UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede

LECTURE NOTES 19

LORENTZ TRANSFORMATION OF ELECTROMAGNETIC FIELDS (SLIGHT RETURN)

Before continuing on with our onslaught of the development of relativistic electrodynamics via tensor analysis, I want to briefly discuss an equivalent, simpler method of Lorentz  transforming the EM fields EB and from one IRF(S) to another IRF(S'), which also sheds some light (by contrast) on how the EM fields Lorentz transform vs. “normal” 4-vectors.

In P436 Lecture Notes 18.5 {p. 18-22} we discussed the tensor algebra method for Lorentz transformation of the electromagnetic field e.g. in the lab frame IRF(S), represented by the EM field strength tensor F v to another frame IRF(S'), represented by the EM field strength tensor Fv via the relation:

vv T FF    n.b. in matrix form: FF    F  since  is symmetric.

Analytically carrying out this tensor calculation by hand can be tedious and time-consuming. If such calculations are to be carried out repeatedly/frequently, we encourage people to code this up and simply let the computer do the repetitive work, which it excels at.

For 1-dimensional Lorentz transformations (only) there is a simpler, less complicated, perhaps somewhat more intuitive method. Starting with the algebraic rules for Lorentz-   transforming { EB and } in one IRF(S) to { EB and } in another IRF(S') e.g. moving with  relative velocity vvx ˆ with respect to IRF(S):

2  component(s): EEx  x Bx  Bx  11

 components: EEcByyz ByyzBEc   vc

EEcBzzy  Bzzy BEc

We can write these relations more compactly and elegantly by resolving them into their  and   components relative to the boost direction: here,  is along vvx ˆ and  is perpendicular   to v , defined as follows {n.b in general, v could be  e.g. to xˆˆ,,yzˆ or r ˆ}:

  EE   vc   EEvBEcB     11 2

B B 11      B BvEB2 E cc

     Now since vvx ˆ {here} then: EE x , B  Bx and: B Byyzˆ Bzˆ , E Eyyzˆ Ezˆ {and similarly for corresponding quantities in IRF(S')}.

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 1 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede

   Since vvx ˆ and EE x then: vE vExx ˆ 0   And likewise, since B  Bx then: vB vBxx ˆ 0 .

   Thus, we can {safely} write: vE vE and vB vB, as long as v is always   to one of the components of E and B - e.g. xˆˆ or yz or ˆ .

Then we can write the Lorentz transformation of EM fields as:

EE   vc     EEvBEcB    vc B B 11    2 B BvEB2 E  11 cc

This can be written more compactly in 2-D matrix form as:

EM Fields: “Normal” 4-Vector: EE     → EE and BB  compare to  x  x cB cB  EE10  EE x x    ↔       cB 01  cB cB    cB ct    ct

Unit Matrix Operator Matrix Scalar Matrix

Thus, we see that for the EM fields vs. the 3-D space-part of a “normal” 4-vector, the  vs.   components are switched, B transforms “sort of” like time t, but 2  2 Lorentz boost matrices  for ( EB and ) vs. 4-vectors are not the same (they are similar, but they are not identical).

We can also write compact inverse Lorentz transformations (e.g. from IRF(S') rest frame → IRF(S) lab frame):

EM Fields: “Normal” 4-Vector: EE     → E EBB and  compare to  x  x cB cB  EE10  EE   x x     ↔          cB 01  cB cB   cB  ct   ct 

Unit Matrix Operator Matrix Scalar Matrix

2 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede  For a general Lorentz transformation (i.e. no restriction on the orientation of v {arbitrary}):

A.) Lorentz transformation from IRF(S) → IRF(S'):

  2    EEcB   E   vc   vc  1   2  1  2 B BE  B  11 c  1 Or:    1 EE  1    cB  cB    1     1  operator matrix

B.) Inverse Lorentz transformation from IRF(S') → IRF(S):

  2  EEcB  E    1 Switch   , E  E and 2  1    B  B in above relations B BE  B  c  1 Or:    1 EE 1      cB cB    x 1     1  operator matrix

Electrodynamics in Tensor Notation

So now that we know how to represent the EM field in relativistic tensor notation (as FGvv or ), we can also reformulate all laws of electrodynamics (e.g. Maxwell’s equations, the Lorentz force law, the continuity equation {expressing electric charge conservation}, etc. . . ) in the mathematical language of tensors.

In order to begin this task, we must first determine how the sources of the EM fields  – the electric charge density  (a scalar quantity) and the electric current density J (a 3-D vector quantity) Lorentz transform.

The electric charge density   QV = charge per unit volume (Coulombs/m3)

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 3 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede

Imagine a cloud of electric charge drifting by. Concentrate on an infinitesimal volume V  containing charge Q moving at (ordinary) velocity u :

Then:   QV = charge density (Coulombs/m3)

  And: Ju  = current density (Amps/m2).

A subtle, but important detail:

If there is only one species (i.e. kind / type) of charge carrier contained within the  infinitesimal volume V, then all charge carriers travel at the same (average / mean) speed u .

However, if there are multiple species (kinds / types) of charge carriers (e.g. with different masses) and/or different signs of charge carriers contained within the the infinitesimal volume V  2 2 (e.g. electrons e with rest masses mec and protons p with rest masses mpc ) then the different constituents / species must be treated separately in the following:

If  N species:   th Current density Juiii  for the i species (i = 1, . . ., N), the electric charge density ii QV NN And: JJiii u ii11  We also need to express  and J in terms of the proper charge density 0 = volume charge density defined in the rest frame of the charge Q, IRF(S0).

The infinitesimal rest volume / proper volume = V0 {defined in the rest/proper frame IRF(S0)}

The proper charge density:   QV ← Recall that electric charge Q (like c) is 00 a Lorentz invariant scalar quantity

Because the longitudinal direction of motion undergoes Lorentz contraction from the rest frame IRF(S0) in the Lorentz transformation → another reference frame, e.g. lab frame IRF(S)

1 1 u Then: VV 0 where: Vwd  and: Vwd  , where:   and:    0000 u 2 u c u 1 u

If the Lorentz transformation is along (i.e. || to) the length, 0 of the infinitesimal volumes 1 Then:  0 and the  components of the volumes are unchanged: ww0  , dd0  .  u

1 QQ    Then if: VV 0 →  uu  0  Ju uu00 u    u  u VV0    d   Recall that the 3-D vector associated with the proper velocity is: uu   J  0 d

4 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede

dx0 dt The zeroth (i.e. temporal/scalar) component of the proper 4-velocity is:  0 cc  ddu

The corresponding zeroth (i.e. temporal/scalar) component of the current density 4-vector J  is:

00 Jcccc00uu  0  

  0 2 The current density 4-vector is: JJJcJcJJJ,,,,,x yz (SI units: Amps/m )

 0   0   Then: J 000, where: ,,uuc u  u cu ,   u cu ,,, xyz u u constant scalar quantity

 J  is a proper four vector, i.e. J  = proper current density 4-vector.

  Thus: JJ JJ is a Lorentz invariant quantity. Is it ???

2 22 1 u cu 222 JJJJ 2   222222  cuuu   2   c c   c 00  uxyz 0uu22 0  0 1122  u2 cc 

2 22   22 2 JJ JJ00 c   0   0 c Since:   c

  Yes, JJ JJ is a Lorentz invariant quantity!

The 3-D continuity equation mathematically expresses local conservation of electric charge (using differential vector calculus):

   rt,   Jrt,  rt, = scalar point function, JJrt , = 3-D vector point function t

  xˆˆ yz  ˆ (in Cartesian coordinates) x yz

We can also express the continuity equation in 4-vector tensor notation: n.b. 3 i 00 repeated  J J y J J  11 c JJ J x  z  and:  ( Jc0   ) indices x yz  xi tc t ct  x0 implies i1 summation!

    33JJii00  J  J 3 JJ    0 0 Then: J  J 0  ii00 =   t t ii11xx  x  x  0 xx     J  Thus: J or J 0   0 Continuity equation (local charge conservation) t t x

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 5 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede

J  Physically, note that is the 4-dimensional space-time divergence of the current density x   J  4-vector JcJ  , . The 4-current density JcJ  , is divergenceless because  0 .   x  The 4-vector operator is called the 4-D gradient operator, (a.k.a the quad operator  x   or “quad” for short). However, because the 4-D gradient operator functions like a covariant x 4-vector, e.g. when it operates on contravariant J  (or any other contravariant 4-vectors), it is  often alternatively given the shorthand notation   . Similarly, because the 4-D x   gradient operator functions like a contravariant 4-vector, e.g. when it operates on x    covariant J (or any other covariant 4-vectors), it is given the shorthand notation . x {See/work thru Griffiths Problem 12.55 (p. 543) for more details.}

1  The contravariant quad/gradient 4-vector operator:  ,    xct

 1  The covariant quad/gradient 4-vector operator: ,    xct 

22   1  22 Then: 22      xx  xx  xx  ct  = D’Alembertian 4-vector operator = 4-D Laplacian operator = Lorentz-invariant quantity!

The 4-vector product of any two (bona-fide) relativistic 4-vectors is a Lorentz invariant quantity (i.e. the same value in any/all IRF’s)!

We can equivalently write the relativistic 4-D continuity equation as:

J    JJ0  n.b. A Lorentz-invariant quantity!!! x 

 Electric charge is (locally) conserved in any/all IRF’s (as it must be!!!)

6 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede

Maxwell’s Equations in Tensor Notation

Maxwell’s Equations:  1 1) Gauss’ law: E  o  2) No Magnetic Monopoles: B 0   B 3) Faraday’s law: E  t   1 E 4) Ampere’s law: BJ   with Maxwell’s Displacement Current o ct2 

Can be written as 4-derivatives of the relativistic EM field strength tensors F v and Gv :

F v Gv FJv  and: G v 0 voxv v xv

n.b. summation over v = 0:3 is implied

 1 F v F 0v E  FJv  00v 1) Gauss’ law . If μ = 0 in: vov , i.e. voFJv  , v = 0:3 o x x

Physically, μ = 0 is the temporal/scalar component of any space-time 4-vector.

FFFFF0v 00010203 Then:  μ = 0 (first row of F v ) xv xxxx0123

Row #

0 Ecxyz Ec Ec  Ecxzy0 B B F v   Ecyz B0 B x  Eczyx B B 0 Column #

0v E  F 11Ex y Ez 0  0 0    E and: ooJc  xcxyzc

   1 2 2 1 1  E oc or: Eco  but: oc   E  c o o

Gauss’ law arises from the   0 (scalar / temporal) component of the 4-vector relation:

F v FJv   voxv

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 7 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede   1 E F v 4) Ampere’s law: BJ   If  1 in FJv   o ct2  voxv

FFFFF1v 10111213 Then:    1 (second row of F v ) xv xxxx0123  1v B  FE11Ex Bz y 1  0     B and: 0 JJ ox xctv 22 yzct x  FE1v 1    B oxJ xctv 2 x

Then for   2 and  3 (third and fourth rows of F v ), likewise we find that:   FE2v 1  FE3v 1   B oyJ and:  B ozJ xctv 2 xctv 2 y z  F v 1 E      J   B  J xv 1:3o 1:3 ct2  o

3-D spatial components of 4-vector J    1 E Or: BJ   Ampere’s law with Maxwell’s displacement current term !!! o ct2 

 Ampere’s law arises from the  1:3 (3-D spatial / vector component) of 4-vector relation:

 1 E  (0  temporal / scalar component)  F v o FJv   v xv 0  1 E  B    J (1:3  3-D spatial / vector component) ct2  o

Thus, Gauss’ law and Ampere’s law form a 4-vector:

0 o J      v  11 1EFv oovJE ,  B2  J v  F cc ct x   o   

Gauss’ law temporal / scalar Ampere’s law 3-D spatial / vector  0   component 0 J of 4-vector o J component 0 J of 4-vector o J

22 And: JJ JJ0 c = Lorentz invariant quantity {from above, page 5}.

8 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede

 G v 2) B 0 no magnetic monopoles / no magnetic charges. If   0 in G v 0 v xv Gv    0 is the temporal (scalar) component of space-time “null” 4-vector G v 00,00 v xv  GGGGG0v 00010203 Then: 0   0 (First row of Gv ) xxxxxv 0123

Row #

0 BBBxyz  Bxzy0 Ec Ec Gv   ByzEc0 Ec x  BEcEczy x 0 Column #

G0v B B B   G0v  000x y z B  B 00  xxyzv xv   B Gv 3) Faraday’s law: E If  1 in G v 0 t v xv

GGGGG110111213v Then: 0   1 (Second row of Gv ) xxxxxv 0123  1v GB1111B E Ey  x 00 z   E  xctcyczctv x  G1v B    0 gives E 0 xv t x Likewise, for   2 and  3 (third and fourth rows of Gv )   G2v B  G3v B   0 gives E 0 and:  0 gives E 0 xv t xv t y z   Gv B   B   0 gives E 0 or: E  xv 1:3 t t  B 0 (   0 temporal / scalar component) Gv Thus:  G v 0 v v  x  B E  (  1:3 3-D spatial / vector component) t Gv Arise from temporal (   0 ) and spatial (  1:3) component of the “null” 4-vector G v 0 v xv

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 9 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede

Thus, in relativistic 4-vector / tensor notation, Maxwell’s 4 equations {written in language of 3-D differential vector calculus}:

Maxwell’s Equations:  1 1) Gauss’ law: E   o  2) No Magnetic Monopoles: B 0   B 3) Faraday’s law: E  t   1 E 4) Ampere’s law: BJ   with Maxwell’s displacement current o ct2  are elegantly represented by two simple 4-vector equations:

 1   0 temporal/scalar component: E  1) Gauss’ law  o F v FJv   voxv  1 E   1:3 spatial/vector component: B    J 4) Ampere’s law ct2  o With Maxwell’s displacement current

   0 temporal/scalar component: B 0 2) No Magnetic Charges G v G v 0 v xv  B  1:3 spatial/vector component: E  0 3) Faraday’s law t

Griffiths Problem 12.53:   B We can show that Maxwell’s two equations B 0 and E  0 that are contained in t G v G v 0 can also be obtained from (the more cumbersome/inelegant relation): v xv

G v FFF n.b. note the cyclic G v 0  FFF v v  0  permutations of v xv vvv    xxxv indices ,  and v.

Since there are 3 indices in the latter equation   0 :3, v 0 :3,   0 :3 , there are actually 64 (= 43) separate equations!!! However, many of these 64 equations are either trivial or redundant.

10 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede

Suppose two indices are the same (e.g.   v )

FFF Then:    0 . But the EM field strength tensor F (like F v ) is anti-symmetric. xxx v

 F  0 and: FF  . Thus,  2 indices the same gives the trivial relation: 0 = 0.

Thus, in order to obtain a / any non-trivial result, μ, v, and λ must all be different from each other.

1) The indices μ, v, and λ could all be spatial indices, such as: μ = 1 (x), v = 2 (y), λ = 3 (z) (or permutations thereof).

Or:

2) One index could be temporal, and two indices could be spatial, such as: μ = 0, v = 1, λ = 2 (or permutations thereof), or: μ = 0, v = 1, λ = 3

1) For the case(s) of all spatial indices, e.g. μ = 1, v = 2, λ = 3:

B  F12 FF23 31 Bz Bx y 0 = 0 = B 0 xxx312 zxy  All other permutations involving the all-spatial indices {1, 2, 3} yield the same relation B 0  or minus it: i.e. B  0 .

2) For the case of one temporal and two spatial indices, e.g. μ = 0, v = 1, λ = 2:

E E FF01F12 20 111Ex Bz y 11Bz y Ex 2010 =  0 = 0 xxx cy ct cx ct c x y  B  = E 0 t z Other Permutations:

For v = 0, μ & λ = 1:3 and λ = 0, μ & v = 1:3 get redundant results (same as above).

If μ = 0, v = 1, λ = 3 get y – component of above relation! If μ = 0, v = 2, λ = 3 get x – component of above relation!  FFF   B  v v 0 contains: B 0 and: E  xxxv t

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 11 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede

Duality Transformation of the Relativistic EM Field Strength Tensors F v and Gv :  The duality transformation for the specific case of space-time “rotating” EcB and  o vv cB E ( duality = 90 ) takes FG , and can be mathematically represented in tensor notation as: vv1 GF 2  

 where: F is the {doubly} covariant form of the contravariant tensor F . and:  v is the totally anti-symmetric rank-four tensor.

+1 for all even permutations of μ = 0, v = 1, λ = 2, σ = 3  v = 0 if any two indices are equal/identical/the same. 1 for all odd permutations of μ = 0, v = 1, λ = 2, σ = 3

Since  v is a rank-four tensor (= 4-dimensional “matrix”) we can’t write it down on 2-D paper all at once!  v has (μ, v, λ, σ = 0:3) → 44 elements = 256 elements!!!

We could write out 16 {44} matrices – e.g. one μ-v matrix for each unique combination of λ and σ:

000 vvv    0 v   012(4 4) (4 4) (4 4)  3 (4 4) 111 vvv   1 v   v 012(4 4) (4 4) (4 4)  3 (4 4)    222 vvv    2 v   (4 4) (4 4) (4 4) (4 4) 01 2 3 333 vv   vv   3  012(4 4) (4 4) (4 4) 3 (4 4)

Define  = 4×4 totally anti-symmetric rank-two tensor (aka the 4×4 Levi-Cività symbol):

 0111  1 0 1 1     11 01   111 0

Then, we can define  v in terms of the product of two   's:  vv   

12 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede

The Minkowski / Proper Force on a Point Electric Charge

The Minkowski force (a.k.a. proper force) K  acting on a point electric charge q can be written in 4-vector / tensor notation in terms of the EM field strength tensor F v and the proper 4-velocity   . Recall that:

dp dp dp KF   where the ordinary force: F   and:  11 2 ddt uu dt uu

 v However, we can equivalently write the Minkowski/proper force as: KqF v v where v is the covariant form of the contravariant proper 4-velocity . v i.e. we contract the EM field strength tensor F with the covariant proper 4-velocity v .

 2 Since: KF  u and: vuv  u , where: uu11 and: u  uc

 v  v  v  KqF v  uuvFquF or: FquF v  ordinary force on point charge

1 1v 010111212 313 If μ = 1 (i.e. row #1): KqFqv  F F F F

  2 ucuv  , ↔ vuuuvcu,  u where:  uu11 and:   uc

Row # 0 0 Ecxyz Ec Ec K 0 Ecxyz Ec Ec0   Ec0 B B K1 Ec0 B B  F v xzyqxzy1  2  Ecyz B0 B x K Ecyz B0 B x  2 Ec B B 0 3 Ec B B 0 zyx K zyx 3 Column #  1   Kquxxyzzyuxyzzyu cEcu0 uBuB qEuBuBqEuB     x

  1   KqEuB u Kq u EuB ← Minkowski 3-D Force law x 

Similarly, for μ = 2, μ = 3:   2  KqEuB u But: KF  u y    3   KqEuB u  FqEuB ← Lorentz 3-D Force law z 

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 13 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede

For μ = 0 (the temporal / scalar component) {see/work Griffiths Problem 12.54, page 541}:  0  Kquxxyxzxu c0 uEcuEcuEcquEc   

000 101 202 303 0v  FF    F   FF v

n.b. this relation explicitly shows that Ex , Ey , Ez are temporal-spatial (or spatial-temporal) v v components of F , whereas Bx , By , Bz are pure spatial-spatial components of F !!!

dp0 11 dE dt dE 1 dE   1 dE We also know that: K 0     KquEc0  dcdcddtcdt u uucdt

dE   or: PquE = {ordinary} relativistic power delivered to point charged particle (> 0) q dt 

dE dW Time rate of change of work done PquEqEuFu    = q dt dt on changed particle by EM field

  Note: The {ordinary} Lorentz force: FqEquB 

    Fu quE q u  u B quE      

   But: uB u  uuB0  magnetic forces do no work !!!

v v Thus we have the relations: KF  u = KqF v and also: FquF v with vuv  u .

The Relativistic 4-Vector Potential A  We know that the electric and magnetic fields EB and can be expressed in terms of a scalar  potential V and a vector potential A as:

  Art,  Ert,, Vrt   and: Brt,, Art  t  Thus, it should not be surprising to realize that the scalar potential V and the vector potential A form the temporal and spatial components (respectively) of the relativistic 4-vector potential A :

  SI Units: Newtons/Ampere = “p/q” The 4-Vector Potential: AVcAVcAAA,,,,x yz  {= momentum per Coulomb!}

Newton- meters V N-- m m N sec Newtons n.b. SI units of V: Volts  then:  == Coulomb c Coul sec Coul Amp

14 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede

The EM field strength tensor F v can be written in terms of covariant space-time derivatives of the 4-vector potential field A as:

AAv  FAAvvv      n.b. covariant differentiation here!! x xv n.b. We must change the sign of the temporal/scalar component of the covariant derivatives

 v   and  relative to that of the contravariant derivatives   and v v . x xv x x  Explicitly evaluate a few terms: AVcA  ,

For μ = 0 and v = 1:  10  01 AAAE Vc 11  A  A FVVx x x xctxctc     tc 01  xx

Likewise, for (μ = 0, v = 2) and (μ = 0, v = 3) we obtain:     02 1 A Ey 03 1 A E FV  and: FV z ctc ctc y z

For μ = 1 and v = 2:

21   12 AAAy Ax FAB  z z xx12  xy

Likewise, for (μ = 1, v = 3) and (μ = 2, v = 3) we obtain:     13 23 FAB   y and: FAB  x y x

Note that the relativistic 4-potential formulation automatically takes care of the  G v   B homogeneous Maxwell equation G v 0 {it gives B 0 and: E  } v xv t G v FFF because G v 0 is equivalent to v v 0 . v xv xxxv

{See/read pages 10-11 of these lecture notes – also see/work Griffiths Problem 12.53, page 541}.

AAv  A A FAAvvv     FAA  v   And since:   vvv  v x xv x x

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 15 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede

FFF Thus: v v 0 xxxv

AAAAvvAA  = vv    v    0 xx  x  xx  x  xx   x

AAAAAA = vv0 xx  xx vv  xx   xx   xx vv   xx 

AAvv AA    AA =  vvvv      0 xx  xx  xx  xx   xx  xx

      =  AAv vv    vv  A0 xx  xx   xx  xx   xx  xx  0 0 0

  But:  i.e. can change the order of differentiation – has no effect! x xxx  22 =   00 x xxx  AAv  F v   The relativistic 4-potential formulation  does indeed automatically satisfy x xv G v FFF Gv G v 0 because v v 0 {shown to be equivalent to  0 } v xv xxxv xv AAv  FAAvvv     is satisfied / obeyed for . x xv AAv  FAAvvv     Does the relativistic 4-potential formulation  x xv F v satisfy the inhomogeneous Maxwell relation FJv   ??? voxv

FAAAAvv22 v FJv       v vv v v o xxxxvv  xxxx

Switching the order of derivatives: vv v FAA   voFJvvv     xxxxx  v 

This is an intractable equation, as it stands now…

16 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede

However, from our formulation of F v in terms of (differences) in space-time derivatives of AAv   FAAvvv     the 4-vector potential A :  it is clear that we can add to the x xv  4-vector potential A the space-time gradient of any scalar function λ: AAA* !!! x   The scalar and vector potentials V and A are not uniquely determined by the EM fields E and B .

Thus:

AA**vv  A  A  AA v 2 2 F *v      xxvvvvvv  xxxxxxxx  xxxv x   0 AAv  F v !!!  xx v

  FF*vv  by AAA* is gauge invariance associated with the EM field F v !!! x

We can exploit the gauge invariant properties of F v to simplify the seemingly intractable relation:

vv v FAA   voFJvvv     xxxxx v 

  1 V   1 V A Using the Lorenz gauge condition: A  A 0  A 0 ct2  ct2   x

0 !! FAAvv  2 A 2 A v   v AJ    Thus: voFJvvv     = v o J or: vov xxxxx v  xxv xxv

22 2 vv 2 vvvv2 1  But: vv v ,   and: vv v v vv 22 x xv xvvxxx  ct  D’Alembertian operator (4-dimensional Laplacian operator)

2 A Single 4-vector equation! 2  v   = the most elegant and simple formulation of Maxwell’s   AAJ vo  = v o J  xxv equations – it contains all four of Maxwell’s equations!!

 Taken together with the continuity equation (charge conservation): JJ 0 , these two relations compactly describe virtually all of {non-matter/free space} EM phenomena!!!

© Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 17 2005-2015. All Rights Reserved. UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 19 Prof. Steven Errede   Note that the choice of the (instantaneous) Coulomb gauge A 0 is a bad one for use in   relativistic electrodynamics, because A {alone} is not a Lorentz invariant quantity!

   1 Vrt,, A  rt However: Art,,0  A  rt  is a Lorentz invariant quantity ct2  x  because it is the product of two relativistic 4-vectors:  and A .  x

  n.b. A 0 is “destroyed” by any Lorentz transformation from one IRF(S) to another IRF(S') !!!

   In order to restore A 0 , one must perform an appropriate gauge transformation for each new inertial system entered, in addition to carrying out the Lorentz transformation itself !!!

   In the Coulomb gauge A is not a “true” relativistic 4-vector, because AA AA is actually not a Lorentz invariant quantity in the Coulomb gauge !!!

  n.b. The Coulomb gauge A 0 is useful when vc , i.e. for non-relativistic problems.

18 © Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005-2015. All Rights Reserved.