Lecture 3' Relativistic Dynamics

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Lecture 3. Relativistic Dynamics

• Outline
• Relativistic Momentum
• Relativistic Kinetic Energy
• Total Energy
• Momentum and Energy in Relativistic Mechanics

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Comment on 3- and 4-vectors
The length of a 3 vector is invariant under
G.Tr.
Is there a combination of (x,y,z,t1) which
remains invariant under L.Tr.? Indeed, such a
combination exists
This is the square of the distance between two
events (ict1x1,y1,z1) and (,ict2,x2,y2,z2) in the
4-dimensional space.
This quantity (a.k.a. the interval) is invariant
under L.Tr. (please show this explicitly at home
using L.Tr.), for two arbitrary events it might
acquire any (zero, positive or negative) value,
unlike the distance in 3D space (Appendix II).
L.Tr. correspond to a rotation of a 4-dimensional
RF through a fixed angle these rotations
preserve the length of 4-vectors.
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Comment on 3- and 4-vectors (contd)
Thus, if one can express some physical law in a
form 4-vector A4-vector B, this would
guarantee the invariance of this law under L.Tr!
Def. A 4-vector is any set of four components
which transforms in the same manner as the
space-time vector under L.Tr.
Examples of 4-vectors
- the vector and scalar potentials in
Electrodynamics
- the momentum and total energy in relativistic
mechanics
Note that both force and acceleration are
3-vectors. Thus, one should not expect that the
2nd Law, being expressed in terms of force and
acceleration, remains invariant under Lorentz
Transformations. This is the reason why force
and acceleration are not popular in
relativistic mechanics. Well discuss
relativistic dynamics in terms of the momentum
and total energy.
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Relativistic Momentum
- the momentum of a particle in classical
mechanics, m is invariant (does not depend on the
velocity)
- expressed in terms of 3-vectors, invariant
under G.Tr. (but not L.Tr.!)
Newtons 2nd Law
Relativistic form of the momentum (introduced by
Einstein)
- definition of the momentum in relativistic
mechanics
Example Calculate the momentum of an electron
moving with a speed of 0.98c.
By ignoring relativistic effects, one would get
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Relativistic Kinetic Energy
2 v vf
Lets calculate the kin. energy gained by an
accelerated particle
1 v 0
- kinetic energy of a particle of the mass m
moving with speed v
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Total and Rest Energies
We expect this result to be reduced to the
classical KE at low speed
?
Lets rewrite the expression for K in the form
the rest energy
the total energy
Limit of small speed
- we must use Relativistic Mechanics when K and
E0 become of the same order of magnitude.
The energy and momentum are conserved (the
consequence of uniform and isotropic space).
For an isolated system of particles
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Electron-Volt convenient unit of energy
In relativistic mechanics, most of the time we
deal with tiny particles like an electron or
proton (after all, its hard to accelerate a
macroscopic body to vc). In this case, the most
convenient unit of energy is an electron-volt,
the kinetic energy acquired by an electron
accelerated through a potential difference of
1Volt.
For example, the rest energy of an electron
Thus, when the electrons are accelerated across a
pot. difference 10kV in a TV tube, they still
can be considered as non-relativistic particles
(Kltltm0c2)
The rest energy of a proton
The rest energy of a neutron
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Problem
An electron whose speed relative to an observer
in a lab RF is 0.8c is also being studied by an
observer moving in the same direction as the
electron at a speed of 0.5c relative to the lab
RF. What is the kinetic energy (in MeV) of the
electron to each observer?
The electron speed v as seen by the moving
observer K
K
K
In the lab IRF K
In the moving IRF K
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Energy and Mass in Cl. M. and Rel. M.
Caution some textbooks use the
velocity-dependent mass m?m0 and the rest mass
m0.
Einstein It is not good to introduce the
concept of the mass m?m0 of a moving body for
which no clear definition can be given. It is
better to introduce no other mass concept than
the rest mass m0. Instead of introducing m it
is better to mention the expression for the
momentum and energy of a body in
motion. Relativistic mass m?m0 just
another name for the energy (Occams Resor
Entities must not be multiplied beyond
necessity). See also Okuns paper on our Webpage.
Important for a system of many particles, the
mass M includes the potential energy of all
interactions between the particles.
The potential energy U for the particles that
attract each other is negative (for repelling
particles positive). Thus, for the stability of
a body, its mass should be smaller than the sum
of masses of all particles that constitute the
body.
Mass and energy are different aspects of the same
thing, they become interchangeable matter can
