Relativistic Classical Mechanics XIX century crisis in physics

Relativistic Classical Mechanics

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 James Clerk Maxwell (1831 -1879) Albert Abraham Michelson (1852 – 1931) Edward Williams Morley (1838 – 1923)

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 7. 1

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) 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) 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. 4 7. 5

3 D Euclidian Cartesian coordinates • Contravariant infinitesimal coordinate vector: • Metric • Covariant infinitesimal coordinate vector: • Magnitude: 7. 4 7. 5

3 D Euclidian spherical coordinates • Contravariant infinitesimal coordinate vector: • Metric • Covariant infinitesimal coordinate vector: • Magnitude: 7. 4 7. 5

Hilbert space of quantum-mechanical wavefunctions • Contravariant vector (ket): • Covariant vector (bra): • Magnitude: David Hilbert (1862 – 1943) • Metric: 7. 4 7. 5

4 D spacetime • Contravariant infinitesimal coordinate 4 -vector: • Metric • Covariant infinitesimal coordinate vector: 7. 4 7. 5

4 D 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 7. 4 7. 5

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)

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 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 7. 2

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‘: 7. 3

Velocity addition 7. 3 • The Lorentz transformation from the reference frame S to the reference frame S‘‘: • On the other hand:

Four-velocity 7. 4 • 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:

Four-velocity • Magnitude of four-velocity 7. 4

7. 1 Minkowski spacetime • Lorentz transformations for parallel axes: t t’ • How do x’ and t’ axes look in the x and t axes? Hermann Minkowski (1864 - 1909) x’ • t’ axis: x • x’ axis:

7. 1 Minkowski spacetime • When t • How do x’ and t’ axes look in the x and t axes? • t’ axis: x • x’ axis:

7. 1 Minkowski spacetime • Let us synchronize the clocks of the S and S’ frames t at the origin t’ • Let us consider an event • In the S frame, the event is to the right of the origin x’ x • In the S‘ frame, the event is to the left of the origin

7. 1 Minkowski spacetime • Let us synchronize the clocks of the S and S’ frames t at the origin t’ • Let us consider an event • In the S frame, the event is after the synchronization x’ x • In the S‘ frame, the event is before the synchronization

Minkowski spacetime 7. 1

Four-momentum 7. 4 • For an object moving relative to a laboratory system, we define a contravariant vector of four-momentum: • Magnitude of four-momentum

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 7. 4

Four-momentum 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 7. 4

Non-covariant Lagrangian formulation of relativistic mechanics • As a starting point, we will try to find a noncovariant Lagrangian formulation (the time variable is still separate) • The equations of motion should look like 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”: 7. 9

Non-covariant Lagrangian formulation of relativistic mechanics • Example: 1 D relativistic motion in a linear potential • The equations of motion: • Acceleration is hyperbolic, not parabolic 7. 9

Useful results

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 7. 9 8. 4

Non-covariant Hamiltonian formulation of relativistic mechanics • We have to express the Hamiltonian as a function of momenta and coordinates: 7. 9 8. 4

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

Non-covariant Hamiltonian formulation of relativistic mechanics • Example: 1 D relativistic harmonic oscillator • The Lagrangian is not an explicit function of time • The quadrature involves elliptic integrals 7. 9 8. 4

Covariant Lagrangian formulation of relativistic mechanics: plan A 7. 10 • So far, our canonical formulations were not Lorentzinvariant – 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

Covariant Lagrangian formulation of relativistic mechanics: plan A 7. 10 • 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:

Covariant Lagrangian formulation of relativistic mechanics: plan A 7. 10 • 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:

Covariant Lagrangian formulation of relativistic mechanics: plan A 7. 10 • From Euler’s theorem on homogeneous functions it follows that • Let us consider the following sum

Covariant Lagrangian formulation of relativistic mechanics: plan A 7. 10 • If three out of four equations of motion are satisfied, the fourth one is satisfied automatically

Example: a free particle • We start with a non-covariant Lagrangian 7. 10

Example: a free particle • Equations of motion 7. 10

Example: a free particle • Equations of motion of a free relativistic particle 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 4 D space • Therefore, such Lagrangian formulation has an inherent constraint • We will impose this constraint only after obtaining the equations of motion 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 7. 10

Covariant Lagrangian formulation of relativistic mechanics: plan B 7. 10 • 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

Covariant Lagrangian formulation of relativistic mechanics: plan B 7. 10 • Example: 1 D 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

Example: relativistic particle in an electromagnetic field • For an electromagnetic field, the covariant Lagrangian has the following form: • The corresponding equations of motion: 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) 7. 10 7. 6

Covariant Lagrangian formulation of relativistic mechanics: plan B 7. 10 • 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

Covariant Hamiltonian formulation of relativistic mechanics: plan A 8. 4 • 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 !!!

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 fourcoordinates and four-momenta • Don’t forget about the constraint: 8. 4

Covariant Hamiltonian formulation of relativistic mechanics: plan B • For a free particle: 8. 4

Covariant Hamiltonian formulation of relativistic mechanics: plan B • For a particle in an electromagnetic field: 8. 4

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 7. 8

Relativistic angular momentum • Evolution of the relativistic angular momentum is determined by: From the equations of motion • For open systems, we have to define generalized relativistic torques in a covariant form 7. 8

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 fourmomentum of the system is conserved • Conservation of a four-momentum includes conservation of a linear momentum and conservation of energy 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 fourmomentum 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 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: 7. 7

Relativistic kinematics of collisions 7. 7

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

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

Example: electron-positron pair annihilation

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