Download Relativity

H1 Introduction to Relativity
A place where a coordinate system can be used to
make measurements of space and time– x,y,z axes and a clock
H.1.1 Describe what is meant by a frame of reference.
H.1.2 Describe what is meant by a Galilean transformation.
A Galilean transformation links displacement,
velocity or time between frames of reference which have relative motion.
H.1.3 Solve problems involving relative velocities, using the Galilean transformation equations.
𝑢𝑥′ = velocity of object in x-direction as measured in a moving frame
ux = velocity of object in x-direction as measured in a stationary frame
v = velocity of frame 2 in x-direction as measured in a stationary frame
1. ux = 𝒖′𝒙 + v
ux = 𝑢𝑥′ + v
2. 𝒙 = 𝒙′ + 𝒗𝒕
(multiply the terms in the above transformation by time)
Eg. Suppose the man walks forward 5 metres in 4 seconds on the back of the truck. What is his
displacement in the frame of reference of the ground based observer?
3. If times are involved t = t ' + ∆𝒕 (∆𝑡 is the time difference between the starting of the watches)
This was fine until in the 1860s Maxwell combined the theories of wave motion and electromagnetism. (Maxwell's
four equations are studied in university physics). According to this the speed of light in vacuum would only depend on
certain electric and magnetic constants (the permittivity and permeability), namely:
00 = 1/c2
So the speed of light depends only on these constants, not on the speed of the observer!!
So the Galilean transformations aren't needed!!
H2 Concepts and postulates of special relativity
H.2.1 Describe what is meant by an inertial frame of reference.
a frame of reference in which Newton’s laws of motion are valid, (in other words if there are no unbalanced forces
an object will remain at rest or move at a constant velocity) ... so the frame is – non-accelerating
H.2.2 State the two postulates of the special theory of relativity.
1. The laws of physics are the same for all inertial observers.
Consequence: There is no preferred frame of reference.
2. The speed of light in a vacuum is the same for all inertial observers.
Consequence: The speed of light is independent of the speed
of its source or the speed of any observer.
H.2.3 Discuss the concept of simultaneity. Two events occurring at different points in space and which are
simultaneous for one observer cannot be simultaneous for another observer in a different frame of reference.
H3 Relativistic Kinematics
H.3.1 Describe the concept of a light clock.
Only a very simple description is required here. For example, a beam of light
reflected between two parallel mirrors may be used to measure time.
H.3.2 Define proper time interval. "the
time between events as measured
in the frame where the events take place"
H.3.3 Derive the time dilation formula. Students should be able to construct a simple derivation of the time dilation
formula based on the concept of the light clock and the postulates of relativity. 𝒕 = 𝜸𝒕𝟎
H.3.4 Sketch and annotate a graph showing the variation with
relative velocity of the Lorentz factor.
𝜸=
𝟏
𝟐
√𝟏− 𝒗𝟐
𝒄
H.3.5 Solve problems involving time dilation.
Length contraction
H.3.6 Define proper length
l0 = the length of an object measured in the frame where it is at rest
H.3.7 Describe the phenomenon of length contraction. The derivation of the length contraction formula is not required.
length contraction: according to a stationary observer, moving objects contract (shrink) in the
direction of motion but not in perpendicular directions.
H.3.8 Solve problems involving length contraction.
𝒍=
𝒍𝟎
𝜸
H4 Some consequences of Special Relativity
The twin paradox
H.4.1 Describe how the concept of time dilation leads to the “twin paradox”.
Since one of the twins makes an outward and return journey, this is no longer a symmetrical situation for the twins.
(They experience a physical acceleration, which changes their frame of reference.)
H.4.2 Discuss the Hafele–Keating experiment.
In 1971 Hafele and Keating made flights around the world in both directions, each circuit taking about three days.
They carried with them four cesium beam atomic clocks, accurate to within ± 10 -9 s. The researchers expected that the
relative motion of the clocks would produce a measurable time dilation effect (“moving clocks run slow”). In a frame
of reference at rest with respect to the center of the earth, the clock aboard the plane moving eastward, in the direction
of the earth's rotation, is moving faster than a clock that remains on the ground, while the clock aboard the plane
moving westward, against the earth's rotation, is moving slower.
When they returned, they compared
their clocks with a ground based clock.
The time intervals measured by the
clocks on the aircraft differed from
those measured by the ground based
clocks and provided confirmation of
the time dilation effects of relativity.
