Symmmetry & Invariance

MIT Course 8.033, Symmmetry & Invariance 1 Topics • The notion of symmetry in physics. Key concepts: – Frame – Inertial frame – Transformation – Invariant – Invariance – Symmetry – Relativity • Examples of possible symmetry: – Translation – Rotation – Parity – Galilean – Lorentz – Diffeomorphism – Gauge • We will study the symmetry of: – Classical mechanics — both initial conditions (v, E, etc.) and laws – Electromagnetism — in particular ∗ The wave equation ∗ Observed properties of light — does speed depend on wavelength? On motion of source? On motion of observer (frame)? • Key people: – Galileo Galilei, 1564-1642 – Emmy Noether, 1882-1935 – Michaelson & Morley 2 Formula summary: transformations • Translation: � r� = r + Δr t� = t + Δt • Rotation: � • Parity � • Galilean: • Lorentz: � r� = Rr t� = t r� = −r t� = t r� = r − vt t� = t x� = Λx, i.e., for v in x-direction, ⎧ � x = γ(x − vt) ⎪ ⎪ ⎨ � y =y , z� = z ⎪ ⎪ ⎩ � 2 t = γ(t − vx/c ) γ≡� 1 1− v2 c2 . 3 Symmetry in physics 3.1 Glossary of key concepts • Frame: A prescription for measuring the physical quantities (e.g., r, t) that appear in our equations (an operational definition of them). • Inertial frame: A particular kind of frame in which Newton’s 1st law holds. • Transformation: The mathematical operation converting the quantities measured in one frame into the quantities measured in another. To avoid confusion, we usually put primes on the quanti­ ties in one of the two frames. • Invariant: A quantity or equation that is left unchanged by a transformation. A quantity Q is invariant if Q� = Q. An equation is invariant if it stays true in the new frame (i.e., when you put primes on all quantities). • Invariance: Being invariant. For example, the equations of clas­ sical mechanics are said to be Galilean invariant because they are invariant under Galilean transformations. • Symmetry: In physics, essentially a synonym for invariance. • Relativity: Essentially a synonym for invariance and symmetry, but usually restricted to the symmetries of spacetime. 3.2 Invariance under translation • No experiment within your lab can determine whether it’s been shifted sideways. In other words, the laws of nature appear to be translationally invariant. • Original frame: masses at r1 and r2 . F = GmM |r2 − r1 |2 • Primed frame: masses at r�1 ≡ r1 + a and r�2 ≡ r2 + a. F� = GmM GmM GmM = = =F |r�2 − r�1 |2 |(r2 + a) − (r1 + a)|2 |r2 − r1 |2 • We have time translationak invariance in time as well as in space. 3.3 Invariance under rotation • No experiment within your spaceship can determine whether it’s been rotated. In other words, the laws of nature appear to be rotationally invariant. • If you’re not cool with 3×3 matrices, please read the matrix primer handout. • Primed frame: masses at r�1 ≡ Rr1 and r�2 ≡ Rr2 F� = GmM GmM GmM = = =F |Rr2 − Rr1 |2 |R(r2 − r1 )|2 |r2 − r1 |2 • Another example: Maxwell’s equations in vacuum imply �2 E = 1 Ë. c2 Since only differences in position and time enter, it’s translationally invariant. Here it’s infinitesimal differences (derivatives), above it was a finite difference (r2 − r1 ). • �2 is invariant under rotation (remember Gauss’ theorem) • At MIT: • Near Australia: ⎛ ⎞ 0 1 ¨ ⎝ 0 ⎠. E = 2E = c 1 ⎛ ⎞ 0 1 ¨ ⎝ 0 ⎠. E = 2E = c −1 • So both observer’s agree that Maxwell was right, i.e., the wave equation is translationally and rotationally invariant. 