Lecture 1' Relativistic kinematics

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Lecture 1. Relativistic kinematics
Prof. Michael Gershenson, Office Serin Physics
122W Phone 5-3180, E-mail gersh_at_physics.rutgers.e
du Office hours Mon. 230-330 PM and by
appointment (e-mailed questions are
encouraged) Textbook A.Beiser, Concepts of
Modern Physics, 6th Ed. Additional reading T.A.
Moore, Six Ideas that Shaped Physics, Unit Q
(Particles behave like waves).

• Outline
• Structure of the course Modern Physics
• Classical Mechanics, Galilean Principle of
  Relativity
• Special Relativity, Einsteins Principle of
  Relativity
• Lorentz Transformations
• Time Dilation

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Structure of the course Modern Physics
v/c
Relativistic mechanics, El.-Mag. (1905)
Relativistic quantum mechanics (1927-)
Maxwells Equations of electromagnetism (1873)
Classical physics
Quantum mechanics (1920s-)
S the actionmomentum?distance, units g?cm2/s
h/s
Newtonian Mechanics, Thermodynamics Statistical
Mechanics

• Relativistic Mechanics (kinematics/dynamics)
• Quantum (non-relativistic) Mechanics
• Intro to Statistical Mechanics of bosons and
  fermions

Course Modern Physics
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Galilean Principle of Relativity
Reference frame a set of 3 spatial coordinates
(e.g. x,y,z) and a time coordinate t.
Space and Time in Classical Mechanics Space
uniform and isotropic. Time uniform and
absolute (the same in both IRFs and non-IRFs).
Inertial reference frames (IRFs) (experimental
observation, their existence is postulated by the
1st Newtons Law) the frames where bodies
removed from interaction with other bodies will
maintain their state of rest or of uniform
straight-line motion.
Practical definition of IRF a frame that moves
with a constant velocity relative to very distant
objects, e.g. distant stars.
(velocity vector (!), speed the length of
this vector)
Significance of IRFs Newtons Laws have the same
form in all these frames (the laws are invariant
under the coordinate transformations that
transform one IRF into another one).
Galilean Principle of Relativity (1632) The laws
of classical mechanics are invariant in all
inertial reference frames
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Newtons Laws (1687)

• First Law
• It is possible to select a set of reference
  frames, called inertial reference frames,
  observed from which a particle moves without any
  change in velocity if no net force acts on it.
  This law is often simplified into the sentence "A
  particle will stay at rest or continue at a
  constant velocity unless acted upon by an
  external unbalanced force."

• Second Law
• Observed from an inertial reference frame, the
  net force on a particle is proportional to the
  time rate of change of its linear momentum F d
  (mv) / dt. Momentum mv is the product of mass and
  velocity. Force and momentum are vector
  quantities and the resultant force is found from
  all the forces present by vector addition. This
  law is often stated as "F ma the net force on
  an object is equal to the mass of the object
  multiplied by its acceleration."
• Third Law
• Whenever a particle A exerts a force on another
  particle B, B simultaneously exerts a force on A
  with the same magnitude in the opposite
  direction. The strong form of the law further
  postulates that these two forces act along the
  same line. This law is often simplified into the
  sentence "Every action has an equal and opposite
  reaction."

