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last edited 12 years ago by Bill Page |
Edit detail for SandBoxLorentzTransformation revision 27 of 30
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Time: 2013/10/07 06:09:41 GMT+0 |
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| Note: ω | ||
changed: - An object P has a unique relative velocity w(P,Q) with respect An object P has a unique relative velocity ω(P,Q) with respect changed: -w(P,Q)==-P/dot(P,Q)-Q -w(P,Q) -w(S,T) ω(P,Q)==-P/dot(P,Q)-Q ω(P,Q) ω(S,T) changed: -map(x+->Simplify x,w(U,V)) map(x+->Simplify x,ω(U,V)) changed: -Is?(% = w(U,V)) Is?(% = ω(U,V)) changed: -Observer P measures velocity w(Q,P). w(Q,P) is space-like Observer P measures velocity ω(Q,P). ω(Q,P) is space-like changed: -dot(w(P,vect [1,0,0,0]),w(P,vect [1,0,0,0])) dot(ω(P,vect [1,0,0,0]),ω(P,vect [1,0,0,0])) changed: -dot(P,w(Q,P)) -possible dot(w(Q,P),w(Q,P))::EXPR Float -dot(Q,w(P,Q)) -possible dot(w(P,Q),w(P,Q))::EXPR Float 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 changed: -w(vect [u0,u1,u2,u3],vect [1,0,0,0]) -w(R,vect [1,0,0,0]) -w(S,vect [1,0,0,0]) -map(Simplify, w(U,vect [1,0,0,0])) 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])) changed: -w(vect [1,0,0,0],S) -w(vect [1,0,0,0],R) -is?(dot(w(P,Q),w(P,Q))=dot(w(Q,P),w(Q,P))) 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))) changed: -Is?(L(P,Q)*w(P,Q) = -w(Q,P)) Is?(L(P,Q)*ω(P,Q) = -ω(Q,P)) changed: -is?(dot(LRPQ*w(S,Q),LRPQ*w(S,Q))=dot(L(R,Q)*w(S,Q),L(R,Q)*w(S,Q))) is?(dot(LRPQ*ω(S,Q),LRPQ*ω(S,Q))=dot(L(R,Q)*ω(S,Q),L(R,Q)*ω(S,Q)))
Lorentz transformations.Book by T. Matolcsi
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) \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)) \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}
In two dimensions \begin{axiom} map(x+->simplify expandhtrigs2 Simplify x, L(U,V)) \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/9176722114772041233-25px.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
1 0 0 0 0 2 2340 0 0 0 0 0 0 0 1.1 75907680 48859920 2342 1 1.3114
2 0 0 0 0 3 6867 0 0 0 0 0 0 0 0.9 223157968 6291456 1587 0 1.1047
3 0 0 0 0 1 8218 1 0 0 1 0 0 0 0.8 267105136 6291456 2 0 0.0591
4 0 4661 0 4890 111 725 77 0 16 8 0 0 0 1.1 30350784 2000000 78 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 28725 26788 29237 565 18326 141 5 64 25 0 0 74 1.0 623063504 [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 469011 tid 469011:
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)
(31) -> vect [1,0,0,0]
$$
\left[
\begin{array}{c}
1 \
0 \
0 \
0
\end{array}
\right]
\leqno(31)
$$
Type: Matrix(Expression(Integer))
dot(%,%)
$$
-1
\leqno(32)
$$
Type: Expression(Integer)
(33) -> 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))
$$
\left[
\begin{array}{c}
{\frac{{{{\left( {p3 \ q3}+{p2 \ q2}+{p1 \ q1}
\right)}
\ {\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}}+{{\left(
-{{q3} \sp {2}} -{{q2} \sp {2}} -{{q1} \sp {2}}
\right)}
\ {\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}}}}}{{{{\sqrt
{{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \ {\sqrt {{{{q3} \sp
{2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q3} -{p2 \ q2} -{p1 \
q1}}}} \
{\frac{{{q1 \ {\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \
{\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q1 \ q3}
-{p2 \ q1 \ q2} -{p1 \ {{q1} \sp {2}}} -p1}}{{{{\sqrt {{{{p3} \sp
