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

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

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)

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 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)

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

3D Euclidian Cartesian coordinates 7.4 7.5 3D Euclidian Cartesian coordinates Contravariant infinitesimal coordinate vector: Metric Covariant infinitesimal coordinate vector: Magnitude:

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

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

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

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)

Lorentz transformations 7.2 Lorentz transformations

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

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

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‘:

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

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:

7.4 Four-velocity Magnitude of four-velocity

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 (1864 - 1909) t x t’ x’

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

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’

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’

7.1 Minkowski spacetime

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

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

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

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

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”:

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

Useful results

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

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:

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

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

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

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:

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:

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

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

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

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

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

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

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

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

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

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:

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)

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

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 !!!

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:

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

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:

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

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

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

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

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:

Relativistic kinematics of collisions 7.7 Relativistic kinematics of collisions

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