2399:
2530:
1026:
1284:
1166:
1567:
1424:
48:
that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any basis and, in particular, are non-numerical. Thus it should not be confused with the
2058:
2249:
680:
1639:
2168:
1174:
3399:
1851:
1702:
2410:
1924:
877:
1056:
866:
800:
179:
3586:
1435:
2241:
1295:
1733:
495:
2212:
447:
343:
294:
1753:
730:
609:
550:
405:
125:
1939:
1788:
703:
582:
523:
374:
243:
223:
199:
94:
501:. However, as the indices are non-numerical, it does not imply summation: rather it corresponds to the abstract basis-independent trace operation (or
1929:
Abstract index notation handles braiding as follows. On a particular tensor product, an ordering of the abstract indices is fixed (usually this is a
3532:
3044:
2571:
132:
70:
3433:
3343:
2394:{\displaystyle \omega _{}:={\frac {1}{3!}}\sum _{\sigma \in \mathrm {S} _{3}}(-1)^{{\text{sgn}}(\sigma )}\omega _{\sigma (a)\sigma (b)\sigma (c)}}
303:
of the slots with Latin letters, which have no significance apart from their designation as labels of the slots (i.e., they are non-numerical):
2650:
2625:
2600:
617:
3507:
3228:
2907:
871:
The last two expressions denote the same object as the first. Tensors of this type are denoted using similar notation, for example:
2687:
1933:). The braid is then represented in notation by permuting the labels of the indices. Thus, for instance, with the Riemann tensor
1587:
2069:
3577:
3109:
1797:
3678:
3442:
1042:
In general, whenever one contravariant and one covariant factor occur in a tensor product of spaces, there is an associated
2525:{\displaystyle \omega _{(abc)}:={\frac {1}{3!}}\sum _{\sigma \in \mathrm {S} _{3}}\omega _{\sigma (a)\sigma (b)\sigma (c)}}
3537:
2960:
2892:
498:
58:
3522:
1647:
497:
over its last two slots. This manner of representing tensor contractions by repeated indices is formally similar to the
3497:
3419:
2985:
2566:
1859:
352:(or trace) between two tensors is represented by the repetition of an index label, where one label is contravariant (an
1021:{\displaystyle h^{a}{}_{bc}{}^{d}{}_{e}\in V^{a}{}_{bc}{}^{d}{}_{e}=V\otimes V^{*}\otimes V^{*}\otimes V\otimes V^{*}.}
3336:
3223:
3581:
1289:
These trace operations are signified on tensors by the repetition of an index. Thus the first trace map is given by
3601:
3562:
3034:
2854:
2541:
1279:{\displaystyle \mathrm {Tr} _{15}:V\otimes V^{*}\otimes V^{*}\otimes V\otimes V^{*}\to V^{*}\otimes V^{*}\otimes V}
1161:{\displaystyle \mathrm {Tr} _{12}:V\otimes V^{*}\otimes V^{*}\otimes V\otimes V^{*}\to V^{*}\otimes V\otimes V^{*}}
3517:
3502:
3367:
2706:
3208:
3290:
3162:
2869:
1791:
811:
738:
138:
3260:
2947:
2864:
3641:
3512:
3329:
3218:
3074:
3029:
1562:{\displaystyle \mathrm {Tr} _{15}:h{}^{a}{}_{b}{}_{c}{}^{d}{}_{e}\mapsto h{}^{a}{}_{b}{}_{c}{}^{d}{}_{a}.}
1419:{\displaystyle \mathrm {Tr} _{12}:h{}^{a}{}_{b}{}_{c}{}^{d}{}_{e}\mapsto h{}^{a}{}_{a}{}_{c}{}^{d}{}_{e}}
685:
Label each factor in this tensor product with a Latin letter in a raised position for each contravariant
3300:
3255:
2735:
2680:
1763:
31:
2217:
3647:
3275:
3203:
3089:
2955:
2917:
2849:
2561:
1711:
66:
455:
3383:
3375:
3152:
2975:
2965:
2814:
2799:
2755:
2184:
1930:
2593:
Modern
Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics
413:
3285:
3142:
2995:
2809:
2745:
1756:
1037:
349:
62:
309:
255:
3596:
3567:
3280:
3188:
3049:
3024:
2839:
2750:
2730:
2646:
2621:
2596:
2546:
1738:
3673:
3629:
3572:
3552:
3295:
3193:
2970:
2937:
2922:
2804:
2673:
1767:
708:
587:
528:
383:
103:
3635:
3591:
3542:
3527:
3487:
3454:
3424:
3265:
3213:
3157:
3137:
3039:
2927:
2794:
2765:
2053:{\displaystyle R=R_{abc}{}^{d}\in V_{abc}{}^{d}=V^{*}\otimes V^{*}\otimes V^{*}\otimes V,}
1705:
502:
2178:
A general tensor may be antisymmetrized or symmetrized, and there is according notation.
