1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
|
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<title>
A Simplified Presentation of Einstein's Unified Field Equations | Project Gutenberg
</title>
<link rel="icon" href="images/cover.jpg" type="image/x-cover">
<style>
body {
margin-left: 10%;
margin-right: 10%;
}
/* General headers */
h1 {
text-align: center;
clear: both;
}
h2, h3 {
text-align: center;
font-weight: bold;
margin-top: 1em;
margin-bottom: 1em;
}
p {
margin-top: .51em;
text-align: justify;
margin-bottom: .49em;
text-indent: 1.5em;
}
.nind {text-indent:0;}
.nindc {text-align:center; text-indent:0;}
.space-above2 { margin-top: 2em; }
.space-below2 { margin-bottom: 2em; }
hr {
width: 33%;
margin-top: 2em;
margin-bottom: 2em;
margin-left: 33.5%;
margin-right: 33.5%;
clear: both;
}
hr.chap {width: 65%; margin-left: 17.5%; margin-right: 17.5%;}
@media print { hr.chap {display: none; visibility: hidden;} }
hr.r5 {width: 5%; margin-top: 1em; margin-bottom: 1em; margin-left: 47.5%; margin-right: 47.5%;}
div.chapter {page-break-before: always;}
h2.nobreak {page-break-before: avoid;}
.allsmcap {font-variant: small-caps; text-transform: lowercase;}
/* Images */
img {max-width: 100%; width: 100%; height: auto;}
.width500 {max-width: 500px;}
.x-ebookmaker .width500 {width: 100%;}
.figcenter {
margin: auto;
text-align: center;
page-break-inside: avoid;
max-width: 100%;
}
/* Footnotes */
.footnotes {border: 1px dashed;}
.footnote {margin-left: 10%; margin-right: 10%; font-size: 0.9em;}
.footnote .label {position: absolute; right: 84%; text-align: right;}
.fnanchor {
vertical-align: super;
font-size: .8em;
text-decoration:
none;
}
/* css needed in m2svg output: displayed equations and prevention of bad breaks*/
.align-center {
display: block;
text-align: center;
text-indent: 0;
margin-top: 1em;
margin-bottom: 1em;
}
.nowrap {
white-space: nowrap;
}
</style>
</head>
<body>
<div style='text-align:center'>*** START OF THE PROJECT GUTENBERG EBOOK 78343 ***</div>
<figure class="figcenter width500" id="cover" style="width: 1623px;">
<img src="images/cover.jpg" width="1623" height="2560" alt="This book
is a technical monograph that provides a more accessible mathematical
treatment of Einstein's attempts to unify gravitational and
electromagnetic fields into a single geometric framework.">
</figure>
<hr class="chap x-ebookmaker-drop">
<div class="chapter">
<h1>A SIMPLIFIED PRESENTATION<br>
<span class="allsmcap">OF</span>
<br>
EINSTEIN'S<br>
UNIFIED FIELD EQUATIONS</h1>
<p class="nindc space-above2">
<span class="allsmcap">By</span><br>
TULLIO LEVI-CIVITA</p>
<p class="nindc space-below2">Professor of Rational Mechanics in the University of Rome<br>
Fellow of R. Accademia Nazionale del Lincei</p>
<p class="nindc space-above2 space-below2">
<i>Authorized Translation by</i><br>
JOHN DOUGALL, <span class="allsmcap">M.A., D.Sc.</span></p>
<p class="nindc space-above2 space-below2">
BLACKIE & SON LIMITED<br>
LONDON AND GLASGOW<br>
1929
</p>
</div>
<hr class="chap x-ebookmaker-drop">
<div class="chapter">
<p class="nindc space-above2 space-below2">
<span class="allsmcap">BLACKIE & SON LIMITED</span><br>
<i>50 Old Bailey, London</i><br>
<i>17 Stanhope Street, Glasgow</i><br>
<span class="allsmcap">BLACKIE & SON (INDIA) LIMITED</span><br>
<i>Warwick House, Fort Street, Bombay</i><br>
<span class="allsmcap">BLACKIE & SON (CANADA) LIMITED</span><br>
<i>1118 Bay Street, Toronto</i><br>
</p>
<p class="nindc space-above2 space-below2">
<i>Printed in Great Britain by Blackie & Son, Ltd., Glasgow</i><br>
</p>
</div>
<hr class="chap x-ebookmaker-drop">
<div class="chapter">
<h2 class="nobreak" id="A_SIMPLIFIED_PRESENTATION_OF">A SIMPLIFIED PRESENTATION OF
EINSTEIN'S
UNIFIED FIELD EQUATIONS</h2>
</div>
<hr class="r5">
<p>In his recent paper, "Zur einheitlichen Feldtheorie",<a id="FNanchor_1" href="#Footnote_1" class="fnanchor">[1]</a> Einstein made
use of the fundamental idea that it is both possible and useful to give
a geometrical interpretation of the complete system of the sixteen
field equations (consisting of Einstein's celebrated gravitational
equations and Maxwell's equations) in such a way as to include the
definition (and the definition only) of an orthogonal quadruplet<a id="FNanchor_2" href="#Footnote_2" class="fnanchor">[2]</a>
embedded in the space-time world.</p>
<p>Conversely, the sixteen parameters determining a quadruplet are to give
a complete definition not only of the Riemannian metric of space (as is
well known, this takes place automatically), but of the phenomena of
electromagnetism as well.</p>
<p>For this purpose the eminent author introduced covariant derivatives
with respect to the quadruplet, and suggested relationships between
them which to a first approximation lead to the required co-ordination
of gravitational and electromagnetic phenomena.</p>
<p>It appears to me, however, that the root problem raised by Einstein can
be solved in a simpler and more general way by making use of perfectly
familiar methods of the absolute differential calculus on the one hand,
while, on the other hand, retaining unaltered all results previously
obtained.</p>
<p class="nindc space-above2">
<b>1. Geometrical and formal preliminaries.<a id="FNanchor_3" href="#Footnote_3" class="fnanchor">[3]</a></b></p>
<p>Let \(x^{\nu}(\nu=0,1, \ldots, n-1)\) be general co-ordinates of a
Riemannian space \(R_{n}\), and \(\lambda_{i}^{\nu} (i=0,1, \ldots, n-1)\)
the parameters of \(n\) congruences, which define a lattice of
lines in \(R_{n}\) and an \(n\)-uplet<a id="FNanchor_4" href="#Footnote_4" class="fnanchor">[4]</a> at every point.</p>
<p>Following Einstein's example I shall use Greek letters for co-ordinate
indices (such as \(\nu\)), and Roman letters, on the other hand, for
indices referring to the \(n\)-uplet (such as \(i\)). I shall leave out
signs of summation with respect to Greek indices (provided they occur
once above and once below), but other \(\Sigma\)'s will be retained.</p>
<p>As usual, let the quantities \(\lambda_{i \mid \nu}\) be the elements
reciprocal to \(\lambda_{i}^{\nu}\) (normalized cofactors). For every
\(i\) they form a covariant system (moments of the \(n\)-uplet in
question). By composition with the quantities \(\lambda_{i}^{\nu}\),
\(\lambda_{i \mid \nu}\) we obtain, from every mixed tensor of rank
\(p + q\) with the components
\[
A_{\mu_{1} \mu_{2} \ldots \mu_{p}}^{\nu_{1} \nu_{2} \ldots \nu_{q}} \quad\left(\mu_{1}, \ldots, \mu_{p}, \nu_{1},
\ldots, \nu_{q}=0,1, \ldots, n-1\right),
\]
an "\(n\)-uplet tensor",<a id="FNanchor_5" href="#Footnote_5" class="fnanchor">[5]</a> the components of which are defined by the
formulæ
\[
\eqalign
{A_{i_{1}} \cdots i_{p} k_{1} \cdots k_{q}=A_{\mu_{1}}^{\nu_{1}} \cdots \mu_{p}{ }_{p}
\lambda_{i_{1}}^{\mu_{1}} \ldots \lambda_{i_{p}}^{\mu_{p}} \lambda_{k_{1} \mid \nu_{1}} \ldots \lambda_{k_{q}
\mid \nu_{q}}, \qquad \text{(1)}}
\]
and conversely, since these formulæ can be solved for the co-ordinate
components in the form
\[
A_{\mu_{1} \ldots \mu_{p}}^{\nu_{1} \ldots \nu{q}} = \sum_{i_{1}}^{n-1} \cdots_{i_{p} k_{1}} \cdots_{k_{q}} A_{i_{1}}
\cdots_{i_{p} k_{1}} \cdots_{k_{q}} \lambda_{i_{1} \mid \mu_{1}} \ldots \lambda_{i_{p} \mid \mu_{p}}
\lambda_{k_{1}}^{\nu_{1}} \ldots \lambda_{k_{q}}^{\nu_{q}}.
