix Expression AlgebraicNumber = zero(nrows(lhs(eq)),ncols(lhs(eq)))$Matrix Expression AlgebraicNumber )::Boolean

Function declaration Is? : Equation(Matrix(Expression(Integer))) -> Boolean has been added to workspace. Type: Void (13) -> G:=diagonalMatrix [-1,1,1,1]

$$ \left[ \begin{array}{cccc} -1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{array} \right] \leqno(13) $$

Type: Matrix(Integer) (14) -> g(x) == transpose(x)*G

Type: Void g(vect [a0,a1,a2,a3])

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

$$ \left[ \begin{array}{cccc} -a0 & a1 & a2 & a3 \end{array} \right] \leqno(15) $$

Type: Matrix(Expression(Integer)) (16) -> dot(x,y) == (g(x)*y)::Scalar

Type: Void dot(vect [a0,a1,a2,a3], vect [b0,b1,b2,b3])

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

$$ {a3 \ b3}+{a2 \ b2}+{a1 \ b1} -{a0 \ b0} \leqno(17) $$

Type: Expression(Integer) (18) -> tensor(x,y) == x*g(y)

Type: Void tensor(vect [a0,a1,a2,a3], vect [b0,b1,b2,b3])

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

$$ \left[ \begin{array}{cccc} -{a0 \ b0} & {a0 \ b1} & {a0 \ b2} & {a0 \ b3} \ -{a1 \ b0} & {a1 \ b1} & {a1 \ b2} & {a1 \ b3} \ -{a2 \ b0} & {a2 \ b1} & {a2 \ b2} & {a2 \ b3} \ -{a3 \ b0} & {a3 \ b1} & {a3 \ b2} & {a3 \ b3} \end{array} \right] \leqno(19) $$

Type: Matrix(Expression(Integer)) (20) -> P:=vect [sqrt(p1^2+p2^2+p3^2+1),-p1,-p2,-p3];

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

$$ -1 \leqno(21) $$

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

Type: Matrix(Expression(Integer)) R:=vect [sqrt(r1^2+r2^2+r3^2+1),-r1,-r2,-r3];

Type: Matrix(Expression(Integer)) S:=1/sqrt(1-s1^2-s2^2-s3^2)*vect [1,-s1,-s2,-s3]

$$ \left[ \begin{array}{c} {\frac{1}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \ -{\frac{s1}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \ -{\frac{s2}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \ -{\frac{s3}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \end{array} \right] \leqno(24) $$

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

$$ -1 \leqno(25) $$

Type: Expression(Integer) T:=1/sqrt(1-t1^2-t2^2-t3^2)*vect [1,-t1,-t2,-t3]

$$ \left[ \begin{array}{c} {\frac{1}{{\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}} \ -{\frac{t1}{{\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}} \ -{\frac{t2}{{\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}} \ -{\frac{t3}{{\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}} \end{array} \right] \leqno(26) $$

Type: Matrix(Expression(Integer)) U:=vect [cosh(u),sinh(u),0,0]

$$ \left[ \begin{array}{c} {\cosh \left( {u} \right)} \ {\sinh \left( {u} \right)} \ 0 \ 0 \end{array} \right] \leqno(27) $$

Type: Matrix(Expression(Integer)) simplify dot(U,U)

$$ -1 \leqno(28) $$

Type: Expression(Integer) V:=vect [cosh(v),sinh(v),0,0]

$$ \left[ \begin{array}{c} {\cosh \left( {v} \right)} \ {\sinh \left( {v} \right)} \ 0 \ 0 \end{array} \right] \leqno(29) $$

Type: Matrix(Expression(Integer)) Simplify dot(U,V)