1704:). In general, the braiding maps are in one-to-one correspondence with elements of the
3623:
3617:
3557:
3547:
3472:
3391:
3305:
3270:
3250:
3167:
3000:
2990:
2980:
2902:
2874:
2859:
2844:
2760:
2551:
1773:
1578:
688:
567:
561:
508:
359:
228:
208:
184:
79:
50:
1644:
interchanges the two tensor factors (so that its action on simple tensors is given by
3667:
3438:
3352:
3242:
3147:
3059:
2932:
202:
54:
3652:
3477:
3310:
3114:
3099:
3064:
2912:
2897:
2181:
We demonstrate the notation by example. Let's antisymmetrize the type-(0,3) tensor
97:
3198:
3172:
3094:
2783:
2722:
40:(also referred to as slot-naming index notation) is a mathematical notation for
17:
3079:
128:
2643:
Spinors and Space-Time, Volume 1: Two-Spinor
Calculus and Relativistic Fields
3492:
3054:
3005:
3084:
3069:
2778:
2740:
3104:
2696:
2556:
1577:
To any tensor product on a single vector space, there are associated
45:
41:
675:{\displaystyle V\otimes V^{*}\otimes V^{*}\otimes V\otimes V^{*}.}
69:
in modern abstract tensor notation, while preserving the explicit
2618:
The Road to
Reality: A Complete Guide to the Laws of the Universe
3321:
3325:
2669:
3400:
Fashion, Faith, and
Fantasy in the New Physics of the Universe
1634:{\displaystyle \tau _{(12)}:V\otimes V\rightarrow V\otimes V}
2163:{\displaystyle R_{abc}{}^{d}+R_{cab}{}^{d}+R_{bca}{}^{d}=0.}
1171:
is the trace on the first two spaces of the tensor product.
2665:
1735:
to denote the braiding map associated to the permutation
1846:{\displaystyle V^{*}\otimes V^{*}\otimes V^{*}\otimes V}
1708:, acting by permuting the tensor factors. Here, we use
225:. In other words, it is a function of two arguments in
2413:
2252:
2220:
2187:
2072:
1942:
1862:
1800:
1776:
1741:
1714:
1650:
1590:
1438:
1298:
1177:
1059:
880:
814:
741:
711:
705:
factor, and in a lowered position for each covariant
691:
620:
590:
570:
531:
511:
458:
416:
386:
362:
312:
258:
231:
211:
187:
141:
106:
82:
3610:
3465:
3410:
3359:
3241:
3181:
3130:
3123:
3015:
2946:
2883:
2827:
2774:
2721:
2714:
1697:{\displaystyle \tau _{(12)}(v\otimes w)=w\otimes v}
2524:
2393:
2235:
2206:
2162:
2052:
1919:{\displaystyle R+\tau _{(123)}R+\tau _{(132)}R=0.}
1918:
1845:
1782:
1747:
1727:
1696:
1633:
1561:
1418:
1278:
1160:
1020:
860:
794:
724:
697:
674:
603:
576:
544:
517:
489:
441:
399:
368:
337:
288:
237:
217:
193:
173:
119:
88:
1853:. The first Bianchi identity then asserts that
560:A general homogeneous tensor is an element of a
61:to compensate for the difficulty in describing
3337:
2681:
2591:Kip S. Thorne and Roger D. Blandford (2017).
732:position. In this way, write the product as
27:Mathematical notation for tensors and spinors
8:
3513:Penrose interpretation of quantum mechanics
2641:Roger Penrose and Wolfgang Rindler (1984).
3344:
3330:
3322:
3127:
2718:
2688:
2674:
2666:
2243:is the symmetric group on three elements.
1286:is the trace on the first and last space.
57:as a way to use the formal aspects of the
2483:
2471:
2466:
2458:
2439:
2418:
2412:
2352:
2332:
2331:
2310:
2305:
2297:
2278:
2257:
2251:
2227:
2222:
2219:
2192:
2186:
2148:
2146:
2133:
2120:
2118:
2105:
2092:
2090:
2077:
2071:
2035:
2022:
2009:
1996:
1994:
1981:
1968:
1966:
1953:
1941:
1895:
1873:
1861:
1831:
1818:
1805:
1799:
1775:
1740:
1719:
1713:
1655:
1649:
1595:
1589:
1550:
1548:
1541:
1539:
1532:
1530:
1523:
1521:
1514:
1512:
1499:
1497:
1490:
1488:
1481:
1479:
1472:
1470:
1463:
1461:
1448:
1440:
1437:
1410:
1408:
1401:
1399:
1392:
1390:
1383:
1381:
1374:
1372:
1359:
1357:
1350:
1348:
1341:
1339:
1332:
1330:
1323:
1321:
1308:
1300:
1297:
1264:
1251:
1238:
1219:
1206:
1187:
1179:
1176:
1152:
1133:
1120:
1101:
1088:
1069:
1061:
1058:
1009:
990:
977:
958:
956:
949:
947:
937:
935:
928:
915:
913:
906:
904:
894:
892:
885:
879:
861:{\displaystyle V^{a}{}_{bc}{}^{d}{}_{e}.}
849:
847:
840:
838:
828:
826:
819:
813:
795:{\displaystyle V^{a}V_{b}V_{c}V^{d}V_{e}}
786:
776:
766:
756:
746:
740:
716:
710:
690:
663:
644:
631:
619:
595:
589:
569:
536:
530:
510:
481:
479:
469:
457:
433:
431:
421:
415:
391:
385:
361:
323:
311:
257:
230:
210:
186:
165:
152:
140:
111:
105:
81:
3045:Covariance and contravariance of vectors
2572:Covariance and contravariance of vectors
1766:, for instance, in order to express the
3434:The Large, the Small and the Human Mind
2583:
174:{\displaystyle h\in V^{*}\otimes V^{*}}
1755:(represented as a product of disjoint
245:which can be represented as a pair of
2174:Antisymmetrization and symmetrization
131:. Consider, for example, an order-2
7:
3533:Penrose–Hawking singularity theorems
299:Abstract index notation is merely a
2908:Tensors in curvilinear coordinates
2467:
2306:
2223:
1444:
1441:
1304:
1301:
1183:
1180:
1065:
1062:
556:Abstract indices and tensor spaces
25:
1581:. For example, the braiding map
505:) between tensor factors of type
53:. The notation was introduced by
3578:Orchestrated objective reduction
3451:White Mars or, The Mind Set Free
2236:{\displaystyle \mathrm {S} _{3}}
376:) and one label is covariant (a
1762:Braiding maps are important in
1728:{\displaystyle \tau _{\sigma }}
2645:. Cambridge University Press.