\]</p>
<p>The components of the \(n\)-uplet tensor are pure invariants with
respect to transformations of co-ordinates; they essentially depend
on the \(n\)-uplet considered, but, as is easily verified, they also
behave like a tensor when the quantities \(\lambda_{i}^{\mu}\) and
\(\lambda_{k \mid \nu}\) are simultaneously subjected to orthogonal
transformations.</p>
<p>If we put
\[
\eqalign
{g_{\mu \nu}=\sum_{0}^{n-1} \lambda_{i \mid \mu} \lambda_{i \mid \nu}, \quad(\mu, \nu=0,1, \ldots, n-1) \qquad \text{(2)}}
\]
a definite metric
\[
\eqalign
{d s^{2}=g_{\mu \nu} d x^{\mu} d x^{\nu} \qquad \text{(3)}}
\]
(for real values of the quantities involved) is introduced into
\(R_{n}\) in such a way that our \(n\)-uplet turns out orthogonal.
Later (§3) I shall give the (unimportant) modifications required
to transfer the \(n\)-uplet theory, avoiding any appearance of
imaginaries, to an indefinite metric (with a given index of inertia).</p>
<p>Meanwhile I suppose that the covariant derivatives of the moments
\(\lambda_{i \mid \nu}\) have been introduced, and, following Ricci, I
take the coefficients of rotation
\[
\eqalign
{\gamma_{i k l}=\lambda_{i \mid \nu \rho} \lambda_k^{\nu} \lambda_l^{\rho} \qquad \text{(4)}}
\]</p>
<p>In virtue of the identities
\[
\eqalign
{\gamma_{i k l}+\gamma_{k i l}=0 \qquad \text{(5)}}
\]
(which result from the relationships between parameters and moments),
Ricci's quantities \(\gamma\) form \(n \frac{n(n-1)}{2}\) invariants
with respect to transformations of co-ordinates, which of course
essentially depend on the given \(n\)-uplet and necessarily include
all its geometrical differential properties of the first order. With
respect to orthogonal transformations <i>with constant coefficients</i>
the quantities \(\gamma\) behave like a tensor of the third rank. In
order to emphasize the limitation to transformations with constant
coefficients I shall call such systems <i>local</i> \(n\)-<i>uplet
tensors</i>. True \(n\)-uplet tensors behave as invariants with respect
to all orthogonal transformations whose coefficients can vary in any
way with the quantities \(x\).</p>
<p>Perhaps it is not superfluous to remark that the explicit expressions
for the coefficients of rotation, \(\gamma\), can also be obtained
directly by ordinary differentiation without making use of the
covariant derivatives of the quantities \(\lambda_{i \mid \nu}\).</p>
<p>In order to do this, we have to introduce either the Pfaffian
expressions
\[
\psi_i=\lambda_{i \mid \nu} d x^{\nu},
\]
or the operators
\[
\frac{d f}{d s_i}=X_i f=\sum_0^{n-1} \lambda_i^{\nu} \frac{\partial f}{\partial x^{\nu}}
\]
(derivatives of a function \(f\left[x^{0}, \ldots, x^{n-1}\right]\) in
the direction of the lines of the congruences), and then to form the
corresponding bilinear covariants or Poisson brackets. We can, however,
attain the desired result even more rapidly by using (4) and noticing
that, according to the definition of covariant differentiation, we have
the identity
\[
\lambda_{i \mid \nu \rho}-\lambda_{i \mid \rho \nu}=\frac{\partial \lambda_{i \mid \nu}}{\partial x^{\rho}}-\frac{\partial \lambda_{i \mid \rho}}{\partial
x^{\nu}} .
\]
We thus obtain
\[
\gamma_{i k l}-\gamma_{i l k}=\sum_{0}^{n-1} \lambda_{k}^{\nu} \lambda_{l}^{\rho}\left\{
\frac{\partial \lambda_{i \mid \nu}}
{\partial x^{\rho}}
-\frac{\partial \lambda_{i \mid \rho}}{\partial x^{\nu}}\right\},
\]
and all the quantities \(\gamma\) are uniquely determined by these
equations together with (5).</p>
<p>Equations (4) can be solved for the quantities \(\lambda_{i \mid \nu\rho}\),
giving
\[
\eqalign
{\lambda_{i \mid v \rho}=\sum_{0}^{n-1} \gamma_{i j h} \lambda_{j \mid \nu} \lambda_{h \mid \rho} \qquad (4')}
\]
from which we obtain the conditions of integrability of \((4')\) by
repeated covariant differentiation and formation of differences. For
this we require the commutation-formula
\[
\eqalign
{\lambda_{i \mid v \rho \sigma}-\lambda_{i \mid v \tau \rho}=R_{\mu v, \rho \sigma} \lambda_{i}^{\mu} \qquad \text{(6)}}
\]
where \(R_{\mu \nu, \rho \sigma}\) denotes the Riemannian tensor. In
this way we obtain
\[
\eqalign
{\gamma_{i j, h k}=R_{\mu \nu, \rho \sigma} \lambda_{i}^{\mu} \lambda_{j}^{\nu} \lambda_{h}^{\rho} \lambda_{k}^{\sigma}
\qquad \text{(7)}}
\]
where for brevity we write
\[
\eqalign
{\gamma_{i j, h k}=
\frac{d \gamma_{i j h}}{d s_{k}}-\frac{d \gamma_{i j k}}{d
s_{h}}+\Sigma_{0}^{n-1}\left[\gamma_{i j l}\left(\gamma_{l h k}-\gamma_{l k h}\right)+\gamma_{l i k}
\gamma_{l j h}-\gamma_{l i h} \gamma_{l j k}\right] . \qquad \text{(8)}}
\]
From (7) we conclude that the 4-index symbols, \(\gamma\), form a
(true) \(n\)-uplet tensor. In virtue of the well-known identities
satisfied by the Riemannian symbols the formulæ (7) lead to similar
identities for the 4-index symbols, \(\gamma\), namely
\[
\left.\begin{array}{l}
\gamma_{i j, h k} = -\gamma_{j i, h k} = -\gamma_{i j, k h} = \gamma_{h k, i j} \\
\gamma_{i j, h k}+\gamma_{i h, k j}+\gamma_{i k, j h} = 0
\end{array}\right\}\qquad \text{(9)}
\]</p>
<p>Now for the Einstein tensor
\[
G_{\mu \sigma} = R_{\mu \nu, \rho \sigma} g^{\nu \rho} .
