Space Time Algebra¶
Intro¶
This notebook demonstrates how to use clifford
to work with Space
Time Algebra. The Pauli algegra of space \(\mathbb{P}\), and Dirac
algebra of space-time \(\mathbb{D}\), are related using the
spacetime split. The split is implemented by using a BladeMap
,
which maps a subset of blades in \(\mathbb{D}\) to the blades in
\(\mathbb{P}\). This split allows a spacetime bivector \(F\)
to be broken up into relative electric and magnetic fields in space.
Lorentz transformations are implemented as rotations in
\(\mathbb{D}\), and the effects on the relative fields are computed
with the split.
Setup¶
First we import clifford
, instantiate the two algebras, and populate
the namespace with the blades of each algebra. The elements of
\(\mathbb{D}\) are prefixed with \(d\), while the elements of
\(\mathbb{P}\) are prefixed with \(p\). Although unconventional,
it is easier to read and to translate into code.
In [ ]:
from clifford import Cl, pretty
pretty(precision=1)
# Dirac Algebra `D`
D, D_blades = Cl(1,3, firstIdx=0, names='d')
# Pauli Algebra `P`
P, P_blades = Cl(3, names='p')
# put elements of each in namespace
locals().update(D_blades)
locals().update(P_blades)
The Space Time Split¶
To two algebras can be related by the spacetime-split. First, we create
a BladeMap
which relates the bivectors in \(\mathbb{D}\) to the
vectors/bivectors in \(\mathbb{P}\). The scalars and psuedo-scalars
in each algebra are equated.
In [ ]:
from IPython.display import SVG
SVG('_static/split.svg')
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from clifford import BladeMap
bm = BladeMap([(d01,p1),
(d02,p2),
(d03,p3),
(d12,p12),
(d23,p23),
(d13,p13),
(d0123, p123)])
Spliting a space-time vector (an event)¶
A vector in \(\mathbb{D}\), reprents a unique place in space and time, i.e. an event. To illustrate the split, create a random event \(X\).
In [ ]:
X = D.randomV()*10
X
This can be split into time and space components by multiplying with the time-vector \(d_0\),
In [ ]:
X*d0
and applying the BladeMap
, which results in a scalar+vector in
\(\mathbb{P}\)
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bm(X*d0)
The space and time components can be seperated by grade projection,
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x = bm(X*d0)
x(0) # the time component
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x(1) # the space component
We therefor define a split()
function, which has a simple condition
allowing it to act on a vector or a multivector in \(\mathbb{D}\).
Spliting a spacetime bivector will be treated in the next section.
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def split(X):
return bm(X.odd*d0+X.even)
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split(X)
The split can be inverted by applying the BladeMap
again, and
multiplying by \(d_0\)
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x = split(X)
bm(x)*d0
Splitting a Bivector¶
Given a random bivector \(F\) in \(\mathbb{D}\),
In [ ]:
F = D.randomMV()(2)
F
\(F\) splits into a vector/bivector in \(\mathbb{P}\)
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split(F)
If \(F\) is interpreted as the electromagnetic bivector, the Electric and Magnetic fields can be seperated by grade
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E = split(F)(1)
iB = split(F)(2)
E
In [ ]:
iB
Lorentz Transformations¶
Lorentz Transformations are rotations in \(\mathbb{D}\), which are implemented with Rotors. A rotor in G4 will, in general, have scalar, bivector, and quadvector components.
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R = D.randomRotor()
R
In this way, the effect of a lorentz transformation on the electric and magnetic fields can be computed by rotating the bivector with \(F \rightarrow RF\tilde{R}\)
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F_ = R*F*~R
F_
Then spliting into \(E\) and \(B\) fields
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E_ = split(F_)(1)
E_
In [ ]:
iB_ = split(F_)(2)
iB_
Lorentz Invariants¶
Since lorentz rotations in \(\mathbb{D}\), the magnitude of elements of \(\mathbb{D}\) are invariants of the lorentz transformation. For example, the magnitude of electromagnetic bivector \(F\) is invariant, and it can be related to \(E\) and \(B\) fields in \(\mathbb{P}\) through the split,
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i = p123
E = split(F)(1)
B = -i*split(F)(2)
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F**2
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split(F**2) == E**2 - B**2 + (2*E|B)*i