Journal of Modern Physics 017 8 944-963 http://www.scirp.or/journal/jmp ISSN Online: 153-10X ISSN Print: 153-1196 Property of Tensor Satisfyin Binary Law Koji Ichidayama 716-000 Okayama Japan How to cite this paper: Ichidayama K. (017) Property of Tensor Satisfyin Binary Law. Journal of Modern Physics 8 944-963. https://doi.or/10.436/jmp.017.86060 Received: pril 7 017 ccepted: May 8 017 Published: May 31 017 Copyriht 017 by author and Scientific Research Publishin Inc. This work is licensed under the Creative Commons ttribution International License (CC BY 4.0). http://creativecommons.or/licenses/by/4.0/ Open ccess bstract I report the reason why Tensor satisfyin Binary Law has relations toward physics in this article. Q: The n th-order covariant derivative of the Vector : n = 1 satisfyin Binary Law. R: The n th-order covariant deriva- { } ( ) tive of the Vector { } :( n ) satisfyin Binary Law. I have reported in other articles about Q. I report R in this article. I obtained the followin results in this. I ot the conclusion that derived function became 0. The derived function becomin 0 in the n th-order covariant derivative of the covariant vector here in the case of n =. Similarly in the n th-order covariant derivative of the contravariant vector in the case of n = 4. Keywords Tensor Covariant Derivative 1. Introduction Definition 1 1 λ 1 λ λ λ λ λ = + + λ λ x x x 1 λ 1 λ τλ λ τ τλ λ 1 λ τλ λ τ + λ + + λ + τ λ 1 1 1 τλ λ λ τλ λ λ λ λ τλ τ λ τ λ τ λ + x x x x is established [1]. Definition x x x x x = x x = x is established []. I named x x x x x = x x = x Binary Law []. Definition 3 If x x x x x = x x = x is established x = x is established []. Definition 4 If x x x x x = x x = x is established x = x is established []. DOI: 10.436/jmp.017.86060 May 31 017
Definition 5 If x x x x x = x x = x is established x = x is established []. Definition 6 If x x x x x = x x = x is established x = x is established []. Definition 7 If all coordinate systems x x x x λ satisfies x x x x x = x x = x all coordinate systems x x x x λ shifts to only two of x x []. Definition 8 1 τλ λ λ 1 τλ λ λ = τ + λ λ τ x x x x is established [3]. Definition 9 = e e is established [4]. Definition 10 = 1 0: ( ) = is establishment [3]. Definition 11 1 λ ελ λ ε 1 λ ελ λ ε = + x x x + + + λ x x x λ 1 τλ λ λ 1 τλ λ λ 1 λ ελ τλ ετ λ τ λ τ + + x x x x λ τ 1 λ λ τλ τ 1 λ λ τλ τ 1 τλ ελ λ ε + τ λ τ λ λ + x x x x + x x x is established [3]. Definition 1 3 1 κ κ κ λ = + λ λ + κ x x x 1 κ 1 κ κ κ κ κ + + κ + λ λ + κ x x x x x x x x α 1 κ κ κ 1 κ ακ κ α + + κ + λ λ x x x x + α κ α 1 κ ακ κ α + α κ λ 1 1 κ κ ακ κ α ακ κ + λ α κ κ x 1 ακ κ κ 1 κ ακ κ α + λ + κ x x x x + α κ 1 1 κ κ ακ κ α ακ κ + α κ κ λ 1 ακ κ κ 1 ακ κ κ + λ κ x α x + κ α λ 1 1 κ ακ κ κ κ ακ α λ κ α κ x 1 κ κ ακ α 1 ακ κ κ λ + + α κ κ 945