be created or destroyed, but when it happens, an
equivalent amount of energy vanishes or comes
into being.
Classical mechanics conservation of mass and
energy Relativistic mechanics conservation of
energy
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Mass Defect
For the stability of a body, its mass defect
should be negative.
Mass defect
the mass of a composite body (system)
the sum of masses of its constituents
Well consider the binding energy in detail when
we consider nuclei and nuclear reactions.
Binding energy
Why was it difficult to notice in Cl. M.? Because
in all processes of chemical transformations (the
most violent processes of the pre-20-century
physics) the energy release is tiny in comparison
with the rest energy of reactants.
Example dynamite explosion
When 1kg of the TNT explodes, the energy release
is 5.4MJ. At the same time, the rest energy of
1kg is
- its very difficult to detect this mass change
The mass loss for the Sun. The power of solar
radiation P 4 1026 J/s (the power per 1m2 on
the Earths surface (1400 W/m2) being
multiplied by the area of a sphere with radius
1.51011 m (the Sun-Earth distance) .
the mass loss per one second
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Examples
An elementary particle (e.g. a free electron)
cannot absorb/emit a photon. (Hint use the
reference frame in which the electron was at rest
before the collision/after absorption).
A composite particle can decay into two (or more)
fragments if the mass of all fragments is less
than the particle mass.
Decay of a neutron a free neutron is unstable
(the lifetime 15 min). It decays into a proton,
an electron, and an electron anti-neutrino
The masses involved
- otherwise, the neutron would be stable and most
of the protons and electrons in the early
Universe would have combined to form neutrons,
leaving little hydrogen to fuel the stars.
However, neutrons are stable in nuclei
- the energy conservation prevents the neutron
from decaying in nuclei.
Without neutrons we would not have the heavier
elements needed for building complex systems such
as life.
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An important relationship between E and p
this combination of E and p does not depend on
the IRF!
(In fact, it is the length2 of a 4-vector formed
by the components of p and i(E/c))
Def. A 4-vector is any set of four components
which transforms in the same manner as the
space-time vector under L.Tr.
Examples of 4-vectors
- the vector and scalar potentials in
Electrodynamics
- the momentum and total energy in relativistic
mechanics
- the wave vector and angular frequency of a
plane harmonic wave
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Ultra-relativistic and massless particles
In the ultra-relativistic case (Kgtgtm0c2)
This equation works for the particles with zero
rest mass (photons). They must move with the
speed of light vc (otherwise, both p and E are
0)
Massless particles the photon (carrier of the
electromagnetic interaction), the gluon (carrier
of the strong interaction, never observed as a
free particle), and, perhaps, the graviton
(carrier of gravitational interaction, remains to
be discovered).
External forces can bend the trajectories of
massless particles (e.g., photons in
gravitational fields), but cannot accelerate
(decelerate) them (v is always c).
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Problems
1. What is the momentum of an electron with K
mc2?
2. How fast is a proton traveling if its kinetic
energy is 2/3 of its total energy?
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Problem
An electron initially moving with momentum pmc
is passed through a retarding potential
difference of V volts which slows it down it
ends up with its final momentum being mc/2. (a)
Calculate V in volts. (b) What would V have to be
in order to bring the electron to rest?
pmc
(a)
pm0c/2
Thus, the retarding potential difference
(b)
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Problem (Relativistic Dynamics)
Beiser 38. A moving electron collides with a
stationary electron and an electron-positron pair
comes into being as a result. When all four
particles have the same velocity after the
collision, the kinetic energy required for this
process is a minimum. Use a relativistic
calculation to show that Kmin6mc2, where m is
the electron mass.
energy conservation
after
before
momentum conservation
In the center-of-mass RF
relative speed
before
after
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Problem (Relativistic Dynamics)
Find the minimum energy a proton must have to
initiate the reaction
(production of anti-protons (Berkeley Bevatron
1954) the energy and, thus, the cost of the
accelerator, must be minimized)
The minimum energy when the products of
reaction are at rest in the center of mass
reference frame all the incoming energy is
transformed into the rest energy.
The invariant has the same value in all RFs!
Lets use the lab RF
For colliding beam accelerators (e.g., LHC),
the center-of-mass frame and the lab frame are
the same (each proton should have E2mc2)
Homework 2 Beiser Ch. 1, Problems 29, 30, 33,
34, 41, 42, 45, 49, 54, 56.