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last edited 10 years ago by Bill Page |
Edit detail for SandBoxLorentzTransformation revision 17 of 30
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Editor: Bill Page
Time: 2013/09/25 00:11:18 GMT+0 |
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| Note: tidy | ||
added: Observer "at rest" \begin{axiom} vect [1,0,0,0] dot(%,%) \end{axiom} added: Velocity with respect to observer "at rest" \begin{axiom} 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]) \end{axiom} removed: -Velocity with respect to observer "at rest" -\begin{axiom} -v(vect [u0,u1,u2,u3],vect [1,0,0,0]) -v(S,vect [1,0,0,0]) -\end{axiom} -
Lorentz transformations.
Book by T. Matolcsi
Mathematical Preliminaries
A vector is represented as a nx1 matrix (column vector)
fricas
vect(x:List Expression Integer):Matrix Expression Integer == matrix map(y+->[y],x)
Function declaration vect : List(Expression(Integer)) -> Matrix( Expression(Integer)) has been added to workspace.
Type: Void
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vect [a0,a1, a2, a3]
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Compiling function vect with type List(Expression(Integer)) ->
Matrix(Expression(Integer))![\label{eq1}\left[
\begin{array}{c}
a 0
\
a 1
\
a 2
\
a 3](images/4443196779762138950-16.0px.png "\label{eq1}\left[
\begin{array}{c}
a 0
\
a 1
\
a 2
\
a 3") | (1) |
Type: Matrix(Expression(Integer))
Identity
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ID:=diagonalMatrix([1,1, 1, 1])
![\label{eq2}\left[
\begin{array}{cccc}
1 & 0 & 0 & 0
\
0 & 1 & 0 & 0
\
0 & 0 & 1 & 0
\
0 & 0 & 0 & 1](images/2198114584885539847-16.0px.png "\label{eq2}\left[
\begin{array}{cccc}
1 & 0 & 0 & 0
\
0 & 1 & 0 & 0
\
0 & 0 & 1 & 0
\
0 & 0 & 0 & 1") | (2) |
Type: Matrix(Integer)
Verification
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possible(x)==subst(x,map(y+->(y=(random(100) - random(100))), variables x) )
Type: Void
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is?(eq:Equation EXPR INT):Boolean == (lhs(eq)-rhs(eq)=0)::Boolean
Function declaration is? : Equation(Expression(Integer)) -> Boolean has been added to workspace.
Type: Void
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Is?(eq:Equation(Matrix(EXPR(INT)))):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
Lorentz Form (metric)
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G:=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/7358876913742565659-16.0px.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)
applied to a vector produces a co-vector (represent as a 1xn matrix or row vector)
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g(x)==transpose(x)*G
Type: Void
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g(vect [a0,a1, a2, a3])
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Compiling function g with type Matrix(Expression(Integer)) -> Matrix
(Expression(Integer)) | (4) |
Type: Matrix(Expression(Integer))
Scalar product
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dot(x,y)== (g(x)*y)::Expression Integer
Type: Void
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dot(vect [a0,a1, a2, a3], vect [b0, b1, b2, b3])
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Compiling function dot with type (Matrix(Expression(Integer)),Matrix (Expression(Integer))) -> Expression(Integer)
 | (5) |
Type: Expression(Integer)
Tensor product
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tensor(x,y) == x*g(y)
Type: Void
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tensor(vect [a0,a1, a2, a3], vect [b0, b1, b2, b3])
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Compiling function tensor with type (Matrix(Expression(Integer)),Matrix(Expression(Integer))) -> Matrix(Expression(Integer))
![\label{eq6}\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/6739546463157801091-16.0px.png "\label{eq6}\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}") | (6) |
Type: Matrix(Expression(Integer))
Massive Objects
A material object (also referred to as an observer) is represented by a time-like 4-vector
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P:=vect [sqrt(p1^2+p2^2+p3^2+1),-p1, -p2, -p3];
Type: Matrix(Expression(Integer))
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dot(P,P)
 | (7) |
Type: Expression(Integer)
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Q:=vect [sqrt(q1^2+q2^2+q3^2+1),-q1, -q2, -q3];
Type: Matrix(Expression(Integer))
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R:=vect [sqrt(r1^2+r2^2+r3^2+1),-r1, -r2, -r3];
Type: Matrix(Expression(Integer))
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S:=1/sqrt(1-s1^2-s2^2-s3^2)*vect [1,-s1, -s2, -s3]
![\label{eq8}\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/4739323798050234047-16.0px.png "\label{eq8}\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}}}") | (8) |
Type: Matrix(Expression(Integer))
fricas
dot(S,S)
 | (9) |
Type: Expression(Integer)
Observer "at rest"
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vect [1,0, 0, 0]
![\label{eq10}\left[
\begin{array}{c}
1
\
0
\
0
\
0](images/9188217390512376258-16.0px.png "\label{eq10}\left[
\begin{array}{c}
1
\
0
\
0
\
0") | (10) |
Type: Matrix(Expression(Integer))
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dot(%,%)
 | (11) |
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
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v(P,Q)==-P/dot(P, Q)-Q
Type: Void
Observer P measures velocity v(Q,P). v(Q,P) is space-like and in 
.
