CF7001

Eletrodinâmica Clássica I

Prof. Dante H. Mosca

2018

Relativity

Chap. 11 and 12

Propagation of light and Michelson - Morley experiment

Special Relativity

Relativistic Doppler effect

Minkowski formalism

Relativistic mass and rest energy

Lorentz group

Covariant formulation of Clasical Electrodynamics

The spinor and spin

Thomas precession (BMT equations)

Charged particle in the electromagnetic field

Covariant Lagrangian formulation

Stress-energy tensor and angular momentum of the field

Asymptotic realms of physics

Michelson-Morley experiments (from 1881 to 1887)

Coherent lenght

http://en.wikipedia.org/wiki/Coherence (physics)

30 km/s

no changes !

Plane electromagnetic wave in vacuum as observed in inertial frames K and K´

point B´ differs from B

Phase changes in Galilean referentials K and K'

Ritz conjecture:

Galilean relativity

Connections between the coordinates in K and K´ :

Wave equation is not invariant under Galilean transformation :

Special Relativity

Second postulate (invariance of c):The light propagates in the vacuum with a constant velocity c which is independent of the motion states of the sources and the observers.

First postulate (principle of relativity):The physical laws must be the same in all inertial frames.

Einstein, Albert (1905), "Zur Elektrodynamik bewegterKörper", Annalen der Physik 322 (10): 891–921.

E o Brasil nisso ...

https://sofiamoutinhoportfolio.files.wordpress.com/2015/05/einstein-01.jpgEinstein no então Instituto Oswaldo Cruz, com Aldolpho Lutz, Carlos Chagas e outros cientistas. (foto: Fiocruz)

08 de maio de 1925

Observação do eclipse total do sol de 29 de maio de 1919 em Sobral no Ceará.

Albert Einstein (no centro e à frente) com a equipe do Observatório Nacional, no Rio de Janeiro. (Foto: Divulgação/Observatório Nacional)

09 de maio de 1925

Escreveu o primeiro artigo da Revista da Academia Brasileira de Ciências: foi publicado no volume 1, número 1, nas páginas 1 a 3, em 1926 : Observações sobre a situação atual da teoria da luz

Relativity

http://fr.wikipedia.org/wiki/Transformation_conforme

z

z z2

Experimental confirmation of the invariance of the light speed in vacuum and in all inertial frames

The Lorentz factor and the speed parameter

Lorentz factor Speed parameter

Therefore:

When an event is relativistic ...

Expanding in series ...

Lorentz transformations

Valid for any physically possible speed.

S S’y y’

x x’

x’vt

x

event

v

Lorentz, Hendrik Antoon (1892), "La Théorie electromagnétique de Maxwell et son application aux corps mouvants at the Internet Archive", Archives néerlandaises des sciences exactes et naturelles 25: 363–552

Lorentz transformation and Relativity principle

Space, time, mass, energy ...space-time, mass-energy, ...

Visible Universe

?Física na Veia! - Uol

Relativistic Doppler effect

Consider the situation below.

Show that the speed transformations are given by:

So ...

Exercise 1

Exercise 2Show that the acceleration transformations are given by:

Remark: Consider a coordinate system where the particle is instantaneously at rest

Minkowski formalism

Registering an event

4 coordinates: 3 space 1 time

Events: collision between two particles, lighting a lamp, passage of a light pulse, evolution of life, ...

Relativity determines the relationship between the coordinates assigned to the same event by two observers moving relative to one another.

World lines

timelike separation

spacelike separation

lightlike separation

Two events with a spacelike separation in one coordinate system have a spacelikeseparation in all coordinate systems and they are not causually connected.

Light cone of Minkowski (observer under high acceleration)

Four-vectors ou 4-vectors

Scalar product: invariant

Norm of a four-vector

Sugestão 1:

Expresse as equações do escocês J. C. Maxwell no formalismo original usado por ele na publicação de 1873.

