Relativistic Classical Mechanics

1
Relativistic Classical Mechanics
2

• XIX century crisis in physics
• some facts
• Maxwell equations of electromagnetism are not
  invariant under Galilean transformations
• Michelson and Morley the speed of light is the
  same in all inertial systems

3
7.1

• Postulates of the special theory
• 1) The laws of physics are the same to all
  inertial observers
• 2) The speed of light is the same to all
  inertial observers
• Formulation of physics that explicitly
  incorporates these two postulates is called
  covariant
• The space and time comprise a single entity
  spacetieme
• A point in spacetime is called event
• Metric of spacetime is non-Euclidean

4
7.5

• Tensors
• Tensor of rank n is a collection of elements
  grouped through a set of n indices
• Scalar is a tensor of rank 0
• Vector is a tensor of rank 1
• Matrix is a tensor of rank 2
• Etc.
• Tensor product of two tensors of ranks m and n
  is a tensor of rank (m n)
• Sum over a coincidental index in a tensor
  product of two tensors of ranks m and n is a
  tensor of rank (m n 2)

5
7.5

• Tensors
• Tensor product of two vectors is a matrix
• Sum over a coincidental index in a tensor
  product of two tensors of ranks 1 and 1 (two
  vectors) is a tensor of rank 1 1 2 0
  (scalar) scalar product of two vectors
• Sum over a coincidental index in a tensor
  product of two tensors of ranks 2 and 1 (a matrix
  and a vector) is a tensor of rank 2 1 2 1
  (vector)
• Sum over a coincidental index in a tensor
  product of two tensors of ranks 2 and 2 (two
  matrices) is a tensor of rank 2 2 2 2
  (matrix)

6
7.4 7.5

• Metrics, covariant and contravariant vectors
• Vectors, which describe physical quantities, are
  called contravariant vectors and are marked with
  superscripts instead of a subscripts
• For a given space of dimension N, we introduce a
  concept of a metric N x N matrix uniquely
  defining the symmetry of the space (marked with
  subscripts)
• Sum over a coincidental index in a product of a
  metric and a contravariant vecor is a covariant
  vector or a 1-form (marked with subscripts)
• Magnitude square root of the scalar product of
  a contravariant vector and its covariant
  counterpart

7
7.4 7.5

• 3D Euclidian Cartesian coordinates
• Contravariant infinitesimal coordinate vector
• Metric
• Covariant infinitesimal coordinate vector
• Magnitude

8
7.4 7.5

• 3D Euclidian spherical coordinates
• Contravariant infinitesimal coordinate vector
• Metric
• Covariant infinitesimal coordinate vector
• Magnitude

9
7.4 7.5

• Hilbert space of quantum-mechanical wavefunctions
• Contravariant vector (ket)
• Covariant vector (bra)
• Magnitude
• Metric

10
7.4 7.5

• 4D spacetime
• Contravariant infinitesimal coordinate 4-vector
• Metric
• Covariant infinitesimal coordinate vector

11
7.4 7.5

• 4D spacetime
• Magnitude
• This magnitude is called differential interval
• Interval (magnitude of a 4-vector connecting two
  events in spacetime)
• Interval should be the same in all inertial
  reference frames
• The simplest set of transformations that
  preserve the invariance of the interval relative
  to a transition from one inertial reference frame
  to another Lorentz transformations

12
7.2

• Lorentz transformations
• We consider two inertial reference frames S and
  S relative velocity as measured in S is v
• Then Lorentz transformations are
• Lorentz transformations can be written in a
• matrix form

13
7.2
Lorentz transformations
14
7.2

• Lorentz transformations
• If the reference frame S moves parallel to the
  x axis of the reference frame S
• If two events happen at the same location in S
• Time dilation

15
7.2

• Lorentz transformations
• If the reference frame S moves parallel to the
  x axis of the reference frame S
• If two events happen at the same time in S
• Length contraction

16
7.3

• Velocity addition
• If the reference frame S moves parallel to the
  x axis of the reference frame S
• If the reference frame S moves parallel to the
  x axis of the reference frame S

17
7.3

• Velocity addition
• The Lorentz transformation from the reference
  frame S to the reference frame S
• On the other hand

18
7.4

• Four-velocity
• Proper time is time measured in the system where
  the clock is at rest
• For an object moving relative to a laboratory
  system, we define a contravariant vector of
  four-velocity

19
7.4

• Four-velocity
• Magnitude of four-velocity

20
7.1

• Minkowski spacetime
• Lorentz transformations for parallel axes
• How do x and t axes look in
• the x and t axes?
• t axis
• x axis

21
7.1

• Minkowski spacetime
• When
• How do x and t axes look in
• the x and t axes?
• t axis
• x axis

22
7.1

• Minkowski spacetime
• Let us synchronize the clocks of the S and S
  frames at the origin
• Let us consider an event
• In the S frame, the event is
• to the right of the origin
• In the S frame, the event is
• to the left of the origin

23
7.1

• Minkowski spacetime
• Let us synchronize the clocks of the S and S
  frames at the origin
• Let us consider an event
• In the S frame, the event is
• after the synchronization
• In the S frame, the event is
• before the synchronization

24
7.1
Minkowski spacetime
25
7.4

• Four-momentum
• For an object moving relative to a laboratory
  system, we define a contravariant vector of
  four-momentum
• Magnitude of four-momentum

26
7.4

• Four-momentum
• Rest-mass mass measured in the system where the
  object is at rest
• For a moving object
• The equation has units of energy squared
• If the object is at rest

27
7.4
Four-momentum
28
7.4

• Four-momentum
• Rest-mass energy energy of a free object at
  rest an essentially relativistic result
• For slow objects
• For free relativistic objects, we introduce
  therefore the kinetic energy as

29
7.9

• Non-covariant Lagrangian formulation of
  relativistic mechanics
• As a starting point, we will try to find a
  non-covariant Lagrangian formulation (the time
  variable is still separate)
• The equations of motion should look like

30
7.9

• Non-covariant Lagrangian formulation of
  relativistic mechanics
• For an electromagnetic potential, the Lagrangian
  is similar
• The equations of motion should look like
• Recall our derivations in Lagrangian
  Formalism

31
7.9

• Non-covariant Lagrangian formulation of
  relativistic mechanics
• Example 1D relativistic motion in a linear
  potential
• The equations of motion
• Acceleration is hyperbolic, not parabolic

32
Useful results
33
7.9 8.4

• Non-covariant Hamiltonian formulation of
  relativistic mechanics
• We start with a non-covariant Lagrangian
• Applying a standard procedure
• Hamiltonian equals the total energy of the object

34
7.9 8.4

• Non-covariant Hamiltonian formulation of
  relativistic mechanics
• We have to express the Hamiltonian as a function
  of momenta and coordinates

35

• More on symmetries
• Full time derivative of a Lagrangian
• Form the Euler-Lagrange equations
• If

36
7.9 8.4

• Non-covariant Hamiltonian formulation of
  relativistic mechanics
• Example 1D relativistic harmonic oscillator
• The Lagrangian is not an explicit function of
  time
• The quadrature involves elliptic integrals

37
7.10

• Covariant Lagrangian formulation of relativistic
  mechanics plan A
• So far, our canonical formulations were not
  Lorentz-invariant all the relationships were
  derived in a specific inertial reference frame
• We have to incorporate the time variable as one
  of the coordinates of the spacetime
• We need to introduce an invariant parameter,
  describing the progress of the system in
  configuration space
• Then

38
7.10

• Covariant Lagrangian formulation of relativistic
  mechanics plan A
• Equations of motion
• We need to find Lagrangians producing equations
  of motion for the observable behavior
• First approach use previously found Lagrangians
  and replace time and velocities according to the
  rule

39
7.10

• Covariant Lagrangian formulation of relativistic
  mechanics plan A
• Then
• So, we can assume that
• Attention regardless of the functional
  dependence, the new Lagrangian is a homogeneous
  function of the generalized velocities in the
  first degree

40
7.10

• Covariant Lagrangian formulation of relativistic
  mechanics plan A
• From Eulers theorem on homogeneous functions it
  follows that
• Let us consider the following sum

41
7.10

• Covariant Lagrangian formulation of relativistic
  mechanics plan A
• If three out of four equations of motion are
  satisfied, the fourth one is satisfied
  automatically

42
7.10

• Example a free particle
• We start with a non-covariant Lagrangian

43
7.10

• Example a free particle
• Equations of motion

44
7.10

• Example a free particle
• Equations of motion of a free relativistic
  particle

45
7.10

• Covariant Lagrangian formulation of relativistic
  mechanics plan B
• Instead of an arbitrary invariant parameter, we
  can use proper time
• However
• Thus, components of the four-velocity are not
  independent they belong to three-dimensional
  manifold (hypersphere) in a 4D space
• Therefore, such Lagrangian formulation has an
  inherent constraint
• We will impose this constraint only after
  obtaining the equations of motion

46
7.10

• Covariant Lagrangian formulation of relativistic
  mechanics plan B
• In this case, the equations of motion will look
  like
• But now the Lagrangian does not have to be a
  homogeneous function to the first degree
• Thus, we obtain freedom of choosing Lagrangians
  from a much broader class of functions that
  produce Lorentz-invariant equations of motion
• E.g., for a free particle we could choose

47
7.10

• Covariant Lagrangian formulation of relativistic
  mechanics plan B
• If the particle is not free, then interaction
  terms have to be added to the Lagrangian these
  terms must generate Lorentz-invariant equations
  of motion
• In general, these additional terms will
  represent interaction of a particle with some
  external field
• The specific form of the interaction will depend
  on the covariant formulation of the field theory
• Such program has been carried out for the
  following fields electromagnetic, strong/weak
  nuclear, and a weak gravitational

48
7.10

• Covariant Lagrangian formulation of relativistic
  mechanics plan B
• Example 1D relativistic motion in a linear
  potential
• In a specific inertial frame, the non-covariant
  Lagrangian was earlier shown to be
• The covariant form of this problem is
• In a specific inertial frame, the interaction
  vector will be reduced to

49
7.10 7.6

• Example relativistic particle in an
  electromagnetic field
• For an electromagnetic field, the covariant
  Lagrangian has the following form
• The corresponding equations of motion

50
7.10 7.6

• Example relativistic particle in an
  electromagnetic field
• Maxwell's equations follow from this covariant
  formulation (check with your EM class)

51
7.10

• Covariant Lagrangian formulation of relativistic
  mechanics plan B
• What if we have many interacting particles?
• Complication 1 How to find an invariant
  parameter describing the evolution? (If proper
  time, then of what object?)
• Complication 2 How to describe covariantly the
  interaction between the particles? (Information
  cannot propagate faster than a speed of light
  action-at-a-distance is outlawed)
• Currently, those are the areas of vigorous
  research

52
8.4

• Covariant Hamiltonian formulation of relativistic
  mechanics plan A
• In Plan A, Lagrangians are homogeneous
  functions of the generalized velocities in the
  first degree
• Let us try to construct the Hamiltonians using
  canonical approach (Legendre transformation)
• Plan A a bad idea !!!

53
8.4

• Covariant Hamiltonian formulation of relativistic
  mechanics plan B
• In Plan B instead of an arbitrary invariant
  parameter, we use proper time
• We have to express four-velocities in terms of
  conjugate momenta and substitute these
  expressions into the Hamiltonian to make it a
  function of four-coordinates and four-momenta
• Dont forget about the constraint

54
8.4

• Covariant Hamiltonian formulation of relativistic
  mechanics plan B
• For a free particle

55
8.4

• Covariant Hamiltonian formulation of relativistic
  mechanics plan B
• For a particle in an electromagnetic field

56
7.8

• Relativistic angular momentum
• For a single particle, the relativistic angular
  momentum is defined as an antisymmetric tensor of
  rank 2 in Minkowski space
• This tensor has 6 independent elements 3 of
  them coincide with the components of a regular
  angular momentum vector in non-relativistic limit

57
7.8

• Relativistic angular momentum
• Evolution of the relativistic angular momentum
  is determined by
• For open systems, we have to define generalized
  relativistic torques in a covariant form

From the equations of motion
58
7.7

• Relativistic kinematics of collisions
• The subject of relativistic collisions is of
  considerable interest in experimental high-energy
  physics
• Les us assume that the colliding particle do not
  interact outside of the collision region, and are
  not affected by any external potentials and
  fields
• We choose to work in a certain inertial
  reference frame in the absence of external
  fields, the four-momentum of the system is
  conserved
• Conservation of a four-momentum includes
  conservation of a linear momentum and
  conservation of energy

59
7.7

• Relativistic kinematics of collisions
• Usually we know the four-momenta of the
  colliding particles and need to find the
  four-momenta of the collision products
• There is a neat trick to deal with such
  problems
• 1) Rearrange the equation for the conservation
  of the four-momentum of the system so that the
  four-momentum for the particle we are not
  interested in stands alone on one side of the
  equation
• 2) Write the magnitude squared of each side of
  the equation using the result that the magnitude
  squared of a four-momentum is an invariant

60
7.7

• Relativistic kinematics of collisions
• Let us assume that we have two particles before
  the collision (A and B) and two particles after
  the collision (C and D)
• Conservation of the four-momentum of the system
  
• 1) Rearrange the equation (supposed we are not
  interested in particle D)
• 2) Magnitude squared of each side of the
  equation

61
7.7
Relativistic kinematics of collisions
62

• Example electron-positron pair annihilation
• Annihilation of an electron and a positron
  produces two photons
• Conservation of the four-momentum of the system
• Let us assume that the positron is initially at
  rest
• 1) Rearrange the equation

63

• Example electron-positron pair annihilation
• 2) Magnitude squared of each side of the
  equation

64
Example electron-positron pair annihilation
65

• Example electron-positron pair annihilation
• The photon energy will be at a maximum when
  emitted in the forward direction, and at a
  minimum when emitted in the backward direction