NOTE: 2005 is the World Year of Physics! In 2005, there will be a world-wide celebration of the centennial of Einstein's famous 1905 papers on Relativity, Brownian Motion, & the Photoelectric Effect (for which he won the Nobel Prize!). A web page telling you more:

Chapter 7: Special Relativity Sect. 7.1: Basic Postulates Special Relativity: One of 2 major (revolutionary!) advances in understanding the physical world which happened in the 20 th Century! Other is Quantum Mechanics of course! Of the 2, Quantum Mechanics is more relevant to everyday life & also has spawned many more physics subfields. However ( my personal opinion ), Special Relativity is by far the most elegant & “beautiful” of the 2. In a (relatively) simple mathematical formalism, it unifies mechanics with E&M! The historical reasons Einstein developed it & the history of its development & eventual acceptance by physicists are interesting. But (due to time) we will discuss this only briefly. The philosophical implications of it, the various “paradoxes” it seems to have, etc. are interesting. But (due to time) we’ll discuss this only briefly.

Newton’s Laws: Are valid only in an Inertial Reference Frame: Defined by Newton’s 1 st Law: –A frame which isn’t accelerating with respect to the “stars”. –Any frame moving with constant velocity with respect to an inertial frame is also an inertial frame! Galilean Transformation: (Galilean Relativity!) 2 reference frames: S, time & space coordinates (t,x,y,z) & S´, time & space Coordinates (t´,x´,y´,z´). S´ moving relative to S with const velocity v in the +x direction. Figure. Clearly: t´= t, x´ = x - vt, y´= y, z´ = z  Galilean Transformation

Newton’s 2 nd Law: Unchanged by a Galilean Transformation (t´= t, x´ = x - vt, y´= y, z´ = z) F = (dp/dt)  F´ = (dp´/dt´) Implicit Newtonian assumption: t´= t. In the equations of motion, the time t is an independent parameter, playing a different role in mechanics than the coordinates x, y, z. Newtonian mechanics: S´ is moving relative to S with constant velocity v in the +x direction; u = velocity of a particle in S, u´ = velocity of particle in S´.  u´ = u - v Contrast: in Special Relativity, the position coordinates x, y, z & time t are on an equal footing.

Electromagnetic Theory (Maxwell’s equations): Contain a universal constant c = The speed of Light in Vacuum. –This is inconsistent with Newtonian mechanics! –Einstein: Either Newtonian Mechanics or Maxwell’s equations need to be modified. He modified Newtonian Mechanics.  2 Basic Postulates of Special Relativity 1. THE POSTULATE OF RELATIVITY: The laws of physics are the same to all inertial observers. This is the same as Newtonian mechanics! 2. THE POSTULATE OF THE CONSTANCY OF THE SPEED OF LIGHT: The speed of light, c, is independent of the motion of its source. A revolutionary idea! Requires modifications of mechanics at high speeds.

2 Basic Postulates 1. RELATIVITY 2. CONSTANT LIGHT SPEED Covariant  A formulation of physics which satisfies 1 & 2 2.  The speed of light c is the same in all coordinate systems. 1 & 2  Space & Time are considered 2 aspects ( coordinates ) of a single Spacetime. = A 4d geometric framework (“Minkowski Space”)  The division of space & time is different for different observers. The meaning of “simultaneity” is different for different observers. Space & time get “mixed up” in transforming from one inertial frame to another.

Event  A point in 4d spacetime. –To make all 4 dimensions have the same units, define the time dimension as ct. The square of distance between events A = (ct 1,x 1,y 1,z 1 ) & B = (ct 2,x 2,y 2,z 2 ) in 4d spacetime: (Δs) 2  c 2 (t 2 -t 1 ) 2 - (x 2 -x 1 ) 2 - (y 2 -y 1 ) 2 - (z 2 -z 1 ) 2 or: (Δs) 2  c 2 (Δt) 2 - (Δx) 2 - (Δy) 2 - (Δz) 2 (1) –Note the different signs of time & space coords! Now, go to differential distances in spacetime: (1)  (ds) 2  c 2 (dt) 2 - (dx) 2 - (dy) 2 - (dz) 2 (2) A body moving at v: (dx) 2 + (dy) 2 + (dz) 2 = (dr) 2 = v 2 (dt) 2  (ds) 2 = [c 2 - v 2 ](dt) 2 > 0 Bodies, moving at v 0

