Download Special relativity - background The principle of relativity:

Special relativity - background
Just a few brief comments to set the stage:
• From the time of Aristotle to that of Galileo, with few exceptions people believed
that motion of any sort was caused by the application of a force.
some sense, "obvious":
This was, in
if you are sliding a book across a table, and you remove
your hand, the book begins to slow down and stop.
It was generally thought possible
to deduce the behavior of nature by engaging in a Socratic dialog or (in more
modern times) by reading the Bible.
The idea of confirming or refuting such deductions
by doing experiments dates from the Renaissance.
• Newton’s first law, that a body not subject to outside forces will move in a straight
line with constant velocity, was a major intellectual achievement.
presented as an obvious notion in textbooks, but this is wrong.
It’s often
It took 1500
years for someone to "notice" this.
• Newton’s second law states that the acceleration of a body is proportional to
the force acting on the body.
the application of a force.
of the body.
That is, a change in the state of motion requires
The constant of proportionality is called the mass
We write F = ma.
The principle of relativity:
Suppose we have an observer A who finds that Newton’s first law holds - that is, objects
that are at rest relative to the observer remain so and experience no acceleration.
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We call this theoretical being an inertial observer, and the rectilinear coordinate
system attached to this observer is called an inertial reference frame.
By Newton’s
laws, another observer B, moving with constant velocity v relative to A, is likewise
an inertial observer.
Their coordinates are related by
xB = xA − vtA
t B = tA
where the subscripts denote the coordinates used by A and B. The meaning of this is
that if observer A assigns the coordinates (tA , xA ) to some event E, then observer B
will assign the coordinates (tB , xB ) to this event.
Time is universal (i.e., independent
of the observer) in Newtonian physics, but we include it in the transformation laws
just to be clear about it.
Exercise: The two coordinate systems are related by a linear transformation called
a Galilean transformation.
Write down the matrix of this transformation.
Suppose
observer C moves with velocity w relative to observer B. What is the Galilean transformation
relating the inertial frames of C and A? How is this related to the transformations
relating A and B, and B and C?
Now suppose an object with mass m is in motion.
It’s described by some curve xA (tA )
in A’s frame, and by the corresponding curve xB (tB ) in B’s frame.
We have
xB (tB ) = xA (tA ) − vtA .
Making use of the fact that tA = tB , and taking derivatives, we find that
m
d2 xB
d2 xA
=m 2 .
2
dtB
dtA
That is, the forces and accelerations are the same in any inertial frame.
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(This is
not true, for instance, if one frame is inertial and the other is rotating.
Observers
in the rotating frame experience forces (Coriolis forces) even when they’re at rest
in the frame.
This is because they’re undergoing constant acceleration with respect
to any inertial frame.)
On the earth, the Coriolis force is responsible for the counter-
clockwise rotation of low pressure systems in the northern hemisphere.
The principle of relativity states that the laws of physics are the same in any inertial
frame of reference.
Provided that the relative velocities are not too large, this
principle holds for Newton’s laws of mechanics.
It’s an experimental fact.
The principle
of relativity is not some content-free statement like "all motion is relative".
The problem with electrodynamics
It took some time for people to realize, but by the end of the 19th century, some problems
had emerged with this (Newtonian) picture.
It was known that electromagnetic radiation
(infrared, visible, ultraviolet, etc) propagated through empty space at the speed of
light c = 300, 000km/sec.
According to Newton’s laws, if light propagates with a speed
c in one inertial frame, and another frame has a non-zero velocity relative to the
first, then the speed of propagation in this second frame should be diffferent from
c; in fact, it should depend on the direction of propagation, ranging from c+v to c−
v, where v = ||v||.
All the experimental attempts to observe this phenomenon failed.
problem:
Maxwell’s equations, which (correctly!)
And there was another
describe the electromagnetic field,
did not keep the same form under Galilean transformations - the electromagnetic field
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did not appear to obey the principle of relativity.
Lorentz found that the problem
could be cured by introducing some ad hoc modifications in the transformation laws,
but it was not clear why this worked or what it meant.
In 1905, Einstein published his famous paper "On the electrodynamics of moving bodies"
which resolved the problems by introducing what’s known, somewhat peculiarly, as the
special theory of relativity (the general theory of relativity is the modern theory
of gravitation, introduced by Einstein in 1916 - we’ll ignore gravitation here).
Einstein accepted the experimental evidence and converted it to a postulate:
of light is the same in all inertial reference frames.
the speed
We should observe that the
truth of this is confirmed millions of times a day, every day:
the magnetic field
strength needed to keep a charged particle of a given mass moving in a circular orbit
(in a particle accelerator) is computed using Einstein’s postulate.
If it were done
according to Newton, the particles would crash into the walls of the accelerator.
In adopting Einstein’s postulate asserting the constant speed of light, we must realize
that the Galilean transformations relating the coordinates of two inertial frames will
no longer hold.
In a Newtonian world, if I were travelling at a speed c/2 toward you
and shined a light toward you, the photons in the light beam would necessarily be travelling
at a speed of 3c/2.
According to Einstein (and billions of experiments!)
this is false.
You will see the photons coming toward you at the speed c.
When we change the laws relating different observers (the new transformation laws are
called Lorentz transformations), we’ll also have to modify the laws of physics in order
to satisfy the principle of relativity.
These modifications lead to some peculiar,
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unanticipated effects such as length contraction and time dilation.
out in the next couple of sections.
We’ll work this
Bear in mind though, that at speeds less than,
say, 1/1000 the speed of light, which is still pretty fast, the modifications to Newtonian
physics due to special relativity are vanishingly small and can safely be ignored.
We don’t need the theory of relativity to send a rocket to the moon.
need it if we contemplate leaving the solar system.
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We shall undoubtedly