Lehner G. Electromagnetic Field Theory for Engineers and Physicists

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624 Appendices

A.6.3.2 Time Dilatation

Let an object (e.g. a clock) be located at position , which emits signals at

different times with . Then for system Σ’ one has

(A.6.29)

From the perspective of the moving system, the temporal distance appears

expanded by the factor . Therefore, a clock runs slower and slower as it

moves faster and faster. This is the so-called time dilatation. This can be proven

experimentally, e.g., by means of very fast instable particles (radioactive decay),

where its half-life relative to an observer at rest is elongated versus its half-live in

the rest frame. There are so-called µ-mesons contained in high altitude radiation.

They are instable and very clearly show the effect of time dilatation. Time

dilatation is also the root cause of the so-called twin paradox.

A.6.3.3 Relativistic Addition of Velocities

Let a particle in the Σ’ system have the velocity . What is its

velocity in the Σ frame? First, using , gives

and

(A.6.30)

Therefore

(A.6.31)

Furthermore

and with eq. (A.6.30) we obtain

∆tt

–=

∆t' t

' t

'–

-----

–





-----

–





–

1 β

–

------------------------------------------------------------

–

1 β

–

-------------------

∆t

1 β

–

-------------------== = =

∆t'

∆t

1 β

–

-------------------

11β

–⁄

u' u

' u

',,〈〉=

β vc⁄ v

c⁄==

--------

dt'

--------- v

1 β

–

---------------------

dt'

------

⋅==

dt'

------

-----------–

1 β

–

--------------------

1 β

–

------------+

----------------------==

' v

------------+

----------------------

, u

–

------------–

---------------------

--------

dt'

---------

dt'

------

⋅ u

dt'

------

,== = u

dt'

------

A.6 Maxwell’s Equations and Relativity 625

(A.6.32)

Eqs. (A.6.31) and (A.6.32) represent the relativistic theorem for addition of

velocities, which for small velocities ( ) reduces to the classical vector

addition ( ).

Consider a few special cases: For one has also .

If , then , where

and thus . Next, more general, let

then

(A.6.33)

where

(A.6.34)

Whereby

and

(A.6.35)

This means that an object (for example. light), moving with the speed of light, has

the same speed in every reference frame. This is, of course, necessary and was the

starting point of our reflections (Sect. 6.1). Nevertheless, the direction of the

motion (that is the propagation direction of e.g. light) depends on the reference

frame. This is the so-called aberration (of light), which is described by the

relations (A.6.34).

-----------–

1 β

–

--------------------

⋅ u

1 β

–

------------+

----------------------

⋅==

-----------–

1 β

–

--------------------

⋅ u

1 β

–

------------+

----------------------

⋅==

vc«

' v

' u

'=,=,+=

u' c 00,,〈〉= u c 00,,〈〉=

u'0c 0,,〈〉= u v

c 1

-----–0,,〈〉=

1 v

⁄–()+ c

==uc=

u' c α'cos c α'sin 0,,〈〉=

c α'cos v

α'cos

-------------------+

-----------------------------

c α'sin 1

-----–

α'cos

-------------------+

-----------------------------------

0,,〈〉c αcos c αsin 0,,〈〉==

αcos

α'cos

-----+

α'cos

-------------------+

-----------------------------= αsin

α'sin 1

-----–

α'cos

-------------------+

--------------------------------

αcos

αsin

+ α'cos

α'sin

+1==

uu' c==

626 Appendices

For , the velocity is independent of u’. This also is an

immediate and plausible consequence of the invariance of the speed of light.

A.6.3.4 Aberration and Doppler Effect

Consider a moving, point-like light source from which a spherical wave emerges

(Fig. A.6.1). At point P, an observer in the rest frame Σ sees it at the angle α, while

an observer moving with the light source (reference frame Σ’) sees the source at the

angle α’. The quantities measured then in Σ have the phase factor

and in Σ’, the phase factor is

These have to be compatible with the Lorentz transformation and mutually

transform into each other. This requires that

By comparing coefficients of t we find

(A.6.36)

that is, the angular frequency ω depends on the reference system.

This represents the so-called relativistic Doppler effect. Comparing coefficients of

and , while also considering (A.6.36), results in eq. (A.6.34), i.e. the

aberration of light.

c= u c 00,,〈〉=

Fig. A.6.1

α'

i ωt

----

αcos–

----

αsin–





exp

i ω't'

ω'

-----

' α'cos–

ω'

-----

' α'sin–





exp

ω't'

ω'

-----

' α'cos–

ω'

-----

' α'sin–

ω' t

----------–





1 β

–

-----------------------------

ω' x

t–()

c 1 β

–

-----------------------------

α'cos–

ω'

-----

α'sin–=

ωt

----

αcos–

----

α .sin–=

ωω'

-----

α'cos+

1 β

–

-----------------------------

A.6 Maxwell’s Equations and Relativity 627

These results amount to the phase being relativistically invariant,

to which we will return in Sect. A.6.5.2.

