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## Ridgely.ws

**Applying covariant versus contravariant electromagnetic tensors**

to rotating media
Charles T. Ridgely

*2936 Maple Avenue, Fullerton, California 92835*
͑Received 14 July 1998; accepted 28 October 1998͒
When the covariant form of Maxwell’s equations are applied to a rotating reference frame, a choicemust be made to work with either a covariant electromagnetic tensor

*F*␣ or a contravariantelectromagnetic tensor

*F*␣. We argue that which tensor one chooses is ultimately dictated bywhether one chooses to express the electric and magnetic fields in terms of a vector basis or in termsof a one-form basis, dual to the vector basis. We explain that when fields are expressed asone-forms, the covariant electromagnetic tensor is used; and when fields are expressed as vectors,the contravariant tensor is used. Using this formalism, we derive general field equations expressedin terms of vector and one-form fields in the rotating and laboratory frames when matter is present.

Fields in the presence of matter are then related to those in a vacuum by using a covariant form ofMinkowski’s constitutive equations, generalized to noninertial frames. Both vector and one-formfield equations are used to derive the fields observed in the reference frame of a polarizable,permeable cylinder that rotates within an axially directed magnetic field. We find that the vector andone-form field equations both lead to predictions consistent with experimental results. We concludethat the choice between working with a covariant or contravariant electromagnetic tensor dependsupon whether one chooses to express fields as vectors or as one-forms.

*1999 American Associationof Physics Teachers.*
**I. INTRODUCTION**
to provide a mathematical framework with which to explainwhy different field equations arise in a rotating frame, and to
In a recent paper,1 we demonstrated one method by which
shed further light on the correct method by which relativistic
relativistic electrodynamics can be extended to include rotat-
electrodynamics can be applied to rotating media.

ing isotropic, linear media. Covariant field equations were
One important point to recognize is that fields, such as the
used to derive general field equations in a rotating coordinate
electric and magnetic fields, can be expressed either in terms
system. Fields in the presence of matter were then related to
of a vector basis ͕

**e**␣͖ or in terms of a one-form basis ͕

**w**
**˜ **␣͖,

the electric and magnetic fields by use of a covariant form of
dual to the ͕

**e**␣͖ according to

**w**
**˜ **␣(

**e**)ϭ␦ .8–10 As is very

Minkowski’s constitutive equations.2–5 We showed that us-
well known, any contravariant vector

**v **can be expressed as a

ing Minkowski’s covariant constitutive equations in rotating
superposition of basis vectors ͕

**e**␣͖ as

**v**ϭ␣

**e**␣ , where ͕

**e**␣͖

coordinates leads directly to the correct constitutive equa-
are distinct, linearly independent vectors. Similarly, any co-
tions and hence to the correct field equations in the rotating
variant vector

**v**
**˜ **can be expressed as a superposition of basis

reference frame when matter is present.6 We concluded that
one-forms ͕

**w**
**˜ **␣͖ as

**v**
**˜**ϭ␣

**w**
**˜ **␣, where the basis one-forms

relativistic electrodynamics can be used for rotating linear
͕

**w˜**␣͖ are distinct and linearly independent. In general, the

media only when a covariant form of the constitutive equa-
vector components ␣ and the one-form components ␣ are
not equal, but instead are mapped into each other by use of
In the course of performing that analysis, we chose to
the metric tensor. For example, vector components ␣ can be
work with the covariant electromagnetic tensor

*F*␣ , com-
mapped from the vector basis ͕

**e**␣͖ to a one-form basis ͕

**w**
posed of the components of the electric and magnetic fields.

An alternative approach is to define the components of the
␣ϭ

*g*␣. Or conversely, one-
electric and magnetic fields to comprise a contravariant elec-
␣ can be mapped from the one-form basis
͕

**w˜**␣͖ to the vector basis ͕

**e**
tromagnetic tensor

*F*␣. We pointed out that either the co-
variant or contravariant tensor can be used so long as one is
tensor: ␣ϭ

*g*␣ .

consistent throughout the analysis. However, although both
The same relationship holds true for second rank electro-
tensors lead to field equations of the same form in an inertial
magnetic tensors

*F*␣ and

*F*␣ . A basis for all second rank
reference frame, when acceleration is present, such as in a
contravariant tensors is

**e**␣

**e** , where represents the outer

rotating reference frame, the covariant and contravariant ten-
product, and a basis for all second rank covariant tensors is
sors lead to different sets of field equations.7 Different pre-

**˜ **.8 The components of contravariant tensors are vec-

dictions for the fields observed in a rotating reference frame
tor components, and the components of covariant tensors are
are then obtained. This is confusing: all observers at rest in a
one-form components. Thus, whether one works with the
particular reference frame ought to be able to agree on the
covariant or contravariant electromagnetic tensor is actually
form of the electric and magnetic fields observed in that
dependent upon whether one chooses to express the electric
frame. Thus, we expect that rotating observers working with
and magnetic fields as vectors or as one-forms. When fields
a covariant electromagnetic tensor should obtain results that
are expressed as vectors, the contravariant electromagnetic
are consistent with rotating observers working with a contra-
tensor

*F*␣ is composed of the vector components of the
variant electromagnetic tensor. It is an objective of this paper
electric and magnetic fields, ͕

*Ei*,

*Bi*͖, and the covariant ten-
Am. J. Phys.