Compiling body of rule htrigs2exp to compute value of type Ruleset( Integer,Integer,Expression(Integer)) Compiling body of rule sinhcosh to compute value of type Ruleset( Integer,Integer,Expression(Integer)) Compiling function Simplify with type Expression(Integer) -> Expression(Integer)

$$ -{\cosh \left( {{v -u}} \right)} \leqno(30) $$

Type: Expression(Integer) W:=vect [cosh(w),0,sinh(w),0]

$$ \left[ \begin{array}{c} {\cosh \left( {w} \right)} \ 0 \ {\sinh \left( {w} \right)} \ 0 \end{array} \right] \leqno(31) $$

Type: Matrix(Expression(Integer)) Simplify dot(U,W)

$$ \frac{{-{\cosh \left( {{w+u}} \right)} -{\cosh \left( {{w -u}} \right)}}}{2} \leqno(32) $$

Type: Expression(Integer) (33) -> vect [1,0,0,0]

$$ \left[ \begin{array}{c} 1 \ 0 \ 0 \ 0 \end{array} \right] \leqno(33) $$

Type: Matrix(Expression(Integer)) dot(%,%)

$$ -1 \leqno(34) $$

Type: Expression(Integer) (35) -> ω(P,Q)==-P/dot(P,Q)-Q

Type: Void ω(P,Q)

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

$$ \left[ \begin{array}{c} {\frac{{{{\left( {p3 \ q3}+{p2 \ q2}+{p1 \ q1} \right)} \ {\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}}+{{\left( -{{q3} \sp {2}} -{{q2} \sp {2}} -{{q1} \sp {2}} \right)} \ {\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}}}}}{{{{\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \ {\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q3} -{p2 \ q2} -{p1 \ q1}}}} \ {\frac{{{q1 \ {\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \ {\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q1 \ q3} -{p2 \ q1 \ q2} -{p1 \ {{q1} \sp {2}}} -p1}}{{{{\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \ {\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q3} -{p2 \ q2} -{p1 \ q1}}}} \ {\frac{{{q2 \ {\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \ {\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q2 \ q3} -{p2 \ {{q2} \sp {2}}} -{p1 \ q1 \ q2} -p2}}{{{{\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \ {\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q3} -{p2 \ q2} -{p1 \ q1}}}} \ {\frac{{{q3 \ {\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \ {\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ {{q3} \sp {2}}}+{{\left( -{p2 \ q2} -{p1 \ q1} \right)} \ q3} -p3}}{{{{\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \ {\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q3} -{p2 \ q2} -{p1 \ q1}}}} \end{array} \right] \leqno(36) $$

Type: Matrix(Expression(Integer)) ω(S,T)

$$ \left[ \begin{array}{c} {\frac{{{{t3} \sp {2}} -{s3 \ t3}+{{t2} \sp {2}} -{s2 \ t2}+{{t1} \sp {2}} -{s1 \ t1}}}{{{\left( {s3 \ t3}+{s2 \ t2}+{s1 \ t1} -1 \right)} \ {\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}}} \ {\frac{{-{s1 \ {{t3} \sp {2}}}+{s3 \ t1 \ t3} -{s1 \ {{t2} \sp {2}}}+{s2 \ t1 \ t2} -t1+s1}}{{{\left( {s3 \ t3}+{s2 \ t2}+{s1 \ t1} -1 \right)} \ {\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}}} \ {\frac{{-{s2 \ {{t3} \sp {2}}}+{s3 \ t2 \ t3}+{{\left( {s1 \ t1} -1 \right)} \ t2} -{s2 \ {{t1} \sp {2}}}+s2}}{{{\left( {s3 \ t3}+{s2 \ t2}+{s1 \ t1} -1 \right)} \ {\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}}} \ {\frac{{{{\left( {s2 \ t2}+{s1 \ t1} -1 \right)} \ t3} -{s3 \ {{t2} \sp {2}}} -{s3 \ {{t1} \sp {2}}}+s3}}{{{\left( {s3 \ t3}+{s2 \ t2}+{s1 \ t1} -1 \right)} \ {\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}}} \end{array} \right] \leqno(37) $$