\]</p>
<p>Its components \(G_{i k}\), with respect to the two members \(i\),
\(k\) of the \(n\)-uplet are expressed, by (1), by
\[
G_{i k} = G_{\mu \sigma} \lambda_{i}^{\mu} \lambda_{k}^{\sigma} ,
\]
whence, by (7),
\[
\eqalign
{G_{i k} = \sum_{0}^{n-1}{_{h}} \gamma_{i h, h k} \qquad \text{(10)}}
\]</p>
<p>The linear (co-ordinate and \(n\)-uplet) invariant
\[
G = G_{\mu \sigma} g^{\mu \sigma}=\sum_{0}^{n-1}{_k} G_{k k}
\]
consequently takes the form
\[
\eqalign
{G = \sum_{0}^{n-1}{_{k k}} \gamma_{k h, k k} \qquad \text{(11)}}
\]</p>
<p>In conclusion, I shall emphasize one other fact, namely that
contraction of two indices in an \(n\)-uplet tensor leads to a reduced
tensor—of the \((m-2)\)th rank if the original tensor is of the
\(m\)th rank.</p>
<p>As we have already seen, the quantities \(\gamma_{i k l}\) form a
local \(n\)-uplet tensor of the third rank, which in virtue of (5) is
skew-symmetrical with respect to the two first indices \(i\), \(k\).
The same is true for the differences \(\gamma_{l i k}-\gamma_{l k i}\),
which for \(i, k \neq l\) are called <i>anormalities</i> (i.e.
quantities which vanish when the \(l\)th congruence of the \(n\)-uplet
is normal).</p>
<p>If we apply the differential operator \(\dfrac{d}{d s_{j}}\) to the
elements \(A_{(h)}\) (where \((h)\) stands for \(h_{1} h_{2} \ldots h_{m}\))
of a local or true \(n\)-uplet tensor, we obtain a new local
\(n\)-uplet tensor \(\dfrac{d A_{(h)}}{d s_{j}}\), the rank of which
exceeds that of the original tensor by unity. In particular, we obtain
in this way the local \(n\)-uplet tensor of the fourth rank
\[
\frac{d \gamma_{i k l}}{d s_{j}}
\]
which is skew-symmetrical with respect to \(i\) and \(k\). By
contraction we obtain
\[
\eqalign
{\xi_{i k}=\sum_{0}^{n-1}{_{l}} \frac{d \gamma_{i k l}}{d s_{l}} \qquad \text{(12)}}
\]
so that we have obviously formed a skew-symmetrical local \(n\)-uplet
tensor \(\boldsymbol{\xi}\) of the second rank. Its covariant and contravariant
components are respectively
\[
\eqalign
{\xi_{\mu \nu}=\sum_{0}^{n-1}{_{ik}} \xi_{i k} \lambda_{i \mid \mu} \lambda_{k \mid \nu},
\quad \xi^{\mu \nu}=\sum_{0}^{n-1}{_{ik}} \xi_{i k} \lambda_{i}^{\mu} \lambda_{k}^{\nu} \qquad \text{(13)}}
\]</p>
<p>We may mention in addition that the \(n\) quantities
\[
\eqalign
{c_{l}=\sum_{0}^{n-1}{_{j}} \gamma_{j l j} \qquad \text{(14)}}
\]
may be interpreted as mean curvatures of the \(n-1\)-fold sections,
drawn orthogonally to the lines of the \(n\)-uplet. By what we have
said above, they are line-components of a local \(n\)-uplet vector.
From the tensor of the third rank, \(\gamma_{l i k}-\gamma_{l k i}\),
and this vector we obtain by contraction a new local \(n\)-uplet tensor
of the second rank, namely
\[
\eqalign
{\eta_{i k}=\sum_{0}^{n-1}{_{l}} c_{l}\left(\gamma_{l i k}-\gamma_{l k i}\right) \qquad \text{(15)}}
\]
which is also skew-symmetrical.</p>
<p class="nindc space-above2">
<b>2. Formation of divergences. The special case</b> \(n=4\).</p>
<p>If \(v^{\nu}\) are the contravariant components of a vector
\(\mathbf{v}\), its divergence is defined by the invariant
\[
\eqalign
{\operatorname{div} \mathbf{v}=v_{\mid \nu}^{\nu}=\frac{1}{\sqrt{ }|g|} \sum_{0}^{n-1} \frac{\partial\left(\sqrt{|g|} v^{\nu}\right)}{\partial x^{\nu}},
\qquad \text{(16)}}
\]
where, as usual, \(g\) denotes the determinant \(\left\|g_{\mu\nu}\right\|\)
and \(|g|\)is written (instead of simply \(g\)) because
the formula is then valid as it stands even for an indefinite
\(ds^{2}\).</p>
<p>For the divergence of a tensor \(\boldsymbol{\xi}\) of the second rank
with the contravariant components \(\xi^{\mu \nu}\) we obtain a vector
\(\chi\) with the contravariant components
\[
\eqalign
{\chi^{\mu}=\xi_{\mid \nu}^{\mu \nu} \qquad \text{(17)}}
\]</p>
<p>Following von Laue,<a id="FNanchor_6" href="#Footnote_6" class="fnanchor">[6]</a> we shall write simply
\[
\eqalign
{\chi=\operatorname{Div} \xi \qquad (17')}
\]</p>
<p>If we here replace the covariant derivatives \(\xi^{\mu \nu}_{\mid_{\rho}}\)
by their explicit values, we obtain
\[
\eqalign
{\chi^{\mu}=\frac{1}{\sqrt{|g|}} \sum_{0}^{n-1}{_{\nu}} \frac{\partial}{\partial x^{\prime}}\left(\sqrt{|g|} \xi^{\mu \nu}\right) \qquad (17'')}
\]
in the case of a skew-symmetrical tensor \(\left(\xi^{\mu \nu}+\xi^{\nu \mu}=0\right)\); hence, by (16),
\[
\eqalign
{\operatorname{div} \chi=\frac{1}{\sqrt{|g|}} \sum_{0}^{n-1}{_{\mu \nu}} \frac{\partial^{2}}{\partial x^{\mu} \partial x^{\nu}}\left(\sqrt{|g|}
\xi^{\mu \nu}\right) \qquad \text{(18)}}
\]</p>
<p>Owing to the skew-symmetry of the quantities \(\xi^{\mu \nu}\), the
right-hand side vanishes identically.</p>
<p>Thus if we again make use of covariant derivatives, we obtain the
identity
\[
\chi^{\mu}{ }_{\mid \mu}=\xi^{\mu \nu}{ }_{\mid \nu \mu}=0
\]
or finally, in tensor notation,
\[
\eqalign
{\operatorname{div}(\operatorname{Div} \xi)=0 \qquad (18')}
\]</p>
<p><i>That is, in an arbitrary Riemannian space the divergence of
the divergence of a skew-symmetrical tensor of the second rank is
identically zero.</i></p>
<p>In order to express the right-hand sides of (16) and (17) in
\(n\)-uplet tensor components, it is sufficient to apply the operator
\[
\frac{d}{d s_{l}}=\lambda_{l}^{\rho} \frac{\partial}{\partial x^{\rho}}
\]
to the formulæ of definition
\[
\begin{aligned}
v_{k} & =v^{\nu} \lambda_{k \mid \nu}, \\
\xi_{i k} & =\xi^{\mu \nu} \lambda_{i \mid \mu} \lambda_{k \mid \nu}.