**67 **͑5͒, May 1999

1999 American Association of Physics Teachers
sor is obtained by using the metric tensor to map those com-
relativity for the case of uniform motion.1,2,4,5,11 We then
explain that observers in the lab frame can analyze experi-
ϭ

*g*␣

*g*

*F*. On the other hand, when fields are expressed
ments involving rotation by using either pair of constitutive
as one-forms the covariant electromagnetic tensor

*F*
equations in conjunction with Maxwell’s field equations.

composed of the one-form components of the electric and
In Sec. V, we demonstrate the application of both field
equations derived on the basis of one-forms and those de-

*i *,

*B i *, and the contravariant tensor is ob-
rived on the basis vectors in a rotating frame. To do this,
tained by mapping the one-form components to the vector
both sets of field equations are used to derive the fields ob-
basis by using

*F*␣ϭ

*g*␣

*g*

*F* . As pointed out above, the
served in the reference frame of a hollow cylinder with elec-
vector components and one-form components are not equal,
tric permittivity and magnetic permeability that rotates
and thus cannot be directly interchanged. In an inertial ref-
within an external, axially directed magnetic field.1,6,12 We
erence frame, this distinction is easily missed simply because
assume that all fields are static, and that the cylinder is com-
Maxwell’s field equations assume the same form with re-
posed of a material that precludes free charges and currents.

spect to either basis. However, in a noninertial frame
Carrying this out, we show that one-form fields and vector
wherein acceleration is present, expressing fields as vectors
fields differ in form in the rotating frame, but assume the
or as one-forms leads to different sets of field equations. It is
same form in the inertial frame of the laboratory.

our opinion that the confusion surrounding the choice of co-
In the Appendix, we derive field transformations that re-
variant versus contravariant electromagnetic tensors arises
late quantities in the rotating frame to those in the lab frame.

from a failure to fully recognize that the components of co-
We begin by noting that, although the rotating frame has a
variant and contravariant tensors are defined with respect to
rotational velocity relative to the lab frame, at any given
instant a momentarily comoving reference frame ͑MCRF͒ of
In the next section we use the covariant form of Maxwell’s
an observer at rest in the rotating frame has a uniform veloc-
equations in rotating coordinates to derive three-dimensional
ity relative to the lab frame. Using this observation to our
field equations for the rotating frame. We begin by consid-
advantage, we first derive a relationship between fields in the
ering the case when rotating observers express fields as one-
rotating frame and those in the MCRF, and then use a Lor-
forms. The covariant electromagnetic tensor is composed of
entz transformation to relate fields in the MCRF to those in
the one-form components of the electric and magnetic fields,
the lab frame. Upon eliminating MCRF quantities between
and the contravariant electromagnetic tensor is obtained by
the two transformations, we obtain a direct transformation
use of the inverse metric tensor in rotating coordinates. Gen-
between the rotating and lab frames. We then transform the
eral field equations in three-dimensional notation are ob-
covariant and contravariant electromagnetic tensors from the
tained, expressed in terms of one-form fields in the presence
lab frame to the rotating frame. Carrying this out, we arrive
of matter. Next, we turn to the case in which rotating observ-
at transformations relating vector and one-form fields in the
ers express fields as vectors. In this case, the contravariant
rotating frame to those in the lab frame.

electromagnetic tensor is composed of the vector compo-nents of the electric and magnetic fields, and the covarianttensor is obtained by using the metric tensor in rotating co-

**II. FIELD EQUATIONS IN THE ROTATING**
ordinates. General field equations are again obtained, but this

**REFERENCE FRAME**
time, expressed in terms of vector fields in the presence of
Although rotation does not lead to space–time curvature, a
rotating reference frame is not a truly inertial frame of ref-
In Sec. III we relate fields in the presence of matter to
erence due to the presence of inertial forces.6,13–16 Maxwell’s
those in a vacuum by using a covariant form of the consti-
equations can be extended to encompass accelerating frames
tutive equations, first introduced by Minkowski in 1908.2–5
We start by noting that, as with the field equations, one has a
choice in expressing fields as one-forms or as vectors. Wethen point out that in order to maintain consistency through-
out the analysis, we must use the same fields that were used
in deriving the field equations. Adhering to this rule, we
derive one-form and vector constitutive equations, and then
where source charges and currents are given by the current
substitute them into the corresponding set of field equations,
four-vector

*j * and we have used, and will continue to use,
derived in Sec. II. Carrying this out, we obtain one-form and
units in which the speed of light is set equal to unity:

*c*ϭ1.

vector field equations for the rotating frame in the presence
Noting that in rotating coordinates the covariant derivative is
of matter. We then bring Sec. III to a close by using a cova-
exactly equal to the ordinary partial derivative,1 the field
riant form of the polarization and magnetization1 to derive
one-form and vector expressions for the polarization andmagnetization in the rotating frame. We find that as with the
constitutive equations, expressing fields as one-forms or as
vectors leads to different forms for the polarization and mag-
As can be seen, Eq. ͑2a͒ is expressed in terms of a con-
Section IV is devoted to deriving constitutive equations
travariant electromagnetic tensor whereas Eq. ͑2b͒ is ex-
for a rotating material, observed in the laboratory reference
pressed in terms of a covariant electromagnetic tensor. Thus,
frame. This is accomplished by using field transformations to
one can choose to work with either the covariant electromag-
transform the constitutive equations from the rotating frame
netic tensor