3.4 Invariance under reflection (parity)? • Yes for all of classical physics • Considered self-evident and obvious • 1956: Chen Ning Yang & Tsung-Dao Lee propose that weak in­ teractions violate parity; Chien-Shiung Wu demonstrates it with cobalt 60, Leon Lederman with accelerator. (Yang & Lee get 1957 Nobel prize.) 3.5 Symmetry is at the heart of modern physics • Special relativity is all about so-called Lorentz symmetry. • General relativity is about so-called diffeomorphism symmetry. • Key topics in particle physics are C, P and T symmetry and com­ binations like CP and CPT symmetry. • A cornerstone of particle physics is gauge symmetry • In 2007, the Large Hadron Collider at CERN will search for super­ symmetry. 4 The symmetry properties of classical mechanics Under what transformations are the laws of classical mechanics invari­ ant? Answer: Galilean transformations. 4.1 Invariance under Galilean transformation • Demo with colliding carts, ball. • So Newtonian mechanics appears to be invariant - let’s understand exactly what the transformation is, and why this is so. • Inertial frame definition (a = 0 if F = 0) • Are we in an inertial frame? (PS1) • Galilean transformation definition (between 2 inertial frames) • Definition of event: a 4D point (x, y, z, t). Examples? • r� = r − vt • Lengths invariant: Δr� ≡ r�2 − r�1 = (r2 − vt) − (r1 − vt) = Δr • But we must measure r�1 and r2� at the same time! • Which we can, since time is invariant and unambiguous: t� = t 4.2 Transforming velocity • How does u transform under a Galilean transformation? u ≡ u� ≡ dr dt dr� d dr = (r − vt) = −v =u−v dt� dt dt So velocities add/subtract as you’d expect: u� = u − v • But what about the flashlight on the train? 4.3 Transforming acceleration a ≡ a� ≡ du dt du� d du = (u − v) = =a dt� dt dt So acceleration is invariant. 4.4 Transforming F = ma • Consider forces that depend on separation: – Spring: F = k(x2 − x1 ) – Gravity: F = GmM |r2 −r1 |2 They are invariant, since lengths are. • m is invariant • Since F, a and m are all invariant, so is the equation F = ma. • So the physical law is invariant, but not the initial conditions! 4.5 Transforming energy & momentum • Neither is invariant, since v isn’t. • But the conservation laws are invariant: E and p are conserved in any frame (PS1). • Work-energy theorem: W = ΔKE, where work defined as � x2 W = F dx. x1 • Proof: � W � x2 = x1 � = � x2 F dx = x1 v2 m v1 x2 ma dx = m dx dv = m dt � x1 v2 v dv = v1 dv dx dt mv12 mv22 − = ΔKE. 2 2 Only assumption here was F = ma, which is invariant, so the work-energy theorem is also invariant. • W and KE alone are not invariant. 4.6 Transforming trajectories • Is the 3D shape of a trajectory not invariant? • No! Basket ball example: line in frame A is parabola in frame B. 4.7 Key Galilean non-invariants ⎧ ⎨ r� = r − vt u� = u − v ⎩ � p = p − mv Quantity t r Δt Δr |Δr| d/dt � �2 v p a F m Ekin W F = ma Newt. mechanics Electromagnetism Translational? N N Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Invariance Rotational? Y N Y N Y Y N Y N N N N Y Y Y Y Y Y Gallilian? Y N Y Y Y Y Y Y N N Y Y Y N N Y Y N 5 The symmetry properties of electromagnetism Under what transformations are the laws of electromagnetism invariant? Let’s focus on a simple special case: the laws that govern the propagation of light. 