A.6.4 Lorentz Transformation of Maxwell’s equations

The experiments mentioned in Sect. A.6.1 to determine the speed of the earth

within the absolute space (ether) had to fail, because electromagnetic waves in

vacuum propagate in all inertial systems with the same speed, the vacuum-speed of

light c. Ultimately, the reason for this is that the Maxwell equations are not Galilei

invariant, but Lorentz invariant, which was only revealed by, and understood in

light of Einstein’s theory of relativity. Today, we are able to realize that Maxwell’s

equations were, and still have to be, the starting point for relativity and, even

though it was not apparent when Maxwell stated them, his equations always

contained relativity. This is the reason why it is so important to show that the

Maxwell equations are indeed Lorentz invariant, that is, that these equations apply

unchanged in all inertial reference frames, as long as the field quantities (i.e. the

components of E and B) are transformed in a suitable manner.

To show this, we consider the two reference frames Σ and Σ’. In Σ we have

the coordinates r, t and the fields E(r,t), B(r,t), in Σ’ we have the coordinates r’, t’

and the fields E’(r’,t’), B’(r’,t’). We limit our examination to free space. Later we

will apply a generally applicable formalism which allows in an elegant way to treat

the general Maxwell equations. For the frame Σ, Maxwell’s equations apply in the

following form

(A.6.37)

The transformation of r, t by the Lorentz transformation to r’, t’ is not difficult but

tedious and therefore skipped here (this calculation can be found e.g. in Simonyi

[40]) and results again in Maxwell’s equations,

(A.6.38)

The transformation of the field components is as follows:

kr•ωt–

B∇×

-----

t∂

∂E

----- ε

E∇×

t∂

∂B

–=

B∇• 0=

E∇• 0=











∇' B'×

-----

∂E'

∂t'

--------

∇' E'×

∂B'

∂t'

--------–=

∇' B'• 0=

∇' E'• 0=











628 Appendices

(A.6.39)

When using , this can be written in a more elegant form

(A.6.40)

Only the field components perpendicular to the velocity v change, while the

parallel ones remain unchanged. This is just the reverse from what was found for

the components of the location vector. This is no accident, as we shall see more

clearly. The nabla symbols for curl and divergence have a prime as a

reminder that E’ and B’ are functions of r’, t’, and therefore the derivative in eqs.

(A.6.38) is with respect to . Conversely, eqs. (A.6.40) are

independent of any particular coordinate system and apply for arbitrary velocities

The force on a particle in system Σ is f, and f ’ in system Σ’

(A.6.41)

This means that the forces on a particle have the same form in every reference

frame, i.e., they are calculated within their respective reference frame from the

electric and magnetic fields in the same way. Nevertheless . The equations

for the transformation of f will be provided in Sect. A.6.5.2, eq. (A.6.72), which

incidentally, can be used to derive the equations for the transformation of the

electromagnetic fields. The charge Q is scalar, i.e., is not transformed, which

means that . Remarkable and easy to verify is also that the two following

quantities are Lorentz invariant:

(A.6.42)

The reader shall be reminded of Sect. 6.1.2 and particularly eq. (6.6). Comparison

with A.6.40 reveals that there, if we consider that E is the electric field in the

moving frame, we had applied the non-relativistic approximation, i.e. the

approximation for velocities .

' E

,= E

–

1 β

–

-----------------------

,= E

1 β

–

-----------------------=

' B

,= B

-----

1 β

–

------------------------

,= B

-----

–

1 β

–

-----------------------=











v v

00,,〈〉=

' E

,= E

⊥

vB×+

1 β

–

---------------------------=

' B

,= B

⊥

-----

vE×–

1 β

–

--------------------------------=











∇' ×∇' •

r' x

' x

',,〈〉=

v v

,,〈〉=

f Q EuB×+()=

f ' Q E' u' B'×+()=







ff'≠

QQ'=

-----

– B

-----

,–= E' B'• EB•=

c 1 β

–1→,«

A.6 Maxwell’s Equations and Relativity 629

Furthermore, the reader shall be reminded of Appendix A.4. There, we have

introduced to the Liénard-Wiechert potentials and calculated the problem of a

uniformly moving charge as a specific example. Within the moving frame, the

particle exhibits exclusively an electric Coulomb field and no magnetic field. The

just provided transformation allows one to obtain the electric and the magnetic

field, as experienced by the observer at rest. Carrying out the transformation yields

precisely the field obtained in Appendix A.4, which demonstrates the fact that

Maxwell’s equations automatically provide the relativistically correct result. This

result is limited to the special case of constant velocities because the Lorentz

transformation is valid only for inertial systems, but not for accelerated reference

frames. We will return in Sect. A.6.6.4 to the discussion of a charge of uniform

velocity.

