1. General Physics (PHY 2140) Lecture 26 Modern Physics Relativity
Relativistic momentum, energy, …
General relativity
Chapter 26
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2. If you want to know your progress so far, please send me an request at
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3. Lightning Review Last lecture: Modern physics
Time dilation, length contraction
Review Problem: A planar electromagnetic wave is propagating through space. Its electric field vector is given by E = Eo cos(kz – wt) x, where x is a unit vector in the positive direction of Ox axis. Its magnetic field vector is
1. B = Bo cos(kz – wt) y
2. B = Bo cos(ky – wt) z
3. B = Bo cos(ky – wt) x
4. B = Bo cos(kz – wt) z
where y and z are unit vectors in the positive directions of Oy and Oz axes respectively.
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4. Reminder (for those who don’t read syllabus)
Reading Quizzes (bonus 5%):
It is important for you to come to class prepared, i.e. be familiar with the
material to be presented. To test your preparedness, a simple five-minute
quiz, testing your qualitative familiarity with the material to be discussed in
class, will be given at the beginning of some of the classes. No make-up
reading quizzes will be given.
There could be one today…
… but then again…
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5. Problem: relativistic pion
The average lifetime of a p meson in its own frame of reference (i.e., the proper lifetime) is 2.6 × 10–8 s. If the meson moves with a speed of 0.98c, what is
its mean lifetime as measured by an observer on Earth and
the average distance it travels before decaying as measured by an observer on Earth?
What distance would it travel if time dilation did not occur?
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6. (c) What distance would it travel if time dilation did not occur?
The average lifetime of a p meson in its own frame of reference (i.e., the proper lifetime) is 2.6 × 10–8 s. If the meson moves with a speed of 0.98c, what is
(a) its mean lifetime as measured by an observer on Earth and (b) the average distance it travels before decaying as measured by an observer on Earth?
(c) What distance would it travel if time dilation did not occur?
Recall that the time measured by observer on Earth will be longer then the proper time. Thus for the lifetime
Given:
v = 0.98 c
tp = 2.6 × 10–8 s
Find:
t = ?
d = ?
dn =?
Thus, at this speed it will travel
If special relativity were wrong, it would only fly about
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7. Problem: space flight
In 1963 when Mercury astronaut Gordon Cooper orbited Earth 22 times, the press stated that for each orbit he aged 2 millionths of a second less than if he had remained on Earth.
Assuming that he was 160 km above Earth in a circular orbit, determine the time difference between someone on Earth and the orbiting astronaut for the 22 orbits. You will need to use the approximation
for x << 1
(b) Did the press report accurate information? Explain.
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8. Length Contraction
The measured distance between two points depends on the frame of reference of the observer
The proper length, Lp, of an object is the length of the object measured by someone at rest relative to the object
The length of an object measured in a reference frame that is moving with respect to the object is always less than the proper length
This effect is known as length contraction
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9. Problem: weird cube
A box is cubical with sides of proper lengths L1 = L2 = L3= 2 m, when viewed in its own rest frame. If this block moves parallel to one of its edges with a speed of 0.80c past an observer,
what shape does it appear to have to this observer, and
what is the length of each side as measured by this observer?
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10. A box is cubical with sides of proper lengths L1 = L2 = L3= 2 m, when viewed in its own rest frame. If this block moves parallel to one of its edges with a speed of 0.80c past an observer, (a) what shape does it appear to have to this observer, and (b) what is the length of each side as measured by this observer?
Recall that only the length in the direction of motion is contracted, so
Given:
v = 0.8 c
Lip = 2.0 m
Find:
shape
Li=?
Thus, numerically,
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11. Relativistic Definitions
To properly describe the motion of particles within special relativity, Newton’s laws of motion and the definitions of momentum and energy need to be generalized
These generalized definitions reduce to the classical ones when the speed is much less than c
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12. 26.7 Relativistic Momentum
To account for conservation of momentum in all inertial frames, the definition must be modified
v is the speed of the particle, m is its mass as measured by an observer at rest with respect to the mass
When v << c, the denominator approaches 1 and so p approaches mv
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13. Problem: particle decay
An unstable particle at rest breaks up into two fragments of unequal mass. The mass of the lighter fragment is 2.50 × 10–28 kg, and that of the heavier fragment is 1.67 × 10–27 kg. If the lighter fragment has a speed of 0.893c after the breakup, what is the speed of the heavier fragment?
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14. For the heavier fragment,
An unstable particle at rest breaks up into two fragments of unequal mass. The mass of the lighter fragment is 2.50 × 10–28 kg, and that of the heavier fragment is 1.67 × 10–27 kg. If the lighter fragment has a speed of 0.893c after the breakup, what is the speed of the heavier fragment?