Sugestão 2:

Pesquise como o alemão H. C. Hertz demonstrou a propagação eletromagnética “wireless” em 1887/1888.

Sugestão 3:

O que é uma Slerp (Spherical Linear intERPolation) usada em computação gráfica.

Proper time and proper length intervalsProper time elapsed between two events, as measured by a clock which passes through both events. Depends not only on events, but also on the movement between events.

A clock moving fast between two events measures a shorter time interval than a clock stationary relative to the same two events: "twin paradox".

Proper length is measured in an inertial reference system that events are simultaneous. So, if the two events occur on opposite sides of an object, the proper length is the object length measured by an observer who is at rest with respect to the object.

Therefore (4-vectors) :

Then

Notation

=

Addition of relativistic velocities

Exercise 3

Show that:

In such a way that

which is equivalent to

Then

Kinetic moment and relativistic energy of identical particles in head-on collision

In K´ :

b = center of mass in K´

Velocity components in K

Conservations of the energy and the kinetic moment

Along x axis:

For ' = 0 and then:

Whence

are functions of ' and a

Since for a particle at low speed are satisfied:

As a result (see mathematical developments in the Jackson´s textbook):

And, a convenient generalization to obtain the relativistic kinetic energy T:

Rest energy, * In inelastic cases, for example, the decay of mesons K neutrons (oscillations of this neutral particle are due to mixing of states):

Such that

or

then

and

* Mass-energy equivalence arises with Poincaré and Hasenohrl, not Einstein. See Holton in American Journal of Physics 30 (6): 462 (1962).

γE2

)2

(Ksdsdo

S

Exercise 4Consider the situation below.

(a) Show that the mass of an object moving relative to an observer is:

then

Since

We obtain:

If then

Then relativistic mass is equivalent to the relativistic energy / c2.

(b) Discuss the following: what is the meaning of m1 and m2 ?

and

Exercise 5(a) Show that:

(b) Show that:

Action and Reaction ...

Schematic drawings (a) e (b) describe the same interaction between Q and q charges.

Show that for a given force acting upon a particle results:

a)

b)

c) Is reasonable to define longitudinal and transverse mass? Explain.

Exercise 6

Defining the mass (what a mess!) ...

Rigorously, the mass can be defined only in terms of their physical properties. Thus, the mass can be used to determine how a particle (or body) resists a change in its state of motion or velocity (or shape) when a force is applied; i.e., it is a measure of the "difficulty" to accelerate a particle (or bodies) stationary or on the move. Further, the mass is a measure of the amount of energy and matter that a particle (or body) contains. This determines the difficulty of changing its state of motion. The mass as a measure of the interaction of a particle (or body) with a gravitational field is a measure of the curvature of space-time induced by the particle (or body), according to the Theory of General Relativity [Einstein, 1920 ]. In sub-atomic scale, the mass may be a measure of rest energy of a particle, due to the mass-energy equivalence principle. It is also a kinetic energy operator's parameter of the wave field associated to a particle (or body) [de Broglie 1925], as well as the parameter which determines its Compton wavelength.The Higgs mechanism is essential to explain the generation of the "mass" property of gauge bosons [Higgs, 1964]. Is still unanswered the question of the Higgs field gives rise the mass for just some of the basic elementary particles, as in the original formulation of the idea, or is it responsible for the masses of all elementary particles.

Using the Minkowski formalism

Satisfying 4 equations to conserve energy and kinetic moment:

Since and

Then

Four-moment: invariant

Mass-energy equivalence

Remark: The "heuristic" construction of the relativistic dynamics is convenient and validated . However, the concepts of rest mass and relativistic mass are didactically questionable. It is also worth of note that, there is no way to establish the exact total energy of an isolated system (even empty).

Remark:

What is the so-called Komar mass?

