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Special relativity, continued:
Spacetime is 4-dimensional: three space and
one time coordinate. Fancy SR deals with “fourvectors”: (x, y, z, ct) and transforms them with
a fancy variant of a rotation matrix, called the
Lorentz Transformation.
For motions along x, the Ltz. Txf. mixes up x and
ct
Momentum and energy are similarly related and
can be put into four vectors: (px, py, pz, and E/c)
For a moving particle, with velocity along the x
axis relative to an observer (“the lab”), the
Lorentz transformation from the particle’s restframe into the lab mixes up px and E in a
precise way, by a kind of “rotation” in the x-t or
px-E plane.
Also, the energy and momentum of a particle
grow without bound even though the speed of
the particle can never get all the way to c
The factor that governs is the same sqrt factor
that governs time dilation and length-contraction
Momentum
• According to Newton’s mechanics, a particle of
mass m0 moving with speed v has a momentum
given by
p = m0 v
• Conservation of momentum is one of the
fundamental conservation rules in physics and is
believed to be satisfied by all the laws of physics,
including the theory of special relativity
• The momentum of a single particle can also be
written as
Dx
p = m0
Dt
Section 27.7
Relativistic Momentum
• Einstein showed that you should use the proper time
to calculate momentum, where the Δt in the previous
slide is in the “LAB” , or observer’s, frame
• The result from special relativity is
p = m0
m0v
Dx
Dx
= m0
=
Dt0
Dt 1 - v 2 c 2
1- v 2 c 2
• This is the correct expression for momentum and
applies even for a particle moving at high speed,
close to the speed of light
Section 27.7
Newton’s vs. Relativistic Momentum
• As v approaches the
“Brick wall”
speed of light, the
relativistic result is very
different than Newton’s
• There is no limit to how
large the momentum
can be
• However, even when
the momentum is very
large, the particle’s
speed never quite
reaches the speed of
light
Inaccessible
region
Section 27.7
Mass - Energy
• The total energy of a particle is also governed by the
•
•
•
•
•
•
•
“sqrt” factor, and is
E = moc2/√(1-v2/c2)
This E includes both Kinetic Energy and Rest Mass
Energy, and the total grows without bound as vc
In the low-velocity Newton approximation, v/c << 1
E = mc2 + ½ mv2 plus neglect O{(v/c)4 and higher}
In general, E grows without bound (just like
momentum) while v approaches but never quite
reaches c
v=c gives infinite E --- ain’t gonna happen!
v>c gives imaginary number --- nope!
Section 27.8
Mass
• Newton’s second law gives mass, m0, as the
constant of proportionality that relates acceleration
and force
• This works very well as long as the object’s speed is
small compared with the speed of light
• At high speeds, though, Newton’s second law
breaks down
Section 27.8
Relativistic Mass
• When the postulates of special relativity are applied
to Newton’s second law, the mass needs to be
replaced with a relativistic factor
m0 ®
m0
(1 - v
2
c
2
)½
3
2
• The total energy is just this mass times c2
• At low speeds, the relativistic term approaches m0
and the two acceleration equations will be the same
• When v ≈ c, the acceleration is very small even when
the force is very large --- that’s because v can
change only a tiny bit when it’s near c
Section 27.8
Rest Mass
• When the speed of the mass is close to the speed of
light, the particle responds to a force as if it had a
mass larger than m0
• Specifically, with momentum, at high speeds the
particle responds to impulses and forces as if its mass
were larger than m0
• Rest mass is denoted by m0
• This is the mass measured by an observer who is
moving very slowly relative to the particle
• The best way to describe the mass of a particle is
through its rest mass
Section 27.8
Mass and Energy
• Relativistic effects need to be taken into account
when dealing with energy at high speeds
• From special relativity and work-energy,
KE =
m0c 2
1- v 2 c 2
- m0c 2
• For v << c, this gives KE ≈ ½ m0 v2 which is the
expression for kinetic energy from Newton’s results
• And of course, the TOTAL relativistic energy of the
particle is just the first term in the above expression,
without subtracting the rest energy
Section 27.9
Kinetic Energy and Speed
• For small velocities, KE
is given by Newton’s
results
• As v approaches c, the
relativistic result has a
different behavior than
Newtonian
• Although the KE can be
made very large, the
particle’s speed never
quite reaches the speed
of light
Section 27.9
Rest and Total Energies
• The kinetic energy is just the Total Energy minus the
moc2 Rest Energy of the particle
• The initial energy, m0c2, is a constant called the rest
energy of the particle
• A particle will have this much energy even when it is at
rest
• The total energy of the particle is the sum of the
kinetic energies and the rest energy
Section 27.9
Mass as Energy
• The rest energy equation implies that mass is a
•
•
•
•
form of energy
It is possible in principle to convert an amount of
energy (m0c2) into a particle of mass m0
It is possible in principle for a particle of mass m0 to
convert into an amount of energy (m0c2)
The principle of conservation of energy must be
extended to include this type of energy
Whether or not a particle can disappear into energy
or be created from energy depends on certain
attributes carried by a given particle, like electric
charge, lepton number, baryon number, spin, etc.
