In mathematical physics, spacetime algebra (STA) is the application of Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4) 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 (directed quantities describing rotations associated with rotations or particular planes, such as areas, or rotations) or blades (quantities associated with particular hyper-volumes) 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(R) 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 (APS, Pauli algebra) 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, a {\displaystyle a} and b {\displaystyle b} , there is a geometric product a b {\displaystyle ab} , scalar ('inner') product a ⋅ b {\displaystyle a\cdot b} and exterior ('wedge', 'outer') product a ∧ b {\displaystyle a\wedge b} . The vector product is a sum of a scalar and exterior product:
a ⋅ b = a b + b a 2 = b ⋅ a , a ∧ b = a b − b a 2 = − b ∧ a , a b = a ⋅ b + a ∧ b . {\displaystyle a\cdot b={\frac {ab+ba}{2}}=b\cdot a,\quad a\wedge b={\frac {ab-ba}{2}}=-b\wedge a,\quad ab=a\cdot b+a\wedge b.}
The scalar product generates a real number (scalar), and the exterior product generates a bivector. The vectors a {\displaystyle a} and b {\displaystyle b} are orthogonal if their scalar product is zero; vectors a {\displaystyle a} and b {\displaystyle b} are parallel if their exterior product is zero. The orthonormal basis vectors are a timelike vector γ 0 {\displaystyle \gamma _{0}} and 3 spacelike vectors γ 1 , γ 2 , γ 3 {\textstyle \gamma _{1},\gamma _{2},\gamma _{3}} . The Minkowski metric tensor's nonzero terms are the diagonal terms, 1 {\displaystyle {1}} . For μ , ν = 0 , 1 , 2 , 3 {\displaystyle \mu ,\nu =0,1,2,3} :
γ μ ⋅ γ ν = γ μ γ ν + γ ν γ μ 2 = η μ ν , γ 0 ⋅ γ 0 = 1 , γ 1 ⋅ γ 1 = γ 2 ⋅ γ 2 = γ 3 ⋅ γ 3 = − 1 , otherwise γ μ γ ν = − γ ν γ μ {\displaystyle \gamma _{\mu }\cdot \gamma _{\nu }={\frac {\gamma _{\mu }\gamma _{\nu }+\gamma _{\nu }\gamma _{\mu }}{2}}=\eta _{\mu \nu },\quad \gamma _{0}\cdot \gamma _{0}=1,\ \gamma _{1}\cdot \gamma _{1}=\gamma _{2}\cdot \gamma _{2}=\gamma _{3}\cdot \gamma _{3}=-1,\quad {\text{ otherwise }}\ \gamma _{\mu }\gamma _{\nu }=-\gamma _{\nu }\gamma _{\mu }}
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 { 1 } {\displaystyle \{1\}} , four vectors { γ 0 , γ 1 , γ 2 , γ 3 } {\displaystyle \{\gamma _{0},\gamma _{1},\gamma _{2},\gamma _{3}\}} , six bivectors { γ 0 γ 1 , γ 0 γ 2 , γ 0 γ 3 , γ 1 γ 2 , γ 2 γ 3 , γ 3 γ 1 } {\displaystyle \{\gamma _{0}\gamma _{1},\gamma _{0}\gamma _{2},\gamma _{0}\gamma _{3},\gamma _{1}\gamma _{2},\gamma _{2}\gamma _{3},\gamma _{3}\gamma _{1}\}} , four pseudovectors (trivectors) { I γ 0 , I γ 1 , I γ 2 , I γ 3 } {\displaystyle \{I\gamma _{0},I\gamma _{1},I\gamma _{2},I\gamma _{3}\}} and one pseudoscalar { I } {\displaystyle \{I\}} with I = γ 0 γ 1 γ 2 γ 3 {\displaystyle I=\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}} . The pseudoscalar commutes with all even-grade STA elements, but anticommutes with all odd-grade STA elements.
