|
|
|
last edited 10 years ago by Bill Page |
Edit detail for SandBoxLorentzTransformation revision 4 of 30
| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 | ||
|
Editor: Bill Page
Time: 2013/09/22 20:27:06 GMT+0 |
||
| Note: | ||
removed: -Then a row vector is -\begin{axiom} -transpose(vect [a1,a2,a3,a4]) -\end{axiom} changed: -or a scalar Then the dual is a row vector is changed: -g(x,y)== (transpose(x)*diagonalMatrix([-1,1,1,1])*y)::EXPR INT g(vect [a1,a2,a3,a4]) added: And scalar product is \begin{axiom} dot(x,y)== (g(x)*y)::Expression Integer dot(vect [a1,a2,a3,a4], vect [b1,b2,b3,b4]) \end{axiom} changed: -tensor(v,w) == v*g(w) tensor(x,y) == x*g(y) changed: -For difficult verifications it is sometimes convenient to replace -symbols by random numerical values. Verification removed: - removed: -The AlgebraicNumber domain can test for numerical equality of complicated -expressions involving $\sqrt{n}$. -\begin{axiom} -IsPossible?(eq:Equation EXPR INT):Boolean == _ - (possible(lhs(eq)-rhs(eq)) :: Expression AlgebraicNumber=0)::Boolean -IsPossible2?(eq:Equation(Matrix(EXPR(INT)))):Boolean == _ - ( map(possible,(lhs(eq)-rhs(eq))) :: Matrix Expression AlgebraicNumber = _ -zero(nrows(lhs(eq)),ncols(lhs(eq)))$Matrix Expression AlgebraicNumber )::Boolean -\end{axiom} changed: -g(P,P) dot(P,P) changed: -g(Q,Q) dot(Q,Q) changed: -g(R,R) dot(R,R) changed: -w(P,Q)==-Q/g(P,Q)-P w(P,Q)==-Q/dot(P,Q)-P changed: -g(P,u) -v:=w(Q,P) -g(Q,v) -possible(g(u,u))::EXPR Float dot(P,u) possible dot(u,u)::EXPR Float v:=w(Q,P); dot(Q,v) possible dot(v,v)::EXPR Float changed: -L(P,Q) == diagonalMatrix([1,1,1,1]) + tensor(P+Q,P+Q)/(1-g(P,Q)) - 2*tensor(P,Q) L(P,Q) == diagonalMatrix([1,1,1,1]) + tensor(P+Q,P+Q)/(1-dot(P,Q)) - 2*tensor(P,Q) changed: -L(P,Q)*u + v map(x+->possible(x)::EXPR Float,L(P,Q)*u + v) removed: -
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
fricas
vect [a1,a2, a3, a4]
fricas
Compiling function vect with type List(Expression(Integer)) ->
Matrix(Expression(Integer))![\label{eq1}\left[
\begin{array}{c}
a 1
\
a 2
\
a 3
\
a 4](images/2198737460665284646-16.0px.png "\label{eq1}\left[
\begin{array}{c}
a 1
\
a 2
\
a 3
\
a 4") | (1) |
Type: Matrix(Expression(Integer))
Applying the Lorentz form produces a row vector
fricas
g(x)==transpose(x)*diagonalMatrix [-1,1, 1, 1]
Type: Void
Then the dual is a row vector is
fricas
g(vect [a1,a2, a3, a4])
fricas
Compiling function g with type Matrix(Expression(Integer)) -> Matrix
(Expression(Integer)) | (2) |
Type: Matrix(Expression(Integer))
And scalar product is
fricas
dot(x,y)== (g(x)*y)::Expression Integer
Type: Void
fricas
dot(vect [a1,a2, a3, a4], vect [b1, b2, b3, b4])
fricas
Compiling function dot with type (Matrix(Expression(Integer)),Matrix (Expression(Integer))) -> Expression(Integer)
 | (3) |
Type: Expression(Integer)
Tensor product is
fricas
tensor(x,y) == x*g(y)
Type: Void
fricas
tensor(vect [a1,a2, a3, a4], vect [b1, b2, b3, b4])
fricas
Compiling function tensor with type (Matrix(Expression(Integer)),Matrix(Expression(Integer))) -> Matrix(Expression(Integer))
![\label{eq4}\left[
\begin{array}{cccc}
-{a 1 \ b 1}&{a 1 \ b 2}&{a 1 \ b 3}&{a 1 \ b 4}
\
-{a 2 \ b 1}&{a 2 \ b 2}&{a 2 \ b 3}&{a 2 \ b 4}
\
-{a 3 \ b 1}&{a 3 \ b 2}&{a 3 \ b 3}&{a 3 \ b 4}
\
-{a 4 \ b 1}&{a 4 \ b 2}&{a 4 \ b 3}&{a 4 \ b 4}](images/1315331596076872291-16.0px.png "\label{eq4}\left[
\begin{array}{cccc}
-{a 1 \ b 1}&{a 1 \ b 2}&{a 1 \ b 3}&{a 1 \ b 4}
\
-{a 2 \ b 1}&{a 2 \ b 2}&{a 2 \ b 3}&{a 2 \ b 4}
\
-{a 3 \ b 1}&{a 3 \ b 2}&{a 3 \ b 3}&{a 3 \ b 4}
\
-{a 4 \ b 1}&{a 4 \ b 2}&{a 4 \ b 3}&{a 4 \ b 4}") | (4) |
Type: Matrix(Expression(Integer))
Verification
fricas
possible(x)==subst(x,map(y+->(y=(random(100) - random(100))), variables x) )
Type: Void
fricas
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
fricas
Is2?(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 Is2? : Equation(Matrix(Expression(Integer))) -> Boolean has been added to workspace.
