Relativity
Classical” or “Newtonian”
physics does a good job of
describing the behavior of
particles at low speeds.
But as speeds approach light
speed we must use The Special
Theory of Relativity proposed
by Einstein in 1905.
Classical physics represents
the “low speed limit” of
relativistic physics.
What do we have to know about relativity
to solve problems in a correct way?
=> Just change Newton’s laws by
introducing a correction factor for the
mass F = d ( mv ) / dt
m0
m=
2
2
1− v / c
The principle of relativity was
first stated by Newton
“The motions of bodies included in a
given space are the same among
themselves, whether that space is or
moves uniformly forward in a straight
line”
Reference Frames
The frame of reference for a person or object is
the coordinate system which moves with the
person or with the object.
Think of the frame of reference as a set of xyz
axes which are attached to the object.
An inertial reference frame is one in which
Newton’s Laws are valid. All inertial reference
frames move with constant velocity relative to
one another. (= Special Theory of Relativity)
*If frames are accelerated (translational or rotational) then General
Relativity applies (A little bit more complex)
The Postulates of Relativity
The Special Theory of Relativity is based on
two postulates:
• The laws of physics are the same in all
inertial reference frames.
• The speed of light in vacuum is ALWAYS
measured to be 3 × 108 m/s, independent of
the motion of the observer or the motion of
the source of light.
The Speed of Light
If we apply Newtonian Mechanics to describe the
propagation of light, then two observers who are in
two different reference frames will measure a
different value for the speed of light if there is relative
motion between the reference frames. (Recall v13 =
v12 + v23)
However, that does
not happen – all
observers measure
the same speed of
light.
We need Relativity
Theory to explain this!
The problems of
“Classical Physics”
•
•
•
•
Transformation of coordinates
Maxwell equations wrong?
Lorentz transformation
Michelson-Morley experiment
Conclusions:
In Relativity Theory, the distance between
two points and the time interval between
two events depend on the frame of
reference in which they are measured!
There is no absolute measure of time.
There is no absolute measure of length.
Time is Relative
Time interval measurements depend on the
frame in which they are measured.
Events are any physical occurrence that
happens at a specific location at a specific time.
This a definition.
Two events which appear to be simultaneous in
one reference frame are in general not
simultaneous in a second frame moving with
respect to the first.
Time Dilation
Two observers, each in their
own reference frame moving
with a relative velocity will not
agree on how fast time passes.
Each will think the other’s clock
is wrong.
Moving clocks run slow
This effect is known as time
dilation.
2d
∆to =
c
∆t =
2d
c2 − v2
=
2d
v2
c 1− 2
c
The proper time ∆t0 is the time interval between
two events as measured by an observer who
sees the events occur at the same place (same
location x,y,z). Let the “proper” frame move with
velocity v with respect another frame.
Time interval in moving frame:
2d
∆t0 / ∆t =
c
Æ ∆t = γ ∆t0
where
2d / c
1 − v2 / c2
1
γ=
2
2
1− v / c
Example:
The decay of a muon. A muon is a naturally occurring subatomic particle of
nature that is unstable and transforms itself to other parts with a mean time
to decay of 2.20µs (created by cosmic radiation high in Earth’s atmosphere).
The muon(-/+) mass is roughly 310 times that of an electron (positron).
Assume that a muon is created at an altitude of 5.00 km above the surface of the
earth and it travels toward the earth with a speed of 0.995c. The muon the gets a
certain distance into the atmosphere and decays. So there is a time delay between
the two events. (Without time delay a Muon cove 657m before decaying)
Let that time delay be measured form the rest frame of the earth.
2.2 µ s
2.2 µ s
∆t =
=
2
1 − .990025
(.995c)
1−
c2
2.20µ s
∆t =
= 22.02 µ s
−2
9.987492 × 10
The average distance traveled is
(0.995c)x(22.02µs)=6570 m
So the muons make it to the ground.
Example 1
An astronaut at rest on Earth has a heartbeat rate
of 70 beats/min. When the astronaut is traveling
in a spaceship at 0.90c, what will this rate be as
measured by (a) an observer also in the ship and
(b) an observer at rest on the Earth.
