Lecture 2: Relativistic Space-Time• Lorentz Transformations

• Invariant Intervals & Proper Time

• Electromagnetic Unification

• Equivalence of Mass and Energy

• Space-Time Diagrams

• Relativistic Optics

Section 6-7, 19-21, 15-18

Useful Sections in Rindler:

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 − Β x/c[ ]

Lorentz Transformations:

= A t − Β x/c[ ]t

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

Similarly, x = γ [ x - Bct ]

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

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

Must correspond to Galilean transformation, so Bc = v

B = v/c

c =√d2 + (∆x )2

∆t

Recall:

Thus, (c ∆t )2 = d2 +(∆x )2

d2 = (c ∆t )2 - (∆x )2invariant

or, more generally,

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

''Invariant Interval”

choose frame''at rest”

= (c ∆τ)2

“Proper Time”

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 magnitude of a forcelooks 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 Iµ

o/ (2πr)

λlab+ = λ λ

lab− = λ

λq+ = λ/γ λ

q− = λγ

λ´ = λq+ −−−− λ−

E = λ′ / 2πrεo= λγ v2 / (2πrε

oc2)

= λγ v2 µο/ (2πr)

| F ´| = Eq = λγ v2 µοq / (2πr)

λv = I

| F | = | F ´| / γ = qvΙµο / (2πr)

= λγβ2= λ(γ−1/γ)

| F ´| = γ Ιvµοq / (2πr)

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 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.”

E = hν (Planck) p = h/λ (De Broglie)

= hc/λE = pc

absorber emitter

p=E/c

recoil

p=Mv

E/c = Mv

motionstops

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

= c∆t/∆x = c/v = 1/β

object stationaryuntil time t

1

x1

ct1

moves with constant

velocity (β) until t2

ct2

x2

returns to point of origin

slope = (ct2- ct

1)/(x

2-x1)

+ x- x

ct

θ

tanθ = x/ct = v/c = β

tanθmax= 1

θmax= 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

1until time t

1

there is no causal contactuntil they are

“inside the light cone”

+ x- x

ct

“absolute future”

“absolute past”

“absolute elsewhere”

+ x- x

ct

θ

+ x- x

ct

θ

+ x- x

ct

θ

+ x- x

ct

θ

+ x- x

ct

θ

+ x- x

ct

θ

θ

+ x- x

ct

θ

θ

+ x- x

ct

θ

θ

S S´

+ x- x

ctSpacetime

Showdown

Relativistic

Optics

v

∆t = γ ∆t′

f = 1/∆t = 1/γ∆t′ = f′/ γ

Transverse Doppler Reddening

a

a

a

v

a

a v/c

v

a√1 - (v/c)2

(a v/c)2 + (a √1 - (v/c)2 )2 = a2

a v/c

v

a√1 - (v/c)2

(a v/c)2 + (a √1 - (v/c)2 )2 = a2

a√1 - (v/c)2

a

Terrell Rotation(1959)

a v/c

Penrose (1959):

A Sphere By Any Other Frame Is Just As Round

v

d

√h2+d

2

h

v

d

h

More generally, from somewhat off-axis ⇒ hyperbolic curvature

√h2+d

2

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 point

in the jet was produced ∆t = (s-d)/c earlier, thus distorting the apparent orientation of the loops

d

vτ

θ

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 receivedsolid angle

“Headlight Effect”