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last edited 10 years ago by Bill Page |
Edit detail for SandBoxLorentzTransformation revision 25 of 30
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Time: 2013/10/07 05:03:13 GMT+0 |
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| Note: tidy | ||
changed: -is?(eq:Equation Scalar):Boolean == (lhs(eq)-rhs(eq)=0)::Boolean is?(eq:Equation Scalar):Boolean == (simplify(lhs(eq)-rhs(eq))=0)::Boolean changed: -( (lhs(eq)-rhs(eq)) :: Matrix Expression AlgebraicNumber = _ (map(simplify,lhs(eq)-rhs(eq)) :: Matrix Expression AlgebraicNumber = _ added: ?c*exp(a-b)+?c*exp(b-a) == 2*c*cosh(a-b) ?c*exp(a-b)-?c*exp(b-a) == 2*c*sinh(a-b) exp2htrigs(x:Scalar):Scalar == htrigs sinhcosh simplify x 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(2*x) == 2*cosh(x)^2-1 sinh(2*x) == 2*sinh(x)*cosh(x) expandhtrigs2 := rule cosh(2*x+2*y) == 2*cosh(x+y)^2-1 sinh(2*x+2*y) == 2*sinh(x+y)*cosh(x+y) cosh(2*x-2*y) == 2*cosh(x-y)^2-1 sinh(2*x-2*y) == 2*sinh(x-y)*cosh(x-y) changed: -U:=vect [(exp(a)+exp(-a))/2,(exp(a)-exp(-a))/2,0,0] -V:=vect [(exp(b)+exp(-b))/2,(exp(b)-exp(-b))/2,0,0] -map(x+->sinhcosh(x),U) U:=vect [(exp(u)+exp(-u))/2,(exp(u)-exp(-u))/2,0,0] added: V:=vect [(exp(v)+exp(-v))/2,(exp(v)-exp(-v))/2,0,0] map(exp2htrigs,U) exp2htrigs dot(U,V) changed: -v(P,Q)==-P/dot(P,Q)-Q -v(P,Q) -v(S,T) -map(x+->htrigs sinhcosh simplify x,v(U,V)) w(P,Q)==-P/dot(P,Q)-Q w(P,Q) w(S,T) changed: -Observer P measures velocity v(Q,P). v(Q,P) is space-like In two dimensions changed: -dot(v(P,vect [1,0,0,0]),v(P,vect [1,0,0,0])) map(x+->simplify x,w(U,V)) vect [cosh(u)/cosh(u-v)-cosh(v),sinh(u)/cosh(u-v)-sinh(v),0,0] Is?(map(x+->expandhtrigs exp2htrigs x, %-w(U,V)) = vect [0,0,0,0]) added: Observer P measures velocity w(Q,P). w(Q,P) is space-like \begin{axiom} dot(w(P,vect [1,0,0,0]),w(P,vect [1,0,0,0])) \end{axiom} changed: -dot(P,v(Q,P)) -possible dot(v(Q,P),v(Q,P))::EXPR Float -dot(Q,v(P,Q)) -possible dot(v(P,Q),v(P,Q))::EXPR Float 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 changed: -v(vect [u0,u1,u2,u3],vect [1,0,0,0]) -v(R,vect [1,0,0,0]) -v(S,vect [1,0,0,0]) -map(x+->htrigs sinhcosh simplify x, v(U,vect [1,0,0,0])) 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(exp2htrigs, w(U,vect [1,0,0,0])) changed: -v(vect [1,0,0,0],S) -v(vect [1,0,0,0],R) -is?(dot(v(P,Q),v(P,Q))=dot(v(Q,P),v(Q,P))) 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))) changed: -Is?(L(P,Q)*v(P,Q) = -v(Q,P)) -map(x+->htrigs sinhcosh simplify x, L(U,V)) Is?(L(P,Q)*w(P,Q) = -w(Q,P)) added: In two dimensions \begin{axiom} map(x+->simplify expandhtrigs2 exp2htrigs x, L(U,V)) \end{axiom} changed: -is?(dot(LRPQ*v(S,Q),LRPQ*v(S,Q))=dot(L(R,Q)*v(S,Q),L(R,Q)*v(S,Q))) is?(dot(LRPQ*w(S,Q),LRPQ*w(S,Q))=dot(L(R,Q)*w(S,Q),L(R,Q)*w(S,Q))) removed: -map(simplify, L(vect [1,0,0,0], map(x+->sinhcosh x,U))) added: map(simplify, L(vect [1,0,0,0], map(exp2htrigs,U)))
Lorentz transformations.