\end{aligned}
\]</p>
<p>By replacing ordinary differentiation by covariant differentiation
on the right-hand side (which is permissible, as we are dealing with
invariants), we obtain
\[
\begin{align*}
\frac{d v_{k}}{d s_{l}}=v_{\mid \rho}^{\nu} \lambda_{k \mid \nu} \lambda_{l}^{\rho}+v^{\nu} \lambda_{k \mid \nu \rho}
\lambda_{l}^{\rho}, \\
\frac{d \xi_{i k}}{d s_{l}}=\xi^{\mu \nu}{ }_{\mid \rho} \lambda_{i \mid \mu} \lambda_{k \mid \nu} \lambda_{l}^{\rho}+\xi^{\mu \nu}
\lambda_{l}^{\rho}\left(\lambda_{i \mid \mu \rho} \lambda_{k \mid \nu}+\lambda_{i \mid \mu} \lambda_{k \mid \nu \rho}\right),
\end{align*}
\]
whence, by \((4')\), (16), and (17),
\[
\begin{align*}
\sum_{0}^{n-1}{_{k}} \frac{d v_{k}}{d s_{k}}&=\operatorname{div} \mathbf{v}+\sum_{0}^{n-1}{_{hk}} \gamma_{k h k} v_{h},
&\qquad \text{(19)}\\
\sum_{0}^{n-1}{_{k}} \frac{d \xi_{i k}}{d s_{k}}&=\chi_{i}+\sum_{0}^{n-1}{_{hk}}\left(\gamma_{i h k}
\xi_{h k}+\gamma_{k h k} \xi_{i h}\right), &\qquad \text{(20)}
\end{align*}
\]
which give the divergences \(\operatorname{div} \mathbf{v}\) and
\(\operatorname{Div} \boldsymbol{\xi}\) of \(n\)-uplet tensors (of the
first or second rank) directly by means of \(n\)-uplet components and
\(n\)-uplet operations.</p>
<p>For \(n=4\) we have an elementary tensor of the fourth rank at our
disposal, namely the well-known Riccian \(\epsilon\)-system, the
covariant and contravariant components of which, \(\epsilon_{\mu \nu\rho \sigma}\),
\(\epsilon^{\mu \nu \rho \sigma}\) respectively, are
equal to zero if the four indices are not all different. The other
components have the respective values \(\pm \sqrt{|g|}\),
\(\pm d\frac{1}{\sqrt{|g|}}\), the upper or lower sign being taken
according as the permutation \((\mu \nu \rho \sigma)\) is even or odd
with respect to (0123).</p>
<p>Let \(\boldsymbol{\xi}\) again be a skew-symmetrical tensor of the
second rank with the contravariant components \(\xi^{\nu \rho}\). If we
put
\[
\eqalign
{p_{\mu}=\epsilon_{\mu \nu \rho \sigma} \xi^{\nu \rho \mid \sigma}, \qquad \text{(21)}}
\]
which means the same as
\[
\begin{aligned}
p^{\mu}&=\epsilon^{\mu \nu \rho \sigma} \xi_{\nu \rho \mid \sigma}, &\qquad (21')\\
\text{or}\quad \mathbf{p}&=\operatorname{Div}^{*} \boldsymbol{\xi} &\qquad (21'')
\end{aligned}
\]
in von Laue's notation, we are justified in calling the vector
\(\mathbf{p}\) with the above covariant and contravariant components
the Pfaffian divergence of \(\xi\), because the \(p^{\mu}\)'s vanish
identically if, and only if, the \(\xi_{\nu \rho}\)'s coincide with
the coefficients of the bilinear covariants of a Pfaffian expression
\(\phi_{\nu} d x^{\nu}\). This is most easily proved by replacing the
covariant derivatives \(\xi_{\nu \rho \mid \sigma}\) in \((21')\) by their
explicit values and noting that, owing to the skew-symmetry of the
quantities \(\xi_{\nu \rho}\), all that we have left is
\[
\eqalign
{p^{\mu}=\sum_{0}^{3} \sum_{\nu \rho \sigma} \epsilon^{\mu \nu \rho \sigma} \frac{\partial \xi_{\nu \rho}}{\partial x^{\sigma}}
\qquad (21''')}
\]
The right-hand sides obviously vanish if the quantities
\(\dfrac{\partial \xi_{\nu \rho}}{\partial x^{\sigma}}\) are replaced
by the differences \(\dfrac{\partial^{2} \phi_{\nu}}{\partial x^{\rho}
\partial x^{\sigma}}-\dfrac{\partial^{2} \phi_{\rho}}{\partial x^{\nu}
\partial x^{\sigma}}\).</p>
<p>By substituting the expression \((21''')\) for the \(p^{\mu}\)'s in the
second form (16) of the divergence of a vector we immediately obtain
\(\operatorname{div} \mathbf{p}=0\), which, bearing \((21'')\) in mind, may be written
\[
\eqalign
{\operatorname{div}\left(\operatorname{Div}^{*} \xi\right)=0 ; \qquad \text{(22)}}
\]
<i>that is, the divergence of the Pfaffian divergence of a
skew-symmetrical tensor of the second rank in</i> \(\mathrm{R}_{4}\)
<i>vanishes identically</i>.</p>
<p>Further, we shall proceed to represent the vector \(\mathbf{p}\) (the
Pfaffian divergence) directly in terms of the \(n\)-uplet components
\[
\xi_{i k}=\xi_{\mu \nu} \lambda_{i}^{\mu} \lambda_{k}^{\nu}
\]
of the given tensor. Here it suggests itself to start from the solved
form of the equations which we have just written down, namely
\[
\xi_{\nu \rho}=\sum_{0}^{3}{_{h k}} \xi_{h k} \lambda_{h \mid \nu} \lambda_{k \mid \rho}
\]
and to calculate the quantities \(\xi_{\nu \rho \mid \sigma}\) by
covariant differentiation of the right-hand side.</p>
<p>From
\[
\xi_{h k \mid \sigma}=\sum_{0}^{3}{_{l}} \frac{d \xi_{h k}}{d s_{l}} \lambda_{l \mid \sigma}
\]
and \((4')\) we obtain
\[
\xi_{\nu \rho \mid \sigma}=\sum_{0}^{3}{_{hkl}} \frac{d \xi_{h k}}{d s_{l}} \lambda_{h \mid \nu} \lambda_{k \mid \rho}
\lambda_{l \mid \sigma}+\sum_{0}^{3}{_{hkjl}} \xi_{h k} \lambda_{l \mid \sigma}\left\{\gamma_{h j l} \lambda_{j \mid \nu}
\lambda_{k \mid \rho}+\gamma_{k j l} \lambda_{j \mid \rho} \lambda_{h \mid \nu}\right\};
\]
hence, by \((21')\),
\[
\begin{aligned}
p_{i}=p^{\mu} \lambda_{i \mid \mu} & =\epsilon^{\mu \nu \rho \sigma} \xi_{\nu \rho \mid \sigma} \lambda_{i \mid \mu} \\
& =\sum_{0}^{3}{_{hkl}} \epsilon_{i h k l} \frac{d \xi_{h k}}{d s_{l}}+\sum_{0}^{3}{_{jhkl}}
\xi_{h k}\left(\epsilon_{i j k l} \gamma_{h j l}+\epsilon_{i h j l} \gamma_{k j l}\right) \\
& =\sum_{0}^{3}{_{hkl}} \epsilon_{i h k l}\left\{\frac{d \xi_{h k}}{d s_{l}}+\sum_{0}^{3}{_{j}}\left(\gamma_{j h l}
\xi_{j k}+\gamma_{j k l} \xi_{h j}\right)\right\}
\end{aligned}
\]
where for brevity we have put
\[
\begin{align*}
\epsilon_{i k k l}=\epsilon^{\mu \nu \rho \sigma} \lambda_{i \mid \mu} \lambda_{k \mid \nu} \lambda_{k \mid \rho} \lambda_{l \mid \sigma}\\
& =\frac{1}{\sqrt{|g|}}\left|\begin{array}{llll}
\lambda_{0 \mid 0} & \lambda_{0 \mid 1} & \lambda_{0 \mid 2} & \lambda_{0 \mid 3} \\
\lambda_{1 \mid 0} & \lambda_{1 \mid 1} & \lambda_{1 \mid 2} & \lambda_{1 \mid 3} \\
\lambda_{2 \mid 0} & \lambda_{2 \mid 1} & \lambda_{2 \mid 2} & \lambda_{2 \mid 3} \\
