829:
1200:
1367:
551:
1012:
940:
denote respectively the components of the electric and magnetic fields, as measured by an inertial observer (at rest in our coordinates). As usual in relativistic physics, we will find it convenient to work with
824:{\displaystyle F_{ab}=\left({\begin{matrix}0&B_{z}&-B_{y}&E_{x}/c\\-B_{z}&0&B_{x}&E_{y}/c\\B_{y}&-B_{x}&0&E_{z}/c\\-E_{x}/c&-E_{y}/c&-E_{z}/c&0\end{matrix}}\right)}
1206:
365:
1412:. In this case, the invariants reveal that the electric and magnetic fields are perpendicular and that they are of the same magnitude (in geometrised units). An example of a null field is a
1573:
1472:
938:
885:
1504:
427:
The classification theorem characterizes the possible principal null directions of a bivector. It states that one of the following must hold for any nonzero bivector:
1542:
1410:
996:
969:
446:
Furthermore, for any non-null bivector, the two eigenvalues associated with the two distinct principal null directions have the same magnitude but opposite sign,
1195:{\displaystyle P\equiv {\frac {1}{2}}F_{ab}\,F^{ab}=\|{\vec {B}}\|^{2}-{\frac {\|{\vec {E}}\|^{2}}{c^{2}}}=-{\frac {1}{2}}{}^{*}F_{ab}\,{}^{*}F^{ab}}
1362:{\displaystyle Q\equiv {\frac {1}{4}}F_{ab}\,{}^{*}F^{ab}={\frac {1}{8}}\epsilon ^{abcd}F_{ab}F_{cd}={\frac {{\vec {E}}\cdot {\vec {B}}}{c}}}
105:
and the underlying manifold; fortunately, nothing is lost by this specialization, for reasons we discuss as the end of the article.
260:
are linearly independent; the two cases are mutually exclusive. Stated like this, the dichotomy makes no reference to the metric
37:
1677:
1561:
304:
1638:
1672:
1662:
386:
139:
431:
the bivector has one "repeated" principal null direction; in this case, the bivector itself is said to be
1507:
1430:
890:
837:
1608:
1667:
264:, only to exterior algebra. But it is easily seen that the associated skew-symmetric linear operator
942:
86:
222:. Any nonzero bivector over a 4-dimensional vector space either is simple, or can be written as
1578:
1477:
535:
109:
41:
438:
the bivector has two distinct principal null directions; in this case, the bivector is called
1588:
1557:
1521:
1510:
for which either the electric or magnetic field vanishes. (These correspond respectively to
531:
511:
17:
85:
is isometric as a real inner product space to E. That is, it has the same notion of vector
1383:
981:
527:
45:
131:
by defining and examining the so-called "principal null directions". Let us explain this.
8:
1553:
1544:, there exists an inertial frame in which electric and magnetic fields are proportional.
948:
94:
21:
1373:(Fundamental means that every other invariant can be expressed in terms of these two.)
526:
The algebraic classification of bivectors given above has an important application in
196:
We mention a dichotomy drawn from exterior algebra. A bivector that can be written as
1634:
101:
Minkowski spacetime. This tends to blur the distinction between the tangent space at
976:
972:
538:) so we immediately obtain an algebraic classification of electromagnetic fields.
1417:
542:
142:
167:
defined by lowering one index with the metric. It acts on the tangent space at
1656:
1583:
1607:
The rank given here corresponds to that as a linear operator or tensor; the
292:
285:
61:
1413:
420:
A 1-dimensional subspace generated by a null eigenvector is called a
193:
to denote either the bivector or the operator, according to context.
29:
33:
1564:, everything works out exactly the same way on curved manifolds.
534:
is represented by a skew-symmetric second rank tensor field (the
60:(i.e. an event) of a Lorentzian spacetime is represented by a
1628:
1006:
The fundamental invariants of the electromagnetic field are:
273:
has rank 2 in the former case and rank 4 in the latter case.