Book by T. Matolcsi
Mathematical Preliminaries
A vector is represented as a 
matrix (column vector)
fricas
Scalar := Expression Integer
 | (1) |
Type: Type
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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
fricas
vect [a0,a1, a2, a3]
fricas
Compiling function vect with type List(Expression(Integer)) ->
Matrix(Expression(Integer))![\label{eq2}\left[
\begin{array}{c}
a 0
\
a 1
\
a 2
\
a 3](images/1362409580171013215-16.0px.png "\label{eq2}\left[
\begin{array}{c}
a 0
\
a 1
\
a 2
\
a 3") | (2) |
Type: Matrix(Expression(Integer))
Identity
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ID:=diagonalMatrix([1,1, 1, 1])
![\label{eq3}\left[
\begin{array}{cccc}
1 & 0 & 0 & 0
\
0 & 1 & 0 & 0
\
0 & 0 & 1 & 0
\
0 & 0 & 0 & 1](images/8334690762880839438-16.0px.png "\label{eq3}\left[
\begin{array}{cccc}
1 & 0 & 0 & 0
\
0 & 1 & 0 & 0
\
0 & 0 & 1 & 0
\
0 & 0 & 0 & 1") | (3) |
Type: Matrix(Integer)
Verification
fricas
possible(x)==subst(x,map(y+->(y=(random(100) - random(100))), variables x) )
Type: Void
fricas
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
fricas
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
fricas
sinhcosh := rule ?c*exp(a)+?c*exp(-a) == 2*c*cosh(a) ?c*exp(a)-?c*exp(-a) == 2*c*sinh(a) ?c*exp(a-b)+?c*exp(b-a) == 2*c*cosh(a-b) ?c*exp(a-b)-?c*exp(b-a) == 2*c*sinh(a-b)
![\label{eq4}\begin{array}{@{}l}
\displaystyle
\left\{{{{c \ {{e}^{a}}}+{c \ {{e}^{- a}}}+ \%B}\mbox{\rm = =}{{2 \ c \ {\cosh \left({a}\right)}}+ \%B}}, \: \right.
\
\
\displaystyle
\left.{{{c \ {{e}^{a}}}-{c \ {{e}^{- a}}}+ \%C}\mbox{\rm = =}{{2 \ c \ {\sinh \left({a}\right)}}+ \%C}}, \: \right.
\
\
\displaystyle
\left.{{{c \ {{e}^{b - a}}}+{c \ {{e}^{- b + a}}}+ \%D}\mbox{\rm = =}{{2 \ c \ {\cosh \left({b - a}\right)}}+ \%D}}, \: \right.
\
\
\displaystyle
\left.{{-{c \ {{e}^{b - a}}}+{c \ {{e}^{- b + a}}}+ \%E}\mbox{\rm = =}{-{2 \ c \ {\sinh \left({b - a}\right)}}+ \%E}}\right\}](images/8633220776302982309-16.0px.png "\label{eq4}\begin{array}{@{}l}
\displaystyle
\left\{{{{c \ {{e}^{a}}}+{c \ {{e}^{- a}}}+ \%B}\mbox{\rm = =}{{2 \ c \ {\cosh \left({a}\right)}}+ \%B}}, \: \right.
\
\
\displaystyle
\left.{{{c \ {{e}^{a}}}-{c \ {{e}^{- a}}}+ \%C}\mbox{\rm = =}{{2 \ c \ {\sinh \left({a}\right)}}+ \%C}}, \: \right.
\
\
\displaystyle
\left.{{{c \ {{e}^{b - a}}}+{c \ {{e}^{- b + a}}}+ \%D}\mbox{\rm = =}{{2 \ c \ {\cosh \left({b - a}\right)}}+ \%D}}, \: \right.
\
\
\displaystyle
\left.{{-{c \ {{e}^{b - a}}}+{c \ {{e}^{- b + a}}}+ \%E}\mbox{\rm = =}{-{2 \ c \ {\sinh \left({b - a}\right)}}+ \%E}}\right\}") | (4) |
Type: Ruleset(Integer,
fricas
exp2htrigs(x:Scalar):Scalar == htrigs sinhcosh simplify x
Function declaration exp2htrigs : Expression(Integer) -> Expression( Integer) has been added to workspace.
Type: Void
fricas
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(2*x) == 2*cosh(x)^2-1 sinh(2*x) == 2*sinh(x)*cosh(x)
![\label{eq5}\begin{array}{@{}l}
\displaystyle
\left\{{{\cosh \left({y + x}\right)}\mbox{\rm = =}{{{\sinh \left({x}\right)}\ {\sinh \left({y}\right)}}+{{\cosh \left({x}\right)}\ {\cosh \left({y}\right)}}}}, \: \right.
\
\
\displaystyle
\left.{{\sinh \left({y + x}\right)}\mbox{\rm = =}{{{\cosh \left({x}\right)}\ {\sinh \left({y}\right)}}+{{\cosh \left({y}\right)}\ {\sinh \left({x}\right)}}}}, \: \right.
\
\
\displaystyle
\left.{{\cosh \left({2 \ x}\right)}\mbox{\rm = =}{{2 \ {{\cosh \left({x}\right)}^{2}}}- 1}}, \: \right.