In this experiment, both time dilation
due to motion or kinematics
(special relativity) and time dilation
due to gravity (general relativity)
are significant and had
to be taken into account.
Velocity addition
H.4.3 Solve one-dimensional problems involving the relativistic addition of velocities.
𝑣𝑎𝑏 =
The derivation of the velocity addition formula is not required.
𝑣𝑎𝑐 + 𝑣𝑐𝑏
𝑣
𝑣
1+ 𝑎𝑐 2 𝑐𝑏
𝑐
Ex 1.Two bicyclists approach each other at a speed of 0.60c. What is their relative velocity of approach? (0.88c)
Ex 2. A missile is shot at the speed 0.3c from earth and hits a spaceship at 0.6 c relative to the ship. How fast is the
ship moving and in what direction?
v = (0.3c - 0.6c) / (1 + 0.3c x -0.6c/ c2) = - 0.3c / (1 - 0.18) = - 0.37c
Mass and energy
H.4.4 State the formula representing the equivalence of mass and energy.
H.4.5 Define rest mass.
E = mc2
Students should be aware that rest mass is an invariant quantity.
Students should be familiar with the unit 𝑀𝑒𝑉𝑐 −2 for mass.
H.4.6 Distinguish between the energy of a body at rest
E0 = m0c2
m = m0
and its total energy when moving.
H.4.7 Explain why no object can ever attain the speed of light in a vacuum.
The idea of an increasing mass fits well together with the fact that no matter how much work or
energy is used to accelerate an object, it will never reach or pass the speed of light.
H.4.8 Determine the total energy of an accelerated particle.
Be able to calculate the total energy of an electron after acceleration through a known potential difference.
E = m0c2 + Ve
H5 Evidence to support special relativity
H.5.1 Discuss muon decay as experimental evidence to support special relativity.
Muons can be produced by collisions of cosmic radiation
with atoms in upper atmosphere.
Due to their unstable nature (and short half-life) they should only
survive for a short time before decaying – shouldn’t reach surface of earth.
But, measurements of number of muons at top of mountain are
approximately same as at bottom of mountain.
Why do so many reach the ground?
Because when radioactive particles (e.g. muons) are accelerated to a high speed, they decay slower than at
rest and therefore travel further than they should before decay.
From Earth frame of reference: time runs slowly for muon so it has time to reach ground before decaying
From muon’s frame of reference: height of atmosphere contracts so has very little distance to travel
EXAMPLE: A muon having a lifetime of 2.2 μs as measured in its own frame of reference is created in the upper
atmosphere and travels toward Earth at a speed of 0.99c.
1. How far can a muon travel before it decays, as measured in its own frame of reference?
2. What is the lifetime of the muon, as measured from the Earth?
3. How far will the muon travel through the atmosphere, as measured from the Earth?
H.5.2 Solve problems involving the muon decay experiment.
H.5.3 Outline the Michelson–Morley experiment.
Outline the principles of the Michelson interferometer
using a simple sketch of the apparatus.
http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/mmexpt6.htm
In the 1800s it was thought that light waves travelled through an invisible substance, the ether (not the chemical).
Since the Earth, in its rotation around the Sun, should be moving through this ether at a high speed, it should be
possible to detect a difference in the speed of light depending on whether the Earth was moving with or against the
"ether wind" in space.
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monochromatic light is sent from a source and split with a "half-silvered" (the reflecting silver layer at the
"back" side of the glass, to the right in the picture) mirror which lets some light through and reflects some, the
beam splitter (A)
some light (I) goes to an ordinary, movable mirror (D) and on towards an observer (O). (Some light is lost by
reflection back to the source).
some light (II) goes to another ordinary, fixed mirror (C) and back to the beam splitter, from where it is
reflected to the observer O. (some is again lost, this time going through the beam splitter and back into the
source).
a compensator plate (B) means that both I and II pass through the same total thickness of glass
at the observer, there will be some interference between the then parallel light rays I and II, since their
travelled distances can be made different
depending on whether the ether wind is parallel to the AD or the AC dimension, the interference pattern
should be slightly different
by rotating the apparatus 90o (or waiting 6 hours for the Earth to rotate this much) a change in the interference
pattern should be observed
H.5.4 Discuss the result of the Michelson–Morley experiment and its implication.
