PHY 4460 RELATIVITY

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PHY 4460RELATIVITY

• K Young, Physics Department, CUHK
• ?The Chinese University of Hong Kong

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CHAPTER 7PARTICLE DYNAMICS ELECTROMAGNETISM
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Classical
SR
? Last chapter
? This chapter
? This chapter
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Objectives (1)

• Why EM
• Lorentz force law
• magnetic field B, R ?
• parallel force (e.g. E)
• perpendicular force
• 4-force Km 4-vector
• Newton's second law in covariant form

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Why EM
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What force laws?

• Most forces are not relativistic
• they do not assume the same force law in all
  reference frames

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What force laws?

• EM force is relativistic
• same form in all frames
• they exist in vacuum (no "ether")
• We shall concentrate on EM force

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Broader view

• All fundamental forces should be relativistic
• strong interaction
• EM
• weak interaction
• gravitation
• short range no classical theory
• long range, force 1/r2
• there is classical theory

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Lorentz Force Law
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Lorentz force law

• The law
• Magnetic field
• Electric field
• General parallel forces
• General perpendicular forces

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Definition of force and Lorentz force law

• Force is defined as

• t is not invariant
• This is not a 4-vector
• Experimentally

• Examples of motion

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Deflection in magnetic field
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• R ? p ? g v ? g for v ? c
• Experimental proof
• Used to measure p

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Motion under E (p//E)
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• v lt 1 always
• For v ltlt 1, reduce to v a t
• qE ? F (any force)

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Parallel force
e.g. electric force
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Perpendicular force
e.g. magnetic force
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Digression

• It is not true that we can simply replace m ? mg
  everywhere. Do not use "relativistic mass"
• No need to remember details of derivation

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4-Force
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4-Force

• Definition
• 4-vector
• Newton's 2nd law in terms of the 4-force

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• It is not convenient for comparing different
  frames.
• F is not a 4-vector F transforms in a messy
  way

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Covariant force

• Change of 4-momentum per unit proper time

• This is a 4-vector
• Its components are

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2 ways of expressing force law

• F ?
• Correct but not covariant
• Km ?
• Correct and covariant
• What is RHS?
• It must be 4-vector made from
• E and B
• v

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Objectives (1)

• Why EM
• Lorentz force law
• magnetic field B, R ?
• parallel force (e.g. E)
• perpendicular force
• 4-force Km 4-vector
• Newton's second law in covariant form

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Objectives (2)

• Potential Am field Fmn
• Covariant form of Lorentz force law
• Transformation of field
• How one phenomenon is explained in 2 different
  frames
• Two relativistic invariants

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Potential and Field
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4-vector potential field tensor

• The scalar potential and vector potential are
  assumed to transform like a 4-vector

• Define the field tensor

• Antisymmetric
• Explicit form of components?

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• V1 and V2 are different components of a vector V
• Under a transformation V1 can turn into V2 , and
  vice versa
• E and B are different components of a tensor Fmn
• Under a transformation E can turn into B, and
  vice versa

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• Under a transformation E can turn into B, and
  vice versa
• Why?
• E produced by charge
• B produced by current
• Current moving charge

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Covariant FormofLorentz Force Law
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Covariant formof Lorentz force law

• Explicitly covariant
• Right form field ? velocity
• Check that it agrees with Lorentz force law
  experiment

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• Assuming Am transforms like a 4-vector,
• we have guaranteed that all EM
• phenomena can be consistently described in
• any ref frame using the same laws

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Transformation of Fields
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Transformation of fields
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Similarly for B'
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Three LevelsofDiscussing Covariance
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3 levels of discussing covariance

• Phenomena

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3 levels of discussing covariance

• Non-covariant equations

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3 levels of discussing covariance

• Covariant equations

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Relativistic Invariants
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Relativistic invariants
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Dual tensor
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Relativistic invariants
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Objectives (2)

• Potential Am field Fmn
• Covariant form of Lorentz force law
• Transformation of field
• How one phenomenon is explained in 2 different
  frames
• Two relativistic invariants

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Objectives (3)

• The 4-current Jm
• Homogeneous Maxwell's equations in covariant form
• Inhomogeneous Maxwell's equations in covariant
  form
• Gauge transformations
• All of EM in compact notation

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4-Current
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Current
Does it transform like a 4-vector? Charge q at
X(t)
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Problem
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Problem
?
?
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Electromagnetism

• Lorentz force law
• fields act on charges / currents
• Maxwell equations
• charges / currents produce fields

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Maxwell's Equations
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Maxwell's equations

• Homogeneous equations

• Inhomogeneous equations

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Homogeneous equations
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Homogeneous equations
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• Homogenous equation can also be written as

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Inhomogeneous equations
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Inhomogeneous equations
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Maxwell's equations

• Explicitly covariant
• Equivalent to usual form of Maxwell's equations

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Express in Terms ofPotential
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In terms of potential

• Homogeneous

Satisfied automatically
Inhomogeneous
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• Equation coupled
• Unless ?m Am term can be removed

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Gauge transformation
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Gauge transformation
Physics is not changed
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But
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• 4 decoupled equations

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Summary
OR
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Objectives (3)

• The 4-current Jm
• Homogeneous Maxwell's equations in covariant form
• Inhomogeneous Maxwell's equations in covariant
  form
• Gauge transformations
• All of EM in compact notation

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Acknowledgment

• This project is supported in part by the Hong
  Kong University Grants Committee (UGC) Teaching
  Development Grants (TDG) 3203005 and 3201032
• I thank Prof. S.C.Liew for software
• I thank Prof. M.C.Chu and Dr. S.S.Tong for advice
• I thank Miss H.Y.Shik for design