\
\
\displaystyle
\left.{{\sinh \left({2 \ x}\right)}\mbox{\rm = =}{2 \ {\cosh \left({x}\right)}\ {\sinh \left({x}\right)}}}\right\}](images/659511536270971872-16.0px.png "\label{eq5}\begin{array}{@{}l}
\displaystyle
\left\{{{\cosh \left({y + x}\right)}\mbox{\rm = =}{{{\sinh \left({x}\right)}\ {\sinh \left({y}\right)}}+{{\cosh \left({x}\right)}\ {\cosh \left({y}\right)}}}}, \: \right.
\
\
\displaystyle
\left.{{\sinh \left({y + x}\right)}\mbox{\rm = =}{{{\cosh \left({x}\right)}\ {\sinh \left({y}\right)}}+{{\cosh \left({y}\right)}\ {\sinh \left({x}\right)}}}}, \: \right.
\
\
\displaystyle
\left.{{\cosh \left({2 \ x}\right)}\mbox{\rm = =}{{2 \ {{\cosh \left({x}\right)}^{2}}}- 1}}, \: \right.
\
\
\displaystyle
\left.{{\sinh \left({2 \ x}\right)}\mbox{\rm = =}{2 \ {\cosh \left({x}\right)}\ {\sinh \left({x}\right)}}}\right\}") | (5) |
Type: Ruleset(Integer,
fricas
expandhtrigs2 := rule cosh(2*x+2*y) == 2*cosh(x+y)^2-1 sinh(2*x+2*y) == 2*sinh(x+y)*cosh(x+y) cosh(2*x-2*y) == 2*cosh(x-y)^2-1 sinh(2*x-2*y) == 2*sinh(x-y)*cosh(x-y)
![\label{eq6}\begin{array}{@{}l}
\displaystyle
\left\{{{\cosh \left({{2 \ y}+{2 \ x}}\right)}\mbox{\rm = =}{{2 \ {{\cosh \left({y + x}\right)}^{2}}}- 1}}, \: \right.
\
\
\displaystyle
\left.{{\sinh \left({{2 \ y}+{2 \ x}}\right)}\mbox{\rm = =}{2 \ {\cosh \left({y + x}\right)}\ {\sinh \left({y + x}\right)}}}, \: \right.
\
\
\displaystyle
\left.{{\cosh \left({{2 \ y}-{2 \ x}}\right)}\mbox{\rm = =}{{2 \ {{\cosh \left({y - x}\right)}^{2}}}- 1}}, \: \right.
\
\
\displaystyle
\left.{-{\%F \ {\sinh \left({{2 \ y}-{2 \ x}}\right)}}\mbox{\rm = =}-{2 \ \%F \ {\cosh \left({y - x}\right)}\ {\sinh \left({y - x}\right)}}}\right\}](images/1439689746051812881-16.0px.png "\label{eq6}\begin{array}{@{}l}
\displaystyle
\left\{{{\cosh \left({{2 \ y}+{2 \ x}}\right)}\mbox{\rm = =}{{2 \ {{\cosh \left({y + x}\right)}^{2}}}- 1}}, \: \right.
\
\
\displaystyle
\left.{{\sinh \left({{2 \ y}+{2 \ x}}\right)}\mbox{\rm = =}{2 \ {\cosh \left({y + x}\right)}\ {\sinh \left({y + x}\right)}}}, \: \right.
\
\
\displaystyle
\left.{{\cosh \left({{2 \ y}-{2 \ x}}\right)}\mbox{\rm = =}{{2 \ {{\cosh \left({y - x}\right)}^{2}}}- 1}}, \: \right.
\
\
\displaystyle
\left.{-{\%F \ {\sinh \left({{2 \ y}-{2 \ x}}\right)}}\mbox{\rm = =}-{2 \ \%F \ {\cosh \left({y - x}\right)}\ {\sinh \left({y - x}\right)}}}\right\}") | (6) |
Type: Ruleset(Integer,
Lorentz Form (metric)
fricas
G:=diagonalMatrix [-1,1, 1, 1]
![\label{eq7}\left[
\begin{array}{cccc}
- 1 & 0 & 0 & 0
\
0 & 1 & 0 & 0
\
0 & 0 & 1 & 0
\
0 & 0 & 0 & 1](images/8214188165138195097-16.0px.png "\label{eq7}\left[
\begin{array}{cccc}
- 1 & 0 & 0 & 0
\
0 & 1 & 0 & 0
\
0 & 0 & 1 & 0
\
0 & 0 & 0 & 1") | (7) |
Type: Matrix(Integer)
applied to a vector produces a co-vector (represent as a 
matrix or row vector)
fricas
g(x) == transpose(x)*G
Type: Void
fricas
g(vect [a0,a1, a2, a3])
fricas
Compiling function g with type Matrix(Expression(Integer)) -> Matrix
(Expression(Integer)) | (8) |
Type: Matrix(Expression(Integer))
Scalar product
fricas
dot(x,y) == (g(x)*y)::Scalar
Type: Void
fricas
dot(vect [a0,a1, a2, a3], vect [b0, b1, b2, b3])
fricas
Compiling function dot with type (Matrix(Expression(Integer)),Matrix (Expression(Integer))) -> Expression(Integer)
 | (9) |
Type: Expression(Integer)
Tensor product
fricas
tensor(x,y) == x*g(y)
Type: Void
fricas
tensor(vect [a0,a1, a2, a3], vect [b0, b1, b2, b3])
fricas
Compiling function tensor with type (Matrix(Expression(Integer)),Matrix(Expression(Integer))) -> Matrix(Expression(Integer))
![\label{eq10}\left[
\begin{array}{cccc}