Subalgebra
STA's even-graded elements (scalars, bivectors, pseudoscalar) form a subalgebra isomorphic to Clifford algebra Cl3,0(R), 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 ( γ 1 γ 0 , γ 2 γ 0 , γ 3 γ 0 ) {\textstyle (\gamma _{1}\gamma _{0},\gamma _{2}\gamma _{0},\gamma _{3}\gamma _{0})} as ( σ 1 , σ 2 , σ 3 ) {\textstyle (\sigma _{1},\sigma _{2},\sigma _{3})} and the STA bivectors ( γ 3 γ 2 , γ 1 γ 3 , γ 2 γ 1 ) {\textstyle (\gamma _{3}\gamma _{2},\gamma _{1}\gamma _{3},\gamma _{2}\gamma _{1})} as ( I σ 1 , I σ 2 , I σ 3 ) {\textstyle (I\sigma _{1},I\sigma _{2},I\sigma _{3})} . The Pauli matrices, σ ^ 1 , σ ^ 2 , σ ^ 3 {\textstyle {\hat {\sigma }}_{1},{\hat {\sigma }}_{2},{\hat {\sigma }}_{3}} , are a matrix representation for σ 1 , σ 2 , σ 3 {\textstyle \sigma _{1},\sigma _{2},\sigma _{3}} . For any pair of ( σ 1 , σ 2 , σ 3 ) {\textstyle (\sigma _{1},\sigma _{2},\sigma _{3})} , the nonzero scalar products are σ 1 ⋅ σ 1 = σ 2 ⋅ σ 2 = σ 3 ⋅ σ 3 = 1 {\textstyle \sigma _{1}\cdot \sigma _{1}=\sigma _{2}\cdot \sigma _{2}=\sigma _{3}\cdot \sigma _{3}=1} , and the nonzero exterior products are:
σ 1 ∧ σ 2 = I σ 3 σ 2 ∧ σ 3 = I σ 1 σ 3 ∧ σ 1 = I σ 2 {\displaystyle {\begin{aligned}\sigma _{1}\wedge \sigma _{2}&=I\sigma _{3}\\\sigma _{2}\wedge \sigma _{3}&=I\sigma _{1}\\\sigma _{3}\wedge \sigma _{1}&=I\sigma _{2}\\\end{aligned}}}
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[0](1,3)(R) of real space-time spinors in Cl1,3(R) is isomorphic to the Clifford algebra Cl3,0(R) of Euclidean space R3 with basis elements. See the illustration of space-time algebra spinors in Cl[0](1,3)(R) under the octonionic product as a Fano plane.
Division A nonzero vector a {\displaystyle a} is a null vector (degree 2 nilpotent) if a 2 = 0 {\displaystyle a^{2}=0} . An example is a = γ 0 + γ 1 {\displaystyle a=\gamma ^{0}+\gamma ^{1}} . Null vectors are tangent to the light cone (null cone). An element b {\displaystyle b} is an idempotent if b 2 = b {\displaystyle b^{2}=b} . Two idempotents b 1 {\displaystyle b_{1}} and b 2 {\displaystyle b_{2}} are orthogonal idempotents if b 1 b 2 = 0 {\displaystyle b_{1}b_{2}=0} . An example of an orthogonal idempotent pair is 1 2 ( 1 + γ 0 γ k ) {\displaystyle {\tfrac {1}{2}}(1+\gamma _{0}\gamma _{k})} and 1 2 ( 1 − γ 0 γ k ) {\displaystyle {\tfrac {1}{2}}(1-\gamma _{0}\gamma _{k})} with k = 1 , 2 , 3 {\displaystyle k=1,2,3} . 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 (reciprocal) 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 c {\textstyle c} may be possible by multiplication by its inverse, defined as c − 1 = ( c ⋅ c ) − 1 c {\displaystyle c^{-1}=(c\cdot c)^{-1}c} .