5.1 The classical wave equation • Classical wave equation (8.03): �2 E − 1 ¨ E = 0. c2w For example, E could denote: – One of the three component of the electric field – One of the three component of the magnetic field – Air density – Height of water surface (2D) – Deflection of guitar string (1D) • 1D special case: d2 E 1 d2 E − 2 = 0. 2 dx cw dt2 • General solution (show on PS2): y = Af (x − cw t) + Bf (x + cw t), for arbitrary smooth function f and constants A & B. • More complicated in 3D, but wavefronts still propagate with speed cw . 5.2 Symmetry of the wave equation • We learned that classical mechanics was invariant under Galilean transformations. • The wave equation can be derived from classical mechanics. Question: is the classical wave equation invariant under Galilean trans­ formations? 1. Yes 2. No 3. Yes, but only if wave speed cw � c 5.3 Transforming the wave equation • Apply Galilean transformation to 1D wave equation: d2 E 1 d2 E − = 0. dx2 c2 dt2 • Do this on PS2 - hints: – x� = x + vt – t� = t – Use chain rule for derivatives: ∂ ∂x ∂ ∂t = = ∂x� ∂ ∂t� ∂ ∂ + = � � ∂x ∂x ∂x ∂t ∂x� � � ∂x ∂ ∂t ∂ ∂ ∂ + =v � + � ∂t ∂x� ∂t ∂t� ∂x ∂t – Work out 2nd derivatives too • Result: � � v 2 d2 E 1 d2 E v d2 E 1− 2 − − 2 = 0. c dx�2 c2 dt�2 c dx� dt� • Wave equation not invariant under Galilean transformation • Show on PS2: the new equation has solution y = Af (x − [c − v]t) + Bf (x + [c + v]t), i.e., waves travel slower forward than backward. • Just what you’d expect for waves in a substance, “aether” (veloc­ ities add). • How can this be consistent with the wave equation being derived from classical mechanics, which is Galilean invariant? 5.4 Observed properties of speed of light • Does speed depend on wavelength? No! • Does speed depend on motion of source? No! • Does speed depend on motion of observer (frame)? No! For all three cases, let’s now look at the evidence. 5.4.1 Does c depend on wavelength? • Does light speed through glass depend on wavelength? • But what about light speed through vacuum? • Gamma-ray bursts provide great test • Gamma-ray bursts last a few seconds to minutes • Old speculations: nefarious nukes, civilization annihilation, nearby neutron stars • Recently shown to originate at cosmological distances (few billion light years ∼ 1017 light-seconds). • Flash seen also at x-rays and optical wavelengths, all within of order a minute ∼ 102 seconds, so Δt < 102 s = 10−15 . t ∼ 1017 s • c = d/t, so relative speed variation with wavelength is Δc Δt < −15 ≈ 10 . c t ∼ Answer: No, at least not more than about 10−15 c ≈ 300 nm/s. 5.4.2 Does c depend on source motion? • Does speed of a bullet depend on speed of rifle? • Does sound speed of a gun shot depend on speed of rifle? • Binary stars provide great test • If velocities add, then t1 = t2 = d c−v d c+v Δt ≡ t1 − t2 = 2dv dv v ≈2 = 2t ≈ 200 years, say c2 − v 2 cc c (for a pulsar in the Large Magellanic Cloud with v = 300 km/s, d = 100000 lightyears) • But half an orbit takes only 2 days, say • You’d see new “Doppler effect” ∝ a rather than v • You’d see things moving backward in time whenever a > wards you c2 d to­ Answer: No dependence on source motion observed (and should be dramatic). 