The result of the experiment was that no change in the interference pattern was observed under any circumstances.
So there is no ether, and the speed of light is the same regardless of what velocity or component of velocity it has
relative to us.
H.5.5 Outline an experiment that indicates that the speed of light in vacuum is independent of its source.
Students should be familiar with pion decay experiments involving the decay of a fast-moving
pion into two gamma-ray photons.
H6 Relativistic momentum and energy
H.6.1 Apply the relation for the relativistic momentum p = 𝛾 𝑚0 𝑢 of particles.
Students should be familiar with momentum expressed in the unit 𝑀𝑒𝑉𝑐 −1 .
H.6.2 Apply the formula 𝑬 = (𝜸 − 𝟏)𝒎𝟎 𝒄𝟐 for the kinetic energy of a particle.
Ex1. Find the kinetic energy of an electron accelerated to 0.90c? The rest mass of an electron is 0.51 MeV c-2.
Ex2. A proton (938 MeV c-2 ) is accelerated to a speed of 0.95c. Determine its energy, rest energy, and kinetic energy.
Ex3. An electron is accelerated through 2.0 x 106 V. Calculate its energy, kinetic energy, and speed.
H.6.3 Solve problems involving relativistic momentum and energy.
E2 = p2c2 + m02c4
NOT
Ek = p2/2m
Students should be able to calculate, for example, the kinetic energy, total energy, speed and momentum of an
accelerated particle and for particles produced in reactions.
Ex4. A proton is accelerated through a potential difference of 3.0 x 10 9 V.
a) Calculate the energy of the proton after its acceleration.
b) Calculate the final momentum of the proton.
Ex5. “Pair production” is a process by which antimatter pairs of particles are
produced from energy. This can happen when a high energy gamma ray photon
is in the vicinity of a heavy nucleus. For example, if a gamma photon is near a lead
atom, the reaction pictured at right might occur, where the photon creates an
electron-positron pair.
If the energy of the photon is 3.20 MeV, calculate the following quantities.
Neglect the recoil of the lead atom and assume the energy is shared equally
between the particles.)
  e  e
Find for the electron its
(a) total energy,
(b) kinetic energy,
(c) speed,
(d) mass and
(e) momentum.
H7 General relativity
The equivalence principle
H.7.1 Explain the difference between the terms gravitational mass and inertial mass.
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inertial mass: the quantity which determines what force is needed to cause a certain acceleration as m in
F = ma or something similar
gravitational mass: the quantity which determines what force of gravity will act on an object as m in
F = mg or more generally as m1 and m2 in F = Gm1m2/r2
H.7.2 Describe and discuss Einstein’s principle of equivalence.
"There is no way of distinguishing between any gravitational and inertial effects."
Students should be familiar with Einstein’s closed elevator “thought experiment”.
If you are in an elevator at rest or moving at a constant velocity up or down and drop a ball it falls to the floor - because of the force
of gravity, we would say. But if the same elevator was far out in space with no (or extremely little) force of gravity acting on it, but
a rocket engine accelerating the elevator at a = 9.81 ms-2 then the dropped ball would "fall" towards the floor in the same way; if the
acceleration is steady and there are no windows in the elevator, there is no way to distinguish between the two cases.
H.7.3 Deduce that the principle of equivalence predicts bending of light rays in a gravitational field.
A ray is sent from A to B on the opposite wall in an
accelerating ship (or elevator). Since the speed of
light is not infinitely high, B will be a little below A
(extremely little, since light moves fast).
But if the same ship stands at rest on Earth, the force
of gravity should bend the ray in the same way although the light has no mass, so no force of gravity acts on it.
H.7.4 Deduce that the principle of equivalence predicts that time slows down near a massive body.
If an object accelerates to high speed, then time dilation will occur. But the principle of equivalence tells us that
acceleration is indistinguishable from the effects of a strong gravitational field, so time dilation must occur near a
massive body.
H.7.5 Describe the concept of spacetime.
Spacetime: four dimensional coordinates used to describe
any event (three spatial dimensions and time).
Spacetime diagrams
This is basically like a regular time-displacement graph with
the axes reversed. Since the speed of light in vacuum is the maximum possible,
an object in the origin of the diagram can in no way affect something outside
the light cone.
Exercise: The spacetime path of a rocket departing earth is shown.
The rocket now sends a radio signal back to earth.