2595:. Princeton University Press.
2517:
2511:
2505:
2499:
2493:
2487:
2431:
2419:
2404:Similarly, we may symmetrize:
2386:
2380:
2374:
2368:
2362:
2356:
2343:
2337:
2328:
2318:
2270:
2258:
1902:
1896:
1880:
1874:
1679:
1667:
1662:
1656:
1619:
1602:
1596:
1505:
1365:
1244:
1126:
490:{\displaystyle t=t_{ab}{}^{c}}
280:
268:
1:
3538:Riemannian Penrose inequality
2961:Exterior covariant derivative
2893:Tensor (intrinsic definition)
2207:{\displaystyle \omega _{abc}}
2063:the Bianchi identity becomes
499:Einstein summation convention
73:of the expressions involved.
59:Einstein summation convention
3445:and Stephen Hawking) (1997)
3420:The Nature of Space and Time
2986:Raising and lowering indices
2567:Raising and lowering indices
442:{\displaystyle t_{ab}{}^{b}}
380:corresponding to the factor
356:corresponding to the factor
3224:Gluon field strength tensor
3695:
3602:Conformal cyclic cosmology
3563:Penrose graphical notation
3035:Cartan formalism (physics)
2855:Penrose graphical notation
2542:Penrose graphical notation
1794:, regarded as a tensor in
1035:
29:
3503:Weyl curvature hypothesis
2707:Glossary of tensor theory
2703:
452:is the trace of a tensor
338:{\displaystyle h=h_{ab}.}
289:{\displaystyle h=h(-,-).}
201:can be identified with a
67:covariant differentiation
3523:Newman–Penrose formalism
3291:Gregorio Ricci-Curbastro
3163:Riemann curvature tensor
2870:Van der Waerden notation
407:). Thus, for instance,
30:Not to be confused with
3483:Abstract index notation
3261:Elwin Bruno Christoffel
3194:Angular momentum tensor
2865:Tetrad (index notation)
2835:Abstract index notation
1748:{\displaystyle \sigma }
38:Abstract index notation
3642:John Beresford Leathes
3582:Penrose–Lucas argument
3573:Penrose–Terrell effect
3368:The Emperor's New Mind
3075:Levi-Civita connection
2616:Roger Penrose (2007).
2526:
2395:
2237:
2208:
2164:
2054:
1931:lexicographic ordering
1920:
1847:
1784:
1749:
1729:
1698:
1635:
1563:
1420:
1280:
1162:
1050:) map. For instance,
1022:
862:
796:
726:
699:
676:
605:
578:
546:
519:
491:
443:
401:
370:
339:
290:
239:
219:
195:
175:
121:
90:
3679:Mathematical notation
3518:Moore–Penrose inverse
3493:Geometry of spacetime
3301:Jan Arnoldus Schouten
3256:Augustin-Louis Cauchy
2736:Differential geometry
2527:
2396:
2238:
2209:
2165:
2055:
1921:
1848:
1785:
1764:differential geometry
1750:
1730:
1699:
1636:
1564:
1421:
1281:
1163:
1023:
863:
797:
727:
725:{\displaystyle V^{*}}
700:
677:
606:
604:{\displaystyle V^{*}}
579:
547:
545:{\displaystyle V^{*}}
520:
492:
444:
402:
400:{\displaystyle V^{*}}
371:
340:
291:
240:
220:
196:
176:
122:
120:{\displaystyle V^{*}}
91:
32:tensor index notation
3648:Illumination problem
3508:Penrose inequalities
3276:Carl Friedrich Gauss
3209:stress–energy tensor
3204:Cauchy stress tensor
2956:Covariant derivative
2918:Antisymmetric tensor
2850:Multi-index notation
2562:Antisymmetric tensor
2411:
2250:
2218:
2185:
2070:
1940:
1860:
1798:
1774:
1739:
1712:
1648:
1588:
1436:
1296:
1175:
1057:
878:
812:
739:
709:
689:
618:
588:
568:
529:
509:
456:
414:
384:
360:
310:
256:
229:
209:
185:
139:
104:
80:
3384:The Road to Reality
3376:Shadows of the Mind
3153:Nonmetricity tensor
3008:(2nd-order tensors)
2976:Hodge star operator
2966:Exterior derivative
2815:Transport phenomena
2800:Continuum mechanics
2756:Multilinear algebra
1757:cyclic permutations
3286:Tullio Levi-Civita
3229:Metric tensor (GR)
3143:Levi-Civita symbol
2996:Tensor contraction
2810:General relativity
2746:Euclidean geometry
2522:
2478:
2391:
2317:
2233:
2204:
2160:
2050:
1916:
1843:
1780:
1745:
1725:
1694:
1631:
1559:
1429:and the second by
1416:
1276:
1158:
1038:Tensor contraction
1018:
858:
792:
722:
695:
672:
601:
574:
542:
525:and those of type
515:
487:
439:
397:
366:
350:tensor contraction
335:
286:
235:
215:
191:
171:
117:
86:
3661:
3660:
3597:Andromeda paradox
3568:Penrose transform
3498:Cosmic censorship
3319:
3318:
3281:Hermann Grassmann
3237:
3236:
3189:Moment of inertia
3050:Differential form
3025:Affine connection
2840:Einstein notation
2823:
2822:
2751:Exterior calculus
2731:Coordinate system
2547:Einstein notation
2454:
2452:
2335:
2293:
2291:
1783:{\displaystyle R}
698:{\displaystyle V}
577:{\displaystyle V}
518:{\displaystyle V}
369:{\displaystyle V}
238:{\displaystyle V}
218:{\displaystyle V}
194:{\displaystyle h}
89:{\displaystyle V}
16:(Redirected from
3686:
3630:Jonathan Penrose
3587:FELIX experiment
3553:Penrose triangle
3458:
3446:
3443:Nancy Cartwright
3428:
3411:Coauthored books
3346:
3339:
3332:
3323:
3296:Bernhard Riemann
3128:
2971:Exterior product
2938:Two-point tensor
2923:Symmetric tensor
2805:Electromagnetism
2719:
2690:
2683:
2676:
2667:
2657:
2656:
2652:978-0-52133707-6
2638:
2632:
2631:
2627:978-0-67977631-4
2613:
2607:
2606:
2602:978-0-69115902-7
2588:
2531:
2529:
2528:
2523:
2521:
2520:
2477:
2476:
2475:
2470:
2453:
2451:
2440:
2435:
2434:
2400:
2398:
2397:
2392:
2390:
2389:
2347:
2346:
2336:
2333:
2316:
2315:
2314:
2309:
2292:
2290:
2279:
2274:
2273:
2242:
2240:
2239:
2234:
2232:
2231:
2226:
2213:
2211:
2210:
2205:
2203:
2202:
2169:
2167:
2166:
2161:
2153:
2152:
2147:
2144:
2143:
2125:
2124:
2119:
2116:
2115:
2097:
2096:
2091:
2088:
2087:
2059:
2057:
2056:
2051:
2040:
2039:
2027:
2026:
2014:
2013:
2001:
2000:
1995:
1992:
1991:
1973:
1972:
1967:
1964:
1963:
1925:
1923:
1922:
1917:
1906:
1905:
1884:
1883:
1852:
1850:
1849:
1844:
1836:
1835:
1823:
1822:
1810:
1809:
1789:
1787:
1786:
1781:
1768:Bianchi identity
1754:
1752:
1751:
1746:
1734:
1732:
1731:
1726:
1724:
1723:
1703:
1701:
1700:
1695:
1666:
1665:
1640:
1638:
1637:
1632:
1606:
1605:
1568:
1566:
1565:
1560:
1555:
1554:
1549:
1546:
1545:
1540:
1537:
1536:
1531:
1528:
1527:
1522:
1519:
1518:
1513:
1504:
1503:
1498:
1495:
1494:
1489:
1486:
1485:
1480:
1477:
1476:
1471:
1468:
1467:
1462:
1453:
1452:
1447:
1425:
1423:
1422:
1417:
1415:
1414:
1409:
1406:
1405:
1400:
1397:
1396:
1391:
1388:
1387:
1382:
1379:
1378:
1373:
1364:
1363:
1358:
1355:
1354:
1349:
1346:
1345:
1340:
1337:
1336:
1331:
1328:
1327:
1322:
1313:
1312:
1307:
1285:
1283:
1282:
1277:
1269:
1268:
1256:
1255:
1243:
1242:
1224:
1223:
1211:
1210:
1192:
1191:
1186:
1167:
1165:
1164:
1159:
1157:
1156:
1138:
1137:
1125:
1124:
1106:
1105:
1093:
1092:
1074:
1073:
1068:
1027:
1025:
1024:
1019:
1014:
1013:
995:
994:
982:
981:
963:
962:
957:
954:
953:
948:
945:
944:
936:
933:
932:
920:
919:
914:
911:
910:
905:
902:
901:
893:
890:
889:
867:
865:
864:
859:
854:
853:
848:
845:
844:
839:
836:
835:
827:
824:
823:
801:
799:
798:
793:
791:
790:
781:
780:
771:
770:
761:
760:
751:
750:
731:
729:
728:
723:
721:
720:
704:
702:
701:
696:
681:
679:
678:
673:
668:
667:
649:
648:
636:
635:
610:
608:
607:
602:
600:
599:
583:
581:
580:
575:
551:
549:
548:
543:
541:
540:
524:
522:
521:
516:
496:
494:
493:
488:
486:
485:
480:
477:
476:
448:
446:
445:
440:
438:
437:
432:
429:
428:
406:
404:
403:
398:
396:
395:
375:
373:
372:
367:
344:
342:
341:
336:
331:
330:
295:
293:
292:
287:
244:
242:
241:
236:
224:
222:
221:
216:
200:
198:
197:
192:
180:
178:
177:
172:
170:
169:
157:
156:
126:
124:
123:
118:
116:
115:
95:
93:
92:
87:
21:
18:Abstract indices
3694:
3693:
3689:
3688:
3687:
3685:
3684:
3683:
3664:
3663:
3662:
3657:
3636:Shirley Hodgson
3606:
3592:Trapped surface
3543:Penrose process
3528:Penrose diagram
3488:Black hole bomb
3461:
3455:Brian W. Aldiss
3449:
3431:
3425:Stephen Hawking
3417:
3406:
3355:
3350:
3320:
3315:
3266:Albert Einstein
3233:
3214:Einstein tensor
3177:
3158:Ricci curvature
3138:Kronecker delta
3124:Notable tensors
3119:
3040:Connection form
3017:
3011:
2942:
2928:Tensor operator
2885:
2879:
2819:
2795:Computer vision
2788:
2770:
2766:Tensor calculus
2710:
2699:
2694:
2663:
2661:
2660:
2653:
2640:
2639:
2635:
2628:
2615:
2614:
2610:
2603:
2590:
2589:
2585:
2580:
2538:
2479:
2465:
2444:
2414:
2409:
2408:
2348:
2327:
2304:
2283:
2253:
2248:
2247:
2221:
2216:
2215:
2188:
2183:
2182:
2176:
2145:
2129:
2117:
2101:
2089:
2073:
2068:
2067:
2031:
2018:
2005:
1993:
1977:
1965:
1949:
1938:
1937:
1891:
1869:
1858:
1857:
1827:
1814:
1801:
1796:
1795:
1772:
1771:
1737:
1736:
1715:
1710:
1709:
1706:symmetric group
1651:
1646:
1645:
1591:
1586:
1585:
1575:
1547:
1538:
1529:
1520:
1511:
1496:
1487:
1478:
1469:
1460:
1439:
1434:
1433:
1407:
1398:
1389:
1380:
1371:
1356:
1347:
1338:
1329:
1320:
1299:
1294:
1293:
1260:
1247:
1234:
1215:
1202:
1178:
1173:
1172:
1148:
1129:
1116:
1097:
1084:
1060:
1055:
1054:
1040:
1034:
1005:
986:
973:
955:
946:
934:
924:
912:
903:
891:
881:
876:
875:
846:
837:
825:
815:
810:
809:
782:
772:
762:
752:
742:
737:
736:
712:
707:
706:
687:
686:
659:
640:
627:
616:
615:
591:
586:
585:
566:
565:
558:
532:
527:
526:
507:
506:
503:natural pairing
478:
465:
454:
453:
430:
417:
412:
411:
387:
382:
381:
358:
357:
319:
308:
307:
254:
253:
227:
226:
207:
206:
183:
182:
161:
148:
137:
136:
107:
102:
101:
78:
77:
35:
28:
23:
22:
15:
12:
11:
5:
3692:
3690:
3682:
3681:
3676:
3666:
3665:
3659:
3658:
3656:
3655:
3650:
3645:
3639:
3633:
3627:
3624:Oliver Penrose
3621:
3618:Lionel Penrose
3614:
3612:
3608:
3607:
3605:
3604:
3599:
3594:
3589:
3584:
3575:
3570:
3565:
3560:
3558:Penrose stairs
3555:
3550:
3548:Penrose tiling
3545:
3540:
3535:
3530:
3525:
3520:
3515:
3510:
3505:
3500:
3495:
3490:
3485:
3480:
3475:
3473:Twistor theory
3469:
3467:
3463:
3462:
3460:
3459:
3447:
3429:
3414:
3412:
3408:
3407:
3405:
3404:
3396:
3392:Cycles of Time
3388:
3380:
3372:
3363:
3361:
3357:
3356:
3351:
3349:
3348:
3341:
3334:
3326:
3317:
3316:
3314:
3313:
3308:
3306:Woldemar Voigt
3303:
3298:
3293:
3288:
3283:
3278:
3273:
3271:Leonhard Euler
3268:
3263:
3258:
3253:
3247:
3245:
3243:Mathematicians
3239:
3238:
3235:
3234:
3232:
3231:
3226:
3221:
3216:
3211:
3206:
3201:
3196:
3191:
3185:
3183:
3179:
3178:
3176:
3175:
3170:
3168:Torsion tensor
3165:
3160:
3155:
3150:
3145:
3140:
3134:
3132:
3125:
3121:
3120:
3118:
3117:
3112:
3107:
3102:
3097:
3092:
3087:
3082:
3077:
3072:
3067:
3062:
3057:
3052:
3047:
3042:
3037:
3032:
3027:
3021:
3019:
3013:
3012:
3010:
3009:
3003:
3001:Tensor product
2998:
2993:
2991:Symmetrization
2988:
2983:
2981:Lie derivative
2978:
2973:
2968:
2963:
2958:
2952:
2950:
2944:
2943:
2941:
2940:
2935:
2930:
2925:
2920:
2915:
2910:
2905:
2903:Tensor density
2900:
2895:
2889:
2887:
2881:
2880:
2878:
2877:
2875:Voigt notation
2872:
2867:
2862:
2860:Ricci calculus
2857:
2852:
2847:
2845:Index notation
2842:
2837:
2831:
2829:
2825:
2824:
2821:
2820:
2818:
2817:
2812:
2807:
2802:
2797:
2791:
2789:
2787:
2786:
2781:
2775:
2772:
2771:
2769:
2768:
2763:
2761:Tensor algebra
2758:
2753:
2748:
2743:
2741:Dyadic algebra
2738:
2733:
2727:
2725:
2716:
2712:
2711:
2704:
2701:
2700:
2695:
2693:
2692:
2685:
2678:
2670:
2659:
2658:
2651:
2633:
2626:
2608:
2601:
2582:
2581:
2579:
2576:
2575:
2574:
2569:
2564:
2559:
2554:
2552:Index notation
2549:
2544:
2537:
2534:
2533:
2532:
2519:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2482:
2474:
2469:
2464:
2461:
2457:
2450:
2447:
2443:
2438:
2433:
2430:
2427:
2424:
2421:
2417:
2402:
2401:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2351:
2345:
2342:
2339:
2330:
2326:
2323:
2320:
2313:
2308:
2303:
2300:
2296:
2289:
2286:
2282:
2277:
2272:
2269:
