February 9, 1999 PHYSICAL APPLICATIONS OF GEOMETRIC ALGEBRA LECTURE 8 SUMMARY In this lecture we introduce the spacetime algebra, the geometric algebra of spacetime. This forms the basis for most of the remaining course, and is central to the formulation of a powerful gauge theory of gravity. ffl Adding a vector for time the 4-d spacetime algebra and some consequences of a mixed signature metric. ffl Paths, observers and frames. ffl Projective splits for observers. ffl Handling Lorentz transformations with rotors. ffl Photons and redshifts. ffl The structure of the Lorentz group. 1
AN ALGEBRA FOR SPACETIME Aim to construct the geometric algebra of spacetime. Invariant interval is s 2 = c 2 t 2 x 2 y 2 z 2 (The particle physics choice. GR flips signs. No observable consequences). Work in natural units, c =1. Need four vectors fe 0 ;e i g;i =1:::3 with properties 2 e 0 =1; 2 ei = 1 e 0 e i =0; e i e j = ffi ij Summarised by e μ e ν = diag(+ ); μ; ν =0:::3 Bivectors 4 3=2 = 6bivectors in algebra. Two types 1. Those containing e 0, e.g. fe i^e 0 g, 2. Those not containing e 0, e.g. fe i^e j g. For any pair of vectors a and b, a b =0have (a^b) 2 = abab = abba = a 2 b 2 2
The two types have different squares (e i^e j ) 2 = e 2 2 i e j = 1 Spacelike Euclidean bivectors, generate rotations in a plane. (e i^e 0 ) 2 = e 2 2 i e 0 =1 Timelike bivectors. Generate hyperbolic geometry: e ffe 1e0 = 1+ffe 1 e 0 + ff 2 =2! + ff 3 =3! e 1 e 0 + = ch(ff) + sh(ff)e 1 e 0 Crucial to treatment of Lorentz transformations. THE PSEUDOSCALAR Define the pseudoscalar I I = e 0 e 1 e 2 e 3 Still chosen to be right-handed. Projecting onto subspaces have e 2 e 1 Down e 3 Down e 2 e 1 e 3 e 0 e 0 3
Have to be careful with these definitions. Traditionally draw spacetime diagrams as t e 0 e 1 x right-handed volume element for this is e 1 e 0. Since I is grade 4, it has ~I = e 3 e 2 e 1 e 0 = I Compute the square of I : I 2 = I ~ I =(e 0 e 1 e 2 e 3 )(e 3 e 2 e 1 e 0 )= 1 Multiply bivector by I, get grade 4 2=2 another bivector. Provides map between bivectors with positive and negative square: Ie 1 e 0 = e 1 e 0 I = e 1 e 0 e 0 e 1 e 2 e 3 = e 2 e 3 Define B i = e i e 0. Bivector algebra is B i B j = ffl ijk IB k ; (IB i ) (IB j )= ffl ijk IB k (IB i ) B j = ffl ijk B k 4
Have four vectors, and four trivectors in algebra. Interchanged by duality e 1 e 2 e 3 = e 0 e 0 e 1 e 2 e 3 = e 0 I = Ie 0 NB I anticommutes with vectors and trivectors. (In space of even dimensions). I always commutes with even-grade. THE SPACETIME ALGEBRA Putting terms together, get an algebra with 16 terms: 1 ffl μ g ffl μ^fl ν g fifl μ g I 1 4 6 4 1 scalar vectors bivectors trivectors pseudoscalar The spacetime algebra or STA. Use ffl μ g for preferred orthonormal frame. Also define ff i = fl i fl 0 Not used i for the pseudoscalar. Potentially confusing. The ffl μ g satisfy fl μ fl ν + fl ν fl μ =2 μν This is the Dirac matrix algebra (with identity matrix on right). A matrix representation of the STA. Explains notation, but ffl μ g are vectors, not a set of matrices in isospace. 5