\lambda_{3 \mid 0} & \lambda_{3 \mid 1} & \lambda_{3 \mid 2} & \lambda_{3 \mid 3}
\end{array}\right|.\qquad \text{(23)}
\end{align*}
\]
Thus these quantities \(\epsilon_{i k k l}\) are equal to zero if
two of the four indices are equal. If, on the other hand, \(ihkl\)
is a permutation of the numbers 0123, \(\epsilon_{\text {ihkl }}\)
has the value \(\pm 1\), according as the class of the substitution
\(\binom{i h k l}{0123}\) is even or odd. We accordingly see that in
the expression which we have just obtained for the \(p_{i}\)'s the two
last terms are equal to each other, so that we finally obtain
\[
p_{i}=\sum_{0}^{3}{_{h k l}} \epsilon_{i h k l}\left\{
\frac{d \xi_{h k}}{d s_{l}}+2\sum_{0}^{3}{_{j}} \gamma_{j h l} \xi_{j k}\right\} \qquad \text{(24)}
\]</p>
<p class="nindc space-above2">
<b>3. Transformations for an indefinite metric.</b></p>
<p>According to Eisenhart<a id="FNanchor_7" href="#Footnote_7" class="fnanchor">[7]</a> all the formulæ of the \(n\)-uplet theory can
be transferred in a readily intelligible way to indefinite metrics,
without leaving the real region even temporarily.</p>
<p>If we consider an indefinite
\[
ds^{2}=g_{\mu \nu} d x^{\mu} d x^{\nu}
\]
we (as is well known) call a (real) direction \(d x^{\nu}\)
<i>time-like</i> or <i>space-like</i>, according as the corresponding
\(ds^{2}\) turns out greater or less than zero; null directions are
those directions, \(\infty^{n-2}\) in number, for which \(ds^{2}=0\).</p>
<p>In any case we call the ratios
\[
\lambda^{\nu}=\frac{d x^{\nu}}{|ds|} \quad(\nu=0,1, \ldots, n-1)
\]
<i>parameters</i> of a proper (i.e. non-null) direction.</p>
<p>Hence we have
\[
\eqalign
{g_{\mu \nu} \lambda^{\mu} \lambda^{\nu}=\frac{d s^{2}}{\left|ds^{2}\right|}= \pm 1=e, \qquad \text{(25)}}
\]
if we henceforth denote positive or negative unity by \(e\).</p>
<p>As in the definite case we introduce as moments of a given direction
the covariant quantities
\[
\eqalign
{\lambda_{\nu}=g_{\mu \nu} \lambda^{\mu}, \qquad \text{(26)}}
\]
so that the quadratic identity (25) takes the form
\[
\eqalign
{\lambda_{v} \lambda^{\nu}=e. \qquad \text{(27)}}
\]</p>
<p>If the quantities \(\lambda_{i}^{\nu}(i=0,1, \ldots, n-1)\) are the
parameters of an orthogonal \(n\)-uplet consisting of proper directions
only, we have
\[
\lambda_{i \mid \nu} \lambda_{k}^{\nu}=0 \quad(i \neq k)
\]
on account of the orthogonality of the \(n\)-uplet, and also
\[
\lambda_{i \mid \nu} \lambda_{i}^{\nu}= \pm 1=e_{i},
\]
by (27).</p>
<p>The total number of negative (and consequently also of the remaining
positive) quantities \(e_{i}\) for a given \(ds^{2}\) is always equal
to its index of inertia, and hence is always the same no matter what
(proper) \(n\)-uplet is considered.</p>
<p>The two groups of relationships between parameters and moments of an
\(n\)-uplet which we have just written down may be summarized in the
single formula
\[
\eqalign
{\lambda_{i}^{\nu} \lambda_{k \mid \nu}=e_{k} \delta_{i k}=e_{i} \delta_{i k}, \qquad \text{(28)}}
\]
where the symbols \(\delta_{i k}\) have their usual meaning; or, since
\(e_{k}^{2}=1\),
\[
\lambda_{i}^{\nu} e_{k} \lambda_{k \mid \nu}=\delta_{i k} .
\]</p>
<p>From this we conclude that the elements reciprocal to the parameters
\(\lambda_{i}^{\nu}\) are not exactly equal to the moments \(\lambda_{i\mid \nu}\),
but to \(e_{i} \lambda_{i \mid \nu}\). Thus the quantities
\(e_{i} \lambda_{i}^{\nu}\) are the elements reciprocal to the moments
\(\lambda_{i \mid \nu}\). If we imagine the equations (26) written down
for every \(n\)-uplet, we have
\[
\lambda_{i \mid \nu}=g_{\rho \nu} \lambda_{i}^{\rho}
\]
(denoting the index of summation by \(\rho\)). By multiplying by
\(e_{i} \lambda_{i \mid \mu}\) and summing with respect to \(i\) we
obtain
\[
g_{\mu \nu}=\sum_{0}^{n-1}{_{i}} e_{i} \lambda_{i \mid \mu} \lambda_{i \mid \nu},
\]
which replaces formula (2) for the definite case, and so on.</p>
<p>From this point it will suffice if I confine myself to quite brief
hints, and I shall of course write down only those formulæ which do
not remain unaltered throughout. These will be marked with an asterisk
and given the same number as the corresponding formula referring to a
definite metric.</p>
<p>In the first place, \(n\)-uplet components of any given tensor
and coefficients of rotation \(\gamma_{i k l}\) must in any case be
introduced by the equations of definition (1) and (4); the solved
expressions for the quantities \(\lambda_{i \mid \nu \rho}\), on the
other hand, are in general
\[
\eqalign
{\lambda_{i \mid \nu \rho}=\sum_{0}^{n-1} {_{j h}} \gamma_{i j h} e_{j} e_{h} \lambda_{j \mid \nu} \lambda_{h \mid \rho}
\qquad (4'*)}
\]</p>
<p>The covariant equations (6), and also the equations of definition of
the 4-index symbols \(\gamma\) (7) are true without restriction; but
the \(n\)-uplet tensor expressions for the quantities \(\gamma_{i j, h k}\)
suffer a small modification. In fact we must in general put
\[
\begin{align*}
\gamma_{i j, h k}= & \frac{d \gamma_{i j h}}{d s_{k}}-\frac{d \gamma_{i j k}}{d s_{h}} \\
& +\sum_{0}^{n-1}{_{l}} e_{l}\left[\gamma_{i j l}\left(\gamma_{l h k}-\gamma_{l k h}\right)+\gamma_{l i k}
\gamma_{l j h}-\gamma_{l i h} \gamma_{l j k}\right] \qquad (8*)
\end{align*}
\]</p>
<p>Of course these quantities are still connected by the relationships
(9), in virtue of equations (7).</p>
<p>It is essential to note, however, that the local transference from
one \(n\)-uplet to another does not correspond to any orthogonal
transformation, but to a pseudo-orthogonal transformation, i.e. to a
transformation which leaves the quadratic form
\[
Q(z)=\sum_{0}^{n-1}{_{i}} e_{i} z_{i}^{2}
\]
invariant. Thus the coefficients \(\alpha_{i k}\) of a
pseudo-orthogonal transformation of this kind must satisfy the
conditions
\[
\sum_{0}^{n-1}{_{l}} e_{l} \alpha_{i l} \alpha_{k l}=\sum_{0}^{n-1}{_{l}} e_{l} \alpha_{l i}
\alpha_{l k}=\delta_{i k}.