*F*␣ or the contravariant tensor

*F*␣ so long as
to the lab frame. For both one-forms and vectors, we arrive
one is consistent throughout the analysis. As pointed out in
at constitutive equations in agreement with Minkowski’s
the Introduction, which tensor one chooses is ultimately dic-
constitutive equations, first obtained in 1908 by using special
tated by how one chooses to describe fields in the rotating
reference frame. The electric and magnetic fields can be ex-
ٌϫ͑

**E**Јϩ

**v**ϫ

**B**Ј͒ϭ0,

pressed in the usual way as vectors

**E**ϭ

*Ei***e***i *and

**B**ϭ

*Bi***e***i *, or

ٌϫ

**H**Јϭ4

**j**Ј.

the fields can be expressed as one-forms

**E**
**˜ **ϭ

*Ei***w**
**˜ ***i *and

**B**
These are the three-dimensional field equations for the rotat-

*i***w**
**˜ ***i*.8,9

When the electric and magnetic fields are expressed as
ing reference frame, written with respect to a vector basis
one-forms, the covariant field tensor

*F*
iar form asͩ 0

*E*1

*E*2

*E*3

**III. CONSTITUTIVE EQUATIONS IN THE**
**ROTATING FRAME**
As yet, we have not specified a relationship between the
auxiliary fields,

**D **and

**H**, and fundamental fields,

**E **and

**B**.17

We can relate auxiliary fields in the presence of matter to the
and the contravariant tensor

*F*␣ is obtained by raising indi-
fundamental fields by using a covariant form of the consti-
tutive equations, first introduced by Minkowski in 1908:2–5

*F*␣ϭ

*g*␣

*g*

*F* .

*H*

*u*ϭ⑀

*F*

*u* ,
Using the inverse metric tensor in rotating coordinates, given
⑀

*F*

*u*ϭ⑀

*H*

*u* ,
where the electric permitivity ⑀ and magnetic permeability

*Fi*0ϭ͑

**E**
**˜ **Ϫ

**v**ϫ

**B**
are proper quantities defined in the local rest frame of the
material. As in the case of the field equations, however, a

*Fi j*ϭϪ⑀

*ijk*„

**B**
**˜ **Ϫ

**v**ϫ͑

**E**
**˜ **Ϫ

**v**ϫ

**B**
**˜ **Ј͒…

choice must be made between working with a covariant or
contravariant electromagnetic tensor. As pointed out in the
where quantities in the rotating frame carry a prime, and
two previous sections, this choice hinges on whether one
observers at rest in the rotating frame have velocity

**v**
chooses to express fields as one-forms or as vectors. Since
ϭ

*r***e** relative to the laboratory frame. Using Eqs. ͑3͒ and

the constitutive equations will be used in conjunction with
͑5͒ in Eqs. ͑2͒, and limiting ourselves to the case of static
the field equations, in order to maintain consistency through-
fields, we find that the field equations are
out the analysis, we must use the same fields that were used

**˜ **Ϫ

**v**ϫ

**H**
**˜ **Ј…ϭ4Ј,

When observers in the rotating frame express fields as
one-forms, Eqs. ͑11͒ are used in conjunction with the cova-

**˜ **Јϭ0,

riant electromagnetic tensor given by Eq. ͑3͒. Generalizing
ٌϫ

**E˜**Јϭ0,

to noninertial frames, Eqs. ͑11͒ are then

*u*ϭ⑀

*g*␣

*F*

*u*,

**˜ **ЈϪ

**v**ϫ͑

**D**
**˜ **Јϩ

**v**ϫ

**H**
**˜ **Ј͒…ϭ4

**j**Ј.

⑀

*F*

*g*␦

*u*␦ϭ⑀

*H*

*g*␦

*u*␦.

These are the general field equations for the rotating refer-ence frame, written in three-dimensional notation with re-
According to the metric tensor in rotating coordinates, given
spect to a one-form basis ͕

**w**
**˜ **␣͖.

in the Appendix, an observer at rest in the rotating frame has
Conversely, when the electric and magnetic fields are ex-
a four-velocity

*ua*ϭ␥(1, 0, 0, 0), where ␥ϭ1/ͱ1Ϫ2. Using
pressed as vectors, the contravariant form of the field tensor
this four-velocity and the metric tensor in Eqs. ͑12͒, we find
that auxiliary fields are related to fundamental fields accord-ing to

**˜ **Јϭ⑀

**E˜**Ј,

**˜ **Јϭ

**B˜**Јϩ

where quantities in the rotating frame carry a prime. Substi-
and the covariant tensor

*F*␣ is then obtained by lowering
tuting Eqs. ͑13͒ into Eqs. ͑6͒ leads to

*F*␣ϭ

*g*␣

*g*

*F*.