1 1 ακ κ κ κ κ ακ α + κ + α κ λ τ 1 κ τκ λκ τλ + λ τ κ 1 1 κ κ τκ λκ τλ τκ κ + λ τ κ κ 1 1 κ τκ λκ τλ τκ κ κ + + λ τ κ + κ α 1 κ τκ λκ τλ 1 τκ ακ κ α + λ τ κ α κ x x x 1 κ τκ λκ τλ 1 τκ ακ 1 κ α ακ κ κ + + λ τ κ + α κ + κ τ 1 κ τκ λκ τλ 1 ακ κ κ λ τ κ κ α x x x 1 κ τκ λκ τλ 1 ακ κ κ 1 τκ κ ακ α + + λ τ κ κ x x x + α κ 1 τκ κ λκ λ + λ κ τ 1 1 κ τκ κ λκ λ κ τκ τ λ κ τ κ x x x 1 τκ κ λκ λ 1 κ κ τκ τ + λ κ + x x τ κ x α 1 τκ κ λκ λ 1 κ ακ κ α + λ κ α κ τ x x x 1 1 1 τκ κ λκ λ κ ακ κ α ακ κ τκ τ λ κ α κ τ + + x x x κ 1 τκ κ λκ λ 1 ακ τκ κ τ + + λ κ x x x + τ κ α 1 1 1 κ τκ κ λκ λ ακ τκ κ τ κ ακ α + λ κ τ κ α κ x x x 1 τκ κ λκ λ + λ κ τ 1 1 τκ κ λκ λ κ κ κ λ κ τ κ x x x 1 1 κ τκ κ λκ λ κ κ λ κ κ τ α 1 τκ κ λκ λ 1 κ ακ τκ ατ + λ κ x x x + τ α κ 1 1 1 κ κ λκ λ ακ τκ ατ κ λ κ τ α κ κ x x x 1 τκ κ λκ λ 1 ακ κ τκ τ + + λ κ x x + τ κ α 1 1 1 κ κ λκ λ κ τκ τ ακ α + λ κ τ κ α κ x x x τκ κ ακ τκ ακ κ is established [3]. 946
Definition 13 For every coordinate systems there is no immediate reason for preferrin certain systems of co-ordinates to others. Definition 14 The physical law is invariable for all coordinate systems [1]. Definition 15 ll coordinate systems satisfies Definision 13 is established if Definision 14 is established. Definition 16 Definision 14 is established if The physical law is described in Tensor is established [1]. Definition 17 ll coordinate systems satisfies Definision 13 is established if ll coordinate systems satisfies Binary Law is established []..Einstein required establishment of Definision 14 approximately 100 years ao [1]. Furthermore he required establishment of The physical law is described in Tensor based on Definision 16 [1]. However. Einstein does not mention Definision 15 at all [1]. I et the conclusion that ll coordinate systems satisfies Definision 13 must be established if Definision 14 is established accordin to Definision 15. On the other hand I ot that Definision 17 was established []. nd I ot the conclusion that must require establishment of ll coordinate systems satisfies Binary Law if I required establishment of Definision 14 by Definision 17. Scalar and Vector have already satisfied these two demands here []. In other words we can use Scalar and Vector to express a physical law. Therefore I do not mention it for Scalar and Vector. I researched it about the Tensor which had not yet satisfied Binary Law in this article. The first purpose of this article is to rewrite the Tensor which does not satisfy Binary Law in Tensor satisfyin Binary Law. Then the second purpose is to find out the property from Tensor satisfyin Binary Law.. bout Property of Tensor Satisfyin Binary Law: The Second Third Fourth-Order Covariant Derivative of the Vector { } Proposition 1 If x x x x x = x x = x is established = = 0 is established. Proof: I et ( ) = 1 = 0 : (1) from Definision 10 if all coordinate systems x x x x λ satisfies Definision. I et 1 = + x x x 1 1 1 1 + + + 947
1 + 1 1 + + + 1 1 1 = x 1 1 1 1 1 + + = x 1 1 1 λ λ 1 λ 1 1 1 1 λ + + from Definision 1 if all coordinate systems x x x x λ satisfies Definision. By the way we cannot handle () accordin to Definision 7. I simplify () here and et λ 1 1 1 λ =. + (3) However (3) can rewrite 1 1 1 = + (4) if x and x of (3) are chaneable to x or x () each. Because index doesn t exist at all in the third term of the riht side of (3) I can chane dummy index λ of (3) to dummy index. Furthermore (4) can rewrite 1 1 1 =. + (5) Because index doesn t exist at all in the second term of the riht side of (4) I can chane dummy index of (4) to dummy index. nd we can handle (5) accordin to Definision 7. The possible rewrite by or of is x x (6) (7) (8) accordin to Definision 4 Definision 6. Because three covariant Vector of the same index exists in one term I don t handle (6). Two sets are dummy index 948
amon three same index in (7) (8). Therefore we must rewrite () to ( ) = + + 1 1 1 1 1 + 1 1 1 + ( ) = + + 1 1 ( ) 1 1 1 + x x x x 1 1 1 + x x x 1 1 = 1 1 1 + x x x x 1 1 1 + x x x by usin Definision 4 Definision 6 with considerin (7) (8). I et ( ) ( ) = = 1 1 = x in consideration of establishment of = from (9) (10) here. I et in consideration of (1) for (1). nd I et from (13). I et from (9) (10) (11) (1) = 0 (13) = 0 = in consideration of Definision 4 here. I et (14) = (15) = 0 (16) 949
from (14) (15). Therefore I et = (17) x x ( ) = (18) x x ( ) ( ) =. (19) x x from (9) (10) (11) in consideration of (1) (13) (16). nd we can rewrite (17) (18) (19) by usin Definision 4 Definision 6 for = (0) x x. Because the second third term of the riht side of (17) (18) (19) does not exist here we may adopt (17) (18) (19) and (0) description form of which. Furthermore I rewrite (0) by Definision 4 and et ( ) = = 0 in consideration of Proposition. nd I rewrite (1) by Definision 4 and et = = 0. End Proof Because () is established I decide not to handle the third-order covariant derivative of the covariant Vector. (1) () Proposition If x x x x x = x x = x is established S = 0 is established. Proof: I et 1 1 1 1 = x = + (3) from Definision 8 if all coordinate systems x x x x λ satisfies Definision. I et from from (4). I et Definision 3. I et = (4) x x = = (5) 950
from (5) Definision 4. Therefore I et = = (6) = 0 (7) from (3) (6). I et = e e (8) from Definision 9 if all coordinate systems x x x x λ satisfies Definision. I et B = B (9) from B Definision 3 if all coordinate systems x x x x λ satisfies Definision. I et from (9). I et B = B (30) B = B (31) from (8) (30). I et ( B ) B = (3) from covariant derivative of (31). I et S = 0 from (7) (3). End Proof Proposition 3 If x x x x x = x x = x is established = is established. x x Proof: I et 1 = + x x x + 1 1 + x x x x 1 1 + + 1 + + 1 1 + x x x 1 1 1 = + + x 1 1 1 1 1 + + 1 1 1 = + + x λ λ 1 1 1 1 1 λ + + λ x (33) 951
= + + x 1 1 1 λ λ 1 1 1 λ 1 1 λ + + (34) from Definision 11 if all coordinate systems x x x x λ satisfies Definision. By the way we cannot handle (33) (34) accordin to Definision 7. I simplify (33) here and et λ 1 1 1 =. + + + λ x x x (35) However (35) can rewrite 1 1 1 = + + + x x x (36) if x and x of (35) are chaneable to x or x each. Because index doesn t exist at all in the third term of the riht side of (35) I can chane dummy index λ of (35) to dummy index. Furthermore (36) can rewrite 1 1 1 =. + + + x x x (37) Because index doesn t exist at all in the second term of the riht side of (36) I can chane dummy index of (36) to dummy index. nd we can handle (37) accordin to Definision 7. The possible rewrite by or of is x x (38) (39) (40) accordin to Definision 4 Definision 6. Because three contravariant Vector of the same index exists in one term I don t handle (40). Two sets are dummy index amon three same index in (38) (39). Therefore we must rewrite (33) to 1 1 = + + 1 1 1 x x x 1 1 1 + + (41) 95
1 1 1 ( ) = + + x x (4) 1 1 1 1 1 + x x x ( ) = + + 1 1 1 1 1 1 1 1 + x x x x x x by usin Definision 4 Definision 6 with considerin (38) (39). I et ( ) = ( ) = 1 1 = in consideration of establishment of = from (4) (43) here. I et = 0 in consideration of (1) for (44). Therefore I et (43) (44) (45) = (46) x x ( ) = (47) x x ( ) = (48) x x from (41) (4) (43) in consideration of (1) (45). nd we can rewrite (46) (47) (48) by usin Definision 4 Definision 6 for. = (49) x x Because the second third term of the riht side of (46) (47) (48) does not exist here we may adopt (46) (47) (48) and (49) description form of which. Similarly we must rewrite (34) to 1 1 1 = + + x x x x (50) 1 1 1 1 1 + + 953
1 1 1 ( ) = + + x (51) 1 1 1 1 1 + ( ) = + + 1 1 1 1 1 + x x x x x x 1 1 1 by usin Definision 4 Definision 6 with considerin (38) (39). Because (51) includes here I don t handle (51). Therefore I et (46) (48) from (50) (5) in consideration of (1). End Proof Proposition 4 If x x x x x = x x = x is established 3 = = is established. Proof: I et 3 1 = + x x x x x + 1 + x 1 + x 1 + + 1 + x + 1 + 1 1 + x + x x x 1 1 + x + x x x + 1 1 + x x x 1 1 + x x 1 1 x x x x 1 1 x + 1 1 + x x x 1 + (5) 954
1 1 + + x x x x 1 1 + x x x 1 1 + + 1 1 1 + x x x + x x x 1 1 x x x + 1 1 1 x x x + x x x 1 1 x 1 1 1 x x x 1 1 x x x 1 1 1 + x x x + x x x 1 1 + x x x 1 1 1 + + + x x x 1 1 1 + x x x x 1 1 x x x 1 1 x x x 1 1 1 + x x 1 1 + x x x 1 1 1 + + + x x x = + + + x x x x x x 3 1 1 1 955
1 1 1 + + + x x x x x 1 1 1 1 1 1 + + + x x x x x 1 1 1 1 x x x x x x x 1 1 1 1 x 1 + 1 1 1 1 1 1 + + + x x x x 1 1 1 1 1 1 1 1 + x x x x x x 1 1 1 1 1 1 x 1 1 1 1 1 1 + 1 1 1 1 1 + 1 1 1 1 1 1 1 1 1 1 1 1 1 + + = + + + x x x 3 1 1 3 3 1 1 3 + x x x x 5 1 1 1 1 + 1 1 1 1 1 + + = + + + x x x 3 1 1 3 3 λ 1 1 3 + x x x x x λ 5 1 1 1 1 λ + λ x x x x x x λ λ λ λ ε λ λ ε λ 1 1 ε 1 λ 1 ε 1 λ + + λ (53) 956
= + + x x x λ 3 λ 1 + + 1 3 x x λ λ 5 1 λ 1 λ 1 + x x x x x x λ ε λ 1 λ 1 λ 1 ε x x x λ ε λ 1 λ 1 1 ε λ + + 3 1 1 3 (54) from Definision 1 if all coordinate systems x x x x λ satisfies Definision. By the way we cannot handle (53) (54) accordin to Definision 7. I simplify (53) here and et 1 = + + x λ 1 + λ ˆ ε λ 1 1 ε ˆ. + λ 3 ˆ ˆ ˆ However (55) can rewrite 1 = + + x λ 1 + λ ˆ λ 1 1 ˆ + λ 3 ˆ ˆ ˆ (55) (56) if x and x xˆ of (55) are chaneable to x or x each. Because index doesn t exist at all in the fourth term of the riht side of (55) I can chane dummy index ε of (55) to dummy index. Furthermore (56) can rewrite 1 = + + x 1 + ˆ 1 1 ˆ. + 3 ˆ ˆ ˆ Because index doesn t exist at all in the third term of the riht side of (56) I can chane dummy index λ of (56) to dummy index. Furthermore (56) (57) 957
can rewrite = + + 3 1 ˆ ˆ ