Lecture II: Relativistic Kinematics

1
Lecture IIRelativistic Kinematics
2
Natural Units

• In these lectures, we are going to work in a
  world where all of the important constants of
  nature are
• Dimensionless that is, have no units
• Are equal to 1 so we can ignore them

Plancks ConstantSpeed of Light?
RT Boltzmanns Constant
3
Discussion of units

• From Gottfried Weisskopf
• Move from units based on
• Mass, Length, and Time
• To
• Mass, Action ( ), and Velocity ( c )
• So we get a much simpler set of quantites
• Dimensionless or
• Units of mass
• Not so common to state length time in terms of
  (mass)-1, but rather state both length time in
  fm

4
Relativistic Kinematics

• Not an introduction to relativity
• I am assuming that you know the basic ideas
• Speak now, or forever hold your peace!
• The idea is to go over the
• Vocabulary
• Mathematical apparatus
• Basic Techniques
• Tricks for developing intuition
• In PP, relativity is everywhere
• Get used to it!

5
Basic Objects

• Relativistic 4-vectors
• Minkowski Metric
• Matrix notation!
• Relativistic dot product

6
Lorentz Boosts

• In previous courses, you should have seen one
  dimensional Lorentz transformations (boosts)
  presented this way

b
r
7
Relativistic Effects I

• Time Dilation
• When viewed from a moving frame, a clock at rest
  seems to be ticking more slowly
• Proper time t is that measured by the clock at
  rest

Bouncing photon clock!
b
8
Simple derivation

• Derivation from H. Goldstein
• dxm is a four vector

9
Relativistic Effects II

• Length Contraction
• Measure an object by the interval when the front
  and back pass an observer
• The observers clock is moving slower than the
  one in the objects rest frame
• Thus, the object passes in less time ? observer
  measures a shorter length

10
Matrix representation

• This is more easily remembered as a matrix
  operation (lets drop x,y coords)
• And lo it looks like a rotation
• But with a wrong sign
• Hyperbolic rotation!

where
11
Rotational Invariance!

• In normal geometry, the dot product is conserved
  when you rotate both vectors
• The same applies under Lorentz transformations,
  which are really rotations
• Any product of two four vectors is then
  necessarily Lorentz invariant under a common boost

12
Useful Lorentz invariants

• Mass
• Proper-time
• Phase of a wave function

13
Space-like Time-like

• The distance between two events is an interval
• Space-time intervals can be
• Time-like when a light signal could traverse Dz
• Space-like when a light signal could never
  traverse it
• Time-like separated events can never be made
  simultaneous, but space-like can be

t
Dr
z
14
Momentum space

• We can also think of intervals in momentum space
• Time-like when all of the energy can be
  considered to have been at a point
• Space-like when the information must travel
  between two points

15
Relativistic Kinematics

• Two main principles
• Conserve Energy-momentum
• Make sure Sp(in) Sp(out), E(in)E(out)
• Invariants stay invariant under boosts
• Eg. CMS Energy, momentum transfer
• Work in the frame that simplifies things
• Example, 2 body scattering

16
Example

• Momentum transfer in elastic scattering

p
q
p
q
lt 0 ? Spacelike!
17
Momentum Transfer

• Look at what we have learned
• The elastic scattering angle relates very simply
  to the amount of momentum transferred to the
  target
• This gives us an efficient way of understanding
  how much we have probed our target!
• Larger angles deeper probe

18
Example 2

• Particle-antiparticle annihilation

M
gt0 Time-like
19
What have we learned here

• If you want to make massive particles, you need
  an initial state with a large invariant mass,
  or CMS energy
• This is why we make colliders
• But simply having it is not sufficient
• It needs to be available all in one place!
• Thus, only point like processes (e.g. between
  fundamental fermions) can create new things

20
Rapidity

• There is a convenient variable used in high
  energy physics to express the longitudinal
  momentum of particles
• A Lorentz boost along the z-axis adds a
  b-dependent constant to this variable, so
  rapidity intervals are invariant (prove this in
  the problem set!)

21
Invariant phase space

• Rapidity is a useful quantity when considering
  what we call invariant phase space
• This is the volume element in momentum space
  normalized by the energy
• A quantity that is used in many papers (but not
  in Perkins anymore)
• You will show that this is invariant under
  lorentz boosts!

22
Relativistic Kinematics

• Many things you will see in this course requires
  you to have a good grasp of the concepts
  mentioned here
• Lorentz boosts
• Time dilation, Length contraction
• Lorentz invariants (mass, time, phase, momentum
  transfer)
• Worth the time to play around with it to get
  comfortable with the formalism and physical ideas