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Edit detail for SandBoxLorentzTransformation revision 30 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: 2014/01/10 03:22:07 GMT+0 Note: most general boost

changed:
-Book by T. Matolcsi
-
-  - "Space-time without reference frames":http://axiom-wiki.newsynthesis.org/uploads/matolcsi.pdf
References

- "Space-time without reference frames":http://axiom-wiki.newsynthesis.org/uploads/matolcsi.pdf  T. Matolcsi

- "The Lorentz boost-link is not unique":http://arxiv.org/pdf/math-ph/0608062.pdf Zbigniew Oziewicz

Most General Lorentz Boost (Oziewicz, 2006)

is given by three non-coplanar vectors
\begin{axiom}
B(P,Q,X) == ID -                                _
( tensor(                                       _
2*X, dot(P-Q,P-Q)*X - 2*dot(X,P)*(P-Q)      _
) +                                           _
tensor(                                       _
P-Q, 2*dot(X,X)*(P-Q)+4*dot(X,Q)*X          _
)                                             _
) / (                                           _
dot(X,X)*dot(P-Q,P-Q)+4*dot(X,P)*dot(X,Q) _
)
Is?(B(P,P,R) = ID)
Is?(B(P,Q,R)*B(Q,P,R) = ID)
Is?(B(P,Q,R)*P=Q)
--Is?(B(P,Q,R)*ω(P,Q) = -ω(Q,P))
Is?(L(P,Q)=B(Q,P,q*Q+p*P))
\end{axiom}



Lorentz transformations.

Mathematical Preliminaries

A vector is represented as a matrix (column vector)

fricas
Scalar := Expression 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))
 (2)
Type: Matrix(Expression(Integer))

Identity

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ID:=diagonalMatrix([1,1,1,1])
 (3)
Type: Matrix(Integer)

Verification

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htrigs2exp == rule
cosh(a) == (exp(a)+exp(-a))/2
sinh(a) == (exp(a)-exp(-a))/2
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)
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)
Type: Void
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)
Type: Void
fricas
Simplify(x:Scalar):Scalar == htrigs sinhcosh simplify htrigs2exp x
Function declaration Simplify : Expression(Integer) -> Expression(
Integer) has been added to workspace.
Type: Void
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

Lorentz Form (metric)

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G:=diagonalMatrix [-1,1,1,1]
 (4)
Type: Matrix(Integer)

applied to a vector produces a co-vector (represent as a 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])
fricas
Compiling function g with type Matrix(Expression(Integer)) -> Matrix
(Expression(Integer))
 (5)
Type: Matrix(Expression(Integer))

Scalar product

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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)
 (6)
Type: Expression(Integer)

Tensor product

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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))
 (7)
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)
 (8)
Type: Expression(Integer)
fricas
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]
 (9)
Type: Matrix(Expression(Integer))
fricas
dot(S,S)
 (10)
Type: Expression(Integer)
fricas
T:=1/sqrt(1-t1^2-t2^2-t3^2)*vect [1,-t1,-t2,-t3]
 (11)
Type: Matrix(Expression(Integer))
fricas
U:=vect [cosh(u),sinh(u),0,0]
 (12)
Type: Matrix(Expression(Integer))
fricas
simplify dot(U,U)
 (13)
Type: Expression(Integer)
fricas
V:=vect [cosh(v),sinh(v),0,0]
 (14)
Type: Matrix(Expression(Integer))
fricas
Simplify dot(U,V)
fricas
Compiling body of rule htrigs2exp to compute value of type Ruleset(
Integer,Integer,Expression(Integer))
fricas
Compiling body of rule sinhcosh to compute value of type Ruleset(
Integer,Integer,Expression(Integer))
fricas
Compiling function Simplify with type Expression(Integer) ->
Expression(Integer)
 (15)
Type: Expression(Integer)
fricas
W:=vect [cosh(w),0,sinh(w),0]
 (16)
Type: Matrix(Expression(Integer))
fricas
Simplify dot(U,W)
 (17)
Type: Expression(Integer)

Observer "at rest"

fricas
vect [1,0,0,0]
 (18)
Type: Matrix(Expression(Integer))
fricas
dot(%,%)
 (19)
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 ω(P,Q) with respect to object Q given by

fricas
ω(P,Q)==-P/dot(P,Q)-Q
Type: Void
fricas
ω(P,Q)
fricas
Compiling function ω with type (Matrix(Expression(Integer)),Matrix(
Expression(Integer))) -> Matrix(Expression(Integer))
 (20)
Type: Matrix(Expression(Integer))
fricas
ω(S,T)
 (21)
Type: Matrix(Expression(Integer))

In two dimensions

fricas
map(x+->Simplify x,ω(U,V))
 (22)
Type: Matrix(Expression(Integer))
fricas
vect [cosh(u)/cosh(u-v)-cosh(v),sinh(u)/cosh(u-v)-sinh(v),0,0]
 (23)
Type: Matrix(Expression(Integer))
fricas
Is?(% = ω(U,V))
fricas
Compiling function Is? with type Equation(Matrix(Expression(Integer)
)) -> Boolean
 (24)
Type: Boolean
fricas
map(x+->Simplify x,ω(U,W))
 (25)
Type: Matrix(Expression(Integer))

