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NOTE: 2005 is the World Year of Physics! In 2005, there will be a world-wide celebration of

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:

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NOTE: 2005 is the World Year of Physics! In 2005, there will be a world-wide celebration of

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  1. 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: <http://www.physics2005.org/>

  2. Chapter 7: Special RelativitySect. 7.1: Basic Postulates • Special Relativity: One of 2 major (revolutionary!) advances in understanding the physical world which happened in the 20th 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 historicalreasons 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 implicationsof it, the various “paradoxes” it seems to have, etc. are interesting. But (due to time) we’ll discuss this only briefly.

  3. Newton’s Laws:Are valid only in an Inertial Reference Frame: Defined by Newton’s 1st 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

  4. Newton’s 2nd 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.

  5. Electromagnetic Theory (Maxwell’s equations): Contain a universal constantc = 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.

  6. 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 singleSpacetime. = 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.

  7. 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 = (ct1,x1,y1,z1) & B = (ct2,x2,y2,z2) in 4d spacetime: (Δs)2 c2(t2-t1)2 - (x2-x1)2 - (y2-y1)2 - (z2-z1)2 or: (Δs)2 c2(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 c2(dt)2 - (dx)2 - (dy)2 - (dz)2 (2) A body moving at v: (dx)2 + (dy)2 + (dz)2 = (dr)2 = v2(dt)2 (ds)2= [c2- v2](dt)2 > 0 Bodies, moving at v < c, have (ds)2> 0

  8. (ds)2 c2(dt)2 - (dx)2 - (dy)2 - (dz)2 (2) (ds)2> 0  Atimelikeinterval (ds)2< 0  A spacelike interval (ds)2= 0  A lightlike or null interval • For all observers, objects which travel with v < c have (ds)2> 0 Such objects are calledtardyons. • Einstein’s theory & the Lorentz Transformation The maximum velocity allowed isv = c. However, in science fiction, can have v > c. If v > c, (ds)2< 0 Such objects are calledtachyons. • A 4d spacetime with an interval defined by (2) Minkowski Space

  9. 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´

  10. (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´ measuredin 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 = c2(dτ)2 , In S, (ds)2 c2(dt)2 - (dx)2 - (dy)2 - (dz)2 = c2(dt)2 - (dr)2 = c2(dt)2 - v2(dt)2 = c2(dt)2[1-(v2)/(c2)]

  11. Time Dilation (ds)2 (ds´)2 (3) • A body moving with v: In S´, (ds´)2 = c2(dτ)2 , In S, (ds)2= c2(dt)2[1-(v2)/(c2)] • Using these in (3)  c2(dτ)2 = c2(dt)2[1-(v2)/(c2)] Or: dt  γdτ (4) where: γ 1/[1 - β2]½ [1 - β2]-½ , β (v/c) (4)  dτ < dt  “Time dilation”  “Moving clocks (appear to) run slow(ly)”

  12. (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 = c2(dt´)2 - (dx´)2 (3) - (dx)2 = c2(dt´)2 - (dx´)2 Or: c2(dt´)2 = (dx´)2 - (dx)2(invoking the Lorentz transform ahead of time)(dx´)2 = γ2(dx)2  c2(dt´)2 = [γ2 -1] (dx)2Or (algebra)c dt´ = γβdx  The 2 events are not simultaneous in S´

  13. (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)2In S´, (invoking the Lorentz transformation ahead of time) dy´=dz´=0,  In S´, (ds´)2 = c2(dt´)2 - (dx´)2 (3) -(dx)2 = c2(dt´)2 - (dx´)2 Or: (dx´)2 = c2(dt´)2 + (dx)2 (invoking the Lorentz transform ahead of time & using results just obtained)c2(dt´)2 =γ2β2(dx)2  (Algebra) (dx´)2 = γ2(dx)2Or dx´ = γdx.For finite length:L´ = γL or L = (L´)γ-1 < L´ Lorentz-Fitzgerald Length Contraction

  14. (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. thez 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. Thepast & the futureare inside the light cone.

  15. (ds)2 (ds´)2 (3) • Consider event B at time tBsuch that (dsAB)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 tB < tA = 0 in one inertial frame, will be so in all inertial frames.  This region is called THE PAST. • Similarly, consider event C at time tC such that (dsAC)2 > 0, (3)  All inertial observers agree on the time order of events A & C. If tC>tA= 0 in one inertial frame, it will be so in all inertial frames.  This region is called THE FUTURE.

