# 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


## 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

(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