5.4.3 Does c depend on observer motion (frame)? • No 1st order effect had been seen • Michelson-Morley experiment hammered it - let’s see how • Consider interferometer moving with velocity v w.r.t. aether and compute round trip flight times parallel (t� ) and perpendicular (t⊥ ) to v. • For light traveling in direction ±v, ct± = t± = t� = L� ± vt± L� cmp v t+ + t− = L� L� 2L� 2 + = γ ,s c−v c+v c where we have defined the quantity γ≡� 1 1− v2 c2 . • For light traveling perpendicularly to v, � 2 + (vt /2)2 (ct⊥ /2)2 = L⊥ ⊥ t⊥ = 2L⊥ γ c • The difference is Δt ≡ t⊥ − t� = 2L� 2 2L⊥ γ− γ c c • Rotating the interferometer by 90◦ changes this to Δt� = 2L⊥ 2 2L� γ − γ, c c • i.e., changes it by an amount Δt� − Δt = 2L� 2 L� + L⊥ 2L⊥ 2 2L� 2L⊥ γ − γ− γ+ γ = 2γ(γ − 1) . c c c c c • To lowest order in v/c, we have γ Δt� − Δt ≈ � Δt − Δt t 1 � v �2 2 c L� + L⊥ � v � 2 , c c � v �2 . c ≈ 1+ ≈ • v ≈ 30 km/s, so (v/c)2 ∼ 10−8 — tough to measure! • But their L� + L⊥ = 11m was about 2 × 107 wavelengths λ ∼ 500nm, and they could see fringe shifts as small as 0.01λ. • But they saw no fringe shift at all! So c appears not to depend on frame. 5.5 Aether rescue attempts (see Resnick Table 1-2) • Lorentz-Fitzgerald contraction: L� contracts to L� /γ. • Ruled out by Kennedy & Thorndike (1932) using interferometer with L� �= L⊥ • Aether drag hypothesis • Ruled out by stellar aberration • Also by light propagation in moving water (Fizeau 1851) • Emission theories (v depends on source speed) • Ruled out by binary stars (above) • Also ruled out by Michelson-Morley with extraterrestrial light • Also ruled out by measuring speed of γ-rays from CERN particle decays We’ve seen that classical mechanics is invariant under Galilean trans­ formations but electromagnetism isn’t. Question: What is wrong? 1. The idea that all inertial frames are equivalent 2. Our theory of mechanics (8.01) 3. Our theory electromagnetism (8.02) 4. Nothing, because of Bohr’s complementarity principle 6 What are we to make of this? • Parity symmetry applied to some things, not others. • Is it the same with Galilean symmetry? • An experimental question: Is physics the same in all inertial frames? • A: Experiments suggest YES, both for mechanics and electromag­ netism • A theoretical question: How describe this invariance mathe­ matically, i.e., what is the tranformation law that leaves physics invariant? • Galilean transformation? Works for mechanics but fails for E&M • Lorentz transformation? Works for E&M (PS3) but fails for me­ chanics • No transformation works for both E&M and mechanics • So at least one of the two must be wrong! • Changing E&M to be have Galilean invariance is experimentally ruled out • So let’s try changing mechanics to be Lorentz invariant! • BINGO! Not only OK with old experiments, but triumphed with new ones. 7 Derivation of the Lorentz Transformation 7.1 Battle Plan We’ll follow Einstein’s approach and derive everything from two postu­ lates: 1. The laws of physics are the same in all inertial frames. 2. The speed of light is same in all inertial frames. Comments: • 2 follows from 1 if we consider the speed of light one of the laws of physics. • Einstein denoted inertial frame invariance “special relativity” • As opposed to “general relativity”, the generalization to non-inertial frames. 