-{a 0 \ b 0}&{a 0 \ b 1}&{a 0 \ b 2}&{a 0 \ b 3}
\
-{a 1 \ b 0}&{a 1 \ b 1}&{a 1 \ b 2}&{a 1 \ b 3}
\
-{a 2 \ b 0}&{a 2 \ b 1}&{a 2 \ b 2}&{a 2 \ b 3}
\
-{a 3 \ b 0}&{a 3 \ b 1}&{a 3 \ b 2}&{a 3 \ b 3}](images/8531148274414574673-16.0px.png "\label{eq10}\left[
\begin{array}{cccc}
-{a 0 \ b 0}&{a 0 \ b 1}&{a 0 \ b 2}&{a 0 \ b 3}
\
-{a 1 \ b 0}&{a 1 \ b 1}&{a 1 \ b 2}&{a 1 \ b 3}
\
-{a 2 \ b 0}&{a 2 \ b 1}&{a 2 \ b 2}&{a 2 \ b 3}
\
-{a 3 \ b 0}&{a 3 \ b 1}&{a 3 \ b 2}&{a 3 \ b 3}") | (10) |
Type: Matrix(Expression(Integer))
Massive Objects
A material object (also referred to as an observer) is represented by a time-like 4-vector
fricas
P:=vect [sqrt(p1^2+p2^2+p3^2+1),-p1, -p2, -p3];
Type: Matrix(Expression(Integer))
fricas
dot(P,P)
 | (11) |
Type: Expression(Integer)
fricas
Q:=vect [sqrt(q1^2+q2^2+q3^2+1),-q1, -q2, -q3];
Type: Matrix(Expression(Integer))
fricas
R:=vect [sqrt(r1^2+r2^2+r3^2+1),-r1, -r2, -r3];
Type: Matrix(Expression(Integer))
fricas
S:=1/sqrt(1-s1^2-s2^2-s3^2)*vect [1,-s1, -s2, -s3]
![\label{eq12}\left[
\begin{array}{c}
{1 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
-{s 1 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
-{s 2 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
-{s 3 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}](images/6129695386615868967-16.0px.png "\label{eq12}\left[
\begin{array}{c}
{1 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
-{s 1 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
-{s 2 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
-{s 3 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}") | (12) |
Type: Matrix(Expression(Integer))
fricas
dot(S,S)
 | (13) |
Type: Expression(Integer)
fricas
T:=1/sqrt(1-t1^2-t2^2-t3^2)*vect [1,-t1, -t2, -t3]
![\label{eq14}\left[
\begin{array}{c}
{1 \over{\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}
\
-{t 1 \over{\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}
\
-{t 2 \over{\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}
\
-{t 3 \over{\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}](images/1091758118900281032-16.0px.png "\label{eq14}\left[
\begin{array}{c}
{1 \over{\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}
\
-{t 1 \over{\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}
\
-{t 2 \over{\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}
\
-{t 3 \over{\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}") | (14) |
Type: Matrix(Expression(Integer))
fricas
U:=vect [(exp(u)+exp(-u))/2,(exp(u)-exp(-u))/2, 0, 0]
![\label{eq15}\left[
\begin{array}{c}
{{{{e}^{u}}+{{e}^{- u}}}\over 2}
\
{{{{e}^{u}}-{{e}^{- u}}}\over 2}
\
0
\
0](images/3999390877348244638-16.0px.png "\label{eq15}\left[
\begin{array}{c}
{{{{e}^{u}}+{{e}^{- u}}}\over 2}
\
{{{{e}^{u}}-{{e}^{- u}}}\over 2}
\
0
\
0") | (15) |
Type: Matrix(Expression(Integer))
fricas
simplify dot(U,U)
 | (16) |
Type: Expression(Integer)
fricas
V:=vect [(exp(v)+exp(-v))/2,(exp(v)-exp(-v))/2, 0, 0]
![\label{eq17}\left[
\begin{array}{c}
{{{{e}^{v}}+{{e}^{- v}}}\over 2}
\
{{{{e}^{v}}-{{e}^{- v}}}\over 2}
\
0
\
0](images/9114959777155255580-16.0px.png "\label{eq17}\left[
\begin{array}{c}
{{{{e}^{v}}+{{e}^{- v}}}\over 2}
\
{{{{e}^{v}}-{{e}^{- v}}}\over 2}
\
0
\
0") | (17) |
Type: Matrix(Expression(Integer))
fricas
map(exp2htrigs,U)
fricas
Compiling function exp2htrigs with type Expression(Integer) ->
Expression(Integer)![\label{eq18}\left[
\begin{array}{c}
{\cosh \left({u}\right)}
\
{\sinh \left({u}\right)}
\
0
\
0](images/1630795346368502284-16.0px.png "\label{eq18}\left[
\begin{array}{c}
