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Edit detail for SandBoxLorentzTransformation revision 24 of 30

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Editor: page
Time: 2013/10/05 10:09:06 GMT+0
Note: Lorentz transformation in one dimension

added:
map(x+->htrigs sinhcosh simplify x, L(U,V))

changed:
-map(simplify, L(vect [1,0,0,0], U))
map(simplify, L(vect [1,0,0,0], map(x+->sinhcosh x,U)))

Lorentz transformations.

Book by T. Matolcsi

Mathematical Preliminaries

A vector is represented as a n\times 1 matrix (column vector)

fricas
Scalar := Expression Integer

\label{eq1}\hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ })(1)
Type: Type
fricas
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 
(2)
Type: Matrix(Expression(Integer))

Identity

fricas
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 
(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 == (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 == _
( (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)

\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\} 
(4)
Type: Ruleset(Integer,Integer,Expression(Integer))

Lorentz Form (metric)

fricas
G:=diagonalMatrix [-1,1,1,1]

\label{eq5}\left[ 
\begin{array}{cccc}
- 1 & 0 & 0 & 0 
\
0 & 1 & 0 & 0 
\
0 & 0 & 1 & 0 
\
0 & 0 & 0 & 1 
(5)
Type: Matrix(Integer)

applied to a vector produces a co-vector (represent as a 1\times n 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))

\label{eq6}\left[ 
\begin{array}{cccc}
- a 0 & a 1 & a 2 & a 3 
(6)
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)

\label{eq7}{a 3 \  b 3}+{a 2 \  b 2}+{a 1 \  b 1}-{a 0 \  b 0}(7)
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{eq8}\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}
(8)
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)

\label{eq9}- 1(9)
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{eq10}\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}}}
(10)
Type: Matrix(Expression(Integer))
fricas
dot(S,S)

\label{eq11}- 1(11)
Type: Expression(Integer)
fricas
T:=1/sqrt(1-t1^2-t2^2-t3^2)*vect [1,-t1,-t2,-t3]

\label{eq12}\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}}}
(12)
Type: Matrix(Expression(Integer))
fricas
U:=vect [(exp(a)+exp(-a))/2,(exp(a)-exp(-a))/2,0,0]

\label{eq13}\left[ 
\begin{array}{c}
{{{{e}^{a}}+{{e}^{- a}}}\over 2}
\
{{{{e}^{a}}-{{e}^{- a}}}\over 2}
\
0 
\
0 
(13)
Type: Matrix(Expression(Integer))
fricas
V:=vect [(exp(b)+exp(-b))/2,(exp(b)-exp(-b))/2,0,0]

\label{eq14}\left[ 
\begin{array}{c}
{{{{e}^{b}}+{{e}^{- b}}}\over 2}
\
{{{{e}^{b}}-{{e}^{- b}}}\over 2}
\
0 
\
0 
(14)
Type: Matrix(Expression(Integer))
fricas
map(x+->sinhcosh(x),U)

\label{eq15}\left[ 
\begin{array}{c}
{\cosh \left({a}\right)}
\
{\sinh \left({a}\right)}
\
0 
\
0 
(15)
Type: Matrix(Expression(Integer))
fricas
simplify dot(U,U)

\label{eq16}- 1(16)
Type: Expression(Integer)

Observer "at rest"

fricas
vect [1,0,0,0]

\label{eq17}\left[ 
\begin{array}{c}
1 
\
0 
\
0 
\
0 
(17)
Type: Matrix(Expression(Integer))
fricas
dot(%,%)

\label{eq18}- 1(18)
Type: Expression(Integer)

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 w(P,Q) with respect to object Q given by

fricas
v(P,Q)==-P/dot(P,Q)-Q
Type: Void
fricas
v(P,Q)
fricas
Compiling function v with type (Matrix(Expression(Integer)),Matrix(
      Expression(Integer))) -> Matrix(Expression(Integer))

\label{eq19}\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}}}
(19)
Type: Matrix(Expression(Integer))
fricas
v(S,T)

\label{eq20}\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}}}}
(20)
Type: Matrix(Expression(Integer))
fricas
map(x+->htrigs sinhcosh simplify x,v(U,V))

\label{eq21}\left[ 
\begin{array}{c}
{{-{\cosh \left({{2 \  b}- a}\right)}+{\cosh \left({a}\right)}}\over{2 \ {\cosh \left({b - a}\right)}}}
\
{{-{\sinh \left({{2 \  b}- a}\right)}+{\sinh \left({a}\right)}}\over{2 \ {\cosh \left({b - a}\right)}}}
\
0 
\
0 
(21)
Type: Matrix(Expression(Integer))

Observer P measures velocity v(Q,P). v(Q,P) is space-like

fricas
dot(v(P,vect [1,0,0,0]),v(P,vect [1,0,0,0]))

\label{eq22}{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}}\over{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}(22)
Type: Expression(Integer)

and in E_P

fricas
dot(P,v(Q,P))

\label{eq23}0(23)
Type: Expression(Integer)
fricas
possible dot(v(Q,P),v(Q,P))::EXPR Float
fricas
Compiling function possible with type Expression(Integer) -> 
      Expression(Integer)

\label{eq24}0.9999980018_5134053903(24)
Type: Expression(Float)
fricas
dot(Q,v(P,Q))

\label{eq25}0(25)
Type: Expression(Integer)
fricas
possible dot(v(P,Q),v(P,Q))::EXPR Float

\label{eq26}0.9999987845_1330937797(26)
Type: Expression(Float)