ٌ• ⑀

**E˜**ЈϪ

**v**ϫ

**B˜**ЈϪ

**v**ϫ͑

**v**ϫ

**E**
Using the metric tensor in rotating coordinates gives
„

**E**Јϩ

**v**ϫ͑

**B**Јϩ

**v**ϫ

**E**Ј͒…

*i*,

ٌ•

**B˜**Јϭ0,

*i j k ***B**Јϩ

**v**ϫ

**E**Ј ͒

*k*.

ٌϫ

**E˜**Јϭ0,

This time, using Eqs. ͑7͒ and ͑9͒ in Eqs. ͑2͒, and again lim-iting ourselves to static fields, leads to
ٌϫͩ 1 „

**B˜**ЈϪ

**v**ϫ

**E˜**Јϩ

**v**ϫ͑

**v**ϫ

**B˜**Ј͒…ͪϭ4

**j**Ј.

ٌ•

**D**Јϭ4Ј,

These field equations are those used by observers at rest in
ٌ•͑

**B**Јϩ

**v**ϫ

**E**Ј͒ϭ0,

the rotating frame that choose to express fields as one-forms.

Observers in the rotating frame choosing to express fields

*H*␣ϭ

*F*␣Ϫ4

*M *␣,
as vectors use Eqs. ͑11͒ in conjunction with the contravariantelectromagnetic tensor given by Eq. ͑7͒. In this case, Eqs.

in which the contravariant magnetization four-tensor

*M *␣ is

*H*

*g*

*u*ϭ⑀

*F*

*g*

*u*,

*g*␣

*g*

*F*␣

*g*␦

*u*␦ϭ⑀

*g*␣

*g*

*H*␣

*g*␦

*u*␦.͑
Carrying out the same procedure used to obtain Eqs. ͑13͒, we
find that auxiliary and fundamental fields are related by
Substituting Eq. ͑22͒ into each of Eqs. ͑15͒ and then solving

**D**Јϭ⑀

**E**Јϩ

*u*ϭ4 ͑⑀Ϫ1͒

*F*

*g*

*u*,

**H**Јϭ

**B**Ј.

⑀

*g*␣

*g*

*M*␣

*g*␦

*u*␦
Upon substituting these constitutive equations into Eqs. ͑10͒,
we find that the field equations used by rotating observers
⑀

*g*␣

*g*

*F*␣

*g*␦

*u*␦.

that choose to express fields as vectors are
The vector polarization and magnetization in the rotating

**v**ϫ

**B**Јϭ4Ј,

**B**Јϩ

**v**ϫ

**E**Ј͒ϭ0,

4 ͑⑀Ϫ1͒

**E**Јϩ 4͑1Ϫ2͒

ٌϫ͑

**E**Јϩ

**v**ϫ

**B**Ј͒ϭ0,

**B**Јϭ4

**j**Ј.

Upon comparing Eqs. ͑25͒ and ͑21͒, it is apparent that as
At this point, we should mention that Eqs. ͑11͒ can also be
with the constitutive equations, expressing fields as one-
used to obtain expressions for the polarization

**P **and magne-

forms or as vectors leads to different forms for the polariza-
tization

**M **observed in the rotating reference frame.1 Rotat-

tion and magnetization in the rotating frame.

ing observers that work with one-forms begin by taking

**IV. CONSTITUTIVE EQUATIONS IN THE**
*H*␣ϭ

*F*␣Ϫ4

*M *␣ ,

**LABORATORY REFERENCE FRAME**
where

*M *␣ is the magnetization four-tensor composed of the
components of the polarization and magnetization:5
We now turn to the problem of finding constitutive equa-
tions for a rotating medium, observed in the laboratory ref-
erence frame. To do this, we transform the constitutive equa-
tions from the rotating frame to the lab frame. As shown in
the Appendix, one-form fields in the rotating frame are re-
lated to those in the lab frame according to

**˜ **Јϭ

**E˜**ϩ

**v**ϫ

**B˜**Ϫ

͑

**v**•

**E˜**͒

**v**,

Substituting Eq. ͑18͒ into each of Eqs. ͑12͒ and solving for

**˜ **Јϭ␥͑

**B˜**Ϫ

**v**ϫ

**E˜**͒ϩ␥2

**v**ϫ͑

**E˜**ϩ

**v**ϫ

**B˜**͒Ϫ

͑

**v**•

**B˜**͒

**v**,

*u*ϭ4 ͑⑀Ϫ1͒

*g*␣

*F*

*u*,

*M *

*g*␦

*u*␦ϭ

**˜ **ϩ

**v**ϫ

**H**
Following the same steps used to derive the constitutive
equations, Eqs. ͑13͒, we find that the polarization and mag-