{\cosh \left({u}\right)}
\
{\sinh \left({u}\right)}
\
0
\
0") | (18) |
Type: Matrix(Expression(Integer))
fricas
exp2htrigs dot(U,V)
 | (19) |
Type: Expression(Integer)
Observer "at rest"
fricas
vect [1,0, 0, 0]
![\label{eq20}\left[
\begin{array}{c}
1
\
0
\
0
\
0](images/8391009454962740385-16.0px.png "\label{eq20}\left[
\begin{array}{c}
1
\
0
\
0
\
0") | (20) |
Type: Matrix(Expression(Integer))
fricas
dot(%,%)
 | (21) |
Type: Expression(Integer)
Associated with each such vector is the orthogonal 3-d Euclidean subspace
Relative Velocity
An object P has a unique relative velocity w(P,Q) with respect to object Q given by
fricas
w(P,Q)==-P/dot(P, Q)-Q
Type: Void
fricas
w(P,Q)
fricas
Compiling function w with type (Matrix(Expression(Integer)),Matrix( Expression(Integer))) -> Matrix(Expression(Integer))
![\label{eq22}\left[
\begin{array}{c}
{{{{\left({p 3 \ q 3}+{p 2 \ q 2}+{p 1 \ q 1}\right)}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}+{{\left(-{{q 3}^{2}}-{{q 2}^{2}}-{{q 1}^{2}}\right)}\ {\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}}}\over{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ q 3}-{p 2 \ q 2}-{p 1 \ q 1}}}
\
{{{q 1 \ {\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ q 1 \ q 3}-{p 2 \ q 1 \ q 2}-{p 1 \ {{q 1}^{2}}}- p 1}\over{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ q 3}-{p 2 \ q 2}-{p 1 \ q 1}}}
\
{{{q 2 \ {\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ q 2 \ q 3}-{p 2 \ {{q 2}^{2}}}-{p 1 \ q 1 \ q 2}- p 2}\over{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ q 3}-{p 2 \ q 2}-{p 1 \ q 1}}}
\
{{{q 3 \ {\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ {{q 3}^{2}}}+{{\left(-{p 2 \ q 2}-{p 1 \ q 1}\right)}\ q 3}- p 3}\over{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ q 3}-{p 2 \ q 2}-{p 1 \ q 1}}}](images/2763553504490017023-16.0px.png "\label{eq22}\left[
\begin{array}{c}
{{{{\left({p 3 \ q 3}+{p 2 \ q 2}+{p 1 \ q 1}\right)}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}+{{\left(-{{q 3}^{2}}-{{q 2}^{2}}-{{q 1}^{2}}\right)}\ {\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}}}\over{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ q 3}-{p 2 \ q 2}-{p 1 \ q 1}}}
\
{{{q 1 \ {\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ q 1 \ q 3}-{p 2 \ q 1 \ q 2}-{p 1 \ {{q 1}^{2}}}- p 1}\over{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ q 3}-{p 2 \ q 2}-{p 1 \ q 1}}}
\
{{{q 2 \ {\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ q 2 \ q 3}-{p 2 \ {{q 2}^{2}}}-{p 1 \ q 1 \ q 2}- p 2}\over{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ q 3}-{p 2 \ q 2}-{p 1 \ q 1}}}
\
{{{q 3 \ {\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ {{q 3}^{2}}}+{{\left(-{p 2 \ q 2}-{p 1 \ q 1}\right)}\ q 3}- p 3}\over{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}\ {\sqrt{{{q 3}^{2}}+{{q 2}^{2}}+{{q 1}^{2}}+ 1}}}-{p 3 \ q 3}-{p 2 \ q 2}-{p 1 \ q 1}}}") | (22) |
Type: Matrix(Expression(Integer))
fricas
w(S,T)
![\label{eq23}\left[
\begin{array}{c}
{{{{t 3}^{2}}-{s 3 \ t 3}+{{t 2}^{2}}-{s 2 \ t 2}+{{t 1}^{2}}-{s 1 \ t 1}}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}
\
{{-{s 1 \ {{t 3}^{2}}}+{s 3 \ t 1 \ t 3}-{s 1 \ {{t 2}^{2}}}+{s 2 \ t 1 \ t 2}- t 1 + s 1}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}
\
{{-{s 2 \ {{t 3}^{2}}}+{s 3 \ t 2 \ t 3}+{{\left({s 1 \ t 1}- 1 \right)}\ t 2}-{s 2 \ {{t 1}^{2}}}+ s 2}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}
\
{{{{\left({s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ t 3}-{s 3 \ {{t 2}^{2}}}-{s 3 \ {{t 1}^{2}}}+ s 3}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}](images/4371568151294734188-16.0px.png "\label{eq23}\left[