FRAMES AND TRAJECTORIES x( ) a spacetime trajectory. Tangent vector is Two cases to consider: x 0 = x( ) @ Timelike, x 0 2 > 0. Introduce proper time fi : v = @ fi x = _x; v 2 =1 Observers measure this. Unit vector v defines the instantaneous rest frame. Null, x 0 2 =0. Describes a null trajectory. Taken by massless particles, (photons, etc.). Proper distance/time = 0. Photons do carry an intrinsic clock (their frequency), but can tick at arbitrary rate. Now take observer on timelike path with instantaneous velocity v. What do we measure? Construct a rest frame fe i g, e i v =0; i =1:::3 Take point on worldline as spatial origin. Event x has time coordinate t = x v and space coordinates x i = x e i The 3-d vector to a point on the worldline of an object 6
intersecting our rest frame: x i e i = x e μ e μ x e 0 e 0 = x x vv = x^vv Wedge product with v projects onto components of x in rest frame of v. Define relative vector by spacetime bivector x^v: x = x^v With this definitions have xv = x v + x^v = t + x Invariant distance decomposes as x 2 = xvvx =(x v + x^v)(x v + v^x) =(t + x)(t x) =t 2 x 2 Recovers usual result. Built into definition of STA. THE EVEN SUBALGEBRA Each observer sees set of relative vectors. Model these as spacetime bivectors. Take timelike vector fl 0, relative vectors ff i = fl i fl 0. Satisfy ff i ff j = 1 2 (fl ifl 0 fl j fl 0 + fl j fl 0 fl i fl 0 ) = 1 2 ( fl ifl j fl j fl i )=ffi ij Generators for a 3-d algebra! The GA of the 3-d relative space 7
in rest frame of fl 0. Volume element ff 1 ff 2 ff 3 =(fl 1 fl 0 )(fl 2 fl 0 )(fl 3 fl 0 )= fl 1 fl 0 fl 2 fl 3 = I so 3-d subalgebra shares same pseudoscalar as spacetime. Still have 1 2 (ff iff j ff j ff i )=ffl ijk Iff k relative vectors and relative bivectors are spacetime bivectors. Projected onto the even subalgebra of the STA. 1 ffl μ g fff i ;Iff i g fifl μ g I 4 d 1 fff i g fiff i g I 3 d The 6 spacetime bivectors split into relative vectors and relative bivectors. This split is observer dependent. Avery useful technique. Conventions Expression like a^b potentially confusing. ffl Spacetime bivectors used as relative vectors are written in bold. Includes the fff i g. ffl If both arguments bold, dot and wedge symbols drop down to their 3-d meaning. ffl Otherwise, keep spacetime definition. 8
EXAMPLES i. Velocity Observer, with constant velocity v. Measures relative velocity of a particle with proper velocity u(fi ), u 2 =1. Form uv = @ fi (x(fi )v) =@ fi (t + x) So that @ fi t = u v The relative velocity is u = @x @t = @x @fi @fi u^v @t = u v Familiar same as used in projective geometry! Also ensured that projective vectors have positive square. Use this computer vision applications! ii. Momentum and Wave Vectors Observe particle with energy-momentum p. Energy measured = p v, relative momentum = p^v, Recover the invariant pv = p v + p^v = E + p m 2 = p 2 = pvvp =(E + p)(e p) =E 2 p 2 9
Similarly, for a photon wave-vector k, kv = k v + k^v =! + k For photons in empty space k 2 =0so 0=kvvk =(! + k)(! k) =! 2 k 2 Recovers jkj =!. Holds in all frames. LORENTZ TRANSFORMATIONS Usually expressed as a coordinate transformation, e.g. x 0 = fl(x fit) t 0 = fl(t fix) x=fl(x 0 + fit 0 ) t = fl(t 0 + fix 0 ) where fl =(1 fi 2 ) 1=2 and fi is scalar velocity. Vector x decomposed in two frames, fe μ g and fe 0 μg, x = x μ e μ = x μ0 e 0 μ with t = e 0 x; t 0 = e 0 0 x: Concentrating on the 0, 1 components: Derive vector relations te 0 +xe 1 = t 0 e 0 0 +x0 e 0 1 ; e 0 0 = fl(e 0 + fie 1 ); e 0 1 = fl(e 1 + fie 0 ): 10