**˜ **Јϭ␥͑

**H**
**˜ **Ϫ

**v**ϫ

**D**
**˜ **͒ϩ␥2

**v**ϫ͑

**D**
**˜ **ϩ

**v**ϫ

**H**
**˜ **͒Ϫ ␥ϩ ͑

**v**•

**H˜**͒

**v**,

where primes denote quantities in the rotating frame, and ␥
4 ͑⑀Ϫ1͒

**E**
ϭ1/ͱ1Ϫ2. Substituting Eqs. ͑26͒ into the constitutive
equations given by Eqs. ͑13͒, and then solving for

**D**
we find that the constitutive equations for the rotating me-
4͑1Ϫ2͒ Ϫ⑀ ͪ

**v**ϫ

**E**
dium, observed in the laboratory frame, are
On the other hand, when fields are expressed as vectors,

**˜ **ϩͩ⑀Ϫ ͓

**v**ϫ

**B˜**Ϫ͑

**v**•

**E˜**͒

**v**͔ , ͑27a͒

**˜ **ϩ͑

**v**•

**B˜**͒

**v**͔ .

Ϫ⑀2ͪ

**B˜**ϩͩ ⑀Ϫ

And upon rearranging Eqs. ͑27͒, we find that the resultingequations are identical to Minkowski’s constitutive equa-tions, first obtained in 1908 by using special relativity for thecase of uniform motion:1,2,4,5,11
Fig. 1. A side cut-away view of a hollow cylinder of polarizable, permeable

**˜ **ϭ ͑1Ϫ⑀2͒ „⑀

**E˜**͑1Ϫ2͒ϩ͑⑀Ϫ1͓͒

**v**ϫ

**H˜**Ϫ⑀͑

**v**•

**E˜**͒

**v**͔…,

material that rotates within an external, axially directed magnetic field with
velocity

**v**ϭ

*r***e**
relative to the laboratory reference frame. Rotating and
laboratory observers alike detect the presence of an electric field between
inner and outer surfaces of the cylinder.

**˜ **ϭ ͑1Ϫ⑀2͒ „

**H˜**͑1Ϫ2͒Ϫ͑⑀Ϫ1͓͒

**v**ϫ

**E˜**Ϫ͑

**v**•

**H˜**͒

**v**͔….

Then, by using Eqs. ͑30͒, lab frame observers can rewrite the
These constitutive equations are those used by observers in
the laboratory frame that choose to work with one-form

**E**ϩͩ ⑀Ϫ ͓

**v**ϫ

**B**Ϫ͑

**v**•

**E**͒

**v**͔ͬͪ

Also shown in the Appendix, vector fields in the rotating

**E**Јϭ␥2͑

**E**ϩ

**v**ϫ

**B**͒Ϫ␥3

**v**ϫ͑

**B**Ϫ

**v**ϫ

**E**͒Ϫ

ٌ•

**B**ϭ0,

␥ϩ ͑

**v**•

**E**͒

**v**,

ٌϫ

**E**ϭ0,

**B**Јϭ␥͑

**B**Ϫ

**v**ϫ

**E**͒Ϫ

͓

**v**ϫ

**E**ϩ͑

**v**
•

**B**͒

**v**͔ͬͪ

ϩ ͑

**v**•

**B**͒

**v**,

͑1Ϫ2͒ Ϫ⑀2ͪ

**B**ϩͩ ⑀Ϫ

ϭ4

**j**.

**D**Јϭ␥2͑

**D**ϩ

**v**ϫ

**H**͒Ϫ␥3

**v**ϫ͑

**H**Ϫ

**v**ϫ

**D**͒Ϫ ␥ϩ ͑

**v**•

**D**͒

**v**,

An identical set of field equations is obtained when one-form
fields are used in the lab frame. Observers in the lab frame
can use either set of field equations to analyze experiments

**H**Јϭ␥͑

**H**Ϫ

**v**ϫ

**D**͒Ϫ

involving axial rotation so long as consistency is maintained
␥ϩ ͑

**v**•

**H**͒

**v**.

Following the same procedure used to obtain Eqs. ͑28͒, we

substitute Eqs. ͑29͒ into Eqs. ͑16͒, and then solve for

**D **and

**V. DERIVING THE FIELDS OBSERVED IN THE**
**B**. Carrying this out again leads directly to Minkowski’s

**REFERENCE FRAME OF A ROTATING CYLINDER**
As demonstrated in the preceding sections, when observ-
ers in a rotating frame define fields as one-forms, the form of

**D**ϭ ͑1Ϫ⑀2͒ „⑀

**E**͑1Ϫ2͒ϩ͑⑀Ϫ1͓͒

**v**ϫ

**H**Ϫ⑀͑

**v**•

**E**͒

**v**͔…,

the resulting field equations differs from those obtained by
observers defining fields as vectors. In this section, we useboth sets of field equations to derive the fields observed be-
tween inner and outer surfaces of a hollow cylinder with

**B**ϭ ͑1Ϫ⑀2͒ „

**H**͑1Ϫ2͒Ϫ͑⑀Ϫ1͓͒

**v**ϫ

**E**Ϫ͑

**v**•

**H**͒

**v**͔….

electric permittivity ⑀ and magnetic permeability that ro-
tates within an external, axially directed magnetic field.1,6,12Figure 1 shows a cut-away view of a such a cylinder. We
These constitutive equations are used by those lab frame ob-
assume that all fields are static, and that the cylinder is com-
servers that work with vector fields, and are identical to the
posed of a material that precludes free charges and currents.

one-form constitutive equations given by Eqs. ͑28͒. Al-
Rotating observers choosing to express fields as one-forms
though different constitutive equations arise in the rotating
frame, one-form fields and vector fields lead to constitutiveequations of the same form in an inertial frame.

