Transcript The Special Theory of Relativity* Physics I Class 26

Physics I
Class 26
The Special Theory of
Relativity*
*This material is optional.
It will not be on any exam.
Rev. 21-Apr-04 GB
26-1
Newton’s Laws of Motion
1. Newton’s First Law: No net force, no change in motion.


2. Newton’s Second Law: Fnet  m a
3. Newton’s Third Law: All forces come in pairs.
(equal magnitudes and opposite directions)
26-2
Are Newton’s Laws True?
It’s been over 300 years since Newton published Principia Mathematica.
How have his laws done since then?
The First Law is still doing fine. In modern times, many types of very
low-friction motion (space travel, magnetic bearings, air hockey tables,
etc.) make this notion more intuitively appealing than in the past.
The Third Law is also doing fine. All forces currently known to physics
obey this law. Any force not obeying this law would cause big problems
in physics, like getting free mechanical energy from nothing.
However, the Second Law in the form we learn it in Physics I is not
exactly correct. Where did Newton go wrong?
26-3
Where Did Newton Go Wrong?
Isaac Newton, 1642-1727
Newton defined time and space as follows:
“Absolute, true, and mathematical time, of itself and from its own nature,
flows equably without relation to anything external…”
“Absolute space, in its own nature, without relation to anything external,
remains always similar and immovable.”
As the 19th Century drew to a close, it became evident that there was
something wrong with these assumptions.
26-4
Maxwell’s Electromagnetic
Theory - 1873
Maxwell developed a theory of electromagnetism that
explained all the phenomena of electricity and
magnetism known then and predicted something new:
electromagnetic waves. This prediction was confirmed
by Hertz in 1886 and light was soon shown to be a type
of electromagnetic wave.


B
E  
t
James Clerk Maxwell
(1831-1879)

 D  

  D
H  J 
t

 B  0
But a question remained: If light is a wave, what is its
medium of propagation? Most physicists assumed that
there must be one and called it the ether. Ether was
assumed to define a fixed reference frame for the
universe (Newton’s “absolute space”) through which
electromagnetic waves travel at speed c.
26-5
The Michelson-Morley
Experiment - 1887
Michelson’s interferometer was designed to measure slight differences in phase
between two light beams that travel in orthogonal directions. By measuring the
phase differences at various places in earth’s orbit, Michelson and Morley hoped
to measure the speed of earth with respect to the ether.
However, no effect was found. The speed of light is constant for all observers.
26-6
Einstein’s Postulates of the
Special Theory of Relativity
Albert Einstein (1879–1955)
Studying Maxwell’s equations and noting a
remarkable symmetry in them between space and
time, Einstein replaced Newton’s definitions of
space and time with two new postulates that led
directly to the Special Theory of Relativity.
Einstein’s Two Postulates of Special Relativity (1905):
1. The laws of physics are the same in all inertial frames.
2. The speed of light, c, is constant in all inertial frames.
26-7
What is an Inertial Frame?
An inertial frame (of reference) is a real or imaginary set of
devices for measuring position and time that are in motion together
according to Newton’s First Law; in other words, these devices
are not accelerating (or rotating).
Neglecting gravity and the small acceleration of Earth (those are
covered in General Relativity), the track and motion detector that
we use in our activities, along with the clock in your PC when you
run LoggerPro, comprise an inertial reference frame with a special
name: the laboratory reference frame (because this is the frame
we use to make measurements).
If we were to set the same equipment up in the Ferris Wheel we
studied earlier, that would not be an inertial reference frame.
Instead, we call that an accelerated frame.
26-8
Where Did Einstein’s
Postulates Come From?
1. The laws of physics are the same in all inertial frames.
This idea goes back to Galileo. Imagine an experiment like our cart
track and hanging weight being performed in an airliner moving
uniformly in one direction at a constant speed and altitude. (Assume
no turbulence and the altitude is low enough so that the force of
gravity is about the same as on the ground.) Our measurements that
we take with LoggerPro should be the same as what we did in class
if we set things up carefully.
2. The speed of light, c, is constant in all inertial frames.
Few wanted to believe this prior to the Michelson-Morley
experiment in 1887. There are now many measurements confirming
this postulate and none contradicting it. However, do a web search
and you will see that many people still refuse to believe it!
26-9
Postulate #1
0.5 c
The first postulate states that any
experiment performed in any
inertial reference frame (like the
starship Enterprise moving at 0.5 c)
will exhibit the same laws of
physics as in any other inertial
frame (like space station Deep
Space 9 at rest). In fact, there is no
way to know which inertial frame
is moving and which is at rest.
26-10
Postulate #2
0.5 c
c
The second postulate states that no
matter what the state of motion of
any source of electromagnetic
waves (like Enterprise sending a
sensor beam), the speed of the
waves as measured in any inertial
frame (like Deep Space 9 or the
Enterprise) will be exactly c.
26-11
How Is This Possible?
0.5 c
c
Kirk: “Scotty, measure the speed of our
sensor beam relative to Enterprise.”
Scott: “Exactly 299,792,458 m/s, sir!”
Sisko: “O’Brien, measure the speed of Enterprise’s
sensor beam relative to us in Deep Space 9.”
O’Brien: “Exactly 299,792,458 m/s, sir!”
26-12
Consequence #1
Time Dilation (Part 1)
mirror
d0
pulsed laser
source & detector
Engineer Scott on Enterprise constructs a special kind of “clock” that
emits a pulse of light, bounces it off a mirror, and detects the returning
pulse. Scott makes the distance to the mirror exactly 14.9896229 cm, so
the round trip takes exactly one nanosecond. One billion “ticks” of this
clock = one second.
Scott synchronizes all of the clocks on Enterprise to this standard clock.
26-13
Consequence #1
Time Dilation (Part 2)
A little algebra (not rocket science):
Path length:
d0
s  2 d0 
2
vt/2
 v t
1
2
2
Speed of light: s  c t
Solve for t and use t0 = 2 d0 / c
How it looks to Chief O’Brien on Deep Space 9:
The path of light is longer because the device is moving, but light still travels at the
same speed of light. Therefore, O’Brien measures a longer time for each “tick”.
If t0 is the time of a tick as measured on Enterprise, t is the time on Deep Space 9:
t
t0
1  v2 c2
Because of postulate #1, this time dilation effect applies not just to Scott’s special
laser clock, but to all clocks on Enterprise, even the biological clocks of the crew!
26-14
Consequence #2
Length Contraction (Part 1)
light
v
d
v
light
d
Scott has a second clock that is oriented bow-stern instead of up-down.
This clock is otherwise identical to the first and keeps the same time.
How it looks to Chief O’Brien on Deep Space 9:
On the first part of the path, light is “chasing” the mirror. On the way back, the
detector is moving toward the light. The total time of the “tick” is
t  t1  t 2 
d
d
2d c