\]</p>
<p>The most general expression which can be attributed to the coefficients
\(\alpha_{i k}\) in the case of infinitesimal pseudo-orthogonal
transformations follows immediately from the condition that the form
\(Q(z)\) is to be invariant. We have merely to put
\[
\alpha_{i k}=\delta_{i k}+e_{i} \beta_{i k}
\]
and to regard the quantities \(\beta_{i k}\) as indefinitely small. If
in \(Q\) we carry out the substitution
\[
\eqalign
{z_{i}=\sum_{0}^{n-1}{_{k}} \alpha_{i k} z_{k}^{\prime}=z_{i}^{\prime}+e_{i} \sum_{0}^{n-1}{_{k}}
\beta_{i k} z_{k}^{\prime} \qquad \text{(29)}}
\]
and require that \(Q\left(z^{\prime}\right)\) should retain the form
\[
\sum_{0}^{n-1}{_{i}} e_{i} z_{i}^{2},
\]
what we obtain (as in the case of pure orthogonal substitutions) is the
condition of skew-symmetry, namely
\[
\eqalign
{\beta_{i k}+\beta_{k i}=0 \qquad \text{(30)}}
\]</p>
<p>The components of an \(n\)-uplet tensor are systems of numbers which
behave like tensors with respect to pseudo-orthogonal transformations;
for <i>local</i> \(n\)-uplet tensors this behaviour is maintained only
with respect to pseudo-orthogonal transformations with <i>constant</i>
coefficients. The operators
\[
\frac{d f}{d s_{i}}=X_{i} f=\sum_{0}^{n-1}{_{\nu}} \lambda_{i}^{\nu} \frac{\partial f}{\partial x^{\nu}}
\]
behave like \(n\)-uplet vectors.</p>
<p>If (\(i\)) and (\(k\)) denote any group of \(n\)-uplet indices and
\(A_{(i) j}\), \(B_{(k) l}\) two local \(n\)-uplet tensors, then
contraction with respect to \(j\), \(l\) is defined by the formula
\[
\sum_{0}^{n-1}{_{l}} e_{l} A_{(i) l} B_{(k) l}.
\]</p>
<p>We accordingly obtain
\[
\begin{align*}
G_{i k} & =\sum_{0}^{n-1}{_{h}} e_{h} \gamma_{i h, h k}, &\qquad \text{(10*)}\\
G & =\sum_{0}^{n-1}{_{hk}} e_{h} e_{k} \gamma_{k h, h k} &\qquad \text{(11*)}
\end{align*}
\]
instead of (10) and (11).</p>
<p>Further, the formulæ (12), (14), and (15) must be replaced by and
\[
\begin{align*}
\xi_{i k} & =\sum_{0}^{n-1}{_{l}} e_{l} \frac{d \gamma_{i k l}}{d s_{l}} &\qquad \text{(12*)}\\
c_{l} & =\sum_{0}^{n-1}{_{j}} e_{j} \gamma_{j l j} &\qquad \text{(14*)}\\
\text{and}\quad \eta_{i k} & =\sum_{0}^{n-1}{_{l}} e_{l} c_{l}\left(\gamma_{l i k}-\gamma_{l k i}\right) &\qquad \text{(15*)}
\end{align*}
\]
while the expressions (13) for covariant and contravariant components
in terms of the \(n\)-uplet components \(\xi_{i k}\) are to be
deduced from (1), the universally valid definition of the \(n\)-uplet
components of a tensor. Hence they become
\[
\left.\begin{array}{l}
\xi_{\mu \nu}=\sum_{0}^{n-1}{_{ik}} \xi_{i k} e_{i} e_{k} \lambda_{i \mid \mu} \lambda_{k \mid \nu}, \\
\xi^{\mu \nu}=\sum_{0}^{n-1}{_{ik}} \xi_{i k} e_{i} e_{k} \lambda_{i}^{\mu} \lambda_{k}^{\nu}
\end{array}\right\}\qquad \text{(13*)}
\]</p>
<p>As contraction of pseudo-orthogonal \(n\)-uplet tensors is brought
about by inserting the factor \(e\) with the appropriate index, it is
at once clear that (19), (20), and (24) take the forms
\[
\begin{align*}
\operatorname{div} \mathbf{v} & =\sum_{0}^{n-1}{_{k}} e_{k} \frac{d v_{k}}{d s_{k}}-\sum_{0}^{n-1}{_{hk}} e_{h k}
e_{k} \gamma_{k h k} v_{h}, &\qquad \text{(19*)}\\
\chi_{i} & =\sum_{0}^{n-1}{_{h}} e_{k} \frac{d \xi_{i k}}{d s_{k}}-\sum_{0}^{n-1}{_{hk}} e_{h k} e_{h}
e_{l}\left(\gamma_{i h k} \xi_{h k}+\gamma_{k h k} \xi_{i h}\right) &\qquad \text{(20*)}\\
p_{i} & =\sum_{0}^{3} \sum_{h k l}{_{h k l}} e_{h} e_{k} e_{l} \epsilon_{i h k l}\left\{\frac{d \xi_{h k}}{d
s_{l}}+2 \sum_{0}^{3}{_{j}} e_{j} \xi_{j k}\right\} &\qquad \text{(24*)}
\end{align*}
\]</p>
<p>Of course the equations \((18')\) and (22), i.e.
\[
\eqalign
{\operatorname{div}(\operatorname{Div} \xi)=0, \quad \operatorname{div}\left(\operatorname{Div}^{*} \xi\right)=0, \qquad \text{(31)}}
\]
which express invariant properties, always remain valid.</p>
<p class="nindc space-above2">
<b>4. Gravitational equations.</b></p>
<p>As usual, let the covariant components of the energy tensor be denoted
by \(T_{\mu \nu}\). If influences of any origin are admitted, these
quantities \(T_{\mu \nu}\) are to be imagined broken up into two parts,
one of which, \(\tau_{\mu \nu}\), is purely electromagnetic, and the
other, \(\mathbf{T}_{\mu \nu}\), represents the remainder, if any. We
therefore put
\[
\eqalign
{T_{\mu \nu}=\tau_{\mu \nu}+\mathbf{T}_{\mu \nu}, \qquad \text{(32)}}
\]
where \(\boldsymbol{\tau}\) is the well-known Maxwell tensor; further,
for empty space \(\mathbf{T}_{\mu \nu}\) is of course equal to zero.</p>
<p>As is well known, the Einstein equations (without the cosmological
term) are
\[
G_{\mu \nu}-\frac{1}{2} G g_{\mu \nu}=-\kappa T_{\mu \nu}
\]
where the constant of proportionality \(\kappa\) may be expressed in
terms of \(f\), the gravitational constant, and \(c\), the velocity of
light \(\left(\kappa=\frac{8 \pi f}{c^{4}}\right)\).</p>
<p>If we introduce the corresponding \(n\)-uplet tensors in accordance
with the formulæ
\[
\begin{aligned}
G_{i k} & =G_{\mu \nu} \lambda_{i}^{\mu} \lambda_{k}^{i} \\
T_{i k} & =T_{\mu \nu} \lambda_{i}^{\mu} \lambda_{k}^{\nu}, \quad\& \mathrm{c},
\end{aligned}
\]
we have, on the one hand,
\[
\eqalign
{T_{i k}=\tau_{i k}+\mathbf{T}_{i k}, \qquad (32')}
\]
from (32), and (what is most important) the gravitational equations in
the \(n\)-uplet tensor form <a id="FNanchor_8" href="#Footnote_8" class="fnanchor">[8]</a>
\[
\eqalign
{G_{i k}-\frac{1}{2} \delta_{i k} G=-\kappa T_{i k}, \quad(i, k=0,1,2,3) \qquad \text{(I)}}
\]
where, in accordance with (10*) and (11*),
\[
G_{i k}=\sum_{0}^{n-1}{_{h}} e_{h} \gamma_{i h, h k}, \quad G=\sum_{0}^{n}{_{k}} e_{k}
G_{k k}=\sum_{0}^{3}{_{hk}} e_{h} e_{k} \gamma_{k h, h k} .