Field equations in the lab frame can be obtained by use of
ٌ•ͩ⑀

**E˜**ЈϪ

**v**ϫ

**B˜**Јͪϭ0,

either Eqs. ͑28͒ or ͑30͒ in conjunction with Maxwell’s fieldequations. Assuming static vector fields, for example, ob-
ٌ•

**B˜**Јϭ0,

servers in the lab frame begin by writing Maxwell’s fieldequations as
ٌϫ

**E˜**Јϭ0,

ٌ•

**D**ϭ4,

ٌ•

**B**ϭ0,

͑

**B˜**ЈϪ

**v**ϫ

**E˜**Ј͒ͪ ϭ0,

where we have neglected terms of higher order than , and

**v**ϭ

*r***e** is the velocity of the rotating cylinder. Since the

ٌϫ

**H**ϭ4

**j**.

external magnetic field is parallel to the axis of the cylinder,
Eq. ͑33a͒ implies that the normal component of ⑀

**E**
ϫ(1/)

**B˜**Ј is continuous at the surface of the rotating me-

Substituting these fields into Eq. ͑40͒, and noting that Eq.

dium. Assuming that the cylinder is sufficiently long for end
͑39d͒ implies that

**B**Ј ϭ

**B**Ј

effects to be ignored, Eq. ͑33a͒ then implies that

**E**Ј ϭͩ 1

⑀Ϫͪ

**v**ϫ

**B**OUT

where the subscripts denote quantities taken inside or outsidethe material, and we have taken ⑀
This electric field is the one detected by rotating observers
unity. Outside the material, the fields in the lab frame are
that choose to express fields as vectors. Using Eq. ͑42͒ in Eq.

͑29a͒, we find that observers in the lab frame detect an elec-

**˜ **OUT

**0˜**. Using these fields in Eqs. ͑26a͒ and

tric field identical to that given by Eq. ͑38͒:1,6
26b͒ gives the external fields in the rotating frame as

**˜ **Ј ϭ

**v**ϫ

**B**
⑀Ϫͪ

**v**ϫ

**B**OUT.

As expected, one-form and vector fields assume different
Thus, the right-hand side of Eq. ͑34͒ is zero. Simplifying a
forms in the rotating frame, but take on the same form when
transformed to the inertial frame of the laboratory. Therefore,field equations obtained on the basis of one-form fields and
those obtained on the basis of vector fields both lead to pre-
⑀

**v**ϫ

**B˜**IN

dictions that are consistent with known experimentalresults.4,6,11,12
where we have dropped the subscripts from ⑀IN and IN .

Again taking into consideration that the magnetic field is
parallel to the axis of rotation, Eq. ͑33d͒ implies that

**B**
**VI. CONCLUSIONS**
**˜ **Ј . Substituting this relationship into Eq. ͑36͒ gives

We have shown two methods by which Maxwell’s equa-
tions can be applied to rotating linear media. In the first
⑀

**v**ϫ

**B˜**OUT

method, the covariant electromagnetic tensor

*F*␣ was used;
This is the electric field, between inner and outer surfaces of
and in the second method, the contravariant tensor

*F*␣ was
the cylinder, that is detected by observers in the rotating
used. We began by explaining that whether one works with a
frame that work with one-form fields. And upon using Eq.

covariant or contravariant tensor is dictated by whether one
͑37͒ in Eq. ͑26a͒, we find that observers in the lab frame
chooses to express fields in terms of a vector basis or in
terms of a dual one-form basis. Using this formalism, generalfield equations were derived in terms of both vector fields
and one-form fields in the rotating and laboratory frames.

⑀Ϫͪ

**v**ϫ

**B˜**OUT.

Fields in the presence of matter were then related to those ina vacuum by using the covariant form of Minkowski’s con-
On the other hand, when rotating observers work with
stitutive equations,2–5 generalized to noninertial frames.

vectors, the field equations assume the form
Next, the constitutive equations were transformed from the
rotating frame to the lab frame. Carrying this out, we found

**E**Јϩͩ ⑀Ϫ

**v**ϫ

**B**Јϭ0,

that although vector and one-form constitutive equations dif-fer in the rotating frame, in the lab frame both pairs of con-
ٌ•͑

**B**Јϩ

**v**ϫ

**E**Ј͒ϭ0,

stitutive equations are in agreement with the 1908 constitu-tive equations of Minkowski.1,2,4,5,11
ٌϫ͑

**E**Јϩ

**v**ϫ

**B**Ј͒ϭ0,

We then demonstrated the use of vector and one-form field
equations in a rotating frame. Both sets of field equations
were used to derive the fields observed in the reference frame

**B**Јϭ0,

of a polarizable, permeable cylinder that rotates within anaxially directed magnetic field.1,6,12 As expected, we found
where as before we have neglected terms of order higher
that vector and one-form fields take on different forms in the
than . According to Eq. ͑39a͒, when the external magnetic
rotating frame, but assume the same form in the inertial
field is parallel to the axis of rotation, rotating observers can
We conclude that when applying the covariant form of
Maxwell’s equations to a noninertial frame, the choice be-
tween working with a covariant or contravariant electromag-
netic tensor depends upon whether one chooses to express
Taking the external fields in the lab frame to be

**E**
fields in terms of a vector basis or in terms of a dual one-
form basis. More particularly, we conclude that the electric