We will carry out the transformation of the fields in a more systematic and

more elegant manner. Thereby, it will be revealed that the field components are

simply components of an anti-symmetric four-dimensional tensor of rank two

(which possesses just six independent components). From this follows immediately

how to transform the field components. Moreover, this tensor allows one to express

Maxwell’s equations in a particularly compact and elegant form, which

furthermore, makes its Lorentz invariance instantly plausible. This requires to

occupy ourselves with the four-dimensional vectors of the Minkowski space (the

so-called 4-vectors) and their corresponding Four-tensors, i.e., the tensor products

of the 4-vectors.

A.6.5 4-Vectors and 4-Tensors

A.6.5.1 Definitions

When changing the reference frame of a 4-vector

(A.6.43)

where the fourth component is , then the Lorentz transform

(A.6.44)

is applied and it holds that

(A.6.45)

Generally, only vectors that transform in this manner are called 4-vectors

(A.6.46)

Quantities which transform like multiple products of the components of 4-vectors

are called tensors of the respective rank. The expressions

r ict,〈〉x

,,,〈〉=

ict=

()=

' L

k 1=

∑

' L

k 1=

∑

630 Appendices

(A.6.47)

, (A.6.48)

represent a tensor of rank 2, rank 3, etc. In this sense, a 4-vector is a tensor of rank

1. A scalar quantity is a tensor of rank zero, for which

(A.6.49)

The scalar product (corresponding to the summation over a common index, which

is also referred to as rank reduction) reduces the rank of the tensor by one each

time. The scalar product of two 4-vectors represents an invariant scalar

quantity. The scalar product of a 4-vector and a 4-tensor of rank two

yields a 4-vector, etc.

The coefficients of the Lorentz transform for are given by

(A.6.24) and by (A.6.27) for .

A.6.5.2 Some Important 4-Vectors

An important vector in the three-dimensional space is the -vector. In the four-

dimensional space it becomes a 4-vector:

(A.6.50)

Now, we explore the continuity equation (1.58) in this context

which is nothing else than the four-dimensional divergence of the 4-vector of the

current density or short 4-current density

(A.6.51)

It is a very remarkable and consequential result that the three components of the

current density, together with icρ (that is the volume charge density, for the most

part) form a 4-vector. That this is true, can be shown on one hand, but is also a

result of the fact that the divergence of the 4-vector vanishes, i.e. it is invariant.

The Lorentz gauge (eq. (7.186)) represents a four-dimensional divergence

(A.6.52)

m 1=

∑

l 1=

∑

ikl

mnp

p 1=

∑

n 1=

∑

m 1=

∑

dd'=

i 1=

∑

i 1=

∑

v v

00,,〈〉=

v v

,,〈〉=

∇

∂

∂x

--------

∂

∂x

--------

∂

∂x

--------

∂

∂x

--------

,,,〈〉

∂

∂x

--------

∂

∂x

--------

∂

∂x

--------

----

∂

∂t

----

,,,〈〉 ==

∇ g

∂ρ

∂t

------+•

∂

∂x

--------

∂

∂x

--------

∂

∂x

--------

----

∂

∂t

----

icρ()+++ 0==

g icρ,〈〉g

icρ,,,〈〉 =

A∇• µ

t∂

∂ϕ

+ A∇•

-----

t∂

∂ϕ

∂

∂x

--------

∂

∂x

--------

∂

∂x

--------

----

∂

∂t

----

icϕ

--------





+++ 0==

A.6 Maxwell’s Equations and Relativity 631

This brings to light the fact, that the vector potential, together with

as its fourth component forms a 4-potential,

(A.6.53)

The scalar product of the four-dimensional -vector with itself yields the four-

dimensional analogue to the Laplacian

(A.6.54)

This scalar operator is the d'Alembertian and frequently represented by the square

symbol . This allows to combine the two inhomogeneous wave equations

(7.187) and (7.188) into a single equation:

(A.6.55)

This represents the four-dimensional inhomogeneous wave equation for the 4-

potential with the 4-current density as its inhomogeneity. It illustrates in a simple

way that here we have Lorentz-invariant equations.