Type: Matrix(Expression(Integer)) (38) -> map(x+->Simplify x,ω(U,V))

$$ \left[ \begin{array}{c} {\frac{{-{\cosh \left( {{{2 \ v} -u}} \right)}+{\cosh \left( {u} \right)}}}{{2 \ {\cosh \left( {{v -u}} \right)}}}} \ {\frac{{-{\sinh \left( {{{2 \ v} -u}} \right)}+{\sinh \left( {u} \right)}}}{{2 \ {\cosh \left( {{v -u}} \right)}}}} \ 0 \ 0 \end{array} \right] \leqno(38) $$

Type: Matrix(Expression(Integer)) vect [cosh(u)/cosh(u-v)-cosh(v),sinh(u)/cosh(u-v)-sinh(v),0,0]

$$ \left[ \begin{array}{c} {\frac{{-{{\cosh \left( {{v -u}} \right)} \ {\cosh \left( {v} \right)}}+{\cosh \left( {u} \right)}}}{{\cosh \left( {{v -u}} \right)}}} \ {\frac{{-{{\cosh \left( {{v -u}} \right)} \ {\sinh \left( {v} \right)}}+{\sinh \left( {u} \right)}}}{{\cosh \left( {{v -u}} \right)}}} \ 0 \ 0 \end{array} \right] \leqno(39) $$

Type: Matrix(Expression(Integer)) Is?(% = ω(U,V))

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

$$ true \leqno(40) $$

Type: Boolean map(x+->Simplify x,ω(U,W))

$$ \left[ \begin{array}{c} {\frac{{-{\cosh \left( {{2 \ w}} \right)}+1}}{{2 \ {\cosh \left( {w} \right)}}}} \ {\frac{{2 \ {\sinh \left( {u} \right)}}}{{{\cosh \left( {{w+u}} \right)}+{\cosh \left( {{w -u}} \right)}}}} \ -{\sinh \left( {w} \right)} \ 0 \end{array} \right] \leqno(41) $$

Type: Matrix(Expression(Integer)) (42) -> dot(ω(P,vect [1,0,0,0]),ω(P,vect [1,0,0,0]))

$$ \frac{{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}}}{{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}} \leqno(42) $$

Type: Expression(Integer) (43) -> dot(P,ω(Q,P))

$$ 0 \leqno(43) $$

Type: Expression(Integer) possible dot(ω(Q,P),ω(Q,P))::EXPR Float

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

$$ 0.9999995659\_3941083651 \leqno(44) $$

Type: Expression(Float) dot(Q,ω(P,Q))

$$ 0 \leqno(45) $$

Type: Expression(Integer) possible dot(ω(P,Q),ω(P,Q))::EXPR Float

$$ 0.9999994628\_9493005142 \leqno(46) $$

Type: Expression(Float) (47) -> ω(vect [u0,u1,u2,u3],vect [1,0,0,0])

$$ \left[ \begin{array}{c} 0 \ {\frac{u1}{u0}} \ {\frac{u2}{u0}} \ {\frac{u3}{u0}} \end{array} \right] \leqno(47) $$

Type: Matrix(Expression(Integer)) ω(R,vect [1,0,0,0])

$$ \left[ \begin{array}{c} 0 \ -{\frac{r1}{{\sqrt {{{{r3} \sp {2}}+{{r2} \sp {2}}+{{r1} \sp {2}}+1}}}}} \ -{\frac{r2}{{\sqrt {{{{r3} \sp {2}}+{{r2} \sp {2}}+{{r1} \sp {2}}+1}}}}} \ -{\frac{r3}{{\sqrt {{{{r3} \sp {2}}+{{r2} \sp {2}}+{{r1} \sp {2}}+1}}}}} \end{array} \right] \leqno(48) $$

Type: Matrix(Expression(Integer)) ω(S,vect [1,0,0,0])

$$ \left[ \begin{array}{c} 0 \ -s1 \ -s2 \ -s3 \end{array} \right] \leqno(49) $$

Type: Matrix(Expression(Integer)) map(Simplify, ω(U,vect [1,0,0,0]))

$$ \left[ \begin{array}{c} 0 \ {\frac{{\sinh \left( {u} \right)}}{{\cosh \left( {u} \right)}}} \ 0 \ 0 \end{array} \right] \leqno(50) $$