(ds) 2  c 2 (dt) 2 - (dx) 2 - (dy) 2 - (dz) 2 (2) (ds) 2 > 0  A timelike interval (ds) 2 < 0  A spacelike interval (ds) 2 = 0  A lightlike or null interval For all observers, objects which travel with v 0  Such objects are called tardyons. Einstein’s theory & the Lorentz Transformation  The maximum velocity allowed is v = c. However, in science fiction, can have v > c. If v > c, (ds) 2 < 0  Such objects are called tachyons. A 4d spacetime with an interval defined by (2)  Minkowski Space

The interval between 2 events (a distance in 4d Minkowski space) is a geometric quantity.  It is invariant on transformation from one inertial frame S to another, S´ moving relative to S with constant velocity v : (ds) 2  (ds´) 2 (3)  (ds) 2  The invariant spacetime interval. (3)  The transformation between S & S´ must involve the relative velocity v in both space & time parts. Or: Space & Time get mixed up on this transformation! “Simultaneity” has different meanings for an observer in S & an observer in S´

(ds) 2  (ds´) 2 (3) Relatively simple consequences of (3): 1. Time Dilation S  Lab frame, S´  moving frame (3)  Time interval dt measured in the lab frame is different from the time interval dt´ measured in the moving frame.  To distinguish them: Time measured in the rest (not moving!) frame of a body (S´if the body moves with v in the lab frame)  Proper time  τ. Time measured in the lab frame ( S)  Lab time  t. For a body moving with v: In S´, (ds´) 2 = c 2 (dτ) 2, In S, (ds) 2  c 2 (dt) 2 - (dx) 2 - (dy) 2 - (dz) 2 = c 2 (dt) 2 - (dr) 2 = c 2 (dt) 2 - v 2 (dt) 2 = c 2 (dt) 2 [1-(v 2 )/(c 2 )]

Time Dilation (ds) 2  (ds´) 2 (3) A body moving with v: In S´, (ds´) 2 = c 2 (dτ) 2, In S, (ds) 2 = c 2 (dt) 2 [1-(v 2 )/(c 2 )] Using these in (3)  c 2 (dτ) 2 = c 2 (dt) 2 [1-(v 2 )/(c 2 )] Or: dt  γdτ (4) where: γ  1/[1 - β 2 ] ½  [1 - β 2 ] -½, β  (v/c) (4)  dτ < dt  “Time dilation”  “Moving clocks (appear to) run slow(ly)”

(ds) 2  (ds´) 2 (3) Relatively simple consequences of (3): 2. “Simultaneity” is relative! Suppose 2 events occur simultaneously in S (  the lab frame), but at different space points (on the x axis, for simplicity). Do they occur simultaneously in S´ (  the moving frame)? In S, dt = 0, dy = dz = 0, dx  0.  In S, (ds) 2 = - (dx) 2 In S´, (invoking the Lorentz transformation ahead of time) dy´=dz´=0,  In S´, (ds´) 2 = c 2 (dt´) 2 - (dx´) 2 (3)  - (dx) 2 = c 2 (dt´) 2 - (dx´) 2 Or: c 2 (dt´) 2 = (dx´) 2 - (dx) 2 (invoking the Lorentz transform ahead of time) (dx´) 2 = γ 2 (dx) 2  c 2 (dt´) 2 = [γ 2 -1] (dx) 2 Or (algebra) c dt´ = γβdx  The 2 events are not simultaneous in S´

(ds) 2  (ds´) 2 (3) Relatively simple consequences of (3): 3. Length Contraction Consider a thin object, moving with v || to x in S. Let S´ be attached to the moving object. Instantaneous measurement of length. In S: dt = 0. For an infinitely thin object: dy = dz = 0.  In S, (ds) 2 = - (dx) 2 In S´, (invoking the Lorentz transformation ahead of time) dy´=dz´=0,  In S´, (ds´) 2 = c 2 (dt´) 2 - (dx´) 2 (3)  -(dx) 2 = c 2 (dt´) 2 - (dx´) 2 Or: (dx´) 2 = c 2 (dt´) 2 + (dx) 2 (invoking the Lorentz transform ahead of time & using results just obtained) c 2 (dt´) 2 = γ 2 β 2 (dx) 2  (Algebra) (dx´) 2 = γ 2 (dx) 2 Or dx´ = γdx. For finite length: L´ = γL or L = (L´)γ -1 < L´ Lorentz-Fitzgerald Length Contraction

(ds) 2  (ds´) 2 (3) (3)  Spacetime is naturally divided into 4 regions. For an arbitrary event A at x = y = z = t = 0, we can see this by looking at the “light cone” of the event. Figure. the z spatial dimension is suppressed. Light cone = set of (ct,x,y) traced out by light emitted from ct = x = y = 0 or by light that reaches x = y = 0 at ct = 0. The past & the future are inside the light cone.