Gives new frame in terms of the old. Now introduce hyperbolic angle ff, tanhff = fi; (fi <1); Gives Vector e 0 0 fl =(1 tanh 2 ff) 1=2 = coshff: is now e 0 0 = ch(ff)e 0 +sh(ff)e 1 = (ch(ff) + sh(ff)e 1 e 0 )e 0 = e ffe 1e0 e 0 ; Similarly, we have e 0 1 =ch(ff)e 1 +sh(ff)e 0 = e ffe1e0 e 1 : Two other frame vectors unchanged. Relationship between the frames is e 0 μ = Re μ ~ R; e μ 0 = Re μ ~ R; R = e ffe 1e0=2 : Same rotor prescription works for boosts as well as rotations! Spacetime is a unified entity now. EXAMPLES i. Addition of Velocities Two objects separating, velocities v 1 = e ff 1e1e0 e 0 ; v 2 = e ff 2e1e0 e 0 : 11
What is the relative velocity sees for each other? Form v 1^v 2 v 1 v 2 h = e (ff1 + ff2)e1e0 i 2 = sinh(ff 1 + ff 2 )e 1 e 0 : h e (ff 1 + ff2)e1e0 i 0 cosh(ff 1 + ff 2 ) Both observers measure relative velocity tanh(ff 1 + ff 2 )= tanhff 1 + tanhff 2 1 tanhff 1 tanhff 2 Addition of velocities is achieved by adding hyperbolic angles. Recovers familiar formula. ii. Photons and Redshifts Two particles on different worldlines. Particle 1 emits a photon, received by particle 2 v 1 k v 2 Frequency for particle 1 is! 1 = v 1 k, for particle 2 is! 2 = v 2 k. Ratio describes the Doppler effect, often expressed as a redshift: 1+z =! 1 =! 2 12
Can be applied in many ways. If emitter receding in e 1 direction, and v 2 = e 0,have so that k =! 2 (e 0 + e 1 ); v 1 = coshffe 0 sinhffe 1 1+z =! 2(coshff +sinhff)! 2 = e ff Boost of a null vector = dilation. Just as in G n;n! Velocity of emitter in e 0 frame is tanhff, and e ff = 1 + tanhff 1 tanhff 1=2 Aberration formulae obtained same way. THE LORENTZ GROUP Group of transformations preserving lengths and angles. Build from reflections a 7! nan 1 The n 1 needed for both timelike n 2 > 0 and spacelike n 2 < 0. Cannot have null n. Timelike n generates time-reversal. Spacelike n preserve time ordering. Full Lorentz group contains 4 sectors. 13
Space Reflection Time reversal I Proper Orthochronous III I with time reversal II I with space refelection IV I with a 7! a Easily understood in STA. Combine an even numbers of reflections, a 7! ψaψ 1 ψ is an even multivector. Need to ensure that result is a vector. Form ψ ψ ~ =(ψ ψ) ~ ο Even and equal to own reverse. A scalar and a pseudoscalar ψ ψ ~ = ff 1 + Iff 2 = ρ e Ifi with ρ 6= 0. Define rotor R by R = ψ(ρ e Ifi ) 1=2 so that R R ~ = ψ ψ(ρ ~ e Ifi ) 1 =1 as required. Now have ψ = ρ 1=2 e Ifi=2 R; ψ 1 = ρ 1=2 e Ifi=2 ~ R 14
and transformation becomes a 7! e Ifi=2 Ra e Ifi=2 ~ R = e Ifi Ra ~ R Must have fi =0or fi = ß. Gives a 7! ±Ra R ~ Proper Orthochronous Transformations Transformation a 7! Ra R ~ preserves causal ordering. Take fl 0 7! v = Rfl ~ 0 R. Need the fl 0 component of v to be positive, fl 0 v = hfl 0 Rfl 0 ~ Ri > 0 Decomposing in fl 0 frame R = ff + a + Ib + Ifi find that hfl 0 Rfl 0 ~ Ri = ff 2 + a 2 + b 2 + fi 2 > 0 as required. Rotor transformation law define the restricted Lorentz group. Physically most relevant. fi = ß gives class IV transformations. 15