90:
97:. To simplify the notation, we will assume the spacetime
576:
112:
for electromagnetic fields characterizes the bivector
1560:, if we simply replace "inertial frame" above with a
1524:
1480:
1433:
1386:
1209:
1015:
984:
951:
893:
840:
554:
456:, so we have three subclasses of non-null bivectors:
307:
276:
To state the classification theorem, we consider the
1536:
1498:
1466:
1404:
1361:
1194:
990:
963:
932:
879:
823:
545:, the electromagnetic field tensor has components
359:
360:{\displaystyle F^{a}{}_{b}r^{b}=\lambda \,r^{a}.}
1654:
1547:
51:
1109:
1093:
1078:
1062:
521:
40:. It is used in the study of solutions of
1460:
1239:
1169:
1045:
343:
1629:Landau, Lev D.; Lifshitz, E. M. (1973).
26:classification of electromagnetic fields
975:" formalism of special relativity, the
1655:
298:which satisfy the eigenvalue equation
116:in relation to the Lorentzian metric
56:The electromagnetic field at a point
998:is used to raise and lower indices.
218:are linearly independent, is called
1467:{\displaystyle P^{2}+Q^{2}\neq \,0}
44:and has applications in Einstein's
13:
284:, that is, the problem of finding
74:defined over the tangent space at
14:
1689:
933:{\displaystyle B_{x},B_{y},B_{z}}
880:{\displaystyle E_{x},E_{y},E_{z}}
1574:Electromagnetic peeling theorem
1631:The Classical Theory of Fields
1601:
1552:So far we have discussed only
1347:
1332:
1102:
1071:
1:
1622:
1556:. According to the (strong)
1001:
510:where the rank refers to the
536:electromagnetic field tensor
7:
1567:
1548:Curved Lorentzian manifolds
10:
1694:
1414:plane electromagnetic wave
1378:null electromagnetic field
52:The classification theorem
1678:Scientific classification
1499:{\displaystyle P\neq 0=Q}
189:. We will use the symbol
1615:is half that given here.
1594:
1508:inertial reference frame
541:In a cartesian chart on
422:principal null direction
1537:{\displaystyle Q\neq 0}
522:Physical interpretation
514:of the linear operator
1633:. New York: Pergamon.
1609:rank as defined for a
1579:Electrovacuum solution
1538:
1500:
1468:
1406:
1363:
1196:
992:
965:
934:
881:
825:
361:
110:classification theorem
1589:Petrov classification
1558:equivalence principle
1539:
1501:
1469:
1407:
1405:{\displaystyle P=Q=0}
1380:is characterised by
1364:
1197:
993:
991:{\displaystyle \eta }
966:
935:
882:
826:
532:electromagnetic field
370:The skew-symmetry of
362:
81:The tangent space at
18:differential geometry
1673:Lorentzian manifolds
1663:Mathematical physics
1522:
1478:
1431:
1427:is characterised by
1384:
1207:
1013:
982:
949:
891:
838:
552:
528:relativistic physics
305:
46:theory of relativity
1554:Minkowski spacetime
964:{\displaystyle c=1}
543:Minkowski spacetime
95:Minkowski spacetime
42:Maxwell's equations
38:Lorentzian manifold
36:at each point of a
22:theoretical physics
1534:
1506:, there exists an
1496:
1464:
1402:
1359:
1192:
988:
961:
930:
877:
821:
815:
357:
278:eigenvalue problem
32:classification of
1357:
1350:
1335:
1273:
1224:
1145:
1129:
1105:
1074:
1030:
943:geometrised units
424:of the bivector.