The Lorentz gauge resulted in a four-dimensional divergence, which means it

is Lorentz invariant. This poses an advantage over other gauges which are not

Lorentz invariant.

Another important 4-vector, particularly for mechanical applications, is the 4-

momentum. Consider a particle in the reference frame Σ with the rest mass and

the velocity . u is used to distinguish the particle velocity from

the relative velocity v of the two reference frames Σ and Σ’. We assign the

momentum vector

(A.6.56)

to the particle in system Σ. For the reference frame Σ’, the vector shall be

(A.6.57)

The relation between u and u’ is given by eqs. (A.6.31) and (A.6.32), which allow

to derive the interesting and oftentimes useful relation

(A.6.58)

Then reapplying eqs. (A.6.31) and (A.6.32) gives

iϕ()c⁄=

A i

---

,〈〉A

---

,,,〈〉 =

∇

∇∇•

∂

∂x

--------

∂

∂x

--------

∂

∂x

--------

-----

∂

∂t

-------

–++ ∇

-----

∂

∂t

-------

–– ===

iϕ

-----

,〈〉µ

g icρ,〈〉 –=

u u

,,〈〉=

-----–

-------------------=

------–

--------------------=

------–

--------------------

-----–

-------------------

-----------–

-----–

--------------------

⋅=

632 Appendices

(A.6.59)

We continue by introducing the masses m and m’, as well as the energies W and W’

in the frames Σ and Σ’, respectively.

(A.6.60)

with the relation

(A.6.61)

Comparison of eqs. (A.6.59) and (A.6.61) with (A.6.24) reveals that the vector

(A.6.62)

is a 4-vector. We call it the 4-momentum. The quantity m is the velocity dependent

mass, which depends on the reference frame, along with the speed u of the particle.

W is the particle’s energy, which also depends on the reference frame. Eq. (A.6.60)

contains the famous Einstein relation between mass and energy ( ) and

the known relation between and . Now, the background and the reasons for

these relations becomes apparent, which rest in the fact that the vector (A.6.62) is a

4-vector. For very small velocities u, its three spatial components give the classical

momentum , which justifies the name 4-momentum.

The rest mass is invariant, the just defined velocity dependent mass is not,

however. Dividing the 4-momentum by gives another 4-vector, the so-called 4-

velocity

------–

--------------------

-----–

-------------------

-----------–

-----–

--------------------

–

-----------–

--------------------

⋅⋅

-----–

-------------------–

-----–

-------------------------------== =

------–

--------------------

-----–

-------------------

-----------–

-----–

--------------------

-----–

-----------–

--------------------

⋅

-----–

------------------- p

== ==

' p











-----–

-------------------

-----

,== m'

------–

--------------------

------

------–

--------------------

-----–

-------------------

-----------–

-----–

--------------------

⋅

-----

–

-----+

-----–

----------------------------== = =

-------

,〈〉m u ic,()

-----–

-------------------

u ic,〈〉 ==

Wmc

A.6 Maxwell’s Equations and Relativity 633

(A.6.63)

Conversely, dividing the 4-momentum by m gives another vector (u,ic), but this is

not a 4-vector. The velocity u can not be regarded as part of a 4-vector. Eqs.

(A.6.31) and (A.6.32) also reveal that u transforms rather differently than the

spatial part of the 4-vector. Only the factor creates a 4-vector.

The absolute value of the 4-momentum is invariant, of course,

for and we obtain

Rearranging reveals the important relation which we have used before, eq. (A.1.7):

(A.6.64)

When specifically considering electromagnetic radiation, one finds that it also

possesses momentum and energy, where both are proportional to each other.

Expressing both for a light quantum (which is not necessary to do), we get

(A.6.65)

and the 4-vector of the momentum becomes

(A.6.66)

Consequently,

(A.6.67)

is a 4-vector whose fourth component is, essentially, the angular frequency. We

might call it the 4-vector of the wave number whose scalar product is invariant

with the 4-vector of the location.

(A.6.68)

This invariance is nothing else than the phase, whose invariance we have already

noted in A.6.3.4. Furthermore:

(A.6.69)

We obtain the dispersion relation, which is of course invariant as well.

Another 4-vector is the 4-force

-----–

-------------------

u ic,〈〉

1/ 1 u

⁄– ()

-------–invariant=

-------– m

–=

p Ñk ,= E Ñω=

Ñ k

iω

------

,〈〉

iω

------

,〈〉

iω

------

,〈〉r ict,()• kr•ωt–invariant==

iω

------

,〈〉

------– invariant 0== =