7.2 Inertial frames done carefully: rods & clocks • Key concept: the event, a point in spacetime. • Define coordinate system with three perpendicular rigid measuring rods • Define time with local clocks • Synchronize clocks with light pulses • Minkowski diagram of synchronization procedure • Minkowski diagram basics • N.B. Don’t confuse frame simultaneity with seeing things simul­ taneously: If you saw SN 1987A and a camera flash at the same time, did these two flashes go off simultaneously in your inertial frame? • The time t in an inertial frame is also called bookkeeper’s time. Don’t confuse with the time when you see something happen. • 1st shocker: Simultaneity is relative! Must abandon t� = t. 7.3 Transformation derivation, part I • Let’s define 4-vectors that have units of length: ⎛ ⎞ x ⎜ y ⎟ ⎟ x=⎜ ⎝ z ⎠. ct • In this course, we’ll often use units were c = 1 (time measured in meters). • Given v, the new 4-vector x� is some function of x - which function? • Translational invariance implies linearity: x� = Λ(v)x + x0 , for some offset x0 and some Lorentz ⎛ Λ11 Λ12 ⎜ Λ21 Λ22 Λ(v) = ⎜ ⎝ Λ31 Λ32 Λ41 Λ42 • Why? Because matrix Λ13 Λ23 Λ33 Λ43 ⎞ Λ14 Λ24 ⎟ ⎟. Λ34 ⎠ Λ44 x�2 − x�1 = Λ(v)(x2 − x1 ) for linear relation — for any nonlinear (rigorously, non-affine) re­ lation, the difference (x�2 − x�1 ) won’t depend only on the difference (x2 − x1 ). • Notation warning: book uses notation where 4th coordinate is t, not ct, so there things get uglier and not all Λ-coefficients are dimensionless. • Notation warning: book uses a, we use Λ since it’s more standard these days. • Velocity sign convention: velocity of primed frame in unprimed frame is v, so velocity of unprimed frame in primed frame is −v • WLOG no translation: x0 = 0 (we can always translate later), so simply need to find the 4 × 4 matrix Λ(v) • (WLOG=without loss of generality.) • WLOG no rotation (we can always gives identity matrix: ⎛ 1 0 ⎜ 0 1 ⎜ Λ(0) = ⎝ 0 0 0 0 rotate later), so v = 0 case 0 0 1 0 ⎞ 0 0 ⎟ ⎟. 0 ⎠ 1 • WLOG v in x-direction, since we can always rotate to make it so • Our transformation respects rotational symmetry around the x­ axis, so neither x� nor t� can depend on y or z, i.e., we have Λ12 = Λ13 = Λ42 = Λ43 = 0. • The x-axis gets transformed into ⎛ Λ11 0 0 Λ14 ⎜ Λ21 Λ22 Λ23 Λ24 ⎜ ⎝ Λ31 Λ32 Λ33 Λ34 Λ41 0 0 Λ44 ⎞⎛ ⎞ ⎛ 1 Λ11 ⎟ ⎜ 0 ⎟ ⎜ Λ21 ⎟⎜ ⎟ ⎜ ⎠ ⎝ 0 ⎠ = ⎝ Λ31 0 Λ41 ⎞ ⎟ ⎟, ⎠ so we have Λ21 = Λ31 = 0 since, by construction, the spatial part of the x-axis coincides continuously with the x� -axis. • Consider events with x = t = 0. ⎛ � ⎞ ⎛ x Λ11 0 0 ⎜ y� ⎟ ⎜ 0 Λ Λ 22 23 ⎜ � ⎟=⎜ ⎝ z ⎠ ⎝ 0 Λ32 Λ33 t� Λ41 0 0 They get transformed into ⎞⎛ ⎞ ⎛ Λ14 0 0 ⎜ y ⎟ ⎜ Λ22 y + Λ23 z Λ24 ⎟ ⎟⎜ ⎟=⎜ Λ34 ⎠ ⎝ z ⎠ ⎝ Λ32 y + Λ33 z Λ44 0 0 ⎞ ⎟ ⎟. ⎠ So all events in this two-dimensional (y, z)-plane are simultaneous in both frames (with t� = t = 0), making it trivial to compare mea­ suring rods in the two frames since their two endpoints can coincide in space and time. This implies that the 2 × 2 transformation ma­ trix in this plane must be the identity matrix, i.e., Λ23 = Λ32 = 0 and Λ22 = Λ33 = 1. • An object moving uniformly with x = vt in the unprimed frame remains at rest at the origin in the primed frame, so ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ Λ11 0 0 Λ14 vt Λ11 vt + Λ14 ct 0 ⎜ 0 ⎟⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 1 0 Λ Λ ct 24 24 ⎜ ⎟⎜ ⎟=⎜ ⎟=⎜ ⎝ 0 0 1 Λ34 ⎠ ⎝ 0 ⎠ ⎝ Λ34 ct ⎠ ⎝ 0 Λ41 0 0 Λ44 ct Λ41 vt + Λ44 ct ct� ⎞ ⎟ ⎟ ⎠ so we have Λ24 = Λ34 = 0 and Λ14 = −βΛ11 , where we’ve defined β≡ v . c • (When we do research using relativity, we normally use units