Show the spacetime path of the radio signal.
H.7.6 State that moving objects follow the shortest path between
two points in spacetime.
H.7.7 Explain gravitational attraction in terms of the warping of spacetime by matter.
Any mass warps (distorts) spacetime - the greater the mass, the greater the warping. Particles,
such as planets, moving in spacetime follow the shortest path. The path becomes more curved as
the object approaches the central mass. Following this curvature of spacetime is interpreted as a
force. " Mass tells space how to curve.... space tells mass how to move."
Black holes
H.7.8 Describe black holes. "A region of spacetime with extreme curvature
due to the presence of a mass".
H.7.9 Define the term Schwarzschild radius. The
distance from the centre
of the black hole to where the escape velocity is equal to the speed of light
Centre of a Black Hole (singularity) – the single point to which all mass would collapse
Surface of a Black Hole (event horizon) – where the escape speed is equal to c and within this
surface, mass has “disappeared” from the universe
(Once a body has collapsed within the Schwarzschild radius, it will continue collapsing until it reaches zero volume.
This is called a singularity, a point with infinite density.)
H.7.10 Calculate the Schwarzschild radius.
RSch =
𝟐𝐆𝐌
𝒄𝟐
Calculate the radius of a black hole that has the same mass as our Sun (m = 1.99 x 10 30 kg).
H.7.11 Solve problems involving time dilation close to a black hole.
Eg. A person who is a distance 3RS from the event horizon of a black
hole measures an event to last 4.0 s. Calculate how long the event
would appear to last for a person very far from the black hole.
Dt 
(2.9 km)
Dt 0
1
Gravitational red-shift
Rs
r
H.7.12 Describe the concept of gravitational red-shift.
"Students should be aware that gravitational red-shift is a prediction of the general theory of relativity."
Light rising upwards in a gravitational field becomes more red,
with a lower f and higher wavelength.
D𝐟
H.7.13 Solve problems involving frequency shifts in a uniform gravitational field.
𝐟
=
𝐠D𝐡
𝒄𝟐
Eg. A red laser (640 nm) is directed straight up from the ground to the top of a building 50 m high.
Show that the change in frequency of the light is 2.6 Hz.
H8 Evidence to support general relativity
H.8.1 Outline an experiment for the bending of EM waves by a massive object.
An outline of the principles used in Eddington’s measurements during the 1919 eclipse of the Sun is sufficient.
The positions of several stars were measured against a background of fixed stars. Six months later, those stars were
hidden behind the Sun due to Earth’s new position in its yearly revolution. It was predicted by
General Relativity that these stars should still be visible if the gravitational field of the Sun bent the light rays around it
and deflected the light rays toward Earth. However, these “hidden” stars would still not be visible due to the glare of
the Sun. But an expedition led by Sir Arthur Eddington sent to the island of Principe (off the west coast of Africa)
sought to measure the deflection of these light rays during a total eclipse of the Sun on May 29 1919 when the stars
would be briefly visible. He measured the new positions of the stars against the background of fixed stars and found
that they had apparently shifted position. This was experimental evidence that gravitational fields do deflect light rays.
H.8.2 Describe gravitational lensing.
Heavy objects can bend the light from a star
or galaxy behind them to produce multiple
images of them for an observer on earth.
H.8.3 Outline an experiment that provides evidence
for gravitational red-shift.
In the Pound-Rebka experiment (1959) gamma rays
(emitted from iron-57) moving from the lower to the upper end
of Harvard tower of the Jefferson laboratory were observed to
decrease their frequency. The distance was only 22.5 m,
so an isotope which emitted very high energy radiation was needed.
(Unless you have high frequenccy, ∆𝑓 will be too small to measure.)
In the Shapiro time delay effect, or gravitational time delay effect, radar signals passing near a massive object take
slightly longer to travel to a target and longer to return than it would if the mass of the object were not present.
Shapiro bounced radar beams off the surface of Venus and Mercury, and measured the round trip travel time. When
the Earth, Sun, and Venus are most favorably aligned, Shapiro showed that the expected time delay, due to the
presence of the Sun, of a radar signal traveling from the Earth to Venus and back, would be about 200 microseconds,
well within the limitations of 1960s era technology. The first tests, performed in 1966 and 1967 were successful,
matching the predicted amount of time delay. The experiments have been repeated many times since then, with
increasing accuracy.