(ds) 2  (ds´) 2 (3) Consider event B at time t B such that (ds AB ) 2 > 0 (timelike). (3)  All inertial observers agree on the time order of events A & B. We can always choose a frame where A & B have the same space coordinates. If t B < t A = 0 in one inertial frame, will be so in all inertial frames.  This region is called THE PAST. Similarly, consider event C at time t C such that (ds AC ) 2 > 0, (3)  All inertial observers agree on the time order of events A & C. If t C >t A = 0 in one inertial frame, it will be so in all inertial frames.  This region is called THE FUTURE.

(ds) 2  (ds´) 2 (3) Consider an event D at time t D such that (ds AD ) 2 < 0 (spacelike). (3)  There exists an inertial frame in which the time ordering of t A & t D are reversed or even made equal.  This region is called THE ELSEWHERE or THE ELSEWHEN. In the region in which D is located, there exists an inertial frame with its origin at event A in which D & A occur at the same time but in which D is somewhere else (elsewhere) than the location of A. There also exist frames in which D occurs before A & frames in which D occurs after A (elsewhen).

(ds) 2  (ds´) 2 (3) The light cone obviously separates the past-future from the elsewhere (elsewhen). On the light cone, (ds) 2 = 0. Light cone = a set of spacetime points from which emitted light could reach A (at origin) & those points from which light emitted from event A could reach. Any interval between the origin & a point inside the light cone is timelike: (ds) 2 > 0. Any interval between the origin & a point outside the light cone is spacelike: (ds) 2 < 0.

Sect. 7.2: Lorentz Transformation Lorentz Transformation: A “derivation” (not in the text!) Introduce new notation: x 0  ct, x 1  x, x 2  y, x 3  z. Lab frame S & inertial frame S´, moving with velocity v along x axis. We had: (ds) 2  (ds´) 2. Assume that this also holds for finite distances: (Δs) 2  (Δs´) 2 or (in the new notation) ( Δ x 0 ) 2 - ( Δ x 1 ) 2 - ( Δ x 2 ) 2 - ( Δ x 3 ) 2 = ( Δ x 0 ´) 2 - ( Δ x 1 ´) 2 - ( Δ x 2 ´) 2 - ( Δ x 3 ´) 2 Assume, at time t = 0, the 2 origins coincide.  Δx μ = x μ & Δx μ ´ = x μ ´ (μ = 0,1,2,3)  (x 0 ) 2 - (x 1 ) 2 - (x 2 ) 2 - (x 3 ) 2 = (x 0 ´) 2 - (x 1 ´) 2 - (x 2 ´) 2 -(x 3 ´) 2

(x 0 ) 2 - (x 1 ) 2 - (x 2 ) 2 - (x 3 ) 2 = (x 0 ´) 2 - (x 1 ´) 2 - (x 2 ´) 2 -(x 3 ´) 2 (1) Want a transformation relating x μ & x μ ´. Assume the transformation is LINEAR: x μ ´  ∑ μ L μν x ν (2) L μν to be determined ( 2): Mathematically identical (in 4d spacetime) to the form for a rotation in 3d space. We could write (2) in matrix form as x´  L  x Where L is a 4x4 matrix & x, x´ are 4d column vectors. We can prove that L is symmetric & acts mathematically as an orthogonal matrix in 4d spacetime.

(x 0 ) 2 - (x 1 ) 2 - (x 2 ) 2 - (x 3 ) 2 = (x 0 ´) 2 - (x 1 ´) 2 - (x 2 ´) 2 -(x 3 ´) 2 (1) x μ ´  ∑ μ L μν x ν (2) Now, invoke some PHYSICAL REASONING: The motion (velocity v) is along the x axis. Any physically reasonable transformation will not mix up x,y,z (if the motion is parallel to x; that is, it involves no 3d rotation!).  y = y´, z = z´ or x 2 = x 2 ´, x 3 = x 3 ´  (1) becomes: (x 0 ) 2 - (x 1 ) 2 = (x 0 ´) 2 - (x 1 ´) 2 (3) Also: L 22 = L 33 = 1. All others are zero except: L 00, L 11, L 01, & L 10. Further, assume that the transformation is symmetric.  L μν = L νμ (this is not necessary, but it simplifies math. Also, after the fact we find that it is symmetric).

Under these conditions, (2) becomes: x 0 ´ = L 00 x 0 + L 01 x 1 (2a) x 1 ´ = L 01 x 0 + L 11 x 1 (2b) (x 0 ) 2 - (x 1 ) 2 = (x 0 ´) 2 - (x 1 ´) 2 (3) (2a), (2b), (3): After algebra we get: ( L 00 ) 2 - (L 01 ) 2 = 1 (4a); (L 11 ) 2 - (L 01 ) 2 = 1 (4b) (L 00 - L 11 )L 01 = 0 (4c) (4a), (4b), (4c): This looks like 3 equations & 3 unknowns. However, it turns out that solving will give only 2 of the 3 unknowns (the 3 rd equation is redundant!).  We need one more equation!

To get this equation, consider the origin of the S´ system at time t in the S system. (Assume, at time t = 0, the 2 origins coincide.) Express it in the S system:  At x 1 ´ = 0, (2b) gives: 0 = L 01 x 0 + L 11 x 1 We also know: x = vt or x 1 = βx 0 Combining gives: L 01 = - βL 11 (5) Along with (L 00 ) 2 - (L 01 ) 2 = 1 (4a) (L 11 ) 2 - (L 01 ) 2 = 1 (4b); (L 00 - L 11 )L 01 = 0 (4c) This finally gives: L 11 = γ =1/[1 - β 2 ] ½ = [1 - (v 2 /c 2 )] -½ and (algebra): L 01 = - βγ, L 00 = γ

Putting this together, The Lorentz Transformation (for v || x): x 0 ´ = γ(x 0 - βx 1 ), x 2 ´ = x 2 x 1 ´ = γ(x 1 - βx 0 ), x 3 ´ = x 3 The inverse Transformation (for v || x): x 0 = γ(x 0 ´ + βx 1 ´), x 2 = x 2 ´ x 1 = γ(x 1 ´ + βx 0 ´), x 3 = x 3 ´ In terms of ct,x,y,z: The Lorentz Transformation is ct´ = γ(ct - βx) (t´ = γ[t - (β/c)x]) x´ = γ(x - βct), y´ = y,z´ = z, β = (v/c) This reduces to the Galilean transformation for v <<c β << 1, γ  1:  x´= x - vt, t´= t, y´= y, z´= z

Lorentz Transformation (for v || x) in terms of a transformation (“rotation”) matrix in 4d spacetime (a “rotation” in the x 0 -x 1 plane): x´  L  x Or: x 0 ´ γ -βγ 0 0 x 0 x 2 ´ = -γβ γ 0 0 x 1 x 3 ´ x 2 x 4 ´ x 3

The generalization to arbitrary orientation of velocity v is straightforward but tedious! The Lorentz Transformation (for general orientation of v): ct´ = γ(ct - β  r), r´ = r + β -2 (β  r)(γ -1)β - γctβ In terms of the transformation (“rotation”) matrix in 4d spacetime: x´  L  x

Briefly back to the Lorentz Transformation (v || x): x´  L  x, L  A “Lorentz boost” or A “boost” Sometimes its convenient to parameterize the transformation in terms of a “boost parameter” or “rapidity” ξ. Define: β  tanh(ξ)  γ = [1 - β 2 ] -½ = cosh(ξ), βγ = sinh(ξ) Then: x 0 ´ cosh(ξ) -sinh(ξ) 0 0 x 0 x 2 ´ = -sinh(ξ) cosh(ξ) 0 0 x 1 x 3 ´ x 2 x 4 ´ x 3  x 0 ´= x 0 cosh(ξ) - x 1 sinh(ξ), x 1 ´= -x 0 sinh(ξ) + x 1 cosh(ξ) Should reminds you of a rotation in a plane, but we have hyperbolic instead of trigonometric functions. From complex variable theory:  “imaginary rotation angle”!

These transformations map the origins of S & S´ to (0,0,0,0). L = a “rotation” in 4d spacetime. A more general transformation is  The Poincaré Transformation: “Rotation” L in 4d spacetime + translation a x´  L  x + a If a = 0  Homogeneous Lorentz Transformation