Space Time Algebra¶
Intro¶
This notebook demonstrates how to use clifford
to work with Space
Time Algebra. The Pauli algebra 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 [1]:
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 pseudo-scalars
in each algebra are equated.
In [2]:
from IPython.display import SVG
SVG('_static/split.svg')
Out[2]:
In [3]:
from clifford import BladeMap
bm = BladeMap([(d01,p1),
(d02,p2),
(d03,p3),
(d12,p12),
(d23,p23),
(d13,p13),
(d0123, p123)])
Splitting a space-time vector (an event)¶
A vector in \(\mathbb{D}\), represents a unique place in space and time, i.e. an event. To illustrate the split, create a random event \(X\).
In [4]:
X = D.randomV()*10
X
Out[4]:
-(2.1^d0) - (16.1^d1) + (5.4^d2) - (0.9^d3)
This can be split into time and space components by multiplying with the time-vector \(d_0\),
In [5]:
X*d0
Out[5]:
-2.1 + (16.1^d01) - (5.4^d02) + (0.9^d03)
and applying the BladeMap
, which results in a scalar+vector in
\(\mathbb{P}\)
In [6]:
bm(X*d0)
Out[6]:
-2.1 + (16.1^p1) - (5.4^p2) + (0.9^p3)
The space and time components can be separated by grade projection,
In [7]:
x = bm(X*d0)
x(0) # the time component
Out[7]:
-2.1
In [8]:
x(1) # the space component
Out[8]:
(16.1^p1) - (5.4^p2) + (0.9^p3)
We therefor define a split()
function, which has a simple condition
allowing it to act on a vector or a multivector in \(\mathbb{D}\).
Splitting a spacetime bivector will be treated in the next section.
In [9]:
def split(X):
return bm(X.odd*d0+X.even)
In [10]:
split(X)
Out[10]:
-2.1 + (16.1^p1) - (5.4^p2) + (0.9^p3)
The split can be inverted by applying the BladeMap
again, and
multiplying by \(d_0\)
In [11]:
x = split(X)
bm(x)*d0
Out[11]:
-(2.1^d0) - (16.1^d1) + (5.4^d2) - (0.9^d3)
Splitting a Bivector¶
Given a random bivector \(F\) in \(\mathbb{D}\),
In [12]:
F = D.randomMV()(2)
F
Out[12]:
(1.4^d01) + (0.1^d02) - (1.0^d03) + (2.0^d12) - (1.0^d13) - (0.8^d23)
\(F\) splits into a vector/bivector in \(\mathbb{P}\)
In [13]:
split(F)
Out[13]:
(1.4^p1) + (0.1^p2) - (1.0^p3) + (2.0^p12) - (1.0^p13) - (0.8^p23)
If \(F\) is interpreted as the electromagnetic bivector, the Electric and Magnetic fields can be separated by grade
In [14]:
E = split(F)(1)
iB = split(F)(2)
E
Out[14]:
(1.4^p1) + (0.1^p2) - (1.0^p3)
In [15]:
iB
Out[15]:
(2.0^p12) - (1.0^p13) - (0.8^p23)
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.
In [16]:
R = D.randomRotor()
R
Out[16]:
-1.4 - (1.3^d01) + (0.2^d02) - (0.0^d03) - (0.7^d12) - (0.4^d13) + (1.4^d23) + (1.3^d0123)
In this way, the effect of a lorentz transformation on the electric and magnetic fields can be computed by rotating the bivector with \(F \rightarrow RF\tilde{R}\)
In [17]:
F_ = R*F*~R
F_
Out[17]:
(0.5^d01) + (3.9^d02) + (8.1^d03) - (3.6^d12) - (7.2^d13) - (4.3^d23)
Then splitting into \(E\) and \(B\) fields
In [18]:
E_ = split(F_)(1)
E_
Out[18]:
(0.5^p1) + (3.9^p2) + (8.1^p3)
In [19]:
iB_ = split(F_)(2)
iB_
Out[19]:
-(3.6^p12) - (7.2^p13) - (4.3^p23)
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,
In [20]:
i = p123
E = split(F)(1)
B = -i*split(F)(2)
In [21]:
F**2
Out[21]:
-2.5 - (6.1^d0123)
In [22]:
split(F**2) == E**2 - B**2 + (2*E|B)*i
Out[22]:
True