Announcements
Review: Relativistic mechanics
• Reading for Monday: Chapters 3.7-3.12
Relativistic momentum:
• Review session for the midterm: in class on
Wed.
• HW 4 due Wed.
Relativistic force:
• Exam 1 in 6 days. It covers Chapters 1 & 2.
Room: G1B30 (next to this classroom).
Total energy of a
particle with mass ‘m’:
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’.
Kinetic energy
Relativistic kinetic energy
The relativistic kinetic energy K of a particle with a rest
mass m is:
The work done by a force F to move a particle
from position 1 to 2 along a path s is defined by:
K = γmc2 - mc2 = (γ-1)mc2
Note: This is very different from the classical K= ½mv2 .
K1,2 being the particle's kinetic energy at positions
1 and 2, respectively (true only for frictionless
system!).
Using our prior definition for the relativistic force we
can now find the relativistic kinetic energy of the
particle.
For slow velocities the relativistic energy equation gives the
same value as the classical equation! Remember the
binomial approximation for γ: γ ≈ 1+ ½v2/c2
 K = (γ-1)mc2 ≈ (1 + ½v2/c2 – 1)mc2 = ½ mv2

QUIZ: Rest energy of a
particle
E = γmc2 = K + mc2
Total energy
We rewrite the equation for the relativistic
kinetic energy and define the total energy of a
particle as:
E = γmc2 = K + mc2
This definition of the relativistic mass-energy E
fulfills our condition of conservation of total energy.
(Not proven here, but we shall see several
examples where this proves to be correct.)
Relation between Mass and Energy
m
v
-v
m
E2 = γmc2 = K + mc2
E1 = γmc2 = K + mc2
In the particle's rest frame, its energy is its rest
energy, E0. What is the value of E0?
A:
B:
C:
D:
E:
0
c2
mc2
(γ-1)mc2
½ mc2
Equivalence of Mass and Energy
m
Etot = E1+E2 = 2K + 2mc2
v
-v
m
Conservation of the total energy requires that the final
1 is the same as the energy Etot, before
energy Etot,final
2
2
thetot
collision. final
Therefore:
initial
E =γ
Total energy:
Note: This suggests a connection
between mass and energy!
Mc = 2K
+ 2mc
Etot,final = Mc2 ≡ 2K + 2mc2 = Etot,initial
We find that the total mass M of the final system is larger
than the sum of the masses of the two parts! M>2m.
Potential energy inside an object contributes to its mass!!!
The vibrating spring
a) Has a constant mass
b) Has the smallest mass when it is moving
fastest (in the middle of its motion)
c) Has the smallest mass when it is moving
slowest (at the end of its motion)
Example:
Rest energy of an object with 1kg
E0 = mc2 = (1 kg)·(3·108 m/s )2 = 9·1016 J
9·1016 J = 2.5·1010 kWh = 2.9 GW · 1 year
This is a very large amount of energy! (Equivalent to the
yearly output of ~3 very large nuclear reactors.)
Enough to power all the homes in Colorado for a year!
Way to convert mass to energy
Example: Deuterium fusion
Example: Deuterium fusion
Isotopes of Hydrogen:
Isotope mass:
Deuterium: 2.01355321270 u
Helium 4: 4.00260325415 u
(1 u ≈ 1.66·10-27 kg)
 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
Enough to power ~20,000 American households for 1 year!
Relationship of Energy and momentum
Recall:
= γmc2
p = γmu
Total Energy: E
Momentum:
Therefore: p2c2 = γ2m2u2c2 = γ2m2c4 · u2/c2
use:
 p2c2 = γ2m2c4 – m2c4
Mc2
= Σ(mi
c2) –
EB
=E2
This leads us the momentum-energy relation:
or:
E2 = (pc)2 + (mc2)2
E2 = (pc)2 + E02
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 another 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.
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.
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!!