{2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \ {\sqrt {{{{q3} \sp {2}}+{{q2} \sp
{2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q3} -{p2 \ q2} -{p1 \ q1}}}} \
{\frac{{{q2 \ {\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \
{\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q2 \ q3}
-{p2 \ {{q2} \sp {2}}} -{p1 \ q1 \ q2} -p2}}{{{{\sqrt {{{{p3} \sp
{2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \ {\sqrt {{{{q3} \sp {2}}+{{q2} \sp
{2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q3} -{p2 \ q2} -{p1 \ q1}}}} \
{\frac{{{q3 \ {\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \
{\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ {{q3} \sp
{2}}}+{{\left( -{p2 \ q2} -{p1 \ q1}
\right)}
\ q3} -p3}}{{{{\sqrt {{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}+1}}} \
{\sqrt {{{{q3} \sp {2}}+{{q2} \sp {2}}+{{q1} \sp {2}}+1}}}} -{p3 \ q3} -{p2
\ q2} -{p1 \ q1}}}}
\end{array}
\right]
\leqno(34)
$$
Type: Matrix(Expression(Integer))
ω(S,T)
$$
\left[
\begin{array}{c}
{\frac{{{{t3} \sp {2}} -{s3 \ t3}+{{t2} \sp {2}} -{s2 \ t2}+{{t1} \sp {2}}
-{s1 \ t1}}}{{{\left( {s3 \ t3}+{s2 \ t2}+{s1 \ t1} -1
\right)}
\ {\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}}} \
{\frac{{-{s1 \ {{t3} \sp {2}}}+{s3 \ t1 \ t3} -{s1 \ {{t2} \sp {2}}}+{s2
\ t1 \ t2} -t1+s1}}{{{\left( {s3 \ t3}+{s2 \ t2}+{s1 \ t1} -1
\right)}
\ {\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}}} \
{\frac{{-{s2 \ {{t3} \sp {2}}}+{s3 \ t2 \ t3}+{{\left( {s1 \ t1} -1
\right)}
\ t2} -{s2 \ {{t1} \sp {2}}}+s2}}{{{\left( {s3 \ t3}+{s2 \ t2}+{s1 \ t1}
-1
\right)}
\ {\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}}} \
{\frac{{{{\left( {s2 \ t2}+{s1 \ t1} -1
\right)}
\ t3} -{s3 \ {{t2} \sp {2}}} -{s3 \ {{t1} \sp {2}}}+s3}}{{{\left( {s3 \
t3}+{s2 \ t2}+{s1 \ t1} -1
\right)}
\ {\sqrt {{-{{t3} \sp {2}} -{{t2} \sp {2}} -{{t1} \sp {2}}+1}}}}}}
\end{array}
\right]
\leqno(35)
$$
Type: Matrix(Expression(Integer))
(36) -> map(x+->Simplify x,ω(U,V))
$$
\left[
\begin{array}{c}
{\frac{{-{\cosh
\left(
{{{2 \ v} -u}}
\right)}+{\cosh
\left(
{u}
\right)}}}{{2
\ {\cosh
\left(
{{v -u}}
\right)}}}}
\
{\frac{{-{\sinh
\left(
{{{2 \ v} -u}}
\right)}+{\sinh
\left(
{u}
\right)}}}{{2
\ {\cosh
\left(
{{v -u}}
\right)}}}}
\
0 \
0
\end{array}
\right]
\leqno(36)
$$
Type: Matrix(Expression(Integer))
vect [cosh(u)/cosh(u-v)-cosh(v),sinh(u)/cosh(u-v)-sinh(v),0,0]
$$
\left[
\begin{array}{c}
{\frac{{-{{\cosh
\left(
{{v -u}}
\right)}
\ {\cosh
\left(
{v}
\right)}}+{\cosh
\left(
{u}
\right)}}}{{\cosh
\left(
{{v -u}}
\right)}}}
\
{\frac{{-{{\cosh
\left(
{{v -u}}
\right)}
\ {\sinh
\left(
{v}
\right)}}+{\sinh
\left(
{u}
\right)}}}{{\cosh
\left(
{{v -u}}
\right)}}}
\
0 \
0
\end{array}
\right]
\leqno(37)
$$
Type: Matrix(Expression(Integer))
Is?(% = ω(U,V))
Compiling function Is? with type Equation(Matrix(Expression(Integer)
)) -> Boolean
$$
true
\leqno(38)
$$
Type: Boolean
(39) -> dot(ω(P,vect [1,0,0,0]),ω(P,vect [1,0,0,0]))
$$
\frac{{{{p3} \sp {2}}+{{p2} \sp {2}}+{{p1} \sp {2}}}}{{{{p3} \sp {2}}+{{p2}
\sp {2}}+{{p1} \sp {2}}+1}}
\leqno(39)
$$
Type: Expression(Integer)
(40) -> dot(P,ω(Q,P))
$$
0
\leqno(40)
$$
Type: Expression(Integer)
possible dot(ω(Q,P),ω(Q,P))::EXPR Float
Compiling function possible with type Expression(Integer) ->
Expression(Integer)
$$
0.9999995659\_3941083651
\leqno(41)
$$
Type: Expression(Float)
dot(Q,ω(P,Q))
$$
0
\leqno(42)
$$
Type: Expression(Integer)