A. Komar,

Positive-Definite Energy Density and Global Consequences for General Relativity,

Physical Review, vol. 129, Issue 4, pp. 1873–1876 (1963)

Covariant formulation of Classical Electrodynamics

Lorentz group

Lorentz scalar is an invariant

Contravariant and covariant tensors

rank #1

rank #2

rank #0

Inner product

Metric tensor: flat spacetime

Four-dimensional Kroneker-delta

mixed tensor

2D curvilinear non-orthogonal grid

The Free Physics TextBook, http://www.motionmountain.net/

Concept of gaussian curvature circumference of a circle in a curved in space

The circle with circumference C(K) in 2D (S 0) is the local of points at a fixed distance S from a given reference point.

Parallel transport of a vector in a closed path and the local perception of curvature.

Berry phase

Parallel transport and rotation in curved space

Differential manifold

Affine connection it is equivalent to the notion of parallel transport; i.e., a method

for transporting tangent vectors along curves, and may be used to define geodesics on a manifold.

inner product between two vectors Manifolds (topological spaces) can be differentiable continuously, n times,

smoothly (Riemann), holomorphically (via Taylor series expansions), ...

Covariant & contravariant derivatives

Christoffel symbolsnumerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in manifolds; i.e., affine connection coefficients constructed by metric tensor and its derivatives.

set for flat-space spherical coordinate

Differential operators

Four-divergence Four-Laplacian (d´Alembertian)

Matrix representation

Inner product and transpose operation

Obs.: there exist 16 - ( 1 + 2 + 3 ) = 10 equations which are linearly

independent for the 16 elements in A.

Lorentz transformations Proper and improper

Improper transformations

Inversion operators

C (charge conjugation) : e -eP (space inversion) : ( t , r ) ( t , -r )T (time inversion) : ( t , r ) ( -t , r )

(inversão espacial)

(inversão espacial e temporal)

(space inversion)

(space and time inversions)

CPT

http://cerncourier.com/cws/article/cern/29224

CP violation

http://en.wikipedia.org/wiki/Kaon

Ks = K1 + K2

KL = K1 + ’K2

CP = + 1

CP = - 1

Group Theory (translations and rotations)

Rotations in 2D

Rotations in 3DThere are ambiguities and not

commutability.

Rotation sequence: zx'z"

rotationaxis

rotationangle

nodalline

Exercise 7Show that after parametrization :

(a)

(b)

Lie Group : underlie continuous transformations group of theories, being a topological space locally Euclidean and differentiable.

Counterexample: mapping is not differentiable !

Lorentz Group: group of all transformations in the Minkowski spacetime. Provide the invariants to Lorentz transformations (laws of relativistic mechanics, Maxwell's equations, Dirac equation, for example).

Poincaré group: group of isometries (mapping transformations that not change time itself along a trajectory between events) in Minkowski spacetime, that includes all the translations and the Lorentz transformations.

Group Theory

Exercise 8Jackson's book uses a possible basis for Lie Groups algebra to describe the infinitesimal generators of Lorentz transformations. Show that the matrices S generate 3D rotations and matrices K generate boost orthogonal to each other.

(a) Show that properties for one S rotation and one K boost. (b) Show that the square of these matrices is diagonal and they satisfy the commutation relations.

Exercise 9

Show that:

using that with:

Exercise 10

Considering that :

Show that:

(a)

(b)

(c)

Exercise 11

Reformulation of Lorentz force law

Revisiting the Lorentz force law

Reformulation of continuity equation

Charge invariance and infinitesinal 4-volume

c transforms equivalently to xo

Four-potential under Coulomb gauge

_c

CGS

SI

o J

CGS

SI

Electromagnetic field-strength tensor

Exercise 12

Explicitly show that :

Remark: usual definition of the dual field-strength tensor take E B and B -E .

Transformation of the fields in the case v // x

--

Exercise 13(a) Show that when it is valid that:

is obtained :

(b) Answer the questions and discuss.

Useful relations

=

is a pseudo-escalar

Exercise 14

Exercise 15

Exercise 16

(c)

Covariant Maxwell´s equations

or

Covariant Lorentz force law

Macroscopic view

Obs.: As (E,B) and (D,H) are antisymmetric second-order tensors, then P e M are also similar mathematical objects. In electrodynamics of macroscopic media, these physical quantities represent the mean macroscopic properties of atomic sources given in the rest referential frame.