Type: Matrix(Expression(Integer)) (51) -> ω(vect [1,0,0,0],S)

$$ \left[ \begin{array}{c} {\frac{{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}}}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \ {\frac{s1}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \ {\frac{s2}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \ {\frac{s3}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \end{array} \right] \leqno(51) $$

Type: Matrix(Expression(Integer)) ω(vect [1,0,0,0],R)

$$ \left[ \begin{array}{c} {\frac{{-{{r3} \sp {2}} -{{r2} \sp {2}} -{{r1} \sp {2}}}}{{\sqrt {{{{r3} \sp {2}}+{{r2} \sp {2}}+{{r1} \sp {2}}+1}}}}} \ r1 \ r2 \ r3 \end{array} \right] \leqno(52) $$

Type: Matrix(Expression(Integer)) is?(dot(ω(P,Q),ω(P,Q))=dot(ω(Q,P),ω(Q,P)))

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

$$ true \leqno(53) $$

Type: Boolean (54) -> L(P,Q) == ID + tensor(P+Q,P+Q)/(1-dot(P,Q)) - 2*tensor(P,Q)

Type: Void Is?(L(P,P) = ID)

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

$$ true \leqno(55) $$

Type: Boolean Is?(L(P,Q)*L(Q,P) = ID)

$$ true \leqno(56) $$

Type: Boolean Is?(L(P,Q)*Q=P)

$$ true \leqno(57) $$

Type: Boolean Is?(L(P,Q)*ω(P,Q) = -ω(Q,P))

$$ true \leqno(58) $$

Type: Boolean (59) -> B(P,Q,X) == ID - ( tensor( 2X, dot(P-Q,P-Q)X - 2dot(X,P)(P-Q) ) + tensor( _ P-Q, 2dot(X,X)(P-Q)+4dot(X,Q)X ) ) / ( _ dot(X,X)dot(P-Q,P-Q)+4dot(X,P)*dot(X,Q) _ )

Type: Void Is?(B(P,P,R) = ID)

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

$$ true \leqno(60) $$

Type: Boolean Is?(B(P,Q,R)*B(Q,P,R) = ID)

$$ true \leqno(61) $$

Type: Boolean Is?(B(P,Q,R)*P=Q)

$$ true \leqno(62) $$

Type: Boolean --Is?(B(P,Q,R)*ω(P,Q) = -ω(Q,P)) Is?(L(P,Q)=B(Q,P,qQ+pP))

$$ true \leqno(63) $$

Type: Boolean (64) -> map(x+->simplify expandhtrigs2 Simplify x, L(U,V))

Compiling body of rule expandhtrigs2 to compute value of type Ruleset(Integer,Integer,Expression(Integer))

$$ \left[ \begin{array}{cccc} {\cosh \left( {{v -u}} \right)} & -{\sinh \left( {{v -u}} \right)} & 0 & 0 \ -{\sinh \left( {{v -u}} \right)} & {\cosh \left( {{v -u}} \right)} & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{array} \right] \leqno(64) $$

Type: Matrix(Expression(Integer)) map(x+->simplify expandhtrigs expandhtrigs2 Simplify x, L(U,W))

