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last edited 11 years ago by Bill Page |
Edit detail for SandBoxLorentzTransformation revision 30 of 30
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Editor: Bill Page
Time: 2014/01/10 03:22:07 GMT+0 |
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| Note: most general boost | ||
changed: -Book by T. Matolcsi - - - "Space-time without reference frames":http://axiom-wiki.newsynthesis.org/uploads/matolcsi.pdf References - "Space-time without reference frames":http://axiom-wiki.newsynthesis.org/uploads/matolcsi.pdf T. Matolcsi - "The Lorentz boost-link is not unique":http://arxiv.org/pdf/math-ph/0608062.pdf Zbigniew Oziewicz added: Most General Lorentz Boost (Oziewicz, 2006) is given by three non-coplanar vectors \begin{axiom} B(P,Q,X) == ID - _ ( tensor( _ 2*X, dot(P-Q,P-Q)*X - 2*dot(X,P)*(P-Q) _ ) + _ tensor( _ P-Q, 2*dot(X,X)*(P-Q)+4*dot(X,Q)*X _ ) _ ) / ( _ dot(X,X)*dot(P-Q,P-Q)+4*dot(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,q*Q+p*P)) \end{axiom}
Lorentz transformations.References
- Space-time without reference frames T. Matolcsi
- The Lorentz boost-link is not unique Zbigniew Oziewicz
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}
Some or all expressions may not have rendered properly, because Axiom returned the following error:
Error: export HOME=/var/zope2/var/LatexWiki; ulimit -t 600; export LD_LIBRARY_PATH=/usr/local/lib/fricas/target/x86_64-linux-gnu/lib; LANG=en_US.UTF-8 /usr/local/lib/fricas/target/x86_64-linux-gnu/bin/fricas -nosman < /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/7701307392394560546-25px.axm
Heap exhausted during garbage collection: 0 bytes available, 16 requested.
Immobile Object Counts
Gen layout fdefn symbol code Boxed Cons Raw Code SmMix Mixed LgRaw LgCode LgMix Waste% Alloc Trig Dirty GCs Mem-age
2 0 0 0 0 3 10478 1 0 0 1 0 0 0 0.9 340512912 202181376 10483 1 1.3689
3 0 0 0 0 1 6759 1 0 0 0 0 0 0 0.8 219689072 6291456 6675 0 0.0126
4 0 4665 0 4894 113 909 78 0 17 9 0 0 0 1.3 36422448 2000000 79 0 0.0000
5 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0 0 2000000 0 0 0.0000
6 749 24064 26788 24347 448 176 63 5 48 16 0 0 74 2.4 26541936 2000000 18 0 0.0000
Tot 749 28729 26788 29241 565 18322 143 5 65 26 0 0 74 1.0 623166368 [99.0% of 629145600 max]
GC control variables:
GC-INHIBIT = true
GC-PENDING = true
STOP-FOR-GC-PENDING = false
fatal error encountered in SBCL pid 1788095 tid 1788095:
Heap exhausted, game over.
Error opening /dev/tty: No such device or address
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 1.3.12 built with sbcl 2.2.9.debian
Timestamp: Sat 7 Jun 23:54:49 CEST 2025
-----------------------------------------------------------------------------
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)))
Welcome to LDB, a low-level debugger for the Lisp runtime environment.
(GC in progress, oldspace=2, newspace=3)
ldb>
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?