Exercise 17Remembering that:

and

Show that with c = vx = v , :

H ’// = H // H ’┴ = (H ┴ – v x D ┴ )

M ’// = M // M ’┴ = (M ┴ + v x P ┴ )

D ’// = D // D ’┴ = (D ┴ + v x H ┴ / c2)

P ’// = P // P ’┴ = (P ┴ – v x M ┴ / c2)

Electromagnetic field along the trajectory of a charged particle

In K´ :

B3 Biot-Savart law (for 1)

In K ...

Field profiles

EQ BQ

Symmetry of the field lines of a charge moving at low speed

Symmetry of the field lines of a charge moving at high speed

http://www.mysearch.org.uk/website1/html/475.Radiation.html

Equipotentials of a moving charge that gives rise mostly to transverse fields

The electric field is no longer isotropic !

There is no magnetic field either behind or in the front !

Charged particles in the presence of electromagnetic fields

Particle in a uniform magnetostatic field

Charged particle in the presence of uniform electric and magnetic fields

In K'

Note: there is a uniform drift in the direction defined by E x B if E < B.

If E < B and B' = 0

If E > B the electric field is strong enough to promote a continuous accelerationin the direction of E and the energy increases continuously in time.

If the fields E and B are transversal to each other, there exist a velocity filter effect.

Exercise 18(a) Show that if E B 0 both fields exist in K and K'.

(b) Show that if E B = 0 it is always possible to find a reference frame where E = 0 |E| < |B| or B = 0 if |B| < |E|.

(c) It can be stated that: is there only one of the fields and the other is a pure manifestation of the Lorentz transformation?

Charged particle in a non-uniform magnetic field

Since the movement in the direction B is unchanged:

Bvc

e

Exercise 19Consider the configuration below.

(a) Show that:

(b) Show that if the field gradient is small will prevail the orbital motion superimposed on the longitudinal drift motion.

Adiabatic invariants of a charged particle in a closed orbit in a magnetic field

The action integral of periodical canonical coordinates, i.e., a pair of periodic generalized coordinate and conjugated momentum :

Then

n

dl

Calculating

with v // dl

using the Stokes theorem:

Then

once

Adiabatic invariants:

Reflection of a charged particle out of region of strong magnetic field

Because is invariant

= 0

The translational energy is converted into rotational energy until the axial velocity vanishes.

Charged particle are reflected by regions of strong magnetic field, the so-called principle of magnetic mirror.

Exercise 20Show that the condition to occur the magnetic mirror effect is:

Spinor and Spin

Dirac solution

Dirac spinors and Pauli matrices

Pauli equation

Stern-Gerlack term

Pauli spin matrices

Spin

The spin of Pauli electronWolfgang Pauli propose an extension of Schroedinger equation to spinning electronsin an external electromagnetic field. In order to model the spin of a non-relativistic electron Pauli takes a space C2 for the space of its wave function :

Let be the scalar electric potential of the exernal field ans A the vector potential. The potential energy due to the torque produced is:

and the ultimate equation is:

Then, since the matrix Bii is not generally diagonal there will be coupling of the two

components of . However, if B points in the z-direction then the matrix Bii = B33

will be diagonal for this choice and up and down components become independent.

Example

Zeeman effect

)2(

Zeemaneffect

Paschen-Backeffect

B = H= 0

Zeeman effect

Spin in a electromagnetic field: naïf approach

Hydrogenoide atom

Factor 1/2X

Bargmann–Michel–Telegdi (BMT) equation

Phys. Rev. Lett. 2 (1959) 435

In the equation above at right, the spin s is expressed in the rest frame,while the B and E values are the fields in the reference laboratory. Such aparametrization is natural because the spin is an intrinsic physical property, and its true significance is inherent to the particle in the rest frame.