2266:
2263:
2260:
2256:
2230:
2225:
2201:
2198:
2195:
2191:
2175:
2172:
2171:
2170:
2159:
2156:
2151:
2142:
2139:
2136:
2132:
2128:
2123:
2114:
2111:
2108:
2104:
2100:
2095:
2086:
2083:
2080:
2076:
2061:
2060:
2049:
2046:
2043:
2038:
2034:
2030:
2025:
2021:
2017:
2012:
2008:
2004:
1999:
1990:
1987:
1984:
1980:
1976:
1971:
1962:
1959:
1956:
1952:
1948:
1945:
1927:
1926:
1915:
1912:
1909:
1904:
1901:
1898:
1894:
1890:
1887:
1882:
1879:
1876:
1872:
1868:
1865:
1842:
1839:
1834:
1830:
1826:
1821:
1817:
1813:
1808:
1804:
1792:Riemann tensor
1779:
1744:
1722:
1718:
1693:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1669:
1664:
1661:
1658:
1654:
1642:
1641:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1604:
1601:
1598:
1594:
1574:
1571:
1570:
1569:
1558:
1553:
1544:
1535:
1526:
1517:
1510:
1507:
1502:
1493:
1484:
1475:
1466:
1459:
1456:
1451:
1446:
1443:
1427:
1426:
1413:
1404:
1395:
1386:
1377:
1370:
1367:
1362:
1353:
1344:
1335:
1326:
1319:
1316:
1311:
1306:
1303:
1275:
1272:
1267:
1263:
1259:
1254:
1250:
1246:
1241:
1237:
1233:
1230:
1227:
1222:
1218:
1214:
1209:
1205:
1201:
1198:
1195:
1190:
1185:
1182:
1169:
1168:
1155:
1151:
1147:
1144:
1141:
1136:
1132:
1128:
1123:
1119:
1115:
1112:
1109:
1104:
1100:
1096:
1091:
1087:
1083:
1080:
1077:
1072:
1067:
1064:
1033:
1030:
1029:
1028:
1017:
1012:
1008:
1004:
1001:
998:
993:
989:
985:
980:
976:
972:
969:
966:
961:
952:
943:
940:
931:
927:
923:
918:
909:
900:
897:
888:
884:
869:
868:
857:
852:
843:
834:
831:
822:
818:
803:
802:
789:
785:
779:
775:
769:
765:
759:
755:
749:
745:
719:
715:
694:
683:
682:
671:
666:
662:
658:
655:
652:
647:
643:
639:
634:
630:
626:
623:
598:
594:
573:
562:tensor product
557:
554:
539:
535:
514:
484:
475:
472:
468:
464:
461:
450:
449:
436:
427:
424:
420:
394:
390:
365:
346:
345:
334:
329:
326:
322:
318:
315:
297:
296:
285:
282:
279:
276:
273:
270:
267:
264:
261:
234:
214:
190:
168:
164:
160:
155:
151:
147:
144:
114:
110:
85:
51:Ricci calculus
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3691:
3680:
3677:
3675:
3672:
3671:
3669:
3654:
3651:
3649:
3646:
3644:(grandfather)
3643:
3640:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3615:
3613:
3609:
3603:
3600:
3598:
3595:
3593:
3590:
3588:
3585:
3583:
3579:
3576:
3574:
3571:
3569:
3566:
3564:
3561:
3559:
3556:
3554:
3551:
3549:
3546:
3544:
3541:
3539:
3536:
3534:
3531:
3529:
3526:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3506:
3504:
3501:
3499:
3496:
3494:
3491:
3489:
3486:
3484:
3481:
3479:
3476:
3474:
3471:
3470:
3468:
3464:
3456:
3452:
3448:
3444:
3440:
3439:Abner Shimony
3436:
3435:
3430:
3426:
3422:
3421:
3416:
3415:
3413:
3409:
3402:
3401:
3397:
3394:
3393:
3389:
3386:
3385:
3381:
3378:
3377:
3373:
3370:
3369:
3365:
3364:
3362:
3358:
3354:
3353:Roger Penrose
3347:
3342:
3340:
3335:
3333:
3328:
3327:
3324:
3312:
3309:
3307:
3304:
3302:
3299:
3297:
3294:
3292:
3289:
3287:
3284:
3282:
3279:
3277:
3274:
3272:
3269:
3267:
3264:
3262:
3259:
3257:
3254:
3252:
3249:
3248:
3246:
3244:
3240:
3230:
3227:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3207:
3205:
3202:
3200:
3197:
3195:
3192:
3190:
3187:
3186:
3184:
3180:
3174:
3171:
3169:
3166:
3164:
3161:
3159:
3156:
3154:
3151:
3149:
3148:Metric tensor
3146:
3144:
3141:
3139:
3136:
3135:
3133:
3129:
3126:
3122:
3116:
3113:
3111:
3108:
3106:
3103:
3101:
3098:
3096:
3093:
3091:
3088:
3086:
3083:
3081:
3078:
3076:
3073:
3071:
3068:
3066:
3063:
3061:
3060:Exterior form
3058:
3056:
3053:
3051:
3048:
3046:
3043:
3041:
3038:
3036:
3033:
3031:
3028:
3026:
3023:
3022:
3020:
3014:
3007:
3004:
3002:
2999:
2997:
2994:
2992:
2989:
2987:
2984:
2982:
2979:
2977:
2974:
2972:
2969:
2967:
2964:
2962:
2959:
2957:
2954:
2953:
2951:
2949:
2945:
2939:
2936:
2934:
2933:Tensor bundle
2931:
2929:
2926:
2924:
2921:
2919:
2916:
2914:
2911:
2909:
2906:
2904:
2901:
2899:
2896:
2894:
2891:
2890:
2888:
2882:
2876:
2873:
2871:
2868:
2866:
2863:
2861:
2858:
2856:
2853:
2851:
2848:
2846:
2843:
2841:
2838:
2836:
2833:
2832:
2830:
2826:
2816:
2813:
2811:
2808:
2806:
2803:
2801:
2798:
2796:
2793:
2792:
2790:
2785:
2782:
2780:
2777:
2776:
2773:
2767:
2764:
2762:
2759:
2757:
2754:
2752:
2749:
2747:
2744:
2742:
2739:
2737:
2734:
2732:
2729:
2728:
2726:
2724:
2720:
2717:
2713:
2709:
2708:
2702:
2698:
2691:
2686:
2684:
2679:
2677:
2672:
2671:
2668:
2664:
2654:
2648:
2644:
2637:
2634:
2629:
2623:
2619:
2612:
2609:
2604:
2598:
2594:
2587:
2584:
2577:
2573:
2570:
2568:
2565:
2563:
2560:
2558:
2555:
2553:
2550:
2548:
2545:
2543:
2540:
2539:
2535:
2514:
2508:
2502:
2496:
2490:
2484:
2480:
2472:
2462:
2459:
2455:
2448:
2445:
2441:
2436:
2428:
2425:
2422:
2415:
2407:
2406:
2405:
2383:
2377:
2371:
2365:
2359:
2353:
2349:
2340:
2324:
2321:
2311:
2301:
2298:
2294:
2287:
2284:
2280:
2275:
2267:
2264:
2261:
2254:
2246:
2245:
2244:
2228:
2199:
2196:
2193:
2189:
2179:
2173:
2157:
2154:
2149:
2140:
2137:
2134:
2130:
2126:
2121:
2112:
2109:
2106:
2102:
2098:
2093:
2084:
2081:
2078:
2074:
2066:
2065:
2064:
2047:
2044:
2041:
2036:
2032:
2028:
2023:
2019:
2015:
2010:
2006:
2002:
1997:
1988:
1985:
1982:
1978:
1974:
1969:
1960:
1957:
1954:
1950:
1946:
1943:
1936:
1935:
1934:
1932:
1913:
1910:
1907:
1899:
1892:
1888:
1885:
1877:
1870:
1866:
1863:
1856:
1855:
1854:
1840:
1837:
1832:
1828:
1824:
1819:
1815:
1811:
1806:
1802:
1793:
1777:
1769:
1765:
1760:
1758:
1742:
1720:
1716:
1707:
1691:
1688:
1685:
1682:
1676:
1673:
1670:
1659:
1652:
1628:
1625:
1622:
1616:
1613:
1610:
1607:
1599:
1592:
1584:
1583:
1582:
1580:
1579:braiding maps
1572:
1556:
1551:
1542:
1533:
1524:
1515:
1508:
1500:
1491:
1482:
1473:
1464:
1457:
1454:
1449:
1432:
1431:
1430:
1411:
1402:
1393:
1384:
1375:
1368:
1360:
1351:
1342:
1333:
1324:
1317:
1314:
1309:
1292:
1291:
1290:
1287:
1273:
1270:
1265:
1261:
1257:
1252:
1248:
1239:
1235:
1231:
1228:
1225:
1220:
1216:
1212:
1207:
1203:
1199:
1196:
1193:
1188:
1153:
1149:
1145:
1142:
1139:
1134:
1130:
1121:
1117:
1113:
1110:
1107:
1102:
1098:
1094:
1089:
1085:
1081:
1078:
1075:
1070:
1053:
1052:
1051:
1049:
1045:
1039:
1031:
1015:
1010:
1006:
1002:
999:
996:
991:
987:
983:
978:
974:
970:
967:
964:
959:
950:
941:
938:
929:
925:
921:
916:
907:
898:
895:
886:
882:
874:
873:
872:
855:
850:
841:
832:
829:
820:
816:
808:
807:
806:
787:
783:
777:
773:
767:
763:
757:
753:
747:
743:
735:
734:
733:
717:
713:
692:
669:
664:
660:
656:
653:
650:
645:
641:
637:
632:
628:
624:
621:
614:
613:
612:
596:
592:
571:
564:of copies of
563:
555:
553:
537:
533:
512:
504:
500:
482:
473:
470:
466:
462:
459:
434:
425:
422:
418:
410:
409:
408:
392:
388:
379:
363:
355:
351:
332:
327:
324:
320:
316:
313:
306:
305:
304:
302:
283:
277:
274:
271:
265:
262:
259:
252:
251:
250:
248:
232:
212:
204:
203:bilinear form
188:
166:
162:
158:
153:
149:
145:
142:
134:
130:
112:
108:
99:
83:
74:
72:
68:
64:
60:
56:
55:Roger Penrose
52:
47:
43:
39:
33:
19:
3653:Quantum mind
3482:
3478:Spin network
3450:
3432:
3418:
3398:
3390:
3382:
3374:
3366:
3311:Hermann Weyl
3115:Vector space
3100:Pseudotensor
3065:Fiber bundle
3018:abstractions
2913:Mixed tensor
2898:Tensor field
2834:
2705:
2662:
2642:
2636:
2617:
2611:
2592:
2586:
2403:
2180:
2177:
2062:
1928:
1770:. Here let
1761:
1643:
1576:
1428:
1288:
1170:
1047:
1043:
1041:
870:
804:
684:
559:
451:
377:
353:
347:
300:
298:
246:
98:vector space
75:
63:contractions
37:
36:
3251:Élie Cartan
3199:Spin tensor
3173:Weyl tensor
3131:Mathematics
3095:Multivector
2886:definitions
2784:Engineering
2723:Mathematics
2620:. Vintage.
