Time Dilation, Summary


The time interval Δt between two events
measured by an observer moving with
respect to a clock is longer than the time
interval Δtp between the same two events
measured by an observer at rest with
respect to the clock
A clock moving past an observer at speed
v runs more slowly than an identical clock
at rest with respect to the observer by a
factor of 1  v 2 / c 2
Identifying Proper Time

The time interval Δtp is called the
proper time
• The proper time is the time interval
between events as measured by an
observer who sees the events occur at
the same position

You must be able to correctly identify the
observer who measures the proper time
interval
Time Dilation – Generalization

All physical processes slow down
relative to a clock when those
processes occur in a frame moving
with respect to the clock
• These processes can be chemical and
biological as well as physical

Time dilation is a very real
phenomena that has been verified by
various experiments
Time Dilation Verification –
Muon Decays




Muons are unstable particles
that have the same charge as
an electron, but a mass 207
times more than an electron
Muons have a half-life of Δtp =
2.2µs when measured in a
reference frame at rest with
respect to them (a)
Relative to an observer on
earth, muons should have a
longer lifetime (b)
A CERN experiment measured
lifetimes in agreement with the
predictions of relativity
Modification of Newton’s Law
Only one modification is necessary to make Newton’s mechanics
To be consistent with the Principle of Relativity.
Redefinition of Momentum

p
mo
2
1 v / c
2

v
mo is called the rest mass, its mass when the object is at rest.
We can also define the relativistic mass m
m
mo
1 v / c
2
2
Then, the relativistic form of Newtonian mechanics reads:
1. In the absence of forces, a body moves with uniform speed
in a straight line.
2. In the presence of force, the body changes its state of motion
in such a way that


Ft  p


mo
p
v
1 v 2 / c 2
Mass (inertia) depends on the speed of the object.
In the limit of v/c << 1, the relativistic mechanics recovers the old form!!
speed
(m/s)
0
rel. mass (kg)
Note
100
Rest mass
90
100
NASCAR
1,000
100
MACH 3
32,000
100.0000006
1000,000
100.0006
0.98c
502.5
Planetary speed
Decent
accelerator
Ex. 29.1. How fast must an object be moving if m/mo = 1.010?
m/mo = 1/sqrt(1 – v2/c2)
(m/mo)2 = (1.010)2 = 1.020
= 1/(1 – v2/c2)
v2/c2 = 0.02/1.020  v = 0.14c = 4.2 x 107 m/s
We have to apply force to make an object move faster. However,
as the speed of an object gets larger, its mass gets larger. Therefore,
we have to apply larger force to increase its speed at the same rate.
It is impossible, from the point of view of the relativistic equations, to
Accelerate a particle to a speed larger than that of light!!
How about energy and work in the relativistic world?
Modification is needed accordingly to be consistent with the
new definition of momentum.
Kinetic energy
2
1
p
 mo v 2 
2
2mo
= E (total energy without force)
Relativistic case
E  mo c 4  p 2 c 2 
2
mo c 2
1 v2 / c2
Relativistic case
E  mo c 4  p 2 c 2 
mo c 2
2
1 v2 / c2
approximate for small v
2
p
E  mo c 
2mo
2
For a particle at rest,
typical kinetic energy
E  mo c
2
the most famous equation in physics!!
Meaning of
E  mo c
2
“If a body gives off the energy E in the form of radiation, its mass
diminishes by E/c2. The fact that the energy withdrawn from the
body becomes energy of radiation evidently makes no difference, so
that we are led to the more general conclusion that the mass of a body
is a measure of its energy-content; if the energy changes by E, the
mass changes the same sense by E/(9 x 1020), the energy being
measured in ergs, and mass in grams.”
A. Einstein in The Principle of Relativity, Methuen, London, 1923.
It is not impossible that with bodies whose energy-content is variable
to a high degree (e.g. with radium salt) the theory may be successfully
put to test
Ex. 29.2 A kilogram of gasoline (about 1.5 liters) yield an energy of
4.8 x 107 J when burned. Compare this to the mass energy of 1 kg
Of a substance.
mo = 1 kg
E = moc2 = (1 kg) x (3 x 108 m/s)2
= 9 x 1016 J
Compare this with 4.8 x 107 J
This huge difference stresses the importance of nuclear energy.