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.
[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.
[2]:
from IPython.display import SVG
SVG('_static/split.svg')
[2]:
[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 \(X\).
[4]:
X = D.randomV()*10
X
[4]:
(13.9^d0) - (7.9^d1) + (10.9^d2) - (2.4^d3)
This can be split into time and space components by multiplying with the time-vector \(d_0\),
[5]:
X*d0
[5]:
13.9 + (7.9^d01) - (10.9^d02) + (2.4^d03)
and applying the BladeMap
, which results in a scalar+vector in \(\mathbb{P}\)
[6]:
bm(X*d0)
[6]:
13.9 + (7.9^p1) - (10.9^p2) + (2.4^p3)
The space and time components can be separated by grade projection,
[7]:
x = bm(X*d0)
x(0) # the time component
[7]:
13.9
[8]:
x(1) # the space component
[8]:
(7.9^p1) - (10.9^p2) + (2.4^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.
[9]:
def split(X):
return bm(X.odd*d0+X.even)
[10]:
split(X)
[10]:
13.9 + (7.9^p1) - (10.9^p2) + (2.4^p3)
The split can be inverted by applying the BladeMap
again, and multiplying by \(d_0\)
[11]:
x = split(X)
bm(x)*d0
[11]:
(13.9^d0) - (7.9^d1) + (10.9^d2) - (2.4^d3)
Splitting a Bivector¶
Given a random bivector \(F\) in \(\mathbb{D}\),
[12]:
F = D.randomMV()(2)
F
[12]:
(0.4^d01) - (0.6^d02) + (0.2^d03) + (0.5^d12) - (1.7^d13) - (0.1^d23)
\(F\) splits into a vector/bivector in \(\mathbb{P}\)
[13]:
split(F)
[13]:
(0.4^p1) - (0.6^p2) + (0.2^p3) + (0.5^p12) - (1.7^p13) - (0.1^p23)
If \(F\) is interpreted as the electromagnetic bivector, the Electric and Magnetic fields can be separated by grade
[14]:
E = split(F)(1)
iB = split(F)(2)
E
[14]:
(0.4^p1) - (0.6^p2) + (0.2^p3)
[15]:
iB
[15]:
(0.5^p12) - (1.7^p13) - (0.1^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.
[16]:
R = D.randomRotor()
R
[16]:
0.8 - (1.4^d01) - (1.2^d02) - (0.5^d03) + (0.5^d12) - (1.9^d13) - (1.2^d23) - (1.0^d0123)
In this way, the effect of a lorentz transformation on the electric and magnetic fields can be computed by rotating the bivector with \(F \rightarrow RF\tilde{R}\)
[17]:
F_ = R*F*~R
F_
[17]:
(0.8^d01) - (12.4^d02) - (16.1^d03) - (2.2^d12) - (1.6^d13) - (20.2^d23)
Then splitting into \(E\) and \(B\) fields
[18]:
E_ = split(F_)(1)
E_
[18]:
(0.8^p1) - (12.4^p2) - (16.1^p3)
[19]:
iB_ = split(F_)(2)
iB_
[19]:
-(2.2^p12) - (1.6^p13) - (20.2^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,
[20]:
i = p123
E = split(F)(1)
B = -i*split(F)(2)
[21]:
F**2
[21]:
-2.5 - (2.0^d0123)
[22]:
split(F**2) == E**2 - B**2 + (2*E|B)*i
[22]:
True