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Galilean Transformations
The IRF transformations that preserve
invariance of Newtons Laws are known as
Galilean Transformations (G.Tr.) (N. Laws are
invariant under G. Tr.)
- the absolute time
- because the space is uniform and isotropic, the
IRFs can move with respect to one another with
constant velocity
Lets check that the 2nd Newtons Law is
invariant under Galilean Transformations
1. differentiate with respect to the (absolute)
time
(at t0 the origins coincide with one another)
- Galilean velocity addition rule
2. differentiate again
(This is not the case in non-inertial RFs an
example of an accelerated RF)
- acceleration is the same in all IRFs
?
The force in N. mechanics can depend (only!) on
the difference of two radius-vectors and
velocities. Thus
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Importance of Vectors in Classical Mechanics
Invariant Lets formulate some physical law in
the form AB. If a coordinate transformation
affects neither A nor B, we say that this law is
invariant under the transformation.
Galilean transformations do not affect the
relations between vectors.
In particular, G.Tr. do not affect the length of
a vector the length of a vector is invariant
under G.Tr.
That makes vectors so useful in classical
mechanics if one can formulate a law that looks
like vector vector , it automatically means
that this law is invariant under G.Tr.
All laws of classical physics must have the
following forms to be invariant under G.Tr.
scalar A scalar B
vector A vector B
In other words, if one side of an equation is a
scalar (vector), the other side must also be a
scalar (vector) to satisfy Galilean Principle of
Relativity.
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Maxwells Equations challenge to Galilean
PR
In 1873, Maxwell formulated Equations of
Electromagnetism. On one hand, Maxwells
Equations describe very well all observed e.-m.
phenomena, on the other hand, they are not
invariant under G.Tr.!
Some odd things At first glance, there is a
built-in asymmetry a charge in motion produces a
magnetic field, whereas a charge at rest does
not. Also, it follows from M.Eqs that the speed
of light is the same in all IRFs, at odds with
Galilean velocity addition.
This asymmetry brought into being an idea of a
unique stationary RF (ether), with respect to
which all velocities are to be measured, and
where M. Eqs can be written in their usual form.
However, the famous Michelson-Morley experiment
(1887) did not detect any motion of the Earth
with respect to the ether.
What are the options? At least one of the
following statements must be wrong (a) the
principle of relativity applies to both
mechanical and e.-m. phenomena (b) M. Eqs are
correct (c) G. Tr. are correct.
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Einsteins Principle of Relativity
Einstein (1905) assumed that (a) and (b) are
correct, and put forward the following
Einstein's Principle of Relativity (the first
postulate of the Special Theory of Relativity)
The laws of physics are the same (covariant) in
all IRFs.
Covariance is less restrictive than invariance
Let AB. If, under RF transformation, both A and
B are transformed into A and B, but still
AB, than the law is covariant.
One of the consequence of Einsteins Principle of
Relativity (being applied to Maxwells
Equations) the speed of light in vacuum is the
same in all IRFs and doesn't depend on the motion
of the source of light or an observer (in line
with the experimental evidence that the ether
does not exist).
However, this applies to all (not necessarily
e.-m.) phenomena. Thus,
The second postulate The speed of light in
vacuum is the same for all inertial observers,
regardless of the motion of the source.
Thus, Maxwells Equations agree with Einsteins
Principle of Relativity. Conclusion Galilean
Transformations must go. The idea of universal
and absolute time is wrong ! One has to come up
with correct transformations that work for
both mechanical and e.-m. phenomena (any speed up
to c). Consequently, the laws of mechanics have
to be modified to be covariant under new
(correct) transformations.
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Lorentz Transformations
The class of transformations that maintain the
covariance of Maxwells equations was already
derived by Lorentz by that time (1904) (though
Lorentz suggested that the ether wind physically
compresses all matter in just the right way to
conceal the variations of c in Michelson-Morley
experiment, he still believed in absolute time).
L.Tr. for a 1D motion along the x axis
or, if we introduce
Linearity of L.Tr. reflects the fact that the
space is uniform and isotropic, the time
uniform.
Note the symmetry btw the transformations of
space and time coordinates.
For small Vltltc (?ltlt1, ?1) L.Tr. are reduced to
G.Tr.
Minkowski Space of itself, and time of itself
will sink into mere shadows, and only a kind of
union between them will survive.
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Example
Lorentz transformations (similar to G.Tr.) show
how to calculate the coordinates (t,x,y,z)