Velocity with respect to observer "at rest"

fricas
v(vect [u0,u1,u2,u3],vect [1,0,0,0])

\label{eq27}\left[ 
\begin{array}{c}
0 
\
{u 1 \over u 0}
\
{u 2 \over u 0}
\
{u 3 \over u 0}
(27)
Type: Matrix(Expression(Integer))
fricas
v(R,vect [1,0,0,0])

\label{eq28}\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}}}
(28)
Type: Matrix(Expression(Integer))
fricas
v(S,vect [1,0,0,0])

\label{eq29}\left[ 
\begin{array}{c}
0 
\
- s 1 
\
- s 2 
\
- s 3 
(29)
Type: Matrix(Expression(Integer))
fricas
map(x+->htrigs sinhcosh simplify x, v(U,vect [1,0,0,0]))

\label{eq30}\left[ 
\begin{array}{c}
0 
\
{{\sinh \left({a}\right)}\over{\cosh \left({a}\right)}}
\
0 
\
0 
(30)
Type: Matrix(Expression(Integer))

Non-reciprocal velocities

fricas
v(vect [1,0,0,0],S)

\label{eq31}\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}}}
(31)
Type: Matrix(Expression(Integer))
fricas
v(vect [1,0,0,0],R)

\label{eq32}\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 
(32)
Type: Matrix(Expression(Integer))
fricas
is?(dot(v(P,Q),v(P,Q))=dot(v(Q,P),v(Q,P)))
fricas
Compiling function is? with type Equation(Expression(Integer)) -> 
      Boolean

\label{eq33} \mbox{\rm true} (33)
Type: Boolean

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.

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

\label{eq34} \mbox{\rm true} (34)
Type: Boolean
fricas
Is?(L(P,Q)*L(Q,P) = ID)

\label{eq35} \mbox{\rm true} (35)
Type: Boolean
fricas
Is?(L(P,Q)*Q=P)

\label{eq36} \mbox{\rm true} (36)
Type: Boolean
fricas
Is?(L(P,Q)*v(P,Q) = -v(Q,P))

\label{eq37} \mbox{\rm true} (37)
Type: Boolean
fricas
map(x+->htrigs sinhcosh simplify x, L(U,V))

\label{eq38}\left[ 
\begin{array}{cccc}
{{{\cosh \left({{2 \  b}-{2 \  a}}\right)}+{2 \ {\cosh \left({b - a}\right)}}+ 1}\over{{2 \ {\cosh \left({b - a}\right)}}+ 2}}&{{-{\sinh \left({{2 \  b}-{2 \  a}}\right)}-{2 \ {\sinh \left({b - a}\right)}}}\over{{2 \ {\cosh \left({b - a}\right)}}+ 2}}& 0 & 0 
\
{{-{\sinh \left({{2 \  b}-{2 \  a}}\right)}-{2 \ {\sinh \left({b - a}\right)}}}\over{{2 \ {\cosh \left({b - a}\right)}}+ 2}}&{{{\cosh \left({{2 \  b}-{2 \  a}}\right)}+{2 \ {\cosh \left({b - a}\right)}}+ 1}\over{{2 \ {\cosh \left({b - a}\right)}}+ 2}}& 0 & 0 
\
0 & 0 & 1 & 0 
\
0 & 0 & 0 & 1 
(38)
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))

\label{eq39} \mbox{\rm false} (39)
Type: Boolean
fricas
RQ:=a*R+b*Q;
Type: Matrix(Expression(Integer))
fricas
rq:=solve(dot(RQ,RQ)=-1,b); #rq

\label{eq40}2(40)
Type: PositiveInteger?
fricas
RQ1:=eval(RQ,rq.1);
Type: Matrix(Expression(Integer))
fricas
dot(RQ1,RQ1)

\label{eq41}- 1(41)
Type: Expression(Integer)
fricas
Is?(L(R,RQ1)*L(RQ1,Q) = L(R,Q))

\label{eq42} \mbox{\rm true} (42)
Type: Boolean
fricas
RQ2:=eval(RQ,rq.2);
Type: Matrix(Expression(Integer))
fricas
Is?(RQ1=RQ2)

\label{eq43} \mbox{\rm false} (43)
Type: Boolean
fricas
dot(RQ2,RQ2)

\label{eq44}- 1(44)
Type: Expression(Integer)
fricas
Is?(L(R,RQ2)*L(RQ2,Q) = L(R,Q))

\label{eq45} \mbox{\rm true} (45)
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)

\label{eq46} \mbox{\rm true} (46)
Type: Boolean
fricas
is?(dot(LRPQ*v(S,Q),LRPQ*v(S,Q))=dot(L(R,Q)*v(S,Q),L(R,Q)*v(S,Q)))

\label{eq47} \mbox{\rm true} (47)
Type: Boolean

Lorentz boost with respect to observer "at rest"

fricas
map(simplify, L(vect [1,0,0,0], map(x+->sinhcosh x,U)))

\label{eq48}\left[ 
\begin{array}{cccc}
{\cosh \left({a}\right)}& -{\sinh \left({a}\right)}& 0 & 0 
\
-{\sinh \left({a}\right)}&{\cosh \left({a}\right)}& 0 & 0 
\
0 & 0 & 1 & 0 
\
0 & 0 & 0 & 1 
(48)
Type: Matrix(Expression(Integer))
fricas
LT:=L(vect [1,0,0,0],vect [u0,-u1,-u2,-u3])

\label{eq49}\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}}
(49)
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{eq50}\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 
(50)
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))

\label{eq51} \mbox{\rm true} (51)
Type: Boolean