Alan Guth, Black-Body Radiation and the Early History of the Universe, Part 2, 8.286 Lecture 16, November 5, 2013, p. 1.
Summary of Lecture 15:
Relativistic Energy
8.286 Lecture 16
November 5, 2013
BLACK-BODY RADIATION
AND
THE EARLY HISTORY
OF THE UNIVERSE, PART 2
E = mc2
The conversion of 1 kg/hour would supply 1.5 times the world
power production (for 2011).
A 15-gallon tank of gasoline could power the world for 2 1/2 days.
But: when U-235 undergoes fission, only about 0.09% of mass is
converted to energy.
Alan Guth
Massachusetts Institute of Technology
8.286 Lecture 16, November 5
Summary of Lecture 15:
Relativistic Four-Momentum
p0 =
E
, pµ =
c
E
,p
c
Lorentz-invariant square of pµ:
2
E2
2
2
2
p2 ≡ |p| − p0 = |p| − 2 = − (m0 c) .
c
.
pµ transforms from one inertial frame to another in the same
way as xµ = (ct, x).
p = γm0v , E = γm0 c2 =
where
1
γ=
1−
v2
c2
–1–
The Hydrogen Atom:
(m0 c2 )2 + |p|2 c2 ,
mH = mp + me − ∆E/c2 ,
where mH , mp , and me are the masses of the hydrogen atom,
the proton, and the electron, and ∆E = 13.6 eV is the binding
energy.
.
–2–
Alan Guth
Massachusetts Institute of Technology
8.286 Lecture 16, November 5
–3–
Alan Guth, Black-Body Radiation and the Early History of the Universe, Part 2, 8.286 Lecture 16, November 5, 2013, p. 2.
Summary of Lecture 15:
Radiation in an Expanding Universe
Summary of Lecture 15:
The Mass of Radiation
Redshift:
ρ = u/c2 ,
nγ ∝
where ρ = mass density, u = energy density.
1
1
, Eγ ∝
a3 (t)
a(t)
uγ
1
∝ 4
.
c2
a (t)
=⇒
ργ =
=⇒
ρr,0
≈ 3.1 × 10−4 .
ρm,0
The photon has zero rest mass:
Radiation-Dominated Era:
|p |2 −
2
E
=0,
c2
or
E = c|p | .
3
ur = 7.01 × 10−14 J/m
Alan Guth
Alan Guth
–4–
Massachusetts Institute of Technology
8.286 Lecture 16, November 5
–5–
Massachusetts Institute of Technology
8.286 Lecture 16, November 5
Summary of Lecture 15:
Modification of the Friedmann Equations
ρr (t)
a(t0 )
=
× 3.1 × 10−4 ,
ρm (t)
a(t)
ρr (teq )
≡1
ρm (teq )
=⇒
ρ∝
a(t0 )
1
=
≈ 3200 .
a(teq )
3.1 × 10−4
1
ȧ
=⇒ ρ̇ = −3 ρ ,
a
a3
dU = −p dV
=⇒
Massachusetts Institute of Technology
8.286 Lecture 16, November 5
1
ȧ
=⇒ ρ̇ = −4 ρ .
a4 (t)
a
ρ̇ and pressure p:
Assuming that a(t) ∝ t2/3 from teq until t0 (crude approximation),
find teq = (3.1 × 10−4 )3/2 t0 ≈ 75, 000 yr, for t0 = 13.8 Gyr.
Alan Guth
ρ(t) ∝
–6–
Alan Guth
Massachusetts Institute of Technology
8.286 Lecture 16, November 5
d 3 2
d
a ρc = −p (a3 )
dt
dt
ȧ p
ρ̇ = −3
ρ+ 2 .
a
c
=⇒
–7–
Alan Guth, Black-Body Radiation and the Early History of the Universe, Part 2, 8.286 Lecture 16, November 5, 2013, p. 3.
Friedmann equations:
 2
ȧ 
8π
kc2

 
=
Gρ − 2
3
a
a
4π
Gρa ,
3
ȧ
ρ̇ = −3 ρ
a
ä = −
matter-dominated
universe
Becomes inconsistent if
ρ̇ = −3
p
ȧ ρ+ 2 .
c
a
Solution:
ä = −
Alan Guth
Massachusetts Institute of Technology
8.286 Lecture 16, November 5
4π
3p
G ρ+ 2 a .
3
c
–8–
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8.286 The Early Universe
Fall 2013
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