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

fricas
dot(ω(P,vect [1,0,0,0]),ω(P,vect [1,0,0,0]))
 (26)
Type: Expression(Integer)

and in

fricas
dot(P,ω(Q,P))
 (27)
Type: Expression(Integer)
fricas
possible dot(ω(Q,P),ω(Q,P))::EXPR Float
fricas
Compiling function possible with type Expression(Integer) ->
Expression(Integer)
 (28)
Type: Expression(Float)
fricas
dot(Q,ω(P,Q))
 (29)
Type: Expression(Integer)
fricas
possible dot(ω(P,Q),ω(P,Q))::EXPR Float
 (30)
Type: Expression(Float)

Velocity with respect to observer "at rest"

fricas
ω(vect [u0,u1,u2,u3],vect [1,0,0,0])
 (31)
Type: Matrix(Expression(Integer))
fricas
ω(R,vect [1,0,0,0])
 (32)
Type: Matrix(Expression(Integer))
fricas
ω(S,vect [1,0,0,0])
 (33)
Type: Matrix(Expression(Integer))
fricas
map(Simplify, ω(U,vect [1,0,0,0]))
 (34)
Type: Matrix(Expression(Integer))

Non-reciprocal velocities

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ω(vect [1,0,0,0],S)
 (35)
Type: Matrix(Expression(Integer))
fricas
ω(vect [1,0,0,0],R)
 (36)
Type: Matrix(Expression(Integer))
fricas
is?(dot(ω(P,Q),ω(P,Q))=dot(ω(Q,P),ω(Q,P)))
fricas
Compiling function is? with type Equation(Expression(Integer)) ->
Boolean
 (37)
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))
 (38)
Type: Boolean
fricas
Is?(L(P,Q)*L(Q,P) = ID)
 (39)
Type: Boolean
fricas
Is?(L(P,Q)*Q=P)
 (40)
Type: Boolean
fricas
Is?(L(P,Q)*ω(P,Q) = -ω(Q,P))
 (41)
Type: Boolean

Most General Lorentz Boost (Oziewicz, 2006)

is given by three non-coplanar vectors

fricas
B(P,Q,X) == ID -                                _
( tensor(                                       _
2*X, dot(P-Q,P-Q)*X - 2*dot(X,P)*(P-Q)      _
) +                                           _
tensor(                                       _
P-Q, 2*dot(X,X)*(P-Q)+4*dot(X,Q)*X          _
)                                             _
fricas
) / (                                           _
dot(X,X)*dot(P-Q,P-Q)+4*dot(X,P)*dot(X,Q) _
)
Type: Void
fricas
Is?(B(P,P,R) = ID)
fricas
Compiling function B with type (Matrix(Expression(Integer)),Matrix(
Expression(Integer)),Matrix(Expression(Integer))) -> Matrix(
Expression(Integer))
 (42)
Type: Boolean
fricas
Is?(B(P,Q,R)*B(Q,P,R) = ID)
 (43)
Type: Boolean
fricas
Is?(B(P,Q,R)*P=Q)
 (44)
Type: Boolean
fricas
--Is?(B(P,Q,R)*ω(P,Q) = -ω(Q,P))
Is?(L(P,Q)=B(Q,P,q*Q+p*P))
 (45)
Type: Boolean

In two dimensions

fricas
map(x+->simplify expandhtrigs2 Simplify x, L(U,V))
fricas
Compiling body of rule expandhtrigs2 to compute value of type
Ruleset(Integer,Integer,Expression(Integer))
 (46)
Type: Matrix(Expression(Integer))
fricas
map(x+->simplify expandhtrigs expandhtrigs2 Simplify x, L(U,W))
fricas
Compiling body of rule expandhtrigs to compute value of type Ruleset
(Integer,Integer,Expression(Integer))
 (47)
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))
 (48)
Type: Boolean
fricas
RQ:=a*R+b*Q;
Type: Matrix(Expression(Integer))
fricas
rq:=solve(dot(RQ,RQ)=-1,b); #rq
 (49)
Type: PositiveInteger?
fricas
RQ1:=eval(RQ,rq.1);
Type: Matrix(Expression(Integer))
fricas
dot(RQ1,RQ1)
 (50)
Type: Expression(Integer)
fricas
Is?(L(R,RQ1)*L(RQ1,Q) = L(R,Q))
 (51)
Type: Boolean
fricas
RQ2:=eval(RQ,rq.2);
Type: Matrix(Expression(Integer))
fricas
Is?(RQ1=RQ2)
 (52)
Type: Boolean
fricas
dot(RQ2,RQ2)
 (53)
Type: Expression(Integer)
fricas
Is?(L(R,RQ2)*L(RQ2,Q) = L(R,Q))
 (54)
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)
 (55)
Type: Boolean
fricas
is?(dot(LRPQ*ω(S,Q),LRPQ*ω(S,Q))=dot(L(R,Q)*ω(S,Q),L(R,Q)*ω(S,Q)))
 (56)
Type: Boolean

Lorentz boost with respect to observer "at rest"

fricas
LT:=L(vect [1,0,0,0],vect [u0,-u1,-u2,-u3])
 (57)
Type: Matrix(Expression(Integer))
fricas
map(simplify, L(vect [1,0,0,0], map(Simplify,U)))
 (58)
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]]
 (59)
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))
 (60)
Type: Boolean

SandBox Idempotent Observers