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Ridgely.ws

Applying covariant versus contravariant electromagnetic tensors
to rotating media

Charles T. Ridgely2936 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 vϭ␯␣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 ͕␣͖ 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 ␣ϭg␣␤␯␤. Or conversely, one- electric and magnetic fields to comprise a contravariant elec- ␣ can be mapped from the one-form basis ͕␣͖ 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: ␯␣ϭg␣␤␯␤ .
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,Bi͖, 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- ϭg␣␮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␣␤ϭg␣␮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: cϭ1.
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- ٌϫ͑EЈϩvϫBЈ͒ϭ0,
pressed in the usual way as vectors EϭEiei and BϭBiei , or
ٌϫHЈϭ4␲jЈ.
the fields can be expressed as one-forms E
˜ ϭEiw
˜ i and B
These are the three-dimensional field equations for the rotat- iw
˜ 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 E1 E2 E3 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␣␤ϭg␣␮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 ␮ Fi0ϭ͑E
˜ ϪvϫB
are proper quantities defined in the local rest frame of the material. As in the case of the field equations, however, a Fi jϭϪ⑀ijkB
˜ Ϫvϫ͑E
˜ ϪvϫB
˜ 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␻re␾ 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 ˜ ϪvϫH
˜ Ј…ϭ4␲␳Ј,
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 ٌϫЈϭ0,
to noninertial frames, Eqs. ͑11͒ are then ␭␮u␮ϭ⑀g␣␭F␭␮u␮, ˜ ЈϪvϫ͑D
˜ ЈϩvϫH
˜ Ј͒…ϭ4␲jЈ.
⑀␴␭␮␯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 uaϭ␥(1, 0, 0, 0), where ␥ϭ1/ͱ1Ϫ␯2. 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⑀Ј,
˜ Јϭ ␮ 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␣␤ϭg␣␮g␤␯F␮␯.
ٌ• ⑀ЈϪvϫ␮ ЈϪ
vϫ͑vϫE
Using the metric tensor in rotating coordinates gives „EЈϩvϫ͑BЈϩvϫEЈ͒…i,
ٌ•Јϭ0,
i j k BЈϩ vϫ EЈ ͒k.
ٌϫЈϭ0,
This time, using Eqs. ͑7͒ and ͑9͒ in Eqs. ͑2͒, and again lim-iting ourselves to static fields, leads to ٌϫͩ 1␮ „ЈϪvϫЈϩvϫ͑vϫЈ͒…ͪϭ4␲jЈ.
ٌ•DЈϭ4␲␳Ј,
These field equations are those used by observers at rest in ٌ•͑BЈϩvϫEЈ͒ϭ0,
the rotating frame that choose to express fields as one-forms.
Observers in the rotating frame choosing to express fields H␣␤ϭF␣␤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 DЈϭ⑀EЈϩ
␮␯u␯ϭ4␲ ͑⑀Ϫ1͒F␮␭g␮␯u␯, HЈϭ ␮ BЈ.
⑀␴␭␮␯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 vϫBЈ΁ϭ4␲␳Ј,
BЈϩvϫEЈ͒ϭ0,
4␲ ͑⑀Ϫ1͒EЈϩ 4␲͑1Ϫ␯2͒
ٌϫ͑EЈϩvϫBЈ͒ϭ0,
BЈϭ4␲jЈ.
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␣␤ϭF␣␤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 ˜ ЈϭϩvϫϪ
͑v͒v,
Substituting Eq. ͑18͒ into each of Eqs. ͑12͒ and solving for ˜ Јϭ␥͑Ϫvϫ͒ϩ␥2vϫ͑ϩvϫ͒Ϫ
͑v͒v,
␭␮u␮ϭ4␲ ͑⑀Ϫ1͒g␣␭F␮␭u␮, M ␭␮g␯␦u␦ϭ ˜ ϩvϫH
Following the same steps used to derive the constitutive equations, Eqs. ͑13͒, we find that the polarization and mag- ˜ Јϭ␥͑H
˜ ϪvϫD
˜ ͒ϩ␥2vϫ͑D
˜ ϩvϫH
˜ ͒Ϫ ␥ϩ ͑v͒v,
where primes denote quantities in the rotating frame, and ␥ 4␲ ͑⑀Ϫ1͒E
ϭ1/ͱ1Ϫ␯2. 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- 4␲͑1Ϫ␯2͒ ␮Ϫ⑀ ͪ vϫE
dium, observed in the laboratory frame, are On the other hand, when fields are expressed as vectors, ˜ ϩͩ⑀Ϫ ͓vϫϪ͑v͒v͔ , ͑27a͒
˜ ϩ͑v͒v͔ .
␮Ϫ⑀␯2ͪ 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 „⑀͑1Ϫ␯2͒ϩ͑⑀␮Ϫ1͓͒vϫϪ⑀͑v͒v͔…,
material that rotates within an external, axially directed magnetic field with velocity vϭ␻re
␾ 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 „␮͑1Ϫ␯2͒Ϫ͑⑀␮Ϫ1͓͒vϫϪ␮͑v͒v͔….
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 Eϩͩ ⑀Ϫ ͓vϫBϪ͑vE͒v͔ͬͪ
Also shown in the Appendix, vector fields in the rotating EЈϭ␥2͑EϩvϫB͒Ϫ␥3vϫ͑BϪvϫE͒Ϫ
ٌ•Bϭ0,
␥ϩ ͑vE͒v,
ٌϫEϭ0,
BЈϭ␥͑BϪvϫE͒Ϫ
͓vϫEϩ͑v
B͒v͔ͬͪ
ϩ ͑vB͒v,
