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Chapter 37: Relativity
Events and Inertial
Reference Frames
Principles of Einsteins
Special Relativity
Relativity of Simultaneity,Time Intervals, Length
Lorentz Transformation
Relativistic Momentum &Energy
General Relativity
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Albert Einstein and the Special Theoryof RelativityThe Special Theory of Relativity was published by Albert Einstein in 1905
when he was still working in a Swiss patent office.
It stands as one of the greatest intellectual achievement of the 20th century.
(In the same year, two groundbreaking works on the theory of Brownian motion
and the Photoelectric Effect were also published.)
Special Relativity led to a fundamental
rethinking of the concepts of space and time:
The measurement of a time interval The measurement of length
The concept of simultaneity/Causality
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Summary of ChangesBefore Special Relativity: space and time are thought to be as absolute
properties of the background or stage where objects act upon.
Different observers from different inertial reference frames will agree
on all the measurements on the previous slide.
After Special Relativity: These space and time measurements are relative
to the observers frame of reference. A stationary observer with respect to a
moving observer will see:
a moving clock runs slower
a moving meter-stick gets shorter
two events being simultaneous for one inertial observer will not necessary
be simultaneous for the other.
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Reality Check
One should note that these relativistic effects are only dramatic for objects
moving at large relative speeds near c!
For everyday objects moving with ordinary slowerspeeds, these effects are small.
In fact, one can show that Einsteins theory reduces to Newtonian mechanicsin the limit u
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Events
Since relativity deals with the fundamental concepts of space and time,
we need to have a concrete basis for our analysis of these quantities:
Event: An occurrence in the physical universe characterized by its position
andtime. We label each event by (x,y,z,t)
Example: a car crash (the event) occurs at a
particular location and time.
(x,y,z)
tan event
space-time
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Inertial Reference Frames
Inertial Reference Frames:
A coordinate system (x,y,z,t) for labeling events attached to an observer
who is not accelerating and there is no net force acting on it.
e.g., A space ship far away from any stars in deep space stopped or in
constant velocity motion.
An effective Inertial Reference Frames:
g
u
S
SBoth Sand S are not truly
inertial ref. frames because
gravity acts on them!
But since acts on them equally
and is (their relative
velocity), we can treat them as
effective inert. ref. frames.
g
to u
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Relative Motion
u
S
S
u
S
S
Sand S are in relative motion
Swould say that S
moves to the rightwithrespect to him/her at a
constant speed u.
S would equivalentlysay that Smoves to the
leftwith respect to
him/her with the same
constant speed u.
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Galilean Relativity (before Einstein)
Principle of Galilean Relativity: TheLaws of Mechanics must be the same in
all inertial reference frames, i.e., Newtons Laws of Motion apply equally to
all inertial observers in relative motion with constant velocity.
u
S
S
Example: observer in S throw a ball straight up.
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Galilean Relativity (before Einstein)
S
In S inertial ref. frame:
In S inertial ref. frame:
u
Su
Although the observed trajectoriesin S& S are not the same, the same
Newtons Equation
describes the observations in both
situations!
mF a
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Galilean Coordinate Transformation
How can one translate physical quantities from one inertialreference frame to another?
u is the relative speed between S& S
(an event)
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Galilean Coordinate Transformation
At a later time t, (x,y,z,t) & (x,y,z,t) for event P are related by:
'
'
'
'
x x ut
y y
z z
t t
Galilean space-time
transformation equations
Notice that in this classical viewpoint,
't tso that clocks runs at the same rates in all inertial reference frames !
One can show that Newtons 2nd law is invariant
under this coordinate transformation !
Galilean
Relativity
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Galilean Velocity Transformation
Let say there is a particle at point P moving in thex-direction with its
instantaneous velocity vx as measured by an observer in S-frame.
How is the velocity vx measured by an observer in S-frame related to vx?
In S-frame, the particle moves a distance of dx in a time dt, so that
x
dxv
dt
From the Galilean Coordinate Transformation, we have in differential form,
'dx dx udt
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Galilean Velocity Transformation
'dx dx
udt dt Dividing dton both sides of the equation gives,
'x x
v v u Galilean Velocity Transformation
relative velocity
between framesvelocity of
particle as
measured by S
velocity of
particle as
measured by S
'
'
'dx dx dxu u
dt dt dt
Since dt=dt, we have,
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The Constancy of the Speed of Light
Recall theMichelson-Morley experiment: Similar to a boat (light) traveling in a
flowing river (ether), the speed of light was expected to depends on its relativemotion with respect to the ether.
Presumed ether wind direction
u
c-u
c+u
u is the relative speed between
the frames (water & shore)
Result: Speed of light c does not follow the Galilean Velocity Transformation
and c is the same for all inertial observers in relative motion.
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The Constancy of the Speed of Light
Recall fromMaxwells Equations in electromagnetic theory: EM waves can be
shown to travel according to the plane wave equation,
2 2
2 2 2
, 1 ,E B E B
x c t
at the same speed c.
If we believeMaxwells Equations to be correct in all inertial reference frames,then we must accept that EM waves travel at the same speed c in all inertial
reference frames.
(Recall Sec. 32.2)
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Einsteins Postulates for SpecialRelativity
1. All laws of physics must be the same in all inertial reference frames.
Specific observations might be different but the same phenomena mustbe described by the same physical law.
Not just the laws of mechanics (as in the Galilean viewpoint). All laws
of physics include mechanics, EM, thermodynamics, QM, etc.
Same emf is induced
in the coil !
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Einsteins Postulates
2. The speed of light c in vacuum is the same in all inertial reference frames
and is independent of the observer or the source.
These two postulates form the basis of Einsteins Special Theory of Relativity.
(old vel addition rule applies to (slowly moving)
missile and it will be generalized for objects
moving close to c.)
(old vel addition rule does not apply to light.
c is the SAME in all frames ! )
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Stating the Results First
u
S
S
flash a flash b
Time Dilation: (moving clock runs slow)
2 2
0
, 1 1t t u c
Measured
by S
Measured
by S
Length Contraction: (moving ruler get shorter)
0L L
Simultaneity:
Two flashes
simultaneous in S but
not in S.
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Notes on Relative Motion
u
S
S
flash a flash b
flash a flash b
- Both observers in Sand S have their own measurement devices and they
can also measure his/her partners devices and compare with his/her own.
- Although time duration and length might depend on the observers inertial
frame, they will agree on the following three items:
c is the same in all frames
their relative speed u is the same all physical laws apply equally
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Relativity of Time Intervals
Measuring Time Intervals with a light clock:
One time unit is measured by the duration
of two events: laser light leaving (tick)
laser light return (tock)
mirror
laser
Consider a boxcar moving with respect to the ground and we are
interested in the measurement of an interval of time by both Sand S
from a clock placed in the boxcar.
u
S
S
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Relativity of Time IntervalsIn the S frame: Mavis O is moving with the boxcar
the clock is stationary with Mavis O
distance travelled by light = 2d
speed of light = c
According to Mavis O,
0
2d
t c (measured in S)
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Relativity of Time Intervals
Now, consider the observation from Stanleys S frame (stationary frame),
Note: speed of light is still c in this frame but it now travels further !
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Relativity of Time Intervals
Thus, if tis the timebetween the bounces of the
laser light in S frame, then
it must be longerthan t0.
From the given geometry, we can explicitly calculate t:
2 2
2 2
2 2 22 2 2
22
2 2
2 2
2 2 4
4
u t c t d l
c t u t t d c u
dt
c u
2 2
2
1
dt
c u c
where
2 2
1 1 u c
0t t
2t l c
(light travels atthe same speed
c in S!)
1
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Time Dilation
0
2 21ttu c
This is the time dilation formula in SR.
Since u is strictly less than c,
and
2 21 1 1u c
0t t always !
(Note: both observers Sand S will agree on this
relationship between time intervals as long as they are
both looking at the same clock in S.)
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Proper Time
0
t t
is called the proper time and it is a special (or proper) time interval
since it is the time interval of the clock measured by an observer stationary
with respect to that clock, i.e., the two events (tick & tock) occur at the same
location.
0t
All other time intervals, , are measured by observers in relative motion
with respect to the clock.
t
All observers have his/her own proper time and all other observersmeasuring other observers clocks will notnecessary beproper.
The proper time will always be the shortesttime interval among all observers.
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Unreality Check
2 22 ut l c
If speed of light changes
according to Galilean VelocityTransformation, then
u
c2 2u c
Then,
Following the same calculation as previously, we have
2 2
2 2 2 2
2 2
2 2
u t td l u c
d u
2 2
2 2
2 2
t tc u
2 2
4
c t
22 2
02
4dt t
c
(no time dilation: all inertial
observers measure the same
time interval.)
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Relativity of Length
Distance between two points on a rigid body P & Q can be measured by a
light signals round trip time.
PQ
mirror
laser
l
l can be measured by the time interval: t2t1, 2 1 2 12 ( ) ( )2
cl c t t l t t
As we have seen, t will be different for different inertial observers, l willalso !