possible dot(ω(P,Q),ω(P,Q))::EXPR Float
$$
0.9999994628\_9493005142
\leqno(43)
$$
Type: Expression(Float)
(44) -> ω(vect [u0,u1,u2,u3],vect [1,0,0,0])
$$
\left[
\begin{array}{c}
0 \
{\frac{u1}{u0}} \
{\frac{u2}{u0}} \
{\frac{u3}{u0}}
\end{array}
\right]
\leqno(44)
$$
Type: Matrix(Expression(Integer))
ω(R,vect [1,0,0,0])
$$
\left[
\begin{array}{c}
0 \
-{\frac{r1}{{\sqrt {{{{r3} \sp {2}}+{{r2} \sp {2}}+{{r1} \sp {2}}+1}}}}} \
-{\frac{r2}{{\sqrt {{{{r3} \sp {2}}+{{r2} \sp {2}}+{{r1} \sp {2}}+1}}}}} \
-{\frac{r3}{{\sqrt {{{{r3} \sp {2}}+{{r2} \sp {2}}+{{r1} \sp {2}}+1}}}}}
\end{array}
\right]
\leqno(45)
$$
Type: Matrix(Expression(Integer))
ω(S,vect [1,0,0,0])
$$
\left[
\begin{array}{c}
0 \
-s1 \
-s2 \
-s3
\end{array}
\right]
\leqno(46)
$$
Type: Matrix(Expression(Integer))
map(Simplify, ω(U,vect [1,0,0,0]))
$$
\left[
\begin{array}{c}
0 \
{\frac{{\sinh
\left(
{u}
\right)}}{{\cosh
\left(
{u}
\right)}}}
\
0 \
0
\end{array}
\right]
\leqno(47)
$$
Type: Matrix(Expression(Integer))
(48) -> ω(vect [1,0,0,0],S)
$$
\left[
\begin{array}{c}
{\frac{{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}}}{{\sqrt {{-{{s3} \sp
{2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \
{\frac{s1}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \
{\frac{s2}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}} \
{\frac{s3}{{\sqrt {{-{{s3} \sp {2}} -{{s2} \sp {2}} -{{s1} \sp {2}}+1}}}}}
\end{array}
\right]
\leqno(48)
$$
Type: Matrix(Expression(Integer))
ω(vect [1,0,0,0],R)
$$
\left[
\begin{array}{c}
{\frac{{-{{r3} \sp {2}} -{{r2} \sp {2}} -{{r1} \sp {2}}}}{{\sqrt {{{{r3} \sp
{2}}+{{r2} \sp {2}}+{{r1} \sp {2}}+1}}}}} \
r1 \
r2 \
r3
\end{array}
\right]
\leqno(49)
$$
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
$$
true
\leqno(50)
$$
Type: Boolean
(51) -> 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))
$$
true
\leqno(52)
$$
Type: Boolean
Is?(L(P,Q)*L(Q,P) = ID)
$$
true
\leqno(53)
$$
Type: Boolean
Is?(L(P,Q)*Q=P)
$$
true
\leqno(54)
$$
Type: Boolean
Is?(L(P,Q)*ω(P,Q) = -ω(Q,P))
$$
true
\leqno(55)
$$
Type: Boolean
(56) -> map(x+->simplify expandhtrigs2 Simplify x, L(U,V))
Compiling body of rule expandhtrigs2 to compute value of type
Ruleset(Integer,Integer,Expression(Integer))
$$
\left[
\begin{array}{cccc}
{\cosh
\left(
{{v -u}}
\right)}
& -{\sinh
\left(
{{v -u}}
\right)}
& 0 & 0 \
-{\sinh
\left(
{{v -u}}
\right)}
& {\cosh
\left(
{{v -u}}
\right)}
& 0 & 0 \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1
\end{array}
\right]
\leqno(56)
$$
Type: Matrix(Expression(Integer))
(57) -> Is?(L(R,P)*L(P,Q) = L(R,Q))
$$
false
\leqno(57)
$$
Type: Boolean
RQ:=aR+bQ;
Type: Matrix(Expression(Integer))
rq:=solve(dot(RQ,RQ)=-1,b); #rq
$$
2
\leqno(59)
$$
Type: PositiveInteger
RQ1:=eval(RQ,rq.1);
Type: Matrix(Expression(Integer))
dot(RQ1,RQ1)
$$
-1
\leqno(61)
$$
Type: Expression(Integer)
Is?(L(R,RQ1)*L(RQ1,Q) = L(R,Q))
$$
true
\leqno(62)
$$
Type: Boolean
RQ2:=eval(RQ,rq.2);
Type: Matrix(Expression(Integer))
Is?(RQ1=RQ2)
$$
false
\leqno(64)
$$
Type: Boolean
dot(RQ2,RQ2)
$$
-1
\leqno(65)
$$
Type: Expression(Integer)
Is?(L(R,RQ2)*L(RQ2,Q) = L(R,Q))
$$
true
\leqno(66)
$$
Type: Boolean
(67) -> LRPQ := L(R,P)*L(P,Q);
Type: Matrix(Expression(Integer))
Is?(LRPQQ = L(R,Q)Q)
$$
true
\leqno(68)
$$
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=1, newspace=2)
ldb>
Some or all expressions may not have rendered properly, because Latex returned the following error:
! Missing $ inserted.