\]</p>
<p>As the space-time manifold on which the general theory of relativity is
to be based possesses an indefinite metric with an index of inertia 3,
we have to put
\[
\eqalign
{e_{0}=1,\quad e_{1}=e_{2}=e_{3}=-1. \qquad \text{(33)}}
\]</p>
<p>The quantities \(\gamma_{i j, h k}\) are introduced by the equations
(8*) as lattice differential elements of the second order. Their
combinations \(G_{i k}\) behave like tensors with respect to all
pseudo-orthogonal (i.e. in the present case Lorentz) transformations
(even if the coefficients are permitted to vary in any way with
position).</p>
<p>Accordingly, as indeed is clear from the outset, the ten equations (I)
do not, as far as their original form is concerned, favour any special
quadruplet. They are valid in one and the same form for all orthogonal
quadruplets of the relativistic \(R_{4}\), and, as is well-known, serve
to define their metric.</p>
<p>As in every case they give ten relationships between the sixteen
parameters \(\lambda_{i}^{\nu}\), we need only find six other
apparently reasonable conditions connecting the latter, in order
to mark out a special lattice (the world lattice) from among
all the possible quadruplets and lattices corresponding to the
space-time-manifold \(R_{4}\).</p>
<p>We shall shortly (§6) carry out this final step, which is in fact the
only essential one. Meanwhile we may appropriately lead up to it by
putting Maxwell's equations into a suitable form.</p>
<p class="nindc space-above2">
<b>5. Electromagnetic equations.</b></p>
<p>Let \(F_{\mu \nu}, F^{\mu \nu}, F_{i k}\) be the (covariant,
contravariant, and \(n\)-uplet) components of the skew-symmetrical
tensor \(\mathbf{F}\) which defines the electromagnetic field
in the space-time world; let \(\mathbf{S}\) (a vector) be the
current-vector<a id="FNanchor_9" href="#Footnote_9" class="fnanchor">[9]</a> and \(S_{\mu}\), \(\&c.\), its four components, where
all the quantities are understood to be measured in so-called rational
units.</p>
<p>Maxwell's equations (as adopted in the general theory of relativity
after Einstein) then take the forms
\[
\eqalign
{\operatorname{Div} \mathbf{F}=\mathbf{S}, \quad \operatorname{Div}^{*} \mathbf{F}=0 . \qquad \text{(34)}}
\]</p>
<p>Each group contains four equations, so that at first glance one would
take the total number of equations to be eight. But we necessarily
have \(\operatorname{div} \mathbf{S}=0\), so that by (31) there must
exist two identical relationships, namely those which express the
fact that the divergences in question vanish. Thus two equations
of the system (34) may (with appropriate subsidiary conditions) be
regarded as resulting from the other six; and in fact we know that if
\(\mathbf{S}\) is regarded as given or as associated in some other way
with the tensor \(\mathbf{F}\), then the equations (34) merely serve to
determine the six components of \(\mathbf{F}\) for \(x^{0}+d x^{0}\)
uniquely from their values for a given \(x^{0}\) (and any \(x^{1}, x^{2}, x^{3}\) ).</p>
<p>We have still to write down the symmetrical stress-energy tensor
explicitly. As is well known, its covariant components are defined as
follows:
\[
\tau_{\mu \nu}=-g^{\rho \sigma} F_{\mu \rho} F_{\nu \sigma}+\frac{1}{4} g_{\mu \nu} F_{\rho \sigma} F^{\rho \sigma} .
\]</p>
<p>By composition with \(\lambda_{i}^{\mu} \lambda_{l c}^{\nu}\)
(by replacing \(g^{\rho \sigma}\) on the right-hand side by
\(\sum_{0}^{3}{_{l}} e_{l} \lambda_{l}^{\rho} \lambda_{l}^{\sigma}\)
and \(F^{\rho \sigma}\) by \(\sum_{0}^{3}{_{jh}} e_{j} e_{h} F_{j h}\lambda_{j}^{\rho} \lambda_{h}^{\sigma}\))
we obtain the required \(n\)-uplet tensor formula:
\[
\eqalign
{\tau_{i k}=-\sum_{0}^{3}{_{l}} e_{l} F_{i l} F_{k l}+\frac{1}{4} \delta_{i k} \sum_{0}^{3}{_{jh}} e_{j}
e_{h} F_{j h}^{2} . \qquad \text{(35)}}
\]</p>
<p class="nindc space-above2">
<b>6. Interpretation of the electromagnetic tensor in the world
lattice. Purely geometrical formulation of the field equations.</b></p>
<p>A priori we may quite arbitrarily connect the six \(n\)-uplet
components \(\boldsymbol{F}_{i k}\) of the electromagnetic field
with any geometrical properties of a quadruplet (thereby defined)
of the \(R_{4}\). A very simple way of doing this is to make the
quantities \(F_{i k}\) proportional to the corresponding elements of a
(differential) skew-symmetrical local \(n\)-uplet tensor, e.g. to the
differential expressions, of the second or first order respectively,
which are defined by the equations
\[
\begin{aligned}
\xi_{i k}&=\sum_{0}^{3}{_{l}} e_{l} \frac{d \gamma_{i k l}}{d s_{l}} &\qquad \text{(12*)}\\
\text{or}\quad \eta_{i k}&=\sum_{0}^{3}{_{l}} e_{l} c_{l}\left(\gamma_{l i k}-\gamma_{l k i}\right) . &\qquad \text{(15*)}
\end{aligned}
\]
of §3.</p>
<p>As we shall see, the best way is to select the first expression, and we
accordingly put
\[
\eqalign
{F_{i k}=v \xi_{i k} \qquad \text{(P)}}
\]
where \(v\) denotes a constant.</p>
<p>As the Ricci coefficients of rotation \(\gamma_{i k l}\) are merely
ratios of an angle and a length, the quantities \(\xi_{i k}\) are of
dimensions \(l^{-2}\). The quantities \(\boldsymbol{F}_{i k}\), on
the other hand, behave like the square root of an energy-density.
Consequently we have
\[
\left[F_{i k}\right]=l^{-\tfrac{1}{2}} t^{-1} m^{\tfrac{1}{2}} .
\]</p>
<p>Hence the factor of proportionality \(v\) has dimensions
\[
l^{\tfrac{2}{2}} t^{-1} m^{\tfrac{1}{2}},
\]
which are those of an electric charge \(e\), e.g. the electronic
charge, so that we may write
\[
\eqalign
{v=z e, \qquad \text{(36)}}
\]
where the factor of proportionality \(z\) is now a pure number.
Moreover, we may also replace \(e\) in (36) by any other quantity of
the same dimensions; e.g. we may put
\[
\eqalign
{v=i_{1} \sqrt{hc} \qquad (36')}
\]
where \(h\) is Planck's constant, \(c\) the velocity of light in empty
space, and \(i_{1}\) a pure number.</p>
<p>Hence the final forms of the geometrical equations which arise from the
Maxwellian system (34) and our proposed addition (P), are
\[
\eqalign
{\operatorname{Div} \xi=\frac{1}{v} \mathbf{S}, \quad \operatorname{Div}^{*} \boldsymbol{\xi}=0, \qquad \text{(II)}}
\]
where \(\boldsymbol{\xi}\) means the local \(n\)-uplet tensor (12*).