of some physical event in one IRF if one knows
the coordinates of the same event in another IRF.
A flash of light occurs at x 1m, y 1m, z
1m, and ct 1m (so t 3.3?10-9s). Locate this
event in the primed RF, which moves at V/c0.6 to
the right. (At t0, the origins of these RFs
coincide).
V/c ? ? 0.6, ? 1.25, so ?? 0.75
Lorentz Tr.
Galilean Tr.
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The relativity of simultaneity
One of the striking consequences of Einsteins
postulates in the relativity of simultaneity.
(a)
(b)
(a)
(b)
K
K
K the rest reference frame of the car. By
reckoning of an observer in this IRF, light from
the bulb in the middle of the car reaches the car
ends events (a) and (b) simultaneously.
Second postulate the speed of light is the same
in all IRFs!
K an observer on the ground. By this
observers reckoning these two events are not
simultaneous for as the light travels from the
bulb, the train itself moves forward, and thus,
event (a) happens before event (b).
Two events that are simultaneous in one IRF are
not, in general, simultaneous in another.
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Time dilation
The proper time (interval), ?t0 the time
interval between two events occurring at the same
position as measured by a clock at rest (with
respect to these two events).
Light-clock
We want to know the time interval between the
same two events occurring at the same position as
measured by (synchronized) clocks in a moving
reference frame
K
K
mirror
mirror
photon
photon
?
?
?
Second postulate the speed of light is the same
in all IRFs!
The time interval measured in the moving system
K is greater than the time interval measured in
system K where these two events occur at the same
place (the proper time is the minimum time
interval).
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Time dilation (contd)
To observe this effect, the relative speed of the
reference frames should be large. For the fastest
spacecraft, the speed is 10-4c, and the effect
is of an order of 10-8
Of course, the same results stems directly from
L.Tr.
Galilean Tr.
sparkler lit
sparkler goes out
Proper time interval
K0
K0
?
?
Comment Its easier to write L.Tr. for the
proper time interval in the right-hand side
V
K
K
?
but
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Time dilation (contd)
The effect of time dilation is symmetric by
the earths observer reckoning, all processes in
a rocket are slowed down, and vice versa, by the
rockets observer reckoning, all processes on the
Earth are slowed down. There is no
contradiction however the observers measure
different things!
time dilation on Earth
time dilation on the rocket
rocket clock
rocket clock A
rocket clock B
?
?
?
?
?
?
ground clock A
ground clock B
ground clock
The ground observer compares two ground clocks
with one rocket clock.
The rocket observer compares two rocket clocks
with one ground clock.
Clocks that are properly synchronized in one
system will not be synchronized when observed
from another system.
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Twin Paradox
The symmetry will be broken if we consider the
flight back and forth to the Earth. The RF of the
rocket going back and forth is a non-inertial RF!
Non-inertial RFs are considered in the General
Theory of Relativity (Einstein, 1916).
Interestingly, the result of the GTR and that of
a (wrong!) application of the STR coincide (e.g.,
one can consider a circular orbit with a constant
speed (but not velocity, there is a constant
acceleration that rotates the vector of
velocity). As a result, a twin traveling on a
rocket on arrival will be younger than his twin
on Earth (twin paradox).
IVIconst
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Problem
The lifetime of the unstable particle in its rest
frame is 2?10-8s. The particles are generated in
the center of a sphere of a radius of 8 m.
Calculate the minimum speed (in the lab frame)
that the particle should have in order to reach
the spheres surface before it decays.
requirement for the min. speed
R8m
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Example decay of cosmic-ray muons
Muon an electrically charged unstable
elementary particle with a rest energy 207
times greater than the rest energy of an
electron. The muon has an average half-life of
2.2 ?10-6 s.
Muons are created at high altitudes due to
collisions of fast cosmic-ray particles (mostly
protons) with atoms in the Earth atmosphere.
(Most cosmic rays are generated in our galaxy,
primarily in supernova explosions)
N0 the number of muons generated at high altitude
In the muons rest frame
By ignoring relativistic effects (wrong!), we get
the decay length
20 km
altitude
In fact, the decay length is much greater, the
muons can be detected at the sea level!
Because of the time dilation, in the RF of the
lab observer the muons lifetime is
N the number of muons measured in the sea-level
lab
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Homework Assignment
HW 1, due 09/10/2009
Problems 4,5,9,10,11,12,14,18,20,21
Please make sure that all required reference
frames are clearly identified!