Heisenberg equation of motion

Thomas precession

Thomas precession is due to the acceleration that the electron experiences under the action of external force of Coulomb screening.

Geodetic precession(analogue)

http://einstein.stanford.edu/SPACETIME/spacetime4.html

Thomas correction

Promoting a Lorentz transformation without rotation:

slide 71 slide 71

Interpretation of the infinitesimal transformation

The Lorentz transformation of the boost without relative rotation ofthe reference laboratory is equivalent to a reference x''' in which the rest reference is rotated by relative to the laboratory reference subsequently to an infinitesimal boost at speed c.

Thomas precession frequency

Spin-orbit coupling in the laboratory reference frame

Coulombian screening:

The precession of Thomas reduces the spin-orbit coupling.

Thomascorrection

factor

Dirac´s electron theory

p - i ħ

E i ħ /t4222 cmpcE

EH

matemathical object of complex vector space

The spin of Dirac electronP. A. M. Dirac proposes a full relativistic model of the electron and its wave function is a bispinor belonging to space C4 :

using Compton wave number = mc / ħ and Dirac operator:

the ultimate Dirac equation for a free electron is:

and as then

Minimal approach for the electromagnetic coupling of an electron in an external electromagnetic field A = ( , Ai)

Minimal prescription takes:

The Dirac equation becomes:

In this case, the coupling of the spin and the external field is described by thematrix:

When this matrix is applied to 2x2 Dirac wave functions with :

results:

The equations for each Pauli spinors are :

Therefore

Spinor 4-Spin Spin

covariance constraint pseudo-vetor: US = 0 So= U S

2

S

SU

cS

2

1

Levi-Civita symbolU = ( u c , u u )

Note : the determination of the pseudovector S in a rest frame will specify the components of the 4-vector S

in any other inertial frame.

Lorentz transformation for spin

Covariant equation of motion

Lorentz transformation for spin S = (So,S)

By imposing:

Candidates for invariants:

kinematical termtorque

In the absence of electromagnetic field is required:

Since then

As a consequence:

Covariant BMT equation

Covariant Lorentz force law

Exercise 21Show that BMT equation:

correctly gives the Thomas precession:

Remark:

Covariant relativistic mechanics (spinless)

Equation of motion :

Hamilton minimal action principle:

Lorentz Invariant:

Free spinless particle

invariant

Spinless particle in an electromagnetic field

Exercise 22In General Relativity:

By convention:

Then, defining:

(a) Explain the meaning of the last equality above.

(b) Explain the meaning of:

Lagrangian of a spinless charged particle in an electromagnetic field

Canonical momentum:

Hamiltonian:

Hamiltonian of a charged particle in an electromagnetic field

Note: this proposal is not invariant to a gauge transformation ...

Then

By simplicity we may admit that the electromagnetic field is linearly dependent on the 4-velocity:

Thus, Fmust be antisymmetric with six degrees of freedom: E e cB.

where E is the electric part of the electromagnetic field according to an observerwith 4-velocidade v

. Since:

Discussion

(v)Eam

ev μμμ

0 xx and c xx then xd

dxv 2

0(v)vE μμ

The presence of an electromagnetic field is observed by a 4-acceleration aimposedto an electron with 4-velocity v. Then:

0 μνμν

νμνμ vvF que tal vF(v)E with

Since

with V and A the instantaneous velocity and acceleration of the electron relative to an observer with 4-velocity . Now, considering the electron at rest for the observer, such that vo = 0:

4-acceleration

),0(c

c

c

2 A,V)(x

,V)(x

t,R)(x

μ

μ

μ

γγ

γ

AVvx

Vvx

o

o

2)(

)(

γγ

γ

x

)(

)()()(i

io

oo

xFFemA

VeFveFVveFxm

μμμ

If we assume that the electromagnetic field is linear :

x

Covariant Lagrangian formulation (free particle in a generic spacetime)

Equivalently:

Electrodynamics in curved spacetime

The metric tensor g

is no longer constant, varying in the spacetime:

Lagrangian density for classical electrodynamics with sources in vacuum:

)det( μνgg

(a) Show that the equation of motion of a free electron in the covariant formalism can be written to a stationary observer such as:

(b) And, for an observer at an arbitrary coordinate system (not necessarily inertial):

where

is defined as affine connection or Christoffel symbol.