  16. (ds)2 (ds´)2 (3) • Consider an event D at time tDsuch that (dsAD)2 < 0 (spacelike). (3)  There exists an inertial frame in which the time ordering of tA& tDare reversed or even made equal.  This region is called THE ELSEWHERE orTHE 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).

  17. (ds)2 (ds´)2 (3) • The light cone obviouslyseparates 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.

  18. Sect. 7.2: Lorentz Transformation • Lorentz Transformation:A “derivation” (not in the text!) • Introduce new notation: x0 ct,x1 x, x2 y,x3 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) (Δx0)2- (Δx1)2 - (Δx2)2 - (Δx3)2 = (Δx0´)2 - (Δx1´)2 - (Δx2´)2 - (Δx3´)2 • Assume, at time t = 0, the 2 origins coincide.  Δxμ= xμ&Δxμ´= xμ´ (μ = 0,1,2,3)  (x0)2- (x1)2 - (x2)2 - (x3)2 = (x0´)2 - (x1´)2 - (x2´)2 -(x3´)2

  19. (x0)2- (x1)2 - (x2)2 - (x3)2 = (x0´)2 - (x1´)2 - (x2´)2 -(x3´)2 (1) • Want a transformation relating xμ& xμ´. Assume the transformation is LINEAR: xμ´ ∑μLμνxν(2) Lμνto 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.

  20. (x0)2- (x1)2 - (x2)2 - (x3)2 = (x0´)2 - (x1´)2 - (x2´)2 -(x3´)2 (1) xμ´ ∑μLμνxν(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 x2 = x2´, x3 = x3´  (1) becomes: (x0)2 - (x1)2 = (x0´)2 - (x1´)2 (3) Also: L22 = L33 = 1. All others are zero except: L00, L11, L01, & L10. Further, assume that the transformation is symmetric.  Lμν= Lνμ(this is not necessary, but it simplifies math. Also, after the fact we find that it is symmetric).

  21. Under these conditions, (2) becomes: x0´= L00 x0 + L01 x1 (2a) x1´= L01 x0 + L11 x1 (2b) (x0)2 - (x1)2 = (x0´)2 - (x1´)2 (3) • (2a), (2b), (3): After algebra we get: (L00)2 - (L01)2 = 1 (4a); (L11)2 - (L01)2 = 1 (4b) (L00 - L11)L01 = 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 3rd equation is redundant!).  We need one more equation!

  22. 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 x1´= 0, (2b) gives: 0 = L01 x0 + L11 x1 We also know:x = vt or x1 = βx0 Combining gives:L01 = - βL11 (5) Along with(L00)2 - (L01)2 = 1 (4a) (L11)2 - (L01)2 = 1 (4b); (L00 - L11)L01 = 0 (4c) This finally gives: L11 = γ =1/[1 - β2]½ = [1 - (v2/c2)]-½ and (algebra): L01 = - βγ , L00 = γ

  23. Putting this together, The Lorentz Transformation (for v || x):x0´ = γ(x0 - βx1), x2´= x2 x1´= γ(x1-βx0), x3´= x3 • The inverse Transformation (for v || x): x0= γ(x0´ +βx1´), x2= x2´ x1 = γ(x1´ +βx0´), x3 = x3´ • In terms of ct,x,y,z: The Lorentz Transformationis 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

  24. Lorentz Transformation (for v || x) in terms of a transformation (“rotation”) matrix in 4d spacetime (a “rotation” in the x0-x1 plane): x´ Lx Or: x0´ γ-βγ 0 0 x0 x2´ = -γβγ 0 0 x1 x3´ 0 0 1 0 x2 x4´ 0 0 0 1 x3

  25. 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

  26. 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 - β2]-½ = cosh(ξ), βγ = sinh(ξ)Then: x0´ cosh(ξ) -sinh(ξ)0 0 x0 x2´ = -sinh(ξ)cosh(ξ) 0 0 x1 x3´ 0 0 1 0 x2 x4´ 0 0 0 1 x3  x0´= x0 cosh(ξ) - x1 sinh(ξ), x1´= -x0 sinh(ξ) + x1 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”!

  27. 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 PoincaréTransformation: “Rotation” L in 4d spacetime + translation a x´  Lx + a If a = 0  Homogeneous Lorentz Transformation

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