\begin{array}{c}
{{{{t 3}^{2}}-{s 3 \ t 3}+{{t 2}^{2}}-{s 2 \ t 2}+{{t 1}^{2}}-{s 1 \ t 1}}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}
\
{{-{s 1 \ {{t 3}^{2}}}+{s 3 \ t 1 \ t 3}-{s 1 \ {{t 2}^{2}}}+{s 2 \ t 1 \ t 2}- t 1 + s 1}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}
\
{{-{s 2 \ {{t 3}^{2}}}+{s 3 \ t 2 \ t 3}+{{\left({s 1 \ t 1}- 1 \right)}\ t 2}-{s 2 \ {{t 1}^{2}}}+ s 2}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}
\
{{{{\left({s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ t 3}-{s 3 \ {{t 2}^{2}}}-{s 3 \ {{t 1}^{2}}}+ s 3}\over{{\left({s 3 \ t 3}+{s 2 \ t 2}+{s 1 \ t 1}- 1 \right)}\ {\sqrt{-{{t 3}^{2}}-{{t 2}^{2}}-{{t 1}^{2}}+ 1}}}}") | (23) |
Type: Matrix(Expression(Integer))
In two dimensions
fricas
map(x+->simplify x,w(U, V))
![\label{eq24}\left[
\begin{array}{c}
{{-{{e}^{{2 \ v}- u}}+{{e}^{u}}+{{e}^{- u}}-{{e}^{-{2 \ v}+ u}}}\over{{2 \ {{e}^{v - u}}}+{2 \ {{e}^{- v + u}}}}}
\
{{-{{e}^{{2 \ v}- u}}+{{e}^{u}}-{{e}^{- u}}+{{e}^{-{2 \ v}+ u}}}\over{{2 \ {{e}^{v - u}}}+{2 \ {{e}^{- v + u}}}}}
\
0
\
0](images/4637317555360539646-16.0px.png "\label{eq24}\left[
\begin{array}{c}
{{-{{e}^{{2 \ v}- u}}+{{e}^{u}}+{{e}^{- u}}-{{e}^{-{2 \ v}+ u}}}\over{{2 \ {{e}^{v - u}}}+{2 \ {{e}^{- v + u}}}}}
\
{{-{{e}^{{2 \ v}- u}}+{{e}^{u}}-{{e}^{- u}}+{{e}^{-{2 \ v}+ u}}}\over{{2 \ {{e}^{v - u}}}+{2 \ {{e}^{- v + u}}}}}
\
0
\
0") | (24) |
Type: Matrix(Expression(Integer))
fricas
vect [cosh(u)/cosh(u-v)-cosh(v),sinh(u)/cosh(u-v)-sinh(v), 0, 0]
![\label{eq25}\left[
\begin{array}{c}
{{-{{\cosh \left({v - u}\right)}\ {\cosh \left({v}\right)}}+{\cosh \left({u}\right)}}\over{\cosh \left({v - u}\right)}}
\
{{-{{\cosh \left({v - u}\right)}\ {\sinh \left({v}\right)}}+{\sinh \left({u}\right)}}\over{\cosh \left({v - u}\right)}}
\
0
\
0](images/3986632354267595069-16.0px.png "\label{eq25}\left[
\begin{array}{c}
{{-{{\cosh \left({v - u}\right)}\ {\cosh \left({v}\right)}}+{\cosh \left({u}\right)}}\over{\cosh \left({v - u}\right)}}
\
{{-{{\cosh \left({v - u}\right)}\ {\sinh \left({v}\right)}}+{\sinh \left({u}\right)}}\over{\cosh \left({v - u}\right)}}
\
0
\
0") | (25) |
Type: Matrix(Expression(Integer))
fricas
Is?(map(x+->expandhtrigs exp2htrigs x,%-w(U, V)) = vect [0, 0, 0, 0])
fricas
Compiling function Is? with type Equation(Matrix(Expression(Integer)
)) -> Boolean | (26) |
Type: Boolean
Observer P measures velocity w(Q,P). w(Q,P) is space-like
fricas
dot(w(P,vect [1, 0, 0, 0]), w(P, vect [1, 0, 0, 0]))
 | (27) |
Type: Expression(Integer)
and in 
fricas
dot(P,w(Q, P))
 | (28) |
Type: Expression(Integer)
fricas
possible dot(w(Q,P), w(Q, P))::EXPR Float
fricas
Compiling function possible with type Expression(Integer) ->
Expression(Integer) | (29) |
Type: Expression(Float)
fricas
dot(Q,w(P, Q))
 | (30) |
Type: Expression(Integer)
fricas
possible dot(w(P,Q), w(P, Q))::EXPR Float
 | (31) |
Type: Expression(Float)
Velocity with respect to observer "at rest"
fricas
w(vect [u0,u1, u2, u3], vect [1, 0, 0, 0])
![\label{eq32}\left[
\begin{array}{c}
0
\
{u 1 \over u 0}
\
{u 2 \over u 0}
\
{u 3 \over u 0}](images/7639508565520336058-16.0px.png "\label{eq32}\left[
\begin{array}{c}
0
\
{u 1 \over u 0}
\
{u 2 \over u 0}
\
{u 3 \over u 0}") | (32) |
Type: Matrix(Expression(Integer))
fricas
w(R,vect [1, 0, 0, 0])
![\label{eq33}\left[
\begin{array}{c}
0
\
-{r 1 \over{\sqrt{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}+ 1}}}
\
-{r 2 \over{\sqrt{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}+ 1}}}