1790:denote the
1044:contraction
1032:Contraction
805:or, simply
378:lower index
354:upper index
3668:Categories
3080:Linear map
2948:Operations
2578:References
1036:See also:
611:, such as
129:dual space
71:covariance
3632:(brother)
3626:(brother)
3457:) (1999)
3427:) (1996)
3219:EM tensor
3055:Dimension
3006:Transpose
2509:σ
2497:σ
2485:σ
2481:ω
2463:∈
2460:σ
2456:∑
2416:ω
2378:σ
2366:σ
2354:σ
2350:ω
2341:σ
2322:−
2302:∈
2299:σ
2295:∑
2255:ω
2190:ω
2042:⊗
2037:∗
2029:⊗
2024:∗
2016:⊗
2011:∗
1975:∈
1893:τ
1871:τ
1838:⊗
1833:∗
1825:⊗
1820:∗
1812:⊗
1807:∗
1743:σ
1721:σ
1717:τ
1689:⊗
1674:⊗
1653:τ
1626:⊗
1620:→
1614:⊗
1593:τ
1506:↦
1366:↦
1271:⊗
1266:∗
1258:⊗
1253:∗
1245:→
1240:∗
1232:⊗
1226:⊗
1221:∗
1213:⊗
1208:∗
1200:⊗
1154:∗
1146:⊗
1140:⊗
1135:∗
1127:→
1122:∗
1114:⊗
1108:⊗
1103:∗
1095:⊗
1090:∗
1082:⊗
1011:∗
1003:⊗
997:⊗
992:∗
984:⊗
979:∗
971:⊗
922:∈
718:∗
665:∗
657:⊗
651:⊗
646:∗
638:⊗
633:∗
625:⊗
597:∗
538:∗
393:∗
301:labelling
278:−
272:−
167:∗
159:⊗
154:∗
146:∈
133:covariant
113:∗
3638:(sister)
3620:(father)
3466:Concepts
3085:Manifold
3070:Geodesic
2828:Notation
2536:See also
2214:, where
1573:Braiding
181:. Then
3674:Tensors
3611:Related
3182:Physics
3016:Related
2779:Physics
2697:Tensors
135:tensor
46:spinors
42:tensors
3453:(with
3437:(with
3423:(with
3403:(2016)
3395:(2010)
3387:(2004)
3379:(1994)
3371:(1989)
3110:Vector
3105:Spinor
3090:Matrix
2884:Tensor
2649:
2624:
2599:
2557:Tensor
100:, and
3360:Books
3030:Basis
2715:Scope
1048:trace
247:slots
96:be a
2647:ISBN
2622:ISBN
2597:ISBN
1046:(or
584:and
127:its
76:Let
65:and
44:and
2334:sgn
1900:132
1878:123
1759:).
205:on
3670::
3441:,
2437::=
2276::=
2158:0.
1914:0.
1660:12
1600:12
1450:15
1310:12
1189:15
1071:12
552:.
348:A
249::
3580:/
3345:e
3338:t
3331:v
2689:e
2682:t
2675:v
2655:.
2630:.
2605:.
2518:)
2515:c
2512:(
2506:)
2503:b
2500:(
2494:)
2491:a
2488:(
2473:3
2468:S
2449:!
2446:3
2442:1
2432:)
2429:c
2426:b
2423:a
2420:(
2387:)
2384:c
2381:(
2375:)
2372:b
2369:(
2363:)
2360:a
2357:(
2344:)
2338:(
2329:)
2325:1
2319:(
2312:3
2307:S
2288:!
2285:3
2281:1
2271:]
2268:c
2265:b
2262:a
2259:[
2229:3
2224:S
2200:c
2197:b
2194:a
2155:=
2150:d
2141:a
2138:c
2135:b
2131:R
2127:+
2122:d
2113:b
2110:a
2107:c
2103:R
2099:+
2094:d
2085:c
2082:b
2079:a
2075:R
2048:,
2045:V
2033:V
2020:V
2007:V
2003:=
1998:d
1989:c
1986:b
1983:a
1979:V
1970:d
1961:c
1958:b
1955:a
1951:R
1947:=
1944:R
1911:=
1908:R
1903:)
1897:(
1889:+
1886:R
1881:)
1875:(
1867:+
1864:R
1841:V
1829:V
1816:V
1803:V
1778:R
1692:v
1686:w
1683:=
1680:)
1677:w
1671:v
1668:(
1663:)
1657:(
1629:V
1623:V
1617:V
1611:V
1608::
1603:)
1597:(
1557:.
1552:a
1543:d
1534:c
1525:b
1516:a
1509:h
1501:e
1492:d
1483:c
1474:b
1465:a
1458:h
1455::
1445:r
1442:T
1412:e
1403:d
1394:c
1385:a
1376:a
1369:h
1361:e
1352:d
1343:c
1334:b
1325:a
1318:h
1315::
1305:r
1302:T
1274:V
1262:V
1249:V
1236:V
1229:V
1217:V
1204:V
1197:V
1194::
1184:r
1181:T
1150:V
1143:V
1131:V
1118:V
1111:V
1099:V
1086:V
1079:V
1076::
1066:r
1063:T
1016:.
1007:V
1000:V
988:V
975:V
968:V
965:=
960:e
951:d
942:c
939:b
930:a
926:V
917:e
908:d
899:c
896:b
887:a
883:h
856:.
851:e
842:d
833:c
830:b
821:a
817:V
788:e
784:V
778:d
774:V
768:c
764:V
758:b
754:V
748:a
744:V
714:V
693:V
670:.
661:V
654:V
642:V
629:V
622:V
593:V
572:V
534:V
513:V
483:c
474:b
471:a
467:t
463:=
460:t
435:b
426:b
423:a
419:t
389:V
364:V
333:.
328:b
325:a
321:h
317:=
314:h
284:.
281:)
275:,
269:(
266:h
263:=
260:h
233:V
213:V
189:h
163:V
150:V
143:h
109:V
84:V
34:.
20:)
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