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dot(P,v(Q, P))
fricas
Compiling function v with type (Matrix(Expression(Integer)),Matrix( Expression(Integer))) -> Matrix(Expression(Integer))
 | (12) |
Type: Expression(Integer)
fricas
possible dot(v(Q,P), v(Q, P))::EXPR Float
fricas
Compiling function possible with type Expression(Integer) ->
Expression(Integer) | (13) |
Type: Expression(Float)
fricas
dot(Q,v(P, Q))
 | (14) |
Type: Expression(Integer)
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possible dot(v(P,Q), v(P, Q))::EXPR Float
 | (15) |
Type: Expression(Float)
Velocity with respect to observer "at rest"
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v(vect [u0,u1, u2, u3], vect [1, 0, 0, 0])
![\label{eq16}\left[
\begin{array}{c}
0
\
{u 1 \over u 0}
\
{u 2 \over u 0}
\
{u 3 \over u 0}](images/4118336355009335748-16.0px.png "\label{eq16}\left[
\begin{array}{c}
0
\
{u 1 \over u 0}
\
{u 2 \over u 0}
\
{u 3 \over u 0}") | (16) |
Type: Matrix(Expression(Integer))
fricas
v(R,vect [1, 0, 0, 0])
![\label{eq17}\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/2682634178941967254-16.0px.png "\label{eq17}\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}}}") | (17) |
Type: Matrix(Expression(Integer))
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v(S,vect [1, 0, 0, 0])
![\label{eq18}\left[
\begin{array}{c}
0
\
- s 1
\
- s 2
\
- s 3](images/7148747656037869481-16.0px.png "\label{eq18}\left[
\begin{array}{c}
0
\
- s 1
\
- s 2
\
- s 3") | (18) |
Type: Matrix(Expression(Integer))
Lorentz Boost
is a linear bijection 
that preserves
and maps orthogonal compliments into each other.
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L(P,Q) == ID + tensor(P+Q, P+Q)/(1-dot(P, Q)) - 2*tensor(P, Q)
Type: Void
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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 | (19) |
Type: Boolean
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Is?(L(P,Q)*L(Q, P) = ID)
 | (20) |
Type: Boolean
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Is?(L(P,Q)*Q=P)
 | (21) |
Type: Boolean
fricas
Is?(L(P,Q)*v(P, Q) = -v(Q, P))
 | (22) |
Type: Boolean
Composition of two Lorentz boosts is not a Lorentz boost
fricas
Is?(L(R,P)*L(P, Q) = L(R, Q))
 | (23) |
Type: Boolean
Lorentz boost with respect to observer "at rest"
fricas
LT:=L(vect [1,0, 0, 0], vect [u0, -u1, -u2, -u3])
![\label{eq24}\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/4633442835078815028-16.0px.png "\label{eq24}\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}}") | (24) |
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{eq25}\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/8229099039613673979-16.0px.png "\label{eq25}\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") | (25) |
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))
 | (26) |
Type: Boolean