\
-{r 3 \over{\sqrt{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}+ 1}}}](images/1749565925610258872-16.0px.png "\label{eq33}\left[
\begin{array}{c}
0
\
-{r 1 \over{\sqrt{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}+ 1}}}
\
-{r 2 \over{\sqrt{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}+ 1}}}
\
-{r 3 \over{\sqrt{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}+ 1}}}") | (33) |
Type: Matrix(Expression(Integer))
fricas
w(S,vect [1, 0, 0, 0])
![\label{eq34}\left[
\begin{array}{c}
0
\
- s 1
\
- s 2
\
- s 3](images/3859192686881148289-16.0px.png "\label{eq34}\left[
\begin{array}{c}
0
\
- s 1
\
- s 2
\
- s 3") | (34) |
Type: Matrix(Expression(Integer))
fricas
map(exp2htrigs,w(U, vect [1, 0, 0, 0]))
![\label{eq35}\left[
\begin{array}{c}
0
\
{{\sinh \left({u}\right)}\over{\cosh \left({u}\right)}}
\
0
\
0](images/3131737043301514960-16.0px.png "\label{eq35}\left[
\begin{array}{c}
0
\
{{\sinh \left({u}\right)}\over{\cosh \left({u}\right)}}
\
0
\
0") | (35) |
Type: Matrix(Expression(Integer))
Non-reciprocal velocities
fricas
w(vect [1,0, 0, 0], S)
![\label{eq36}\left[
\begin{array}{c}
{{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}}\over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
{s 1 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
{s 2 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
{s 3 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}](images/6731816699764726386-16.0px.png "\label{eq36}\left[
\begin{array}{c}
{{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}}\over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
{s 1 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
{s 2 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}
\
{s 3 \over{\sqrt{-{{s 3}^{2}}-{{s 2}^{2}}-{{s 1}^{2}}+ 1}}}") | (36) |
Type: Matrix(Expression(Integer))
fricas
w(vect [1,0, 0, 0], R)
![\label{eq37}\left[
\begin{array}{c}
{{-{{r 3}^{2}}-{{r 2}^{2}}-{{r 1}^{2}}}\over{\sqrt{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}+ 1}}}
\
r 1
\
r 2
\
r 3](images/157767678132595531-16.0px.png "\label{eq37}\left[
\begin{array}{c}
{{-{{r 3}^{2}}-{{r 2}^{2}}-{{r 1}^{2}}}\over{\sqrt{{{r 3}^{2}}+{{r 2}^{2}}+{{r 1}^{2}}+ 1}}}
\
r 1
\
r 2
\
r 3") | (37) |
Type: Matrix(Expression(Integer))
fricas
is?(dot(w(P,Q), w(P, Q))=dot(w(Q, P), w(Q, P)))
fricas
Compiling function is? with type Equation(Expression(Integer)) ->
Boolean | (38) |
Type: Boolean
Lorentz Boost
is a linear bijection 
that preserves
and maps orthogonal compliments into each other.
fricas
L(P,Q) == ID + tensor(P+Q, P+Q)/(1-dot(P, Q)) - 2*tensor(P, Q)
Type: Void
fricas
Is?(L(P,P) = ID)
fricas
Compiling function L with type (Matrix(Expression(Integer)),Matrix( Expression(Integer))) -> Matrix(Expression(Integer))
 | (39) |
Type: Boolean
fricas
Is?(L(P,Q)*L(Q, P) = ID)
 | (40) |
Type: Boolean
fricas
Is?(L(P,Q)*Q=P)
 | (41) |
Type: Boolean
fricas
Is?(L(P,Q)*w(P, Q) = -w(Q, P))
 | (42) |
Type: Boolean
In two dimensions
fricas
map(x+->simplify expandhtrigs2 exp2htrigs x,L(U, V))
![\label{eq43}\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](images/7589217129323424952-16.0px.png "\label{eq43}\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") | (43) |
Type: Matrix(Expression(Integer))
Composition of two Lorentz boosts is not a Lorentz boost unless all three observers are in the same plane.
fricas
Is?(L(R,P)*L(P, Q) = L(R, Q))
 | (44) |
Type: Boolean
fricas
RQ:=a*R+b*Q;
Type: Matrix(Expression(Integer))
fricas
rq:=solve(dot(RQ,RQ)=-1, b); #rq
 | (45) |
Type: PositiveInteger?