1685:
1668:Electromagnetism
1644:
1616:
1605:
1543:
1541:
1540:
1535:
1505:
1503:
1502:
1497:
1473:
1471:
1470:
1465:
1456:
1455:
1443:
1442:
1411:
1409:
1408:
1403:
1368:
1366:
1365:
1360:
1358:
1353:
1352:
1351:
1343:
1337:
1336:
1328:
1324:
1319:
1318:
1306:
1305:
1293:
1292:
1274:
1266:
1261:
1260:
1248:
1247:
1242:
1238:
1237:
1225:
1217:
1201:
1199:
1198:
1193:
1191:
1190:
1178:
1177:
1172:
1168:
1167:
1155:
1154:
1149:
1146:
1138:
1130:
1128:
1127:
1118:
1117:
1116:
1107:
1106:
1098:
1091:
1086:
1085:
1076:
1075:
1067:
1058:
1057:
1044:
1043:
1031:
1023:
997:
995:
994:
989:
977:Minkowski metric
973:Index gymnastics
970:
968:
967:
962:
939:
937:
936:
931:
929:
928:
916:
915:
903:
902:
886:
884:
883:
878:
876:
875:
863:
862:
850:
849:
830:
828:
827:
822:
820:
816:
804:
799:
798:
781:
776:
775:
758:
753:
752:
733:
728:
727:
711:
710:
696:
695:
679:
674:
673:
662:
661:
645:
644:
625:
620:
619:
608:
607:
593:
592:
567:
566:
503:
486:
455:
403:
381:the eigenvector
366:
364:
363:
358:
353:
352:
336:
335:
326:
325:
320:
317:
316:
243:
209:
188:
166:
130:
73:
1693:
1692:
1688:
1687:
1686:
1684:
1683:
1682:
1653:
1652:
1641:
1625:
1620:
1619:
1606:
1602:
1597:
1570:
1550:
1523:
1520:
1519:
1479:
1476:
1475:
1451:
1447:
1438:
1434:
1432:
1429:
1428:
1418:Minkowski space
1385:
1382:
1381:
1342:
1341:
1327:
1326:
1325:
1323:
1311:
1307:
1298:
1294:
1279:
1275:
1265:
1253:
1249:
1243:
1241:
1240:
1230:
1226:
1216:
1208:
1205:
1204:
1183:
1179:
1173:
1171:
1170:
1160:
1156:
1150:
1148:
1147:
1137:
1123:
1119:
1112:
1108:
1097:
1096:
1092:
1090:
1081:
1077:
1066:
1065:
1050:
1046:
1036:
1032:
1022:
1014:
1011:
1010:
1004:
983:
980:
979:
950:
947:
946:
924:
920:
911:
907:
898:
894:
892:
889:
888:
871:
867:
858:
854:
845:
841:
839:
836:
835:
814:
813:
808:
800:
794:
790:
785:
777:
771:
767:
762:
754:
748:
744:
738:
737:
729:
723:
719:
717:
712:
706:
702:
697:
691:
687:
684:
683:
675:
669:
665:
663:
657:
653:
651:
646:
640:
636:
630:
629:
621:
615:
611:
609:
603:
599:
594:
588:
584:
582:
575:
571:
559:
555:
553:
550:
549:
524:
497:
480:
447:
408:the eigenvalue
390:
348:
344:
331:
327:
321:
319:
318:
312:
308:
306:
303:
302:
272:
223:
197:
184:
172:
165:
153:
145:
143:linear operator
129:
117:
65:
54:
12:
11:
5:
1691:
1681:
1680:
1675:
1670:
1665:
1651:
1650:
1639:
1624:
1621:
1618:
1617:
1599:
1598:
1596:
1593:
1592:
1591:
1586:
1581:
1576:
1569:
1566:
1549:
1546:
1533:
1530:
1527:
1495:
1492:
1489:
1486:
1483:
1463:
1459:
1454:
1450:
1446:
1441:
1437:
1425:non-null field
1401:
1398:
1395:
1392:
1389:
1371:
1370:
1356:
1349:
1346:
1340:
1334:
1331:
1322:
1317:
1314:
1310:
1304:
1301:
1297:
1291:
1288:
1285:
1282:
1278:
1272:
1269:
1264:
1259:
1256:
1252:
1246:
1236:
1233:
1229:
1223:
1220:
1215:
1212:
1202:
1189:
1186:
1182:
1176:
1166:
1163:
1159:
1153:
1144:
1141:
1136:
1133:
1126:
1122:
1115:
1111:
1104:
1101:
1095:
1089:
1084:
1080:
1073:
1070:
1064:
1061:
1056:
1053:
1049:
1042:
1039:
1035:
1029:
1026:
1021:
1018:
1003:
1000:
987:
960:
957:
954:
927:
923:
919:
914:
910:
906:
901:
897:
874:
870:
866:
861:
857:
853:
848:
844:
832:
831:
819:
812:
809:
807:
803:
797:
793:
789:
786:
784:
780:
774:
770:
766:
763:
761:
757:
751:
747:
743:
740:
739:
736:
732:
726:
722:
718:
716:
713:
709:
705:
701:
698:
694:
690:
686:
685:
682:
678:
672:
668:
664:
660:
656:
652:
650:
647:
643:
639:
635:
632:
631:
628:
624:
618:
614:
610:
606:
602:
598:
595:
591:
587:
583:
581:
578:
577:
574:
570:
565:
562:
558:
523:
520:
508:
507:
506:
505:
487:
470:
444:
443:
436:
418:
417:
374:implies that:
368:
367:
356:
351:
347:
342:
339:
334:
330:
324:
315:
311:
268:
180:
161:
149:
140:skew-symmetric
125:
53:
50:
9:
6:
4:
3:
2:
1690:
1679:
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1660:
1658:
1648:
1642:
1640:0-08-025072-6
1636:
1632:
1627:
1626:
1614:
1612:
1604:
1600:
1590:
1587:
1585:
1584:Lorentz group
1582:
1580:
1577:
1575:
1572:
1571:
1565:
1563:
1559:
1555:
1545:
1531:
1528:
1525:
1518:fields.) If
1517:
1516:electrostatic
1513:
1512:magnetostatic
1509:
1493:
1490:
1487:
1484:
1481:
1461:
1457:
1452:
1448:
1444:
1439:
1435:
1426:
1421:
1419:
1415:
1399:
1396:
1393:
1390:
1387:
1379:
1374:
1354:
1344:
1338:
1329:
1320:
1315:
1312:
1308:
1302:
1299:
1295:
1289:
1286:
1283:
1280:
1276:
1270:
1267:
1262:
1257:
1254:
1250:
1244:
1234:
1231:
1227:
1221:
1218:
1213:
1210:
1203:
1187:
1184:
1180:
1174:
1164:
1161:
1157:
1151:
1142:
1139:
1134:
1131:
1124:
1120:
1113:
1099:
1087:
1082:
1068:
1059:
1054:
1051:
1047:
1040:
1037:
1033:
1027:
1024:
1019:
1016:
1009:
1008:
1007:
999:
985:
978:
974:
958:
955:
952:
944:
925:
921:
917:
912:
908:
904:
899:
895:
872:
868:
864:
859:
855:
851:
846:
842:
817:
810:
805:
801:
795:
791:
787:
782:
778:
772:
768:
764:
759:
755:
749:
745:
741:
734:
730:
724:
720:
714:
707:
703:
699:
692:
688:
680:
676:
670:
666:
658:
654:
648:
641:
637:
633:
626:
622:
616:
612:
604:
600:
596:
589:
585:
579:
572:
568:
563:
560:
556:
548:
547:
546:
544:
539:
537:
533:
529:
519:
517:
513:
501:
495:
491:
488:
484:
478:
474:
471:
468:
464:
461:
460:
459:
458:
457:
454:
450:
441:
437:
434:
430:
429:
428:
425:
423:
415:
411:
407:
401:
397:
393:
388:
384:
380:
377:
376:
375:
373:
354:
349:
345:
340:
337:
332:
328:
322:
313:
309:
301:
300:
299:
297:
294:
290:
287:
283:
279:
274:
271:
267:
263:
259:
255:
251:
247:
242:
238:
234:
230:
226:
221:
217:
213:
208:
204:
200:
194:
192:
187:
183:
179:
175:
170:
164:
160:
157:
152:
148:
144:
141:
137:
134:The bivector
132:
128:
124:
120:
115:
111:
106:
104:
100:
96:
92:
88:
84:
79:
77:
72:
68:
63:
59:
49:
47:
43:
39:
35:
31:
27:
23:
19:
1646:
1630:
1610:
1603:
1551:
1515:
1511:
1424:
1422:
1377:
1375:
1372:
1005:
833:
540:
525:
515:
509:
499:
493:
489:
482:
476:
472:
466:
462:
452:
448:
445:
439:
432:
426:
421:
419:
413:
409:
405:
399:
395:
391:
382:
378:
371:
369:
295:
293:eigenvectors
288:
281:
277:
275:
269:
265:
261:
257:
253:
249:
245:
240:
236:
232:
228:
224:
219:
215:
211:
206:
202:
198:
195:
190:
185:
181:
177:
173:
168:
162:
158:
155:
150:
146:
135:
133:
126:
122:
118:
113:
107:
102:
98:
82:
80:
75:
70:
66:
57:
55:
25:
15:
1562:frame field
387:null vector
286:eigenvalues
1657:Categories
1647:section 25
1623:References
1002:Invariants
971:. In the "
490:non-simple
1529:≠
1485:≠
1458:≠
1348:→
1339:⋅
1333:→
1277:ϵ
1245:∗
1214:≡
1175:∗
1152:∗
1135:−
1110:‖
1103:→
1094:‖
1088:−
1079:‖
1072:→
1063:‖
1020:≡
986:η
945:in which
788:−
765:−
742:−
700:−
634:−
597:−
463:spacelike
412:is zero,
341:λ
138:yields a
87:magnitude
64:bivector
34:bivectors
30:pointwise
1568:See also
496:≠ 0 and
479:≠ 0 and
475: :
473:timelike
440:non-null
244:, where
210:, where
1613:-vector
414:or both
1637:
834:where
530:: the
494:ν
477:ν
467:ν
453:ν
449:λ
389:(i.e.
379:either
289:λ
256:, and
220:simple
24:, the
1595:Notes
1474:. If
498:rank
481:rank
402:) = 0
385:is a
91:angle
28:is a
1645:See
1635:ISBN
1514:and
887:and
512:rank
433:null
291:and
280:for
108:The
89:and
62:real
20:and
1416:in
502:= 4
485:= 2
469:= 0
451:= ±
404:),
171:by
93:as
16:In
1659::
1423:A
1420:.
1376:A
518:.
492::
465::
406:or
252:,
248:,
239:∧
235:+
231:∧
227:=
214:,
205:∧
201:=
176:→
163:cb
154:=
127:ab
121:=
99:is
78:.
69:=
48:.
1649:.
1643:.
1611:k
1532:0
1526:Q
1494:Q
1491:=
1488:0
1482:P
1462:0
1453:2
1449:Q
1445:+
1440:2
1436:P
1400:0
1397:=
1394:Q
1391:=
1388:P
1369:.
1355:c
1345:B
1330:E
1321:=
1316:d
1313:c
1309:F
1303:b
1300:a
1296:F
1290:d
1287:c
1284:b
1281:a
1271:8
1268:1
1263:=
1258:b
1255:a
1251:F
1235:b
1232:a
1228:F
1222:4
1219:1
1211:Q
1188:b
1185:a
1181:F
1165:b
1162:a
1158:F
1143:2
1140:1
1132:=
1125:2
1121:c
1114:2
1100:E
1083:2
1069:B
1060:=
1055:b
1052:a
1048:F
1041:b
1038:a
1034:F
1028:2
1025:1
1017:P
959:1
956:=
953:c
926:z
922:B
918:,
913:y
909:B
905:,
900:x
896:B
873:z
869:E
865:,
860:y
856:E
852:,
847:x
843:E
818:)
811:0
806:c
802:/
796:z
792:E
783:c
779:/
773:y
769:E
760:c
756:/
750:x
746:E
735:c
731:/
725:z
721:E
715:0
708:x
704:B
693:y
689:B
681:c
677:/
671:y
667:E
659:x
655:B
649:0
642:z
638:B
627:c
623:/
617:x
613:E
605:y
601:B
590:z
586:B
580:0
573:(
569:=
564:b
561:a
557:F
516:F
504:,
500:F
483:F
442:.
435:,
416:.
410:λ
400:r
398:,
396:r
394:(
392:η
383:r
372:F
355:.
350:a
346:r
338:=
333:b
329:r
323:b
314:a
310:F
296:r
282:F
270:b
266:F
262:η
258:y
254:x
250:w
246:v
241:y
237:x
233:w
229:v
225:F
216:w
212:v
207:w
203:v
199:F
191:F
186:r
182:b
178:F
174:r
169:p
159:η
156:F
151:b
147:F
136:F
123:η
119:η
114:F
103:p
83:p
76:p
71:F
67:F
58:p
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