Section 27.9
Mass as Energy LECTURE QUIZ
• What is the TOTAL energy of a proton (rest mass
energy is 938 MeV) seen moving at v = 0.999999 c
• If you don’t like calculators,
m0c 2
2
KE
=
m
c
0
2
2
• (1-ε)2≈1- 2ε for small ε
1- v c
.
• A) 470 GeV
• B) 550 GeV
• C) 605 GeV
• D) 663 GeV
• E) 890 GeV
v/c = 0.999999
Square it
Subtract that from 1.0
Take the square root of
the resulting small
number
Divide 0.938 GeV by that
square root.
Section 27.9
Whether or not a particle can disappear into energy
or be created from energy depends on certain
attributes carried by a given particle, like electric
charge, lepton number, baryon number, spin, etc.
Some of these quantum numbers are “conserved” –
what you end up with must equal what you start out
with.
Example: γ + γ  e+ + ePhotons (gammas) are electrically neutral. The
final state must therefore be neutral, hence we
need to produce the electron plus its antiparticle,
the positron, to conserve electric charge.
Protons carry “baryon number” (bary means
heavy). Baryon number is apparently conserved,
since protons have not yet been seen to decay
spontaneously. Proton mean life, measured, is
,
>1033 years. That’s lucky for us!
Baryon number must have been violated at some
point in the early Big Bang universe, since theory
has it that equal numbers of protons and
antiprotons must have been originally created.
But the universe is almost entirely baryons
(protons and neutrons) today. We still have no
exact understanding of what went on right after BB
Speed of Light as a Speed Limit
• Several results of special relativity suggest that
speeds greater than the speed of light are not
possible
2
2
• The factor 1  v c that appears in time dilation and
length contraction is imaginary if v > c
• The relativistic momentum of a particle becomes
infinite as v → c
• This suggests that an infinite force or impulse would
be needed for a particle to reach the speed of light
Section 27.9
Speed Limit, cont.
• The total energy of a particle becomes infinite as v
→c
• This suggests that an infinite amount of mechanical
work is required to accelerate a particle to the speed
of light
• The idea that c is a “speed limit” is not one of the
postulates of special relativity, it is a consequence
• Combining the two postulates of special relativity
leads to the conclusion that it is not possible for a
particle to travel faster than the speed of light
• Not only that, massive particles always have v<c
• Only massless particles, like photons, have v=c
Section 27.9
Mass-Energy Conversions
• Conversion of mass into energy is important in
nuclear reactions, but also occurs in other cases
• A chemical reaction occurs when a hydrogen atom is
dissociated
• The mass of a hydrogen atom must be less than the
sum of the masses of an electron and proton
• The electron’s energy is lower by 13.6 eV when it’s
bound in the atom
• Mass is not conserved when a hydrogen atom
dissociates
• Δm0 = 2.4 x 10-35 kg
• This is much less than the mass of a proton and can
generally be ignored
Section 27.9
Conservation Principles
• Conservation of mass
• Mass is a conserved quantity in Newton’s mechanics
•
The total mass of a closed system cannot change
• Special relativity indicates that mass is not conserved
•
The principle of conservation of energy must be extended to
include mass
• Momentum is conserved in collisions – BUT to be
exact
• Use the relativistic expression for momentum
• Electric charge (and a lot of other quantum numbers)
are conserved
• For example, it is not possible to create or annihilate
charges if the total charge changes
Section 27.9
General Relativity
• A noninertial reference frame is one that has a
nonzero acceleration
• Physics in noninertial frames is describe by general
relativity
• General relativity is based on a postulate known as
the equivalence principle
• The equivalence principle states the effects of a
uniform gravitational field are identical to motion with
constant acceleration
Section 27.10
Equivalence Example
• Case A Ted stands in an
elevator at rest on Earth
• He feels the normal force
exerted by the floor on his
feet
• He concludes that he is in
a gravitational field
• Case B The elevator is not
in a gravitational field (out
in space) but has an
acceleration of 1 g
• Since there is an
acceleration, Ted feels the
same force on his feet
Section 27.10
Equivalence Principle, cont.