Length Contraction
(Lorentz- or Fitzgerald Contraction)
The distance between two points depends on the
frame of reference in which it is measured.
The proper length L0 is the length of the object
measured in the frame of reference in which the
object is at rest.
If the object is moving in a reference frame, its
length L will be measured to be less than its proper
length:
L = L0 / γ = L0 1 − v 2 /c 2
This is known as relativistic length contraction.
Example 2
A friend in a spaceship travels past you at a high
speed. He tells you that his ship is 20 m long and
that the identical ship you are sitting in is 19 m long.
According to your observations, (a) how long is your
ship, (b) how long is his ship, and (c) what is the
speed of your friend's ship?
Example 3
The proper length of one spaceship is three times
that of another. The two spaceships are traveling
in the same direction and, while both are passing
overhead, an Earth observer measures the two
spaceships to have the same length. If the slower
spaceship is moving with a speed of 0.35c,
determine the speed of the faster spaceship.
Example 4
A supertrain of proper length 100 m travels at a
speed of 0.95c as it passes through a tunnel
having proper length 50 m. As seen by a
trackside observer, is the train ever completely
within the tunnel? If so, by how much?
Relativistic Addition of Velocities
In relativity theory we must modify the way we
add velocities: The correct relativistic addition
of velocities is:
v1 + v 2
v =
v 1v 2
1 +
2
c
We are moving on a straight line relative to an object
(for example an asteroid) with a speed v1 of with
respect to that object. We fire an object away from us
at speed relative to us v2. If the two objects move on the
same straight line then, the speed of object 2 as seen
from the asteroid v is given by the equation above
A spaceship, Enterprise, moving at 0.99c
away from the earth launches a spaceship,
Picard, back toward the earth with a speed
relative to the Enterprise of 0.6c. What
speed do we see the Picard moving on the
earth?
v=
v1 + v2
0.99c − 0.6c
.36c
0.36c
=
=
=
= 0.8866995c
v1v2
.99c( −.6c ) 1 − .5940 .4060
1+ 2
1+
c
c2
If instead, if the Picard was launched away
from the Enterprise in the same direction as
its motion, the answer would be
v=
v1 + v2
0.99c + 0.6c
1.59c
1.59c
=
=
=
= .99624c
v1v2
.99c(.6c ) 1 + .5940 1.5960
1+ 2
1+
c
c2
as seen from the earth (assumed to be at rest
in these two examples).
Example 5
Spaceship R is moving to the right at a speed of
0.70c with respect to the Earth. A second
spaceship, L, moves to the left at the same
speed with respect to the Earth. What is the
speed of L with respect to R?
Example 6
A spaceship is traveling at 0.95c with
respect to the Earth. Inside the spaceship, a
man shines a flashlight in the direction the
spaceship is moving. What is the velocity of
the light from the flashlight with respect to
Earth?
Relativistic Momentum
In order for momentum to be conserved in
relativistic interactions, we must modify our
classical expression for momentum to read:
p=
m 0v
1 − v 2 /c 2
= γ m 0v
m0 is called the rest mass of the object.
Example 7
Calculate the momentum of a proton
moving with a speed of 0.5c.
Relativistic Energy
The total energy E of an object moving with velocity v is:
2
m
c
0
E = γm0 c 2 =
= mc 2
2 2
1 − v /c
Mass is another form of energy!
The energy of an object at rest is E0 = m0c2, which is known as
the rest energy.
The total energy is comprised of the rest energy and the
kinetic energy:
E= m0c2 + K
In nuclear reactions like in the sun,
we convert four hydrogen atoms
into one helium atom. The mass
before and after the reaction is
different – the mass difference is
released as the energy that drives
the solar furnace.
Example 8
An electron moves with a speed of 0.80c.
Calculate its (a) rest energy, (b) total energy,
and (c) kinetic energy.
Example 9
A proton in a high-energy accelerator is
given a kinetic energy of 50.0 GeV.
Determine (a) the momentum and (b) the
speed of the proton.