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Review: Relativistic mechanics
From last class: total energy
E = γmc2 = K + mc2
Relativistic momentum:
m
Relativistic force:
Total energy of a
particle with mass ‘m’:
v
-v
m
Etot = 2K + 2mc2
Etot = γmc2 = K + mc2
These definitions fulfill the momentum and energy
conservation laws. And for u<<c the definitions for p, F,
and K match the classical definitions. But we found that
funny stuff happens to the proper mass ‘m’.
m
m
Etot = Mc2
Mc2 ≡ 2K + 2mc2
 M > 2m
Example: Deuterium fusion
Isotopes of Hydrogen:
Example: Deuterium fusion
Relationship of Energy and momentum
Recall:
= γmc2
p = γmu
Total Energy: E
Momentum:
Therefore: p2c2 = γ2m2u2c2 = γ2m2c4 · u2/c2
Isotope mass:
Deuterium: 2.01355321270 u
Helium 4: 4.00260325415 u
(1 u ≈ 1.66·10-27 kg)
use:
 p2c2 = γ2m2c4 – m2c4
=E2
This leads us the momentum-energy relation:
 1kg of Deuterium yields ~0.994 kg of Helium 4.
Energy equivalent of 6 grams:
E0 = mc2 = (0.006 kg)·(3·108 m/s )2 = 5.4·1014 J
or:
E2 = (pc)2 + (mc2)2
E2 = (pc)2 + E02
Enough to power ~20,000 American households for 1 year!
Application: Massless particles
From the momentum-energy relation E2 = p2c2 + m2c4
we obtain for mass-less particles (i.e. m=0):
E = pc , (if m=0)
p=γmu and E=γmc2  p/u = E/c2
Using E=pc leads to:
u=c
, (if m=0)
Massless particles travel at the speed of light!!
… no matter what their total energy is!!
Example:
Electron-positron annihilation
Positrons (e+, aka. antielectron) have exactly the same mass
as electrons (e-) but the opposite charge:
me+ = me-= 511 keV/c2 (1 eV ≈ 1.6·10-19J)
E1, p1
eBAM!
e+
E2, p2
At rest, an electron-positron pair has a total energy
E = 2 · 511 keV. Once they come close enough to each
other, they will annihilate one other and convert into two
photons.
Conservation
momentum:
. photons?
1 = -p2
What can of
you
tell about pthose
two
Conservation of energy: E1+E2 = 2mc2 ,  E1 = E2 = 511 keV
Do neutrinos have a mass?
Do neutrinos have a mass? (cont.)
Neutrinos are elementary particles. They come in three
flavors: electron, muon, and tau neutrino (νe,νµ, ντ). The
standard model of particle physics predicted such particles.
The prediction said that they were mass-less.
Bruno Pontecorvo predicted the ‘neutrino oscillation,’ a
quantum mechanical phenomenon that allows the neutriono
to change from one flavor to another while traveling from the
sun to the earth.
The fusion reaction that takes place in the sun (H + H  He)
produces such νe. The standard solar model predicts the
number of νe coming from the sun.
All attempts to measure this number on earth revealed only
about one third of the number predicted by the standard
solar model.
Summary SR
• Classical relativity  Galileo transformation
• Special relativity (consequence of 'c' is the same in all inertial
frames; remember Michelson-Morley experiment)
– Time dilation & Length contraction, events in
spacetime  Lorentz transformation
– Spacetime interval (invariant under LT)
– Relativistic forces, momentum and energy
– Lot's of applications (and lot's of firecrackers)
…
Everything we have discussed to this point will be part of the first
mid-term exam (including reading assignments and homework.)
If you have questions ask as early as possible!!
Why would this imply that the neutrinos have a
mass?
Massless particles travel at the speed of light!
i.e. γ  ∞, and therefore, the time seems to be standing
still for the neutrino:
ΔtEarth = γ · Δtneutrino(“proper”)
In the HW: muon or pion experiments. The half-live time of the
muons/pions in the lab-frame is increased by the factor γ.