**0**, Eqs. ͑29a͒ and ͑29b͒ imply that the external fields

and magnetic fields can be expressed either as vectors or asone-forms so long as one is consistent throughout the analy-

**APPENDIX: DERIVING A DIRECT FIELD**
**TRANSFORMATION FROM THE LABORATORY**
ͩ␥

*x*␥1ϩ2

*A * ͪ

**FRAME TO THE ROTATING FRAME**
We wish to derive a transformation that relates fields ob-
served in a rotating reference frame to those observed in a
laboratory frame. The metric tensor in the rotating coordinatesystem is6,7,19
in which

*A*ϭ(␥Ϫ1)/2. Equating Eqs. ͑A3͒ and ͑A5͒ andsolving for

*F*Ј leads to an expression that transforms quan-
tities from the lab frame directly to the rotating frame:

*F*Јϭ͑

*LR*Ϫ1͒

*TF*͑

*LR*Ϫ1͒.

Working out the matrix multiplication, we find that Eq. ͑A7͒can be rewritten simply as
ϩ2. The inverse metric tensor in rotating coordinates is7
ͩ 1 Ϫ Ϫ

*xy *0 where

*N*isgivenby
We begin by noting that, although the rotating frame has a
velocity

**v**ϭ

*r***e**
relative to the lab frame, at any given in-
stant a momentarily comoving reference frame ͑MCRF͒ of
Upon inserting the covariant electromagnetic tensor,
an observer at rest in the rotating frame has a uniform veloc-

ity

**v **relative to the lab frame. Thus, we can obtain a rela-

tionship between fields in the rotating and lab frames as fol-

lows. We first find a relationship between fields in the
rotating frame and those in the MCRF, and then we use a
Lorentz transformation to relate fields in the MCRF to those
in the lab frame. Upon eliminating MCRF quantities betweenthe two transformations, we find a direct transformation from
into the right-hand side of Eq. ͑A8͒, and noting that the
the lab frame to the rotating frame.

components of

*F*␣ are the components of

**E**
**˜ **, we find

The covariant electromagnetic tensor

*F*␣ can be trans-
that one-form fields in the rotating frame are related to those
formed from the rotating frame to the MCRF by using

**˜ **Јϭ

**E˜**ϩ

**v**ϫ

**B˜**Ϫ ␥ϩ ͑

**v**•

**E˜**͒

**v**,

where

*F*Љ is the electromagnetic tensor in the MCRF,

*RT *isthe transpose of

*R*, and

*R *is a transformation that relates

**˜ **Јϭ␥͑

**B˜**Ϫ

**v**ϫ

**E˜**͒ϩ␥2

**v**ϫ͑

**E˜**ϩ

**v**ϫ

**B˜**͒Ϫ

͑

**v**•

**B˜**͒

**v**,

quantities in the rotating frame to those in the MCRF, given
where quantities in the rotating frame carry a prime. Carry-
ing out the same procedure used to find Eqs. ͑A11͒, analo-
gous transformations for auxiliary fields

**D**
**˜ **are ob-

**˜ **ϩ

**v**ϫ

**H**
**˜ **Ϫ ␥ϩ ͑

**v**•

**D˜**͒

**v**,

in which ␥ϭ1/ͱ1Ϫ2. Similarly, the electromagnetic tensor
can be transformed from the lab frame to the MCRF by using

**˜ **Јϭ␥͑

**H**
**˜ **Ϫ

**v**ϫ

**D**
**˜ **͒ϩ␥2

**v**ϫ͑

**D**
**˜ **ϩ

**v**ϫ

**H**
**˜ **͒Ϫ ␥ϩ ͑

**v**•

**H˜**͒

**v**.

With the transformation for the covariant electromagnetic
where

*F *is the electromagnetic tensor in the lab frame, and

*L*
tensor known, the transformation for the contravariant tensor
is the well-known Lorentz transformation that connects two

*F*␣ can be most simply obtained by taking
frames moving with respect to each other in an arbitrarydirection in the

*x *–

*y *plane,21
where

*F*Ј now represents the contravariant electromagnetic
C. T. Ridgely, ‘‘Applying relativistic electrodynamics to a rotating mate-
tensor in the rotating frame, and

*M *ϭ

*N*Ϫ1.6 Taking the in-
rial medium,’’ Am. J. Phys.

**66**, 114–121 ͑1998͒.

2L. Landau and E. Lifshitz,

*Electrodynamics of Continuous Media *͑Perga-
verse of Eq. ͑A9͒, we find that

*M *is given by
mon, New York, 1984͒, 2nd ed., pp. 260–263.

3H. Weyl,

*Space-Time-Matter *͑Dover, New York, 1952͒, 4th ed., pp. 193–
␥2͑1Ϫ2␥͒ Ϫ ␥͑␥Ϫ␥2͒ Ϫ ␥͑␥Ϫ␥2͒
4W. Pauli,

*Theory of Relativity *͑Pergamon, New York, 1958; reprinted,
Dover, New York, 1981͒, pp. 99–113.

R. Becker,

*Electromagnetic Fields and Interactions *͑Blaisdell, New York,
1964; reprinted, Dover, New York, 1982͒, Vol. 1, pp. 299–304; 374–376.