<inserted text>
$
l.192 ω(P,Q)==-P/dot(P,Q)-Q Missing $ inserted.
<inserted text>
$
l.193 ω(P,Q)
Missing $ inserted.
<inserted text>
$
l.194 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.195 \end{axiom}\newpage
LaTeX Warning: No verbatim text on input line 195.
LaTeX Warning: No verbatim text on input line 195.
Missing $ inserted.
<inserted text>
$
l.195 \end{axiom}\newpage
LaTeX Warning: Characters dropped after `\end{axiom}' on input line 195.
Missing $ inserted.
<inserted text>
$
l.197 map(x+->Simplify x,ω(U,V))
Missing $ inserted.
<inserted text>
$
l.198 ...v)-cosh(v),sinh(u)/cosh(u-v)-sinh(v),0,0]
Missing $ inserted.
<inserted text>
$
l.199 Is?(% = ω(U,V))
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.200 \end{axiom}\newpage
LaTeX Warning: No verbatim text on input line 200.
LaTeX Warning: No verbatim text on input line 200.
Missing $ inserted.
<inserted text>
$
l.200 \end{axiom}\newpage
LaTeX Warning: Characters dropped after `\end{axiom}' on input line 200.
Missing $ inserted.
<inserted text>
$
l.202 ...�(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.203 \end{axiom}\newpage
LaTeX Warning: No verbatim text on input line 203.
LaTeX Warning: No verbatim text on input line 203.
Missing $ inserted.
<inserted text>
$
l.203 \end{axiom}\newpage
LaTeX Warning: Characters dropped after `\end{axiom}' on input line 203.
[4]
Missing $ inserted.
<inserted text>
$
l.206 dot(P,ω(Q,P))
Missing $ inserted.
<inserted text>
$
l.207 possible dot(ω(Q,P),ω(Q,P))::EXPR Float
Missing $ inserted.
<inserted text>
$
l.208 dot(Q,ω(P,Q))
Missing $ inserted.
<inserted text>
$
l.209 possible dot(ω(P,Q),ω(P,Q))::EXPR Float
LaTeX Warning: Characters dropped after `\end{axiom}' on input line 210.
Missing $ inserted.
<inserted text>
$
l.212 ω(vect [u0,u1,u2,u3],vect [1,0,0,0])
Missing $ inserted.
<inserted text>
$
l.213 ω(R,vect [1,0,0,0])
Missing $ inserted.
<inserted text>
$
l.214 ω(S,vect [1,0,0,0])
Missing $ inserted.
<inserted text>
$
l.215 map(Simplify, ω(U,vect [1,0,0,0]))
LaTeX Warning: Characters dropped after `\end{axiom}' on input line 216.
Missing $ inserted.
<inserted text>
$
l.218 ω(vect [1,0,0,0],S)
Missing $ inserted.
<inserted text>
$
l.219 ω(vect [1,0,0,0],R)
Missing $ inserted.
<inserted text>
$
l.220 ...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.221 \end{axiom}\newpage
LaTeX Warning: No verbatim text on input line 221.
LaTeX Warning: No verbatim text on input line 221.
Missing $ inserted.
<inserted text>
$
l.221 \end{axiom}\newpage
LaTeX Warning: Characters dropped after `\end{axiom}' on input line 221.
[5] [6]
Missing $ inserted.
<inserted text>
$
l.229 Is?(L(P,Q)*ω(P,Q) = -ω(Q,P))
Missing $ inserted.
<inserted text>
$
l.229 Is?