Spacetime algebra


In mathematical physics, spacetime algebra is the application of Clifford algebra Cl1,3, or equivalently the geometric algebra of physics. Spacetime algebra provides a "unified, coordinate-free formulation for all of relativistic physics, including the Dirac equation, Maxwell equation and general relativity" and "reduces the mathematical divide between classical, quantum and relativistic physics".
Spacetime algebra is a vector space that allows not only vectors, but also bivectors or blades to be combined, as well as rotated, reflected, or Lorentz boosted. It is also the natural parent algebra of spinors in special relativity. These properties allow many of the most important equations in physics to be expressed in particularly simple forms, and can be very helpful towards a more geometric understanding of their meanings.
In comparison to related methods, STA and Dirac algebra are both Clifford Cl1,3 algebras, but STA uses real number scalars while Dirac algebra uses complex number scalars.
The STA space–time split is similar to the algebra of physical space approach. APS represents spacetime as a paravector, a combined 3-dimensional vector space and a 1-dimensional scalar.

Structure

For any pair of STA vectors, and, there is a geometric product, scalar product and exterior product. The vector product is a sum of an scalar and exterior product:
The scalar product generates a real number, and the exterior product generates a bivector. The vectors and are orthogonal if their scalar product is zero; vectors and are parallel if their exterior product is zero.
The orthonormal basis vectors are a timelike vector and 3 spacelike vectors. The Minkowski metric tensor's nonzero terms are the diagonal terms,. For :
The Dirac matrices share these properties, and STA is equivalent to the algebra generated by the Dirac matrices over the field of real numbers; explicit matrix representation is unnecessary for STA.
Products of the basis vectors generate a tensor basis containing one scalar, four vectors, six bivectors, four pseudovectors and one pseudoscalar with. The pseudoscalar commutes with all even-grade STA elements, but anticommutes with all odd-grade STA elements.

Subalgebra

STA's even-graded elements form a subalgebra isomorphic to Clifford algebra Cl3,0, which is equivalent to the APS or Pauli algebra. The STA bivectors are equivalent to the APS vectors and pseudovectors. The STA subalgebra becomes more explicit by renaming the STA bivectors as and the STA bivectors as. The Pauli matrices,, are a matrix representation for. For any pair of, the nonzero scalar products are, and the nonzero exterior products are:
The sequence of algebra to even subalgebra continues as algebra of physical space, quaternion algebra, complex numbers and real numbers. The even STA subalgebra Cl of real space-time spinors in Cl1,3 is isomorphic to the Clifford algebra Cl3,0 of Euclidean space R3 with basis elements. See the illustration of space-time algebra spinors in Cl under the octonionic product as a Fano plane.

Division

A nonzero vector is a null vector if. An example is. Null vectors are tangent to the light cone. An element is an idempotent if. Two idempotents and are orthogonal idempotents if. An example of an orthogonal idempotent pair is and with. Proper zero divisors are nonzero elements whose product is zero such as null vectors or orthogonal idempotents. A division algebra is an algebra that contains multiplicative inverse elements for every element, but this occurs if there are no proper zero divisors and if the only idempotent is 1. The only associative division algebras are the real numbers, complex numbers and quaternions. As STA is not a division algebra, some STA elements may lack an inverse; however, division by the non-null vector may be possible by multiplication by its inverse, defined as.

Reciprocal frame

Associated with the orthogonal basis is the reciprocal basis set satisfying these equations:
These reciprocal frame vectors differ only by a sign, with, but,,.
A vector may be represented using either the basis vectors or the reciprocal basis vectors with summation over, according to the Einstein notation. The scalar product of vector and basis vectors or reciprocal basis vectors generates the vector components.
The metric and index gymnastics raise or lower indices:

Spacetime gradient

The spacetime gradient, like the gradient in a Euclidean space, is defined such that the directional derivative relationship is satisfied:
This requires the definition of the gradient to be
Written out explicitly with, these partials are

