lecture 2: relativistic space-time · consider light beam moving along positive x-axis: x = ct or x...
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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:
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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:
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= 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
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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”
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Maxwell’s Equations
''Lorentz-Fitzgerald Contraction”
''Aether Drag”
George Francis Fitzgerald
Hendrik Antoon Lorentz
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+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
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+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)
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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
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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.”
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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”:
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+ x- x
ct
-y
+ y
Space-Time:
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+ 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)
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+ x- x
ct
θ
tanθ = x/ct = v/c = β
tanθmax= 1
θmax= 45°
45°
v = c
45°
v = c
light sent backwards
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“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”
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+ x- x
ct
“absolute future”
“absolute past”
“absolute elsewhere”
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+ x- x
ct
θ
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+ x- x
ct
θ
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+ x- x
ct
θ
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+ x- x
ct
θ
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+ x- x
ct
θ
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+ x- x
ct
θ
θ
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+ x- x
ct
θ
θ
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+ x- x
ct
θ
θ
S S´
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+ x- x
ctSpacetime
Showdown
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Relativistic
Optics
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v
∆t = γ ∆t′
f = 1/∆t = 1/γ∆t′ = f′/ γ
Transverse Doppler Reddening
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a
a
a
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v
a
a v/c
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v
a√1 - (v/c)2
(a v/c)2 + (a √1 - (v/c)2 )2 = a2
a v/c
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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
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Penrose (1959):
A Sphere By Any Other Frame Is Just As Round
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v
d
√h2+d
2
h
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v
d
h
More generally, from somewhat off-axis ⇒ hyperbolic curvature
√h2+d
2
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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
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Can fit distance to the source = 5.5 kpc (K. Blundell & M. Bowler)
Can even show evidence of jet speed variations!
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Angular compression towards centre of field-of-view
Intensity = increases towards centrelight receivedsolid angle
“Headlight Effect”
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