c  v c  v 1  v2 c2
This time must be the same as O’Brien measured for the first clock or Postulate #1
would be violated. Doing the algebra, the only possible conclusion is that d as
measured by O’Brien must be different than d 0 as measured by Scott:
d  d0 1  v2 c2
26-15
Consequence #2
Length Contraction (Part 2)
v=0
0.5 c
The length contraction effect
applies only in the direction of
relative motion of an object as
observed from another inertial
frame of reference.
To the equipment and people
0.8 c inside Enterprise, nothing
unusual is noticed. As far as
they are concerned, they are at
rest while everything around
0.95 c them is moving.
26-16
Are There Real Values for Time
and Length?
Evidently, time and length can vary depending on which inertial
frame of reference we use, but can we find the real values of time
and length, as opposed to apparent values? All values that we
measure in a repeatable way are real, but physicists have two special
ways of measuring time and length so that everyone can agree:
Proper Time:
The proper time interval between two events is the time measured in
the inertial frame in which the events happen at the same location.
Proper Length:
The proper length of an object is the length measured in the inertial
frame in which the object is at rest.
26-17
Special Relativity Applied to
Electric and Magnetic Fields
Y
v = 1000 km/s
electron
E = 1000 V/m
B = 0.001 T
X
An electron is moving in the +X direction at 1000 km/s in an electric
field of 1000 V/m in the +Y direction and a magnetic field of 0.001 T
in the +Z direction. What is the net force on the electron?
Take a few minutes to determine the answer before we continue.
26-18
Special Relativity Applied to
Electric and Magnetic Fields
You should have gotten exactly 0 because the electric force cancels
the magnetic force. The electron follows Newton’s First Law.
Now look at the situation from the electron’s inertial frame.
Y
E' = ?
v=0
electron
B' = ?
X'
Since the electron is not moving in this frame, the magnetic force
must be zero no matter what the magnetic field. But the total force
must still be zero since the electron stays at rest in this inertial frame.
What happened to the electric field or electric force?
26-19
Special Relativity Applied to
Electric and Magnetic Fields
The electric and magnetic fields are not separate fields! They are
really components of the same field and these components transform
into each other when we switch reference frames. In this case, the
transformation equations are
Ey 
E y  v Bz
1 v c
2
2
Bz 
Bz  v E y c 2
1  v2 c2
You can verify that E' = 0 for this case. The reason there is no electric
force in the electron’s frame is that the electric field is zero.
26-20
Einstein’s Corrections to
Newtonian Mechanics
The familiar quantities of Newtonian mechanics have new
definitions in Special Relativity:
Momentum:

p
m
1  v2 c2
E  mc  K 
2
Total Energy:

v
m c2
1  v2 c2


1

K  mc
 1
 1  v2 c2



2
Kinetic Energy:
26-21
Einstein’s Corrections to
Newtonian Mechanics
Relation of p and E:
E 2  p 2c 2  m 2c 4
Relation of p and K:
p 2c 2  K 2  2 K m c 2
Newton’s 2nd Law (New Form):




d p d 
mv

Fnet 

d t d t  1  v2 c2 


From the laboratory inertial frame of reference, a particle’s mass
appears to increase as the particle moves faster. This effect is
significant only when the speed of the particle approaches c.
26-22
If “F = m a” isn’t true,
why do we still use it?
The original form of Newton’s Second Law is true to a very good
approximation when dealing with velocities much less than the speed of
light. For most calculations involving ordinary objects, it is close enough
for practical purposes. For some calculations, we need relativity.
“Disintegration of the Persistence of
Memory” by Salvador Dalí, 1931
Art inspired by the
Theory of Relativity?
26-23
What’s so “special” about
Special Relativity?
The “special” part means that it deals only with inertial frames of
reference. Einstein worked another decade to generalize his ideas to
accelerating frames of reference. In 1915, his General Theory of
Relativity revolutionized our ideas about gravity and the nature of
space and time. Unfortunately, we can’t go into the details today.
To find out more, check the website on the next slide.
26-24
http://www.amnh.org/exhibitions/einstein/?src=h_h
26-25
Activity #26
Special Relativity
Objectives of the Activity:
1.
2.
3.
Think about and resolve the “paradoxes” of Special Relativity.
Try some simple calculations using Special Relativity.
Have fun. (Well, for some people this will be fun.)
26-26