Type: Void
Massive Objects
An object (also referred to as an obserser) 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)
 | (5) |
Type: Expression(Integer)
fricas
Q:=vect [sqrt(q1^2+q2^2+q3^2+1),q1, q2, q3];
Type: Matrix(Expression(Integer))
fricas
dot(Q,Q)
 | (6) |
Type: Expression(Integer)
fricas
R:=vect [1,0, 0, 0]
![\label{eq7}\left[
\begin{array}{c}
1
\
0
\
0
\
0](images/2111725994280903269-16.0px.png "\label{eq7}\left[
\begin{array}{c}
1
\
0
\
0
\
0") | (7) |
Type: Matrix(Expression(Integer))
fricas
dot(R,R)
 | (8) |
Type: Expression(Integer)
Associated with each such vector is the orthogonal 3-d Euclidean subspace
Relative Velocity
An object Q has a unique relative velocity w(P,Q) with respect to object P given by
fricas
w(P,Q)==-Q/dot(P, Q)-P
Type: Void
Lorentz factor
fricas
gamma(v)==1/sqrt(1-g(v,v))
Type: Void
Binary Boost
fricas
b(P,v)==gamma(v)*(P+v)
Type: Void
Observer P measures velocity u. u is space-like and in 
.
fricas
u:=w(P,Q);
fricas
Compiling function w with type (Matrix(Expression(Integer)),Matrix( Expression(Integer))) -> Matrix(Expression(Integer))
Type: Matrix(Expression(Integer))
fricas
dot(P,u)
 | (9) |
Type: Expression(Integer)
fricas
possible dot(u,u)::EXPR Float
fricas
Compiling function possible with type Expression(Integer) ->
Expression(Integer) | (10) |
Type: Expression(Float)
fricas
v:=w(Q,P);
Type: Matrix(Expression(Integer))
fricas
dot(Q,v)
 | (11) |
Type: Expression(Integer)
fricas
possible dot(v,v)::EXPR Float
 | (12) |
Type: Expression(Float)
fricas
L(P,Q) == diagonalMatrix([1, 1, 1, 1]) + tensor(P+Q, P+Q)/(1-dot(P, Q)) - 2*tensor(P, Q)
Type: Void
fricas
L(P,P)
fricas
Compiling function L with type (Matrix(Expression(Integer)),Matrix( Expression(Integer))) -> Matrix(Expression(Integer))
![\label{eq13}\left[
\begin{array}{cccc}
1 & 0 & 0 & 0
\
0 & 1 & 0 & 0
\
0 & 0 & 1 & 0
\
0 & 0 & 0 & 1](images/7509693640650835016-16.0px.png "\label{eq13}\left[
\begin{array}{cccc}
1 & 0 & 0 & 0
\
0 & 1 & 0 & 0
\
0 & 0 & 1 & 0
\
0 & 0 & 0 & 1") | (13) |
Type: Matrix(Expression(Integer))
fricas
L(P,Q)*Q-P
![\label{eq14}\left[
\begin{array}{c}
0
\
0
\
0
\
0](images/5647471933272843491-16.0px.png "\label{eq14}\left[
\begin{array}{c}
0
\
0
\
0
\
0") | (14) |
Type: Matrix(Expression(Integer))
fricas
L(R,P)
![\label{eq15}\left[
\begin{array}{cccc}
{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}& - p 1 & - p 2 & - p 3
\
- p 1 &{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+{{p 1}^{2}}+ 1}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}&{{p 1 \ p 2}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}&{{p 1 \ p 3}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}
\
- p 2 &{{p 1 \ p 2}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}&{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+{{p 2}^{2}}+ 1}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}&{{p 2 \ p 3}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}
\
- p 3 &{{p 1 \ p 3}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}&{{p 2 \ p 3}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}&{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+{{p 3}^{2}}+ 1}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}](images/2646131123187380817-16.0px.png "\label{eq15}\left[
\begin{array}{cccc}
{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}& - p 1 & - p 2 & - p 3
\
- p 1 &{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+{{p 1}^{2}}+ 1}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}&{{p 1 \ p 2}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}&{{p 1 \ p 3}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}
\
- p 2 &{{p 1 \ p 2}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}&{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+{{p 2}^{2}}+ 1}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}&{{p 2 \ p 3}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}
\
- p 3 &{{p 1 \ p 3}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}&{{p 2 \ p 3}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}&{{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+{{p 3}^{2}}+ 1}\over{{\sqrt{{{p 3}^{2}}+{{p 2}^{2}}+{{p 1}^{2}}+ 1}}+ 1}}") | (15) |
Type: Matrix(Expression(Integer))
fricas
map(x+->possible(x)::EXPR Float,L(P, Q)*u + v)
![\label{eq16}\left[
\begin{array}{c}
-{70859.0949812301_84461}
\
-{361538.1204534425_9971}
\
-{540768.0117583411_6253}
\
{566080.9433370158_4835}](images/3835259587211157805-16.0px.png "\label{eq16}\left[
\begin{array}{c}
-{70859.0949812301_84461}
\
-{361538.1204534425_9971}
\
-{540768.0117583411_6253}
\
{566080.9433370158_4835}") | (16) |
Type: Matrix(Expression(Float))