Momentum must be conserved, so the momenta of the two fragments must add to zero. Thus, their magnitudes must be equal, or
Given:
v1 = 0.8 c
m1=2.50×10–28 kg
m2=1.67×10–27 kg
Find:
v2 = ?
For the heavier fragment,
which reduces to
and yields
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15. 26.8 Relativistic Addition of Velocities
Galilean relative velocities cannot be applied to objects moving near the speed of light
Einstein’s modification is
The denominator is a correction based on length contraction and time dilation
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16. Problem: more spaceships…
A spaceship travels at 0.750c relative to Earth. If the spaceship fires a small rocket in the forward direction, how fast (relative to the ship) must it be fired for it to travel at 0.950c relative to Earth?
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17. A spaceship travels at 0. 750c relative to Earth
A spaceship travels at 0.750c relative to Earth. If the spaceship fires a small rocket in the forward direction, how fast (relative to the ship) must it be fired for it to travel at 0.950c relative to Earth?
Since vES = -VSE = velocity of Earth relative to ship, the relativistic velocity addition equation gives
Given:
vSE = c
vRE = c
Find:
vRS = ?
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18. 26.9 Relativistic Energy
The definition of kinetic energy requires modification in relativistic mechanics
KE = mc2 – mc2
The term mc2 is called the rest energy of the object and is independent of its speed
The term mc2 is the total energy, E, of the object and depends on its speed and its rest energy
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19. Relativistic Energy – Consequences
A particle has energy by virtue of its mass alone
A stationary particle with zero kinetic energy has an energy proportional to its inertial mass
E = mc2
The mass of a particle may be completely convertible to energy and pure energy may be converted to particles
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20. Energy and Relativistic Momentum
It is useful to have an expression relating total energy, E, to the relativistic momentum, p
E2 = p2c2 + (mc2)2
When the particle is at rest, p = 0 and E = mc2
Massless particles (m = 0) have E = pc
This is also used to express masses in energy units
mass of an electron = 9.11 x kg = MeV
Conversion: 1 u = MeV/c2
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21. QUICK QUIZ
A photon is reflected from a mirror. True or false: (a) Because a photon has a zero mass, it does not exert a force on the mirror. (b) Although the photon has energy, it cannot transfer any energy to the surface because it has zero mass. (c) The photon carries momentum, and when it reflects off the mirror, it undergoes a change in momentum and exerts a force on the mirror. (d) Although the photon carries momentum, its change in momentum is zero when it reflects from the mirror, so it cannot exert a force on the mirror.
False
True
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22. Example 1: Pair Production
An electron and a positron are produced and the photon disappears
A positron is the antiparticle of the electron, same mass but opposite charge
Energy, momentum, and charge must be conserved during the process
The minimum energy required is 2me = 1.04 MeV
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23. Example 2: Pair Annihilation
In pair annihilation, an electron-positron pair produces two photons
The inverse of pair production
It is impossible to create a single photon
Momentum must be conserved
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24. 26.10 General relativity: Mass – Inertial vs. Gravitational
Mass has a gravitational attraction for other masses
Mass has an inertial property that resists acceleration
Fi = mi a
The value of G was chosen to make the values of mg and mi equal
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25. Einstein’s Reasoning Concerning Mass
That mg and mi were directly proportional was evidence for a basic connection between them
No mechanical experiment could distinguish between the two
He extended the idea to no experiment of any type could distinguish the two masses
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26. Postulates of General Relativity
All laws of nature must have the same form for observers in any frame of reference, whether accelerated or not
In the vicinity of any given point, a gravitational field is equivalent to an accelerated frame of reference without a gravitational field
This is the principle of equivalence
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27. Implications of General Relativity
Gravitational mass and inertial mass are not just proportional, but completely equivalent
A clock in the presence of gravity runs more slowly than one where gravity is negligible
The frequencies of radiation emitted by atoms in a strong gravitational field are shifted to lower frequencies
This has been detected in the spectral lines emitted by atoms in massive stars
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28. More Implications of General Relativity
A gravitational field may be “transformed away” at any point if we choose an appropriate accelerated frame of reference – a freely falling frame
Einstein specified a certain quantity, the curvature of time-space, that describes the gravitational effect at every point
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29. Testing General Relativity
General Relativity predicts that a light ray passing near the Sun should be deflected by the curved space-time created by the Sun’s mass
The prediction was confirmed by astronomers during a total solar eclipse
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30. Black Holes
If the concentration of mass becomes great enough, a black hole is believed to be formed
In a black hole, the curvature of space-time is so great that, within a certain distance from its center, all light and matter become trapped
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