fricas
RQ1:=eval(RQ,rq.1);
Type: Matrix(Expression(Integer))
fricas
dot(RQ1,RQ1)
 | (46) |
Type: Expression(Integer)
fricas
Is?(L(R,RQ1)*L(RQ1, Q) = L(R, Q))
 | (47) |
Type: Boolean
fricas
RQ2:=eval(RQ,rq.2);
Type: Matrix(Expression(Integer))
fricas
Is?(RQ1=RQ2)
 | (48) |
Type: Boolean
fricas
dot(RQ2,RQ2)
 | (49) |
Type: Expression(Integer)
fricas
Is?(L(R,RQ2)*L(RQ2, Q) = L(R, Q))
 | (50) |
Type: Boolean
but the composition does preserve observers and magnitudes
fricas
LRPQ := L(R,P)*L(P, Q);
Type: Matrix(Expression(Integer))
fricas
Is?(LRPQ*Q = L(R,Q)*Q)
 | (51) |
Type: Boolean
fricas
is?(dot(LRPQ*w(S,Q), LRPQ*w(S, Q))=dot(L(R, Q)*w(S, Q), L(R, Q)*w(S, Q)))
 | (52) |
Type: Boolean
Lorentz boost with respect to observer "at rest"
fricas
LT:=L(vect [1,0, 0, 0], vect [u0, -u1, -u2, -u3])
![\label{eq53}\left[
\begin{array}{cccc}
u 0 & u 1 & u 2 & u 3
\
u 1 &{{{{u 1}^{2}}+ u 0 + 1}\over{u 0 + 1}}&{{u 1 \ u 2}\over{u 0 + 1}}&{{u 1 \ u 3}\over{u 0 + 1}}
\
u 2 &{{u 1 \ u 2}\over{u 0 + 1}}&{{{{u 2}^{2}}+ u 0 + 1}\over{u 0 + 1}}&{{u 2 \ u 3}\over{u 0 + 1}}
\
u 3 &{{u 1 \ u 3}\over{u 0 + 1}}&{{u 2 \ u 3}\over{u 0 + 1}}&{{{{u 3}^{2}}+ u 0 + 1}\over{u 0 + 1}}](images/1411280327353333722-16.0px.png "\label{eq53}\left[
\begin{array}{cccc}
u 0 & u 1 & u 2 & u 3
\
u 1 &{{{{u 1}^{2}}+ u 0 + 1}\over{u 0 + 1}}&{{u 1 \ u 2}\over{u 0 + 1}}&{{u 1 \ u 3}\over{u 0 + 1}}
\
u 2 &{{u 1 \ u 2}\over{u 0 + 1}}&{{{{u 2}^{2}}+ u 0 + 1}\over{u 0 + 1}}&{{u 2 \ u 3}\over{u 0 + 1}}
\
u 3 &{{u 1 \ u 3}\over{u 0 + 1}}&{{u 2 \ u 3}\over{u 0 + 1}}&{{{{u 3}^{2}}+ u 0 + 1}\over{u 0 + 1}}") | (53) |
Type: Matrix(Expression(Integer))
fricas
map(simplify,L(vect [1, 0, 0, 0], map(exp2htrigs, U)))
![\label{eq54}\left[
\begin{array}{cccc}
{\cosh \left({u}\right)}& -{\sinh \left({u}\right)}& 0 & 0
\
-{\sinh \left({u}\right)}&{\cosh \left({u}\right)}& 0 & 0
\
0 & 0 & 1 & 0
\
0 & 0 & 0 & 1](images/5284046883741507212-16.0px.png "\label{eq54}\left[
\begin{array}{cccc}
{\cosh \left({u}\right)}& -{\sinh \left({u}\right)}& 0 & 0
\
-{\sinh \left({u}\right)}&{\cosh \left({u}\right)}& 0 & 0
\
0 & 0 & 1 & 0
\
0 & 0 & 0 & 1") | (54) |
Type: Matrix(Expression(Integer))
Two dimensional Lorentz Transformation
fricas
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]]
![\label{eq55}\left[
\begin{array}{cccc}
{1 \over{\sqrt{-{{v'}^{2}}+ 1}}}&{v' \over{\sqrt{-{{v'}^{2}}+ 1}}}& 0 & 0
\
{v' \over{\sqrt{-{{v'}^{2}}+ 1}}}&{1 \over{\sqrt{-{{v'}^{2}}+ 1}}}& 0 & 0
\
0 & 0 & 1 & 0
\
0 & 0 & 0 & 1](images/3039517637270043312-16.0px.png "\label{eq55}\left[
\begin{array}{cccc}
{1 \over{\sqrt{-{{v'}^{2}}+ 1}}}&{v' \over{\sqrt{-{{v'}^{2}}+ 1}}}& 0 & 0
\
{v' \over{\sqrt{-{{v'}^{2}}+ 1}}}&{1 \over{\sqrt{-{{v'}^{2}}+ 1}}}& 0 & 0
\
0 & 0 & 1 & 0
\
0 & 0 & 0 & 1") | (55) |
Type: Matrix(Expression(Integer))
fricas
Is?(%=map(x+->eval(x,[u0=1/sqrt(1-v'^2), u1=v'/sqrt(1-v'^2), u2=0, u3=0]), LT))
 | (56) |
Type: Boolean