<i>In conclusion, then, the geometrical definition of the quadruplet
(world lattice) associated with the field is to be taken from the
two systems</i> (I) <i>and</i> (II), <i>which together give sixteen
(apparently eighteen, but in reality only sixteen) differential
equations (of the second and third order respectively) involving the
sixteen</i> \(n\)-<i>uplet parameters</i> \(\lambda_{i}^{\nu}\).</p>
<p class="nindc space-above2">
<b>7. The case of empty space: absence of an electromagnetic field.</b></p>
<p>In empty space \((T_{i k}=0, \mathbf{S}=0)\), (I) reduces in virtue of
(32) to the form
\[
\eqalign
{G_{i k}-\frac{1}{2} \delta_{i k} G=-\kappa \tau_{i k}, \qquad \text{(I)}'}
\]
where the term \(\tau_{i k}\) on the right-hand side is given by
\[
\eqalign
{\tau_{i k}=-v^{2} \sum_{0}^{3}{_{l}} e_{l} \xi_{i l} \xi_{k l}+\frac{1}{4} v^{2} \delta_{i k}
\sum_{0}^{3}{_{jh}} e_{j} e_{l h} \xi_{j h}{ }^{2} \qquad (35')}
\]
by (35) and (P); while the system (II) becomes
\[
\eqalign
{\operatorname{Div} \boldsymbol{\xi}=0, \quad \operatorname{Div}^{*} \boldsymbol{\xi}=0 . \qquad \text{(II)}'}
\]</p>
<p>If the electromagnetic field vanishes in addition to the external
energy tensor \(\mathbf{T}_{i k}\), the quantities \(\xi_{i k}\), and
hence, by \((35')\), the quantities \(\boldsymbol{\tau}_{i k}\) also, are
equal to zero. If this happens everywhere in the space-time world, we
know <a id="FNanchor_10" href="#Footnote_10" class="fnanchor">[10]</a> that the equations \((I')\), which simply become \(G_{i k}=0\),
necessarily imply that the metric of the space is Euclidean or, more
correctly, pseudo-Euclidean.</p>
<p>What, then, is the geometrical meaning of the absence of
electromagnetic phenomena in this limiting case, i.e. what is the
geometrical meaning of the equations
\[
\eqalign
{\boldsymbol{\xi}_{i k}=0 . \qquad \text{(37)}}
\]</p>
<p><i>They simply state the fact that the world lattice is Cartesian or,
more correctly, pseudo-Cartesian.</i></p>
<p>In order to give as concise a proof of this as possible, I shall only
consider quadruplets in which the deviations from a pseudo-Cartesian
lattice are infinitely small.</p>
<p>If, in particular, we take the co-ordinates \(x^{\nu}\) to be Cartesian
co-ordinates with respect to that lattice, we have
\[
\lambda_{i}^{\prime \nu}=\delta_{i \nu}
\]
for the parameters of the corresponding quadruplet.</p>
<p>Let \(\lambda_{i}^{\nu}\) be the parameters of any
neighbouring quadruplet. Since the passage from the quantities
\(\lambda_{i}^{\gamma^{{}\nu}}\) to the quantities \(\lambda_{i}^{\nu}\)
corresponds to an infinitesimal pseudo-orthogonal transformation, the
quantities \(\lambda_{i}^{\nu}\) must, by (29), be expressible as
follows:
\[
\eqalign
{\lambda_{i}^{\nu}=\delta_{i \nu}+e_{i} \sum_{0}^{3}{_{k}} \beta_{i k} \delta_{k \nu}=\delta_{i \nu}+e_{i}
\beta_{i \nu}, \qquad \text{(38)}}
\]
where the quantities \(\beta_{i k}\) <i>form a skew-symmetrical</i>
\(n\)-<i>uplet tensor</i>. From this we can immediately calculate the
reciprocal elements. To a first approximation we obtain
\[
e_{i} \lambda_{i \mid \nu}=\delta_{i \nu}+e_{\nu} \beta_{i \nu},
\]
whence, multiplying by \(e_{i}\),
\[
\eqalign
{\lambda_{i \mid \nu}=\delta_{i \nu} e_{i}+e_{i} e_{\nu} \beta_{i \nu} . \qquad (38')}
\]</p>
<p>On the other hand, if we altogether neglect infinitely small
quantities, the operators
\[
\frac{d}{d s_{l}}=\sum_{0}^{3}{_{\nu}} \lambda_{l}^{\nu} \frac{\partial}{\partial x^{\nu}}
\]
reduce to the simple form
\[
\frac{\partial}{\partial x^{l}},
\]
and the covariant derivatives reduce to their usual forms.</p>
<p>Thus (4), the definition of the rotational invariants \(\gamma\), gives
(except for infinitely small quantities of the second order)
\[
\gamma_{i k l}=e_{i} e_{k} \frac{\partial \beta_{i k}}{\partial x^{l}},
\]
and from (12*) we further obtain
\[
\xi_{i k}=e_{i} e_{k} \sum_{0}^{3}{_{l}} e_{l} \frac{\partial^{2} \beta_{i k}}{\left(\partial x^{l}\right)^{2}}.
\]</p>
<p>The differential operator \(\sum_{0}^{3}{_{l}} e_{l} \frac{\partial^{2}\beta_{i k}}{\left(\partial x^{l}\right)^{2}}\)
is none other than the Dalembertian or Lorentz operator \(\square\).
Thus the equations (37) take the form
\[
\eqalign
{\square \beta_{i k}=0, \qquad (37')}
\]
and together with suitable initial and boundary conditions they give
\[
\beta_{i k}=0,
\]
i.e. <i>the Cartesian</i> (or, more correctly, pseudo-Cartesian)
<i>character of the world lattice</i>. I think that this conclusion
justifies our assumption (P). If we had put, say,
\[
F_{i k}=v^{\prime} \eta_{i k}, \quad\left(v^{\prime}=\text { constant }\right)
\]
where the quantities \(\eta_{i k}\) are given by the expressions (15*),
we should not have obtained any satisfactory result.</p>
<p>A more general assumption, such as
\[
F_{i k}=v \xi_{i k}+v^{\prime} \eta_{i k},
\]
would, on the other hand, be more complicated, though just as
admissible as (A) from the logical point of view. To a first
approximation, in fact, we should obtain the same result, as the
\(\eta\)'s are of higher order than the \(\xi\)'s.</p>
<div class="footnotes"><h3>FOOTNOTES:</h3>
<div class="footnote">
<p class="nind"><a id="Footnote_1" href="#FNanchor_1" class="label">[1]</a>
<i>Berliner Berichte</i>, I, 1929, pp. 1-8.</p>
</div>
<div class="footnote">
<p class="nind"><a id="Footnote_2" href="#FNanchor_2" class="label">[2]</a>
Ger. <i>Vierbein.</i></p>
</div>
<div class="footnote">
<p class="nind"><a id="Footnote_3" href="#FNanchor_3" class="label">[3]</a>
See in particular my <i>Absolute Differential Calculus</i>
(English translation by Miss Long), Chap. III. Blackie & Son, Ltd.,
1927.</p>
</div>
<div class="footnote">
<p class="nind"><a id="Footnote_4" href="#FNanchor_4" class="label">[4]</a>
Ger. <i>n-Bein</i>.</p>
</div>
<div class="footnote">
<p class="nind"><a id="Footnote_5" href="#FNanchor_5" class="label">[5]</a>
Ger. <i>Beintensor</i>.</p>
</div>
<div class="footnote">
<p class="nind"><a id="Footnote_6" href="#FNanchor_6" class="label">[6]</a>
<i>Die Relativitätstheorie</i>, Bd. II (2nd edition,
Vieweg. Brunswick, 1923), § 14.</p>
</div>
<div class="footnote">
<p class="nind"><a id="Footnote_7" href="#FNanchor_7" class="label">[7]</a>
<i>Riemannian Geometry</i>, Princeton University Press,
1926, Chap. III.</p>
</div>
<div class="footnote">
<p class="nind"><a id="Footnote_8" href="#FNanchor_8" class="label">[8]</a>
Given in 1918 by Cisotti (<i>Rend. Acc. Lincei</i>, Vol.
XXVII, pp. 366-371), but confined to the (imaginary) notation of (8),
(10), (11).</p>
</div>
<div class="footnote">
<p class="nind"><a id="Footnote_9" href="#FNanchor_9" class="label">[9]</a>
Ger. <i>Viererstrom</i>.</p>
</div>
<div class="footnote">
<p class="nind"><a id="Footnote_10" href="#FNanchor_10" class="label">[10]</a>
Cf. Serini, <i>Rend. Acc. Lincei</i>, Vol. XX VII, 1918,
pp. 235-238.</p>
</div>
</div>
<div style='text-align:center'>*** END OF THE PROJECT GUTENBERG EBOOK 78343 ***</div>
</body>
</html>
|
