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
Albert Abraham
Michelson
(1852 – 1931)
Edward Williams
Morley
(1838 – 1923)
James Clerk
Maxwell
( )

3. Postulates of the special theory
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: spacetime
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. Tensors Tensor product of two vectors is a matrix
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 – 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 (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 (matrix)

6. Metrics, covariant and contravariant vectors
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. 3D Euclidian Cartesian coordinates
7.4
7.5
3D Euclidian Cartesian coordinates
Contravariant infinitesimal coordinate vector:
Metric
Covariant infinitesimal coordinate vector:
Magnitude:

8. 3D Euclidian spherical coordinates
7.4
7.5
3D Euclidian spherical coordinates
Contravariant infinitesimal coordinate vector:
Metric
Covariant infinitesimal coordinate vector:
Magnitude:

9. 4D spacetime Contravariant infinitesimal coordinate 4-vector: Metric
7.4
7.5
4D spacetime
Contravariant infinitesimal coordinate 4-vector:
Metric
Covariant infinitesimal coordinate vector:

10. 4D spacetime Magnitude: This magnitude is called differential interval
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

11. Lorentz transformations
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
Hendrik Antoon
Lorentz
(1853 – 1928)

12. Lorentz transformations
7.2
Lorentz transformations

13. Lorentz transformations
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

14. Lorentz transformations
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

15. 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‘:

16. 7.3
Velocity addition
The Lorentz transformation from the reference frame S to the reference frame S‘‘:
On the other hand:

17. 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:

18. 7.4
Four-velocity
Magnitude of four-velocity

19. Minkowski spacetime t t’ x’ x
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:
Hermann
Minkowski
( ) t x t’ x’

20. Minkowski spacetime t x When How do x’ and t’ axes look in
7.1
Minkowski spacetime
When
How do x’ and t’ axes look in
the x and t axes?
t’ axis:
x’ axis: t x

21. Minkowski spacetime t t’ x’ x
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 t x x’ t’

22. Minkowski spacetime t t’ x’ x
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 t x x’ t’

23. 7.1
Minkowski spacetime

24. 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

25. 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

26. 7.4
Four-momentum

27. 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

28. Non-covariant Lagrangian formulation of relativistic mechanics
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

29. Non-covariant Lagrangian formulation of relativistic mechanics
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”:

30. Non-covariant Lagrangian formulation of relativistic mechanics
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

31. Useful results

32. Non-covariant Hamiltonian formulation of relativistic mechanics
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

33. Non-covariant Hamiltonian formulation of relativistic mechanics
7.9
8.4
Non-covariant Hamiltonian formulation of relativistic mechanics
We have to express the Hamiltonian as a function of momenta and coordinates:

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

35. Non-covariant Hamiltonian formulation of relativistic mechanics
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

36. Covariant Lagrangian formulation of relativistic mechanics: plan A
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

37. Covariant Lagrangian formulation of relativistic mechanics: plan A
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:

38. Covariant Lagrangian formulation of relativistic mechanics: plan A
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:

39. Covariant Lagrangian formulation of relativistic mechanics: plan A
7.10
Covariant Lagrangian formulation of relativistic mechanics: plan A
From Euler’s theorem on homogeneous functions it follows that
Let us consider the following sum

40. Covariant Lagrangian formulation of relativistic mechanics: plan A
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

41. Example: a free particle
7.10
Example: a free particle
We start with a non-covariant Lagrangian

42. Example: a free particle
7.10
Example: a free particle
Equations of motion

43. Example: a free particle
7.10
Example: a free particle
Equations of motion of a free relativistic particle

44. Covariant Lagrangian formulation of relativistic mechanics: plan B
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

45. Covariant Lagrangian formulation of relativistic mechanics: plan B
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

46. Covariant Lagrangian formulation of relativistic mechanics: plan B
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

47. Covariant Lagrangian formulation of relativistic mechanics: plan B
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

48. Example: relativistic particle in an electromagnetic field
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:

49. Example: relativistic particle in an electromagnetic field
7.10
7.6
Example: relativistic particle in an electromagnetic field
Maxwell's equations follow from this covariant formulation (check with your E&M class)

50. Covariant Lagrangian formulation of relativistic mechanics: plan B
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

51. Covariant Hamiltonian formulation of relativistic mechanics: plan A
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 !!!

52. Covariant Hamiltonian formulation of relativistic mechanics: plan B
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
Don’t forget about the constraint:

53. Covariant Hamiltonian formulation of relativistic mechanics: plan B
8.4
Covariant Hamiltonian formulation of relativistic mechanics: plan B
For a free particle:

54. Covariant Hamiltonian formulation of relativistic mechanics: plan B
8.4
Covariant Hamiltonian formulation of relativistic mechanics: plan B
For a particle in an electromagnetic field:

55. Relativistic angular momentum
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

56. Relativistic angular momentum
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

57. Relativistic kinematics of collisions
7.7
Relativistic kinematics of collisions
The subject of relativistic collisions is of considerable interest in experimental high-energy physics
Let us assume that the colliding particles 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

58. Relativistic kinematics of collisions
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

59. Relativistic kinematics of collisions
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:

60. Relativistic kinematics of collisions
7.7
Relativistic kinematics of collisions

61. 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

62. Example: electron-positron pair annihilation
2) Magnitude squared of each side of the equation:

63. Example: electron-positron pair annihilation

64. 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