where c = 1, so that we can write simply β = v.) • Progress update: ⎛ Λ11 ⎜ 0 Λ(v) = ⎜ ⎝ 0 Λ41 i.e. ⎛ ⎞ ⎛ x� Λ11 ⎜ y� ⎟ ⎜ 0 ⎜ � ⎟=⎜ ⎝ z ⎠ ⎝ 0 ct� Λ41 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 ⎞ −βΛ11 ⎟ 0 ⎟, ⎠ 0 Λ44 ⎞⎛ −βΛ11 x ⎟⎜ y 0 ⎟⎜ ⎠⎝ z 0 Λ44 ct ⎞ (1) ⎛ ⎞ Λ11 (x − vt) ⎟ ⎜ ⎟ y ⎟=⎜ ⎟. ⎠ ⎝ ⎠ z Λ41 x + Λ44 ct (2) 7.4 Galileo and Einstein part ways So far, we haven’t assumed anything about the speed of light, so our results must still include both the Galilean transform and the Lorentz transform. Let’s do the Galilean first: • Assuming that t� = t gives Λ41 = 0 and Λ44 = 1. • Assuming that measuring rods have the same length in both frames implies Λ11 = 1. • This implies the Galilean transformation matrix: ⎛ ⎞ 1 0 0 −β ⎜ 0 1 0 0 ⎟ ⎟ G(v) = ⎜ ⎝ 0 0 1 0 ⎠. 0 0 0 1 7.5 Transformation derivation, part II Let’s revert to equation (1) and assume that light has same speed c in both frames. • Imagine a light flash created at x = (0, 0, 0, 0) expanding with speed c in all directions, creating an expanding spherical wavefront of radii ct and ct� in the two frames. This light cone (a cone in 4D spacetime) is described by and x2 + y 2 + z 2 − (ct)2 = 0 (3) x�2 + y �2 + z �2 − (ct� )2 = 0 (4) in the two frames. • Substiting equation (2) into the last equation gives Λ211 (x − vt)2 + y 2 + z 2 − (Λ41 x + Λ44 ct)2 = 0 • Rearranging terms: 2 (Λ211 −Λ241 )x2 +y 2 +z 2 −(Λ44 −β 2 Λ11 )(ct)2 −2(βΛ211 +Λ41 Λ44 )ctx = 0. • So the light cone is where this quadratic polynomial in x, y, z and t vanishes. • This polynomial will vanish on the same cone as the polynomial of equation (3) only if the two polynomials are identical, i.e., if ⎧ 2 ⎨ Λ11 − Λ241 = 1, Λ2 − β 2 Λ211 = 1, ⎩ 442 βΛ11 + Λ41 Λ44 = 0. • Solve: ⎧ 2 ⎨ Λ41 = Λ211 − 1, Λ2 = β 2 Λ2 + 1, ⎩ 44 2 4 11 2 2 0 = β Λ11 − Λ41 Λ44 = ... = 1 − (1 − β 2 )Λ11 Solution: ⎧ Λ11 ⎪ ⎪ ⎨ Λ44 Λ41 ⎪ ⎪ ⎩ Λ14 = γ, = γ, , = −βγ, = −βγ, where we have defined 1 γ≡� 1− β2 1 =� 1− v2 c2 . • (We don’t care about the 2nd solution with Λ11 = −γ, which corresponds to flipping the sign of t and x, “TP”.) • We’re done! The Lorentz transformation is ⎛ γ 0 0 −γβ ⎜ 0 1 0 0 Λ(v) = ⎜ ⎝ 0 0 1 0 −γβ 0 0 γ i.e., ⎛ � x ⎜ y� ⎜ � ⎝ z ct� ⎞ ⎛ γ ⎟ ⎜ 0 ⎟=⎜ ⎠ ⎝ 0 −γβ 0 1 0 0 0 0 1 0 ⎞⎛ −γβ x ⎜ y 0 ⎟ ⎟⎜ 0 ⎠⎝ z γ ct • Compare to Einstein’s 1905 paper ⎞ ⎞ ⎟ ⎟, ⎠ ⎛ ⎞ γ(x − βct) ⎟ ⎜ ⎟ y ⎟=⎜ ⎟. ⎠ ⎝ ⎠ z γ(ct − βx) 7.6 The inverse Lorentz transform • Since x� = Λ(v)x and x = Λ(−v)x� , we get the consistency re­ quirement x = Λ(−v)x� = Λ(−v)Λ(v)x for any event x, so we must have Λ(−v) = Λ(v)−1 , the matrix inverse of Λ(v). • Is it? ⎛ γ ⎜ 0 Λ(−v)Λ(v) = ⎜ ⎝ 0 γβ 0 1 0 0 0 0 1 0 ⎞⎛ γβ γ ⎜ 0 0 ⎟ ⎟⎜ 0 ⎠⎝ 0 γ −γβ 0 1 0 0 0 0 1 0 ⎞ ⎛ −γβ 1 ⎜ 0 0 ⎟ ⎟=⎜ 0 ⎠ ⎝ 0 γ 0 i.e., yes! 7.7 Spacetime transformation summary • Translation: � r� = r + Δr t� = t + Δt • Rotation: � • Parity � • Galilean “boost”: • Combined: � � r� = Rr t� = t r� = −r t� = t r� = r − vt t� = t r� = ±Rr + Δr − vt t� = t + Δt • Lorentz “boost”: x� = Λx, i.e., for v in x-direction, ⎧ � x = γ(x − vt) ⎪ ⎪ ⎨ � y =y z� = z ⎪ ⎪ ⎩ � t = γ(t − vx/c2 ) • Poincaré: x� = Λx + x0 This is the most general spacetime symmetry transformation of special relativity, with 10 parameters: x0 can give 3 independent translations in space and 1 translation in time), Λ can give 3 inde­ pendent rotations, 3 independent boosts and also reversal of space and/or time. 0 1 0 0 0 0 1 0 ⎞ 0 0 ⎟ ⎟, 0 ⎠ 1