6G. N. Pellegrini and A. R. Swift, ‘‘Maxwell’s equations in a rotating
medium: Is there a problem?’’ Am. J. Phys.

**63**, 694–705 ͑1995͒.

Using the contravariant electromagnetic tensor,
W. Crater, ‘‘General covariance, Lorentz covariance, the Lorentz force,

and the Maxwell equations,’’ Am. J. Phys.

**62**, 923–931 ͑1994͒.

8See, for example, B. F. Schutz,

*A First Course in General Relativity*
͑Cambridge, New York, 1990͒, pp. 62–78.

9See, for example, Ohanian and Ruffini,

*Gravitation and Spacetime*
͑Norton, New York, 1994͒, 2nd ed., p. 103.

10Other names for a one-form are covector, covariant vector, or dual vector.

See, for example, B. F. Schutz, Ref. 8.

11E. G. Cullwick,

*Electromagnetism and Relativity *͑Longmans, London,
in the right-hand side of Eq. ͑A13͒, we find that vector fields
12Such an experiment was originally performed by M. Wilson and H. A.

in the rotating and lab frames are related according to
Wilson in 1913. The Wilsons measured an electric potential, between in-
ner and outer surfaces of the cylinder, which closely agreed with the spe-

**E**Јϭ␥2͑

**E**ϩ

**v**ϫ

**B**͒Ϫ␥3

**v**ϫ͑

**B**Ϫ

**v**ϫ

**E**͒Ϫ

␥ϩ ͑

**v**•

**E**͒

**v**,

cial relativistic prediction ⌬

*V*ϭ

*B*
ample, W. Pauli, Ref. 4; G. N. Pellegrini and A. R. Swift, Ref. 6; and E. G.

13J. Ise and J. Uretsky, ‘‘Vacuum electrodynamics on a merry-go-round,’’

**B**Јϭ␥͑

**B**Ϫ

**v**ϫ

**E**͒Ϫ ␥ϩ ͑

**v**•

**B**͒

**v**.

Am. J. Phys.

**26**, 431–435 ͑1958͒.

14O”. Gro”n, ‘‘Relativistic description of a rotating disk,’’ Am. J. Phys.

**43**,

A similar transformation is easily obtained for auxiliary
fields

**D **and

**H**:

D. L. Webster, ‘‘Schiff’s charges and currents in rotating matter,’’ Am. J.

Phys.

**31**, 590–597 ͑1963͒.

16L. Landau and E. Lifshitz,

*The Classical Theory of Fields *͑Addison–

**D**Јϭ␥2͑

**D**ϩ

**v**ϫ

**H**͒Ϫ␥3

**v**ϫ͑

**H**Ϫ

**v**ϫ

**D**͒Ϫ ␥ϩ ͑

**v**•

**D**͒

**v**,

Wesley, Reading, MA, 1951͒, pp. 60–62; 281–282.

When the field equations are to take the presence of matter into account,
we introduce the electromagnetic tensor

*H*␣, having the same form as

*F*␣ but composed of the components of the electric and magnetic fields in

**H**Јϭ␥͑

**H**Ϫ

**v**ϫ

**D**͒Ϫ ␥ϩ ͑

**v**•

**H**͒

**v**.

the presence of matter,

**D **and

**H**. Here, we note that

**D **and

**H **are well

defined both inside and outside of matter, as are the true electric and
We note that when the speed of rotation is taken to be very
magnetic fields,

**E **and

**B**. Since the fields

**D **and

**H **are simply useful in

much less than the speed of light, the relationship between
calculating

**E **and

**B **when matter is present, to avoid confusion, we will

one-form and vector fields in the rotating frame and those in
refer to

**E **and

**B **as fundamental fields, and to

**D **and

**H **as auxiliary fields.

18H. Ohanian and R. Ruffini, Ref. 9, pp. 322–323.

19L. Schiff, ‘‘A question in general relativity,’’ Proc. Natl. Acad. Sci. USA

**˜ **Јϭ

**E˜**ϩ

**v**ϫ

**B˜**,

**25**, 391–395 ͑1939͒.

20C. T. Ridgely, ‘‘The electrodynamics of a rotating material medium,’’

**˜ **Јϭ

**B˜**,

M.S. thesis, California State University, Long Beach, 1996, p. 30.

21See, for example, A. O. Barut,

*Electrodynamics and Classical Theory of*
*Fields and Particles *͑Dover, New York, 1980͒, p. 19.

22G. Modesitt, ‘‘Maxwell’s equations in a rotating reference frame,’’ Am. J.

**B**Јϭ

**B**Ϫ

**v**ϫ

**E**.

Phys.

**38**, 1487–1489 ͑1970͒.

**NEVER ECONOMIZE**
One way to lessen the risk inherent in undertaking a major project is to make sure that you
spend enough money on it. After a research department or funding agency has invested enough inyour goals, it has a real stake in your success and becomes very reluctant to admit that your projectis not working out. No one ever got ahead in science by saving money.

Peter J. Feibelman,

*A Ph.D. Is Not Enough—A Guide to Survival in Science *͑Addison–Wesley, Reading, MA, 1993͒, p.

105.

Source: http://www.ridgely.ws/inertia/ridgely_ajp_67_414_99.pdf

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