Reciprocal frame Associated with the orthogonal basis { γ 0 , γ 1 , γ 2 , γ 3 } {\displaystyle \{\gamma _{0},\gamma _{1},\gamma _{2},\gamma _{3}\}} is the reciprocal basis set { γ 0 , γ 1 , γ 2 , γ 3 } {\displaystyle \{\gamma ^{0},\gamma ^{1},\gamma ^{2},\gamma ^{3}\}} satisfying these equations:
γ μ ⋅ γ ν = δ μ ν , μ , ν = 0 , 1 , 2 , 3 {\displaystyle \gamma _{\mu }\cdot \gamma ^{\nu }=\delta _{\mu }^{\nu },\quad \mu ,\nu =0,1,2,3}
These reciprocal frame vectors differ only by a sign, with γ 0 = γ 0 {\displaystyle \gamma ^{0}=\gamma _{0}} , but γ 1 = − γ 1 {\displaystyle \gamma ^{1}=-\gamma _{1}} , γ 2 = − γ 2 {\displaystyle \gamma ^{2}=-\gamma _{2}} , γ 3 = − γ 3 {\displaystyle \gamma ^{3}=-\gamma _{3}} . A vector a {\displaystyle a} may be represented using either the basis vectors or the reciprocal basis vectors a = a μ γ μ = a μ γ μ {\displaystyle a=a^{\mu }\gamma _{\mu }=a_{\mu }\gamma ^{\mu }} with summation over μ = 0 , 1 , 2 , 3 {\displaystyle \mu =0,1,2,3} , according to the Einstein notation. The scalar product of vector and basis vectors or reciprocal basis vectors generates the vector components.
a ⋅ γ ν = a ν , ν = 0 , 1 , 2 , 3 a ⋅ γ ν = a ν , ν = 0 , 1 , 2 , 3 {\displaystyle {\begin{aligned}a\cdot \gamma ^{\nu }&=a^{\nu },\quad \nu =0,1,2,3\\a\cdot \gamma _{\nu }&=a_{\nu },\quad \nu =0,1,2,3\end{aligned}}}
The metric and index gymnastics raise or lower indices:
γ μ = η μ ν γ ν , μ , ν = 0 , 1 , 2 , 3 γ μ = η μ ν γ ν , μ , ν = 0 , 1 , 2 , 3 {\displaystyle {\begin{aligned}\gamma _{\mu }&=\eta _{\mu \nu }\gamma ^{\nu },\quad \mu ,\nu =0,1,2,3\\\gamma ^{\mu }&=\eta ^{\mu \nu }\gamma _{\nu },\quad \mu ,\nu =0,1,2,3\end{aligned}}}
Spacetime gradient The spacetime gradient, like the gradient in a Euclidean space, is defined such that the directional derivative relationship is satisfied:
a ⋅ ∇ F ( x ) = lim τ → 0 F ( x + a τ ) − F ( x ) τ . {\displaystyle a\cdot \nabla F(x)=\lim _{\tau \rightarrow 0}{\frac {F(x+a\tau )-F(x)}{\tau }}.}
This requires the definition of the gradient to be
∇ = γ μ ∂ ∂ x μ = γ μ ∂ μ . {\displaystyle \nabla =\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}=\gamma ^{\mu }\partial _{\mu }.}
Written out explicitly with x = c t γ 0 + x k γ k {\displaystyle x=ct\gamma _{0}+x^{k}\gamma _{k}} , these partials are
∂ 0 = 1 c ∂ ∂ t , ∂ k = ∂ ∂ x k . {\displaystyle \partial _{0}={\frac {1}{c}}{\frac {\partial }{\partial t}},\quad \partial _{k}={\frac {\partial }{\partial {x^{k}}}}.}
Space–time split
In STA, a space–time split is a projection from four-dimensional space into (3+1)-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 γ 0 {\displaystyle \gamma _{0}} , which serves to split a four vector into a scalar timelike and a bivector spacelike component, in the reference frame co-moving with γ 0 {\displaystyle \gamma _{0}} . With x = x μ γ μ {\displaystyle x=x^{\mu }\gamma _{\mu }} , we have