Exercise 23

Naïve approach to an electron in gravitational and electromagnetic fields

Electron (spinless) in the presence of gravitational and eletromagnetic fields

Electron (spinlesss) in an electromagnetic field

Generalization of the Lagrangian formalism to describe the electromagnetic field and interaction

between sources

Each point of the spacetime x corresponds to a finite number of values with

discrete index i. The dependent variables are replaced by the values of the continuous field k at each point of spacetime. The equations of motion are obtained from the minimum principle of the action integral.

Action integral of the Lagrangian density

Disregarding the spins and time retardation effects, the Lagrangian for situationsof electrostatic nature or magnetostatic interaction between charged particles is:

SI

CGS

Remark: not curved space-time

Lagrangian formulation of classical electromagnetism

Then

Using Euler-Lagrange equation:

Giving

Two inhomogeneous Maxwell equations are obtained:

Meanwhile,

Homogeneous Maxwell equations:

Bianchi identity:

Exercise 24Show that:

and

Resulting in the equation of motion of the electromagnetic field:

Which also refers to due to continuity equation:

Exercise 25Show through the following settings:

that the lagrangean density is:

Or, equivalently, using non-relativistic vector notation :

Exercise 26The stability and the possibility of a nonzero mass for photon are still matter of debate [see J. Heeck, Phys. Rev. Lett. 111 (2013) 021801]. Also, the question of origin of the photons [http://periodicos.unb.br/index.php/e-bfis/index or eBFIS 4 3102-1(2015].

An exploratory mathematical treatment of the eventual photon mass come from the Lagrangian proposed by Proca in 1930.

Show that from this Lagrangian, the parameter provides a rest mass for the photons and a scalar potential given by:

Note: experiments show that , if not zero!

Exercise 27By taking into account the retarded effect as the first-order correction in classical Lagrangian of interaction between particles, Charles Darwin (biologist grandson) proposes:

Such that:

Get the Lagrangian Darwin assuming low-order corrections in scalar and vectorpotentials under the Coulomb gauge.

Stress-energy tensor and conservation laws in covariant electrodynamics

Changing the Lagrangian formulation for the Hamiltonian formulation, that ismore directly connected to the principle of symmetry of Noether's theorem:

The first term of the sum is the field momemtum which is canonically conjugated to k(x) that multiplies a term which is equivalent to speed. Then, the Lorentz

transformation ansatz suggests the construction of the tensor:

Lagrangian of the free electromagnetic field

Canonical stress tensor:

where

Therefore,

Further,

Conservation of electric charge and the continuity equation

Note: In this context is circumvented the problem of electromagnetic self-energy. Also, the spin is totally disregarded.

Valid in the rest frame in which the fields are specified.

Symmetry of the angular momentum tensor in electrodynamics

Angular momentum density is a third-order tensor (rank #3):

The conservation implies:

0 0

M is symmetrical

Symmetrization of stress tensor

Symmetric stress tensor

By construction:

such that

and

Electromagnetic stress-energy tensor (SI)

Exercise 28(a) Show that = 0:

(b) Show that for = i :

where S is the Poynting vector:

Conservation of the angular momentum (orbital or spinless) of the electromagnetic field

The tensor M0contains the angular momentum density. Thus, in a

covariant generalization of the angular momentum of the field:

with

then

Since and :

Since the contraction of a symmetric tensor with an antisymmetric one is always zero, then:

Resulting

In fact,

4-vector of the Lorentz force density

As a consequence:

The variation of the action integral with respect to particle variables leads to Lorentz force equation, while variation of the field coordinatesgives the Maxwell's equations.