• According to the equivalence principle, there is no
local way for Ted to tell the difference between the
effects of the gravitational field and the accelerated
motion (this is not strictly true if Ted can measure
accelerations over a finite volume, and see the
GRADIENT in gravity due to the Earth’s 1/r2 factor)
• The equivalence principle has the following
consequences
• Inertial mass and gravitational mass are equivalent
• ma = GmM/r2 left m is inertial, right m is gravitational
• Light can be deflected by gravity
Section 27.10
Light and Gravity
• The light beam travels through the elevator while the elevator
is in distant space
• When a = 0, Ted sees the light beam travel in a straight line (A)
• When a ≠ 0, the light beam travels in a curved line relative to
the elevator (B)
• In a gravitational field, case (C) the light beam also curves
Section 27.10
Deflection of Light by Sun
• The gravitational field of the Sun should deflect light
from a star
• Easiest to see during a solar eclipse
• Experiments (Eddington) in 1919 verified that light
passing near the Sun during an eclipse was deflected
by the predicted amount
Section 27.10
Black Holes
• Black holes contain so
much mass that light
inside the “event horizon”
is not able to escape from
the gravitational attraction
• A black hole can be
“seen” by its effect on the
motion of nearby objects
• Stars near a black hole
move in curved
trajectories and so the
mass and location of the
black hole can be
determined
Section 27.10
Gravitational Lensing
• If the black hole is between the star and the Earth, light
from the star can pass by either side of the black hole and
still be bent by gravity and reach the Earth
• The black hole acts as a gravitational lens
• Light from a single star can produce multiple images
• Analysis of the images can be used to deduce the mass
Section 27.10
of the black hole
Gravitational Lensing
• Gravitational lensing by Galaxies, and “microlensing” by
Stars, are powerful tools used by astronomers to map out
the mass distribution of the universe.
• This is one of the ways we have determined that the
visible mass (atoms in stars that shine) is less than the
total mass in the universe. About 4.5% of the universe is
baryonic matter (atomic nuclei). About 25% is the
mysterious “Dark Matter” which also causes stars to orbit
faster around the centers of galaxies than can be
accounted for by the mass of shining stars.
• We have no idea, yet, what the dark matter actually is.
Section 27.10
Relativity and Electromagnetism
• Alice is at rest with the charged line and the point
charge
• Ted sees the line of charge and the point charge in
motion
• The moving charged line acts as a current
Section 27.11
Relativity and EM, cont.
• Ted says that there is an electric force and a
magnetic force on the particle
• Alice says there is only an electric force
• Both are correct
• They will agree on the total force acting on the
particle
• The larger electric force (due to the denser line
charge) observed by Ted due to his motion is canceled
by the magnetic force produced (line current B, also
electron moves parallel to the line so v x B)
• Maxwell’s equations were already consistent with
special relativity
Section 27.11
Importance of Relativity
• The relation between mass and energy and the
possibility that mass can be converted to energy
(and energy to mass) mean that mass is not
conserved
• Instead we have a more general view of energy and its
conservation
• The three absolute conservation principles in physics
are
•
•
•
Conservation of energy
Conservation of momentum (and of angular momentum)
Conservation of charge
• It is believed that all the laws of physics must obey
these three conservation principles
Section 27.12
Importance of Relativity, cont.
• The rest energy of a particle is huge
• This has important consequences for the amount of
energy available in processes such as nuclear
reactions
• Relativity changes our notion of space, time, and
simultaneity
• Our everyday intuition breaks down when applied to
special relativity
Section 27.12
Importance of Relativity, final
• Relativity plays a key role in understanding how the
universe was formed and how it is evolving
• Black holes can’t be understood without relativity
• Relativity shows that Newton’s mechanics is not an
exact description of the physical world
• Instead, Newton’s laws are only an approximation that
works very well in some cases, but not in others
• We shouldn’t discard Newton’s mechanics, but
rather, understand its limits
Section 27.12