Compiling body of rule expandhtrigs to compute value of type Ruleset (Integer,Integer,Expression(Integer))

$$ \left[ \begin{array}{cccc} {\frac{{{{\left( {2 \ {{{\cosh \left( {u} \right)}} \sp {2}}} -1 \right)} \ {{{\cosh \left( {w} \right)}} \sp {2}}}+{{\cosh \left( {u} \right)} \ {\cosh \left( {w} \right)}} -{{{\cosh \left( {u} \right)}} \sp {2}}+1}}{{{{\cosh \left( {u} \right)} \ {\cosh \left( {w} \right)}}+1}}} & {\frac{{{\left( {\cosh \left( {w} \right)}+{\cosh \left( {u} \right)} \right)} \ {\sinh \left( {u} \right)}}}{{{{\cosh \left( {u} \right)} \ {\cosh \left( {w} \right)}}+1}}} & {\frac{{{\left( {{\left( -{2 \ {{{\cosh \left( {u} \right)}} \sp {2}}}+1 \right)} \ {\cosh \left( {w} \right)}} -{\cosh \left( {u} \right)} \right)} \ {\sinh \left( {w} \right)}}}{{{{\cosh \left( {u} \right)} \ {\cosh \left( {w} \right)}}+1}}} & 0 \ {\frac{{{\left( {2 \ {\cosh \left( {u} \right)} \ {{{\cosh \left( {w} \right)}} \sp {2}}}+{\cosh \left( {w} \right)} -{\cosh \left( {u} \right)} \right)} \ {\sinh \left( {u} \right)}}}{{{{\cosh \left( {u} \right)} \ {\cosh \left( {w} \right)}}+1}}} & {\frac{{{{\cosh \left( {u} \right)} \ {\cosh \left( {w} \right)}}+{{{\cosh \left( {u} \right)}} \sp {2}}}}{{{{\cosh \left( {u} \right)} \ {\cosh \left( {w} \right)}}+1}}} & {\frac{{{\left( -{2 \ {\cosh \left( {u} \right)} \ {\cosh \left( {w} \right)}} -1 \right)} \ {\sinh \left( {u} \right)} \ {\sinh \left( {w} \right)}}}{{{{\cosh \left( {u} \right)} \ {\cosh \left( {w} \right)}}+1}}} & 0 \ {\frac{{{\left( -{\cosh \left( {w} \right)} -{\cosh \left( {u} \right)} \right)} \ {\sinh \left( {w} \right)}}}{{{{\cosh \left( {u} \right)} \ {\cosh \left( {w} \right)}}+1}}} & {\frac{{{\sinh \left( {u} \right)} \ {\sinh \left( {w} \right)}}}{{{{\cosh \left( {u} \right)} \ {\cosh \left( {w} \right)}}+1}}} & {\frac{{{{{\cosh \left( {w} \right)}} \sp {2}}+{{\cosh \left( {u} \right)} \ {\cosh \left( {w} \right)}}}}{{{{\cosh \left( {u} \right)} \ {\cosh \left( {w} \right)}}+1}}} & 0 \ 0 & 0 & 0 & 1 \end{array} \right] \leqno(65) $$

Type: Matrix(Expression(Integer)) (66) -> Is?(L(R,P)*L(P,Q) = L(R,Q))

$$ false \leqno(66) $$

Type: Boolean RQ:=aR+bQ;

Type: Matrix(Expression(Integer)) rq:=solve(dot(RQ,RQ)=-1,b); #rq

$$ 2 \leqno(68) $$

Type: PositiveInteger RQ1:=eval(RQ,rq.1);

Type: Matrix(Expression(Integer)) dot(RQ1,RQ1)

$$ -1 \leqno(70) $$

Type: Expression(Integer) Is?(L(R,RQ1)*L(RQ1,Q) = L(R,Q))

$$ true \leqno(71) $$

Type: Boolean RQ2:=eval(RQ,rq.2);

Type: Matrix(Expression(Integer)) Is?(RQ1=RQ2)

$$ false \leqno(73) $$

Type: Boolean dot(RQ2,RQ2)

$$ -1 \leqno(74) $$

Type: Expression(Integer) Is?(L(R,RQ2)*L(RQ2,Q) = L(R,Q))

$$ true \leqno(75) $$

Type: Boolean (76) -> LRPQ := L(R,P)*L(P,Q);

Type: Matrix(Expression(Integer)) Is?(LRPQQ = L(R,Q)Q)

$$ true \leqno(77) $$

Type: Boolean is?(dot(LRPQ*ω(S,Q),LRPQ*ω(S,Q))=dot(L(R,Q)*ω(S,Q),L(R,Q)*ω(S,Q)))


Some or all expressions may not have rendered properly, because Latex returned the following error:
! Missing $ inserted.
<inserted text> 
                $
l.194 ω(P,Q)==-P/dot(P,Q)-Q

Missing $ inserted. <inserted text> $ l.195 ω(P,Q)

Missing $ inserted. <inserted text> $ l.196 ω(S,T)

LaTeX Error: Command \end{axiom} invalid in math mode.