Space–time split

In STA, a space–time split is a projection from four-dimensional space into -dimensional space in a chosen reference frame by means of the following two operations:
  • a collapse of the chosen time axis, yielding a 3-dimensional space spanned by bivectors, equivalent to the standard 3-dimensional basis vectors in the algebra of physical space and
  • a projection of the 4D space onto the chosen time axis, yielding a 1-dimensional space of scalars, representing the scalar time.
This is achieved by left-multiplication or right-multiplication by a timelike basis vector, which serves to split a four vector into a scalar timelike and a bivector spacelike component, in the reference frame co-moving with. With, we have
Space–time split is a method for representing an even-graded vector of spacetime as a vector in the Pauli algebra, an algebra where time is a scalar separated from vectors that occur in 3 dimensional space. The method replaces these spacetime vectors
As these bivectors square to, they serve as a spatial basis. Utilizing the Pauli matrix notation, these are written. Spatial vectors in STA are denoted in boldface; then with and, the -space–time split, and its reverse are:
However, the above formulas only work in the Minkowski metric with signature. For forms of the space–time split that work in either signature, alternate definitions in which and must be used.

Transformations

To rotate a vector in geometric algebra, the following formula is used:
where is the angle to rotate by, and is the bivector representing the plane of rotation normalized so that.
For a given spacelike bivector,, so Euler's formula applies, giving the rotation
For a given timelike bivector,, so a "rotation through time" uses the analogous equation for the split-complex numbers:
Interpreting this equation, these rotations along the time direction are simply hyperbolic rotations. These are equivalent to Lorentz boosts in special relativity.
Both of these transformations are known as Lorentz transformations, and the combined set of all of them is the Lorentz group. To transform an object in STA from any basis to another, one or more of these transformations must be used.
Any spacetime element is transformed by multiplication with the pseudoscalar to form its Hodge dual. Duality rotation transforms spacetime element to element through angle with pseudoscalar is:
Duality rotation occurs only for non-singular Clifford algebra, non-singular meaning a Clifford algebra containing pseudoscalars with a non-zero square.
Grade involution transforms every -vector to :
Reversion transformation occurs by decomposing any spacetime element as a sum of products of vectors and then reversing the order of each product. For multivector
arising from a product of vectors, the reversion is :
Clifford conjugation of a spacetime element combines reversion and grade involution transformations, indicated as :
The grade involution, reversion and Clifford conjugation transformations are involutions.

Classical electromagnetism

Faraday bivector

In STA, the electric field and magnetic field can be unified into a single bivector field, known as the Faraday bivector, equivalent to the Faraday tensor. It is defined as:
where and are the usual electric and magnetic fields, and is the STA pseudoscalar. Alternatively, expanding in terms of components, is defined that
The separate and fields are recovered from using
The term represents a given reference frame, and as such, using different reference frames will result in apparently different relative fields, exactly as in standard special relativity.
Since the Faraday bivector is a relativistic invariant, further information can be found in its square, giving two new Lorentz-invariant quantities, one scalar, and one pseudoscalar:
The scalar part corresponds to the Lagrangian density for the electromagnetic field, and the pseudoscalar part is a less-often seen Lorentz invariant.

Maxwell's equation

STA formulates Maxwell's equations in a simpler form as one equation, rather than the 4 equations of vector calculus. Similarly to the above field bivector, the electric charge density and current density can be unified into a single spacetime vector, equivalent to a four-vector. As such, the spacetime current is given by
where the components are the components of the classical 3-dimensional current density. When combining these quantities in this way, it makes it particularly clear that the classical charge density is nothing more than a current travelling in the timelike direction given by.
Combining the electromagnetic field and current density together with the spacetime gradient as defined earlier, we can combine all four of Maxwell's equations into a single equation in STA.
The fact that these quantities are all covariant objects in the STA automatically ensures Lorentz covariance of the equation, which is much easier to show than when separated into four separate equations.
In this form, it is also much simpler to prove certain properties of Maxwell's equations, such as the conservation of charge. Using the fact that for any bivector field, the divergence of its spacetime gradient is, one can perform the following manipulation:
This equation has the clear meaning that the divergence of the current density is zero, i.e. the total charge and current density over time is conserved.
Using the electromagnetic field, the form of the Lorentz force on a charged particle can also be considerably simplified using STA.