lecture 2: relativistic space-time invariant intervals & proper time lorentz transformations...

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, 15-18

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

v v v v vd

dt

c = dt

-v

t

c /t )2 + v2t tt

1 - (v/c)2

=d/t )2+v2

x = vt

c = d2 +v2t 2

t

Time Dilation:

c = d2 +(x )2

t

Recall:

Thus, (c t )2 = d2 + (x )2

d2 = (c t )2 - (x )2

invariant

or, more generally,

S2 = (c t )2 - [(x )2+ (y )2+ (z )2 ]

''Invariant Interval”

choose frame ''at rest”

= (c

“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 = tx/c(a+ b) 2

(a-b) 2

= tx/c(a+ b) 2

(a-b)(a+b)[ ]

= A tx/c[ ]

Lorentz Transformations:

= A tx/c[ ]t

So, we know that A = t = A t(at fixed x)

Similarly, x = [ x - Bct ]

x = [ x - vt ] t = [ t - (v/c2)x ]

In non-relativistic limit ( 1) : x [ x - Bct ]

Must correspond to Galilean transformation, so Bc = v

B = v/c

Maxwell’s Equations

''Lorentz-Fitzgerald Contraction”

''Aether Drag”

George Francis Fitzgerald

Hendrik Antoon Lorentz

+q

+

+

+

+

vI

B

Lab Frame

F(pure magnetic)

+

+

+

+

+

+q

In Frame ofTest Charge

Lorentzexpanded

Lorentzcontracted

F(pure electrostatic)

Electricity & Magnetismare identically the sameforce, just viewed from different reference frames

UNIFICATION !!(thanks to Lorentz invariance)

Symmetry:The effect of a force looks the same whenviewed from reference frames boosted in the perpendicular direction

+q

+

+

+

+

vI

B

Lab Frame

+

+

+

+

+

+q

In Frame ofTest Charge

Lorentzexpanded

Lorentzcontracted

F(pure magnetic)

F(pure electrostatic)

F = qv B

| F | = qv Io/ (2r)

lab

+ = lab

=

q =

q =

´ = q

+

E = / 2ro = v/ (2r

oc2)

= v/ (2r)

| F ´| = Eq = vq / (2r) v = I

| F | = | F ´| / = qv / (2r)

=

| F ´| = vq / (2r)

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

Planck’s recommendation for Einstein’s nomination to the Prussian Academy in 1913:

“In summary, one can say that there is hardly one amongthe great problems in which modern physics is so rich towhich 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 lightquanta, cannot really be held against him, for it is not possible to introduce really new ideas even in the mostexact sciences without sometimes taking a risk.”

1905

E = h (Planck) p = h/ (De Broglie)

= hc/E = pc

absorber emitterp=E/c

recoilp=Mv

E/c = Mv

motion stops

distance travelled

d = vt = v (L/c)

L

= EL/(Mc2)

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/(Mc2)} = m LMd = mL

“Einstein’s Box”:

+ x- x

ct

-y

+ y

Space-Time:

+ x- x

ct

= ct/x = c/v = 1/

object stationary until time t

1

x1

ct1

moves with constant velocity () until t

2

ct2

x2

returns to point of origin

slope = (ct2 - ct

1)/(x

2-x

1)

+ x- x

ct

tan = x/ct = v/c =

tanmax

= 1

max

= 45°

45°

v = c

45°

v = c

light sent backwards

“absolute past”

+ x- x

ct

“absolute future”

“absolute elsewhere”

x1

ct1

no message sent from theorigin can be received by observers at x

1 until time t

1

there is no causal contact until they are “inside the light cone”

+ x- x

ct

“absolute future”

“absolute past”

“absolute elsewhere”

+ x- x

ct

+ x- x

ct

+ x- x

ct

+ x- x

ct

+ x- x

ct

+ x- x

ct

+ x- x

ct

+ x- x

ct

S S´

+ x- x

ctSpacetimeShowdown

Relativistic Optics

v

t = t

f = 1/t = 1/t = f/

Transverse Doppler Reddening

a

a

a

a

a

v

a

v

a

a v/c

v

(a v/c)2 + (a1 - (v/c)2 )2 = a2

a v/c

a1 - (v/c)2

v

(a v/c)2 + (a 1 - (v/c)2 )2 = a2

a

Terrell Rotation (1959)

a v/c

a1 - (v/c)2

a1

- (v

/c)2

Penrose (1959):

A Sphere By Any Other Frame Is Just As Round

v

d

h2+d2h

h

v

d

More generally, from somewhat off-axis hyperbolic curvature

h2+d2h

h

SS 433

If assumed distance to object increases,so must the distance traversed by jet topreserve 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 pointin the jet was produced t = (s-d)/c earlier, thus distorting the apparent orientation of the loops

d

v

s

Can fit distance to the source = 5.5 kpc (K. Blundell & M. Bowler)

Can even show evidence of jet speed variations!

Angular compression towards centre of field-of-view

Intensity = increases towards centrelight received solid angle

“Headlight Effect”

From “Visualizing Special Relativity” www.anu.edu.au/Physics/Searle