See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ...

l.197 \end{axiom}\newpage

LaTeX Warning: No verbatim text on input line 197.

LaTeX Warning: No verbatim text on input line 197.

Missing $ inserted. <inserted text> $ l.197 \end{axiom}\newpage

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 197.

Missing $ inserted. <inserted text> $ l.199 map(x+->Simplify x,ω(U,V))

Missing $ inserted. <inserted text> $ l.200 ...v)-cosh(v),sinh(u)/cosh(u-v)-sinh(v),0,0]

Missing $ inserted. <inserted text> $ l.201 Is?(% = ω(U,V))

Missing $ inserted. <inserted text> $ l.202 map(x+->Simplify x,ω(U,W))

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 203.

Missing $ inserted. <inserted text> $ l.205 ...�(P,vect [1,0,0,0]),ω(P,vect [1,0,0,0]))

LaTeX Error: Command \end{axiom} invalid in math mode.

See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ...

l.206 \end{axiom}\newpage

LaTeX Warning: No verbatim text on input line 206.

LaTeX Warning: No verbatim text on input line 206.

Missing $ inserted. <inserted text> $ l.206 \end{axiom}\newpage

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 206.

[4] Missing $ inserted. <inserted text> $ l.209 dot(P,ω(Q,P))

Missing $ inserted. <inserted text> $ l.210 possible dot(ω(Q,P),ω(Q,P))::EXPR Float

Missing $ inserted. <inserted text> $ l.211 dot(Q,ω(P,Q))

Missing $ inserted. <inserted text> $ l.212 possible dot(ω(P,Q),ω(P,Q))::EXPR Float

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 213.

Missing $ inserted. <inserted text> $ l.215 ω(vect [u0,u1,u2,u3],vect [1,0,0,0])

Missing $ inserted. <inserted text> $ l.216 ω(R,vect [1,0,0,0])

Missing $ inserted. <inserted text> $ l.217 ω(S,vect [1,0,0,0])

Missing $ inserted. <inserted text> $ l.218 map(Simplify, ω(U,vect [1,0,0,0]))

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 219.

Missing $ inserted. <inserted text> $ l.221 ω(vect [1,0,0,0],S)

Missing $ inserted. <inserted text> $ l.222 ω(vect [1,0,0,0],R)

Missing $ inserted. <inserted text> $ l.223 ...ot(ω(P,Q),ω(P,Q))=dot(ω(Q,P),ω(Q,P)))

LaTeX Error: Command \end{axiom} invalid in math mode.

See the LaTeX manual or LaTeX Companion for explanation. Type H <return> for immediate help. ...

l.224 \end{axiom}\newpage

LaTeX Warning: No verbatim text on input line 224.

LaTeX Warning: No verbatim text on input line 224.

Missing $ inserted. <inserted text> $ l.224 \end{axiom}\newpage

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 224.

[5] [6] Missing $ inserted. <inserted text> $ l.232 Is?(L(P,Q)*ω(P,Q) = -ω(Q,P))

Missing $ inserted. <inserted text> $ l.232 Is?(L(P,Q)*ω(P,Q) = -ω(Q,P))

LaTeX Warning: Characters dropped after `\end{axiom}' on input line 233.

Missing $ inserted. <inserted text> $ l.248 --Is?(B(P,Q,R)*ω(P,Q) = -ω(Q,P))

Missing $ inserted. <inserted text> $ l.248 --Is?




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