x γ 0 = x 0 + x k γ k γ 0 γ 0 x = x 0 − x k γ k γ 0 {\displaystyle {\begin{aligned}x\gamma _{0}&=x^{0}+x^{k}\gamma _{k}\gamma _{0}\\\gamma _{0}x&=x^{0}-x^{k}\gamma _{k}\gamma _{0}\end{aligned}}}
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 ( γ {\displaystyle \gamma } ) As these bivectors γ k γ 0 {\displaystyle \gamma _{k}\gamma _{0}} square to 1 {\displaystyle 1} , they serve as a spatial basis. Utilizing the Pauli matrix notation, these are written σ k = γ k γ 0 {\displaystyle \sigma _{k}=\gamma _{k}\gamma _{0}} . Spatial vectors in STA are denoted in boldface; then with x = x k σ k {\displaystyle \mathbf {x} =x^{k}\sigma _{k}} and x 0 = c t {\displaystyle x^{0}=ct} , the γ 0 {\displaystyle \gamma _{0}} -space–time split x γ 0 {\displaystyle x\gamma _{0}} , and its reverse γ 0 x {\displaystyle \gamma _{0}x} are:
x γ 0 = x 0 + x k σ k = c t + x γ 0 x = x 0 − x k σ k = c t − x {\displaystyle {\begin{aligned}x\gamma _{0}&=x^{0}+x^{k}\sigma _{k}=ct+\mathbf {x} \\\gamma _{0}x&=x^{0}-x^{k}\sigma _{k}=ct-\mathbf {x} \end{aligned}}}
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 σ k = γ k γ 0 {\displaystyle \sigma _{k}=\gamma _{k}\gamma ^{0}} and σ k = γ 0 γ k {\displaystyle \sigma ^{k}=\gamma _{0}\gamma ^{k}} must be used.
Transformations To rotate a vector v {\displaystyle v} in geometric algebra, the following formula is used:
v ′ = e − β θ 2 v e β θ 2 {\displaystyle v'=e^{-\beta {\frac {\theta }{2}}}\ v\ e^{\beta {\frac {\theta }{2}}}} , where θ {\displaystyle \theta } is the angle to rotate by, and β {\displaystyle \beta } is the bivector representing the plane of rotation normalized so that β β ~ = 1 {\displaystyle \beta {\tilde {\beta }}=1} . For a given spacelike bivector, β 2 = − 1 {\displaystyle \beta ^{2}=-1} , so Euler's formula applies, giving the rotation
v ′ = ( cos ( θ 2 ) − β sin ( θ 2 ) ) v ( cos ( θ 2 ) + β sin ( θ 2 ) ) {\displaystyle v'=\left(\cos \left({\frac {\theta }{2}}\right)-\beta \sin \left({\frac {\theta }{2}}\right)\right)\ v\ \left(\cos \left({\frac {\theta }{2}}\right)+\beta \sin \left({\frac {\theta }{2}}\right)\right)} . For a given timelike bivector, β 2 = 1 {\displaystyle \beta ^{2}=1} , so a "rotation through time" uses the analogous equation for the split-complex numbers:
v ′ = ( cosh ( θ 2 ) − β sinh ( θ 2 ) ) v ( cosh ( θ 2 ) + β sinh ( θ 2 ) ) {\displaystyle v'=\left(\cosh \left({\frac {\theta }{2}}\right)-\beta \sinh \left({\frac {\theta }{2}}\right)\right)\ v\ \left(\cosh \left({\frac {\theta }{2}}\right)+\beta \sinh \left({\frac {\theta }{2}}\right)\right)} . 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 (corresponding to a reference frame) to another, one or more of these transformations must be used.
Any spacetime element A {\displaystyle A} is transformed by mu