Lecture 2: Relativistic Space-Time Invariant Intervals & Proper Time Lorentz Transformations Electromagnetic Unification Equivalence of Mass and Energy Space-Time Diagrams Relativistic Optics Section 6-7, 19-21, Useful Sections in Rindler:

Einstein’s Two Postulates of Special Relativity: I. The laws of physics are identical in all inertial frames II. Light propagates in vacuum rectilinearly, with the same speed at all times, in all directions and in all inertial frames Einstein’s postulates

vv vv v d d tt c = d tt -v tt  c /  t ) 2 + v 2 tt tt tt  1 - (v/c) 2 =  d/  t ) 2 +v 2  x = v  t c =  d 2 +v 2  t 2 tt Time Dilation: Time dilation

c =  d 2 +(  x ) 2 tt Recall: Thus, (c  t ) 2 = d 2 + (  x ) 2 d 2 = (c  t ) 2 - (  x ) 2 invariant or, more generally, S 2 = (c  t ) 2 - [(  x ) 2 + (  y ) 2 + (  z ) 2 ] ''Invariant Interval” choose frame ''at rest” = (c   “Proper Time” Invariant interval & proper time

Consider light beam moving along positive x-axis: x = ct or x - ct = 0 Similarly, in the moving frame, we want to have x = ct or x - ct = 0 We can insure this is the case if: x - ct = a(x - ct ) Generally, the factor could be different for motion in the opposite direction: x + ct = b(x + ct ) Subtracting t = t  x/c (a+ b) 2 (a-b) 2 = t  x/c (a+ b) 2 (a-b) (a+b) [ ] = A t  x/c [ ] Lorentz Transformations: Lorentz transformations

= A t  x/c [ ] t So, we know that A =   t = A  t (at fixed x) Similarly, x =  [ x - Bct ] x =  [ x - vt ] t =  [ t - (v/c 2 )x ] In non-relativistic limit (   1) : x  [ x - Bct ] Must correspond to Galilean transformation, so Bc = v B = v/c Lorentz transformations (2)

Maxwell’s Equations ''Lorentz-Fitzgerald Contraction” ''Aether Drag” George Francis Fitzgerald Hendrik Antoon Lorentz Lorentz & Fitzgerald

+q +  +  +  +  v I B Lab Frame F (pure magnetic) +    q In Frame of Test Charge Lorentz expanded Lorentz contracted F (pure electrostatic)  Electricity & Magnetism are identically the same force, just viewed from different reference frames UNIFICATION !! (thanks to Lorentz invariance)  Relativity & Electromagnetism Symmetry: The effect of a force looks the same when viewed from reference frames boosted in the perpendicular direction

+q +  +  +  +  v I B Lab Frame +    q In Frame of Test Charge Lorentz expanded Lorentz contracted F (pure magnetic) F (pure electrostatic) Equivalence of EM forces F = qv  B | F | = qv I  o / (2  r) lab + = lab  = q  =  q  =  ´ = q +    E =  / 2  r  o =  v   / (2  r  o c 2 ) =  v    / (2  r) | F ´| = Eq =  v    q / (2  r) v = I | F | = | F ´| /  = qv   / (2  r)   =  | F ´| =  v   q / (2  r)

Einstein’s The 2 Postulates of Special Relativity: I. The laws of physics are identical in all inertial frames II. Light propagates in vacuum rectilinearly, with the same speed at all times, in all directions and in all inertial frames Einstein’s postulates, again

Planck’s recommendation for Einstein’s nomination to the Prussian Academy in 1913: “In summary, one can say that there is hardly one among the great problems in which modern physics is so rich to which Einstein has not made a remarkable contribution. That he may sometimes have missed the target in his speculations, as, for example, in his hypothesis of light quanta, cannot really be held against him, for it is not possible to introduce really new ideas even in the most exact sciences without sometimes taking a risk.”

1905

E = h (Planck) p = h/ (De Broglie) = hc/ E = pc absorber emitter p=E/c recoil p=Mv E/c = Mv motion stops distance travelled d = vt = v (L/c) = EL/(Mc 2 ) But no external forces, so CM cannot change! Must have done the equivalent of shifting some mass m to other side, such that M {EL/(Mc 2 )} = m L Md = mL “Einstein’s Box”: Einstein’s box

+ x- x ct -y + y Space-Time: Space-time diagram

+ x- x ct = c  t/  x = c/v = 1/  object stationary until time t 1 x1x1 ct 1 moves with constant velocity (  ) until t 2 ct 2 x2x2 returns to point of origin slope = (ct 2 - ct 1 )/(x 2 -x 1 ) Space-time trajectory

+ x- x ct  tan  = x/ct = v/c =  tan  max = 1  max = 45° 45° v = c 45° v = c light sent backwards Velocities in space-time

“absolute past” + x- x ct “absolute future” “absolute elsewhere” x1x1 ct 1 no message sent from the origin can be received by observers at x 1 until time t 1 there is no causal contact until they are “inside the light cone” Space-time cones

+ x- x ct “absolute future” “absolute past” “absolute elsewhere” Space-time cones (2)

+ x- x ct  ST light rays & trajectory

+ x- x ct  From other frame?

+ x- x ct  Distort trajectory space axis

+ x- x ct  Distort trajectory space axis (2)

+ x- x ct  Distort trajectory space axis (3)

+ x- x ct   Constant c in both coords

+ x- x ct   Constant c in both frames (2)

+ x- x ct   S S´S´ Coord grid in both frames

+ x- x ct Spacetime Showdown Spacetime showdown

Relativistic Optics Relativistic Optics

v  t =   t f = 1/  t = 1/  t = f/  Transverse Doppler Reddening Transverse Doppler effect

Shadow of stationary cube a

v Shadow of Galilean moving cube

v

v ( a v/c ) 2 + ( a  1 - (v/c) 2 ) 2 = a 2 Shadow of relativistic cube a  1 - (v/c) 2

v ( a v/c ) 2 + ( a  1 - (v/c) 2 ) 2 = a 2 a Terrell Rotation (1959) Equivalent Terrell rotation a  1 - (v/c) 2

Penrose (1959): A Sphere By Any Other Frame Is Just As Round Penrose sphere

v Simultaneous snapshot?  h 2 +d 2 h

v More generally, from somewhat off-axis  hyperbolic curvature Apparent curvature  h 2 +d 2 h

SS 433 If assumed distance to object increases, so must the distance traversed by jet to preserve same angular scale for “peaks” and, hence, jet velocity must increase. History of jet precession (period = 162 days) Jet orientation fixed by relative Doppler shifts Light observed from a given point in the jet was produced  t = (s-d)/c earlier, thus distorting the apparent orientation of the loops d vv  s SS 433

Jet aberation

Can fit distance to the source = 5.5 kpc (K. Blundell & M. Bowler) Can even show evidence of jet speed variations! SS 433 distance fit

Angular compression towards centre of field-of-view Intensity = increases towards centre light received solid angle “Headlight Effect” Angular compression & headlight effect

From “Visualizing Special Relativity”