͑1Ϫ␯2͒ ␮Ϫ⑀␯2ͪ Bϩͩ ⑀Ϫ ␮
ϭ4␲j.
DЈϭ␥2͑DϩvϫH͒Ϫ␥3vϫ͑HϪvϫD͒Ϫ ␥ϩ ͑vD͒v,
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 HЈϭ␥͑HϪvϫD͒Ϫ
involving axial rotation so long as consistency is maintained ␥ϩ ͑vH͒v.
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 Dϭ ͑1Ϫ⑀␮␯2͒ „⑀E͑1Ϫ␯2͒ϩ͑⑀␮Ϫ1͓͒vϫHϪ⑀͑vE͒v͔…,
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 Bϭ ͑1Ϫ⑀␮␯2͒ „␮H͑1Ϫ␯2͒Ϫ͑⑀␮Ϫ1͓͒vϫEϪ␮͑vH͒v͔….
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 ٌ•ͩ⑀ЈϪvϫ␮ Јͪϭ0,
either Eqs. ͑28͒ or ͑30͒ in conjunction with Maxwell’s fieldequations. Assuming static vector fields, for example, ob- ٌ•Јϭ0,
servers in the lab frame begin by writing Maxwell’s fieldequations as ٌϫЈϭ0,
ٌ•Dϭ4␲␳,
ٌ•Bϭ0,
␮ ͑ЈϪvϫЈ͒ͪ ϭ0,
where we have neglected terms of higher order than ␯, and vϭ␻re␾ is the velocity of the rotating cylinder. Since the
ٌϫHϭ4␲j.
external magnetic field is parallel to the axis of the cylinder, Eq. ͑33a͒ implies that the normal component of ⑀E
ϫ(1/␮)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 BЈ ϭ␮BЈ
effects to be ignored, Eq. ͑33a͒ then implies that EЈ ϭͩ 1
⑀Ϫ␮ͪ vϫBOUT
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 . 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 ϭvϫB
⑀Ϫ␮ͪ vϫBOUT.
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- ⑀␮ vϫIN
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 ⑀ vϫOUT
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.
⑀Ϫ␮ͪ vϫOUT.
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 EЈϩͩ ⑀Ϫ ␮ vϫBЈ΁ϭ0,
that although vector and one-form constitutive equations dif-fer in the rotating frame, in the lab frame both pairs of con- ٌ•͑BЈϩvϫEЈ͒ϭ0,
stitutive equations are in agreement with the 1908 constitu-tive equations of Minkowski.1,2,4,5,11 ٌϫ͑EЈϩvϫBЈ͒ϭ0,
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 ␮ BЈϭ0,
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␯␥1ϩ␯2A ␯␯ ͪ 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 Aϭ(␥Ϫ1)/␯2. Equating Eqs. ͑A3͒ and ͑A5͒ andsolving for FЈ leads to an expression that transforms quan- tities from the lab frame directly to the rotating frame: FЈϭ͑LRϪ1͒TF͑LRϪ1͒.
Working out the matrix multiplication, we find that Eq. ͑A7͒can be rewritten simply as ϩ␯2. The inverse metric tensor in rotating coordinates is7 ͩ 1 Ϫ␯ Ϫ␯xy 0 whereNisgivenby We begin by noting that, although the rotating frame has a velocity vϭ␻re
␾ 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 ˜ ЈϭϩvϫϪ ␥ϩ ͑v͒v,
where FЉ is the electromagnetic tensor in the MCRF, RT isthe transpose of R, and R is a transformation that relates ˜ Јϭ␥͑Ϫvϫ͒ϩ␥2vϫ͑ϩvϫ͒Ϫ
͑v͒v,
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-
˜ ϩvϫH
˜ Ϫ ␥ϩ ͑v͒v,
in which ␥ϭ1/ͱ1Ϫ␯2. Similarly, the electromagnetic tensor can be transformed from the lab frame to the MCRF by using ˜ Јϭ␥͑H
˜ ϪvϫD
˜ ͒ϩ␥2vϫ͑D
˜ ϩvϫH
˜ ͒Ϫ ␥ϩ ͑v͒v.
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 FЈ now represents the contravariant electromagnetic C. T. Ridgely, ‘‘Applying relativistic electrodynamics to a rotating mate- tensor in the rotating frame, and M ϭNϪ1.6 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– ␥2͑1Ϫ␯2␥͒ 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- EЈϭ␥2͑EϩvϫB͒Ϫ␥3vϫ͑BϪvϫE͒Ϫ
␥ϩ ͑vE͒v,
cial relativistic prediction ⌬Vϭ␻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,’’ BЈϭ␥͑BϪvϫE͒Ϫ ␥ϩ ͑vB͒v.
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– DЈϭ␥2͑DϩvϫH͒Ϫ␥3vϫ͑HϪvϫD͒Ϫ ␥ϩ ͑vD͒v,
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 HЈϭ␥͑HϪvϫD͒Ϫ ␥ϩ ͑vH͒v.
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 ˜ Јϭϩvϫ,
25, 391–395 ͑1939͒.
20C. T. Ridgely, ‘‘The electrodynamics of a rotating material medium,’’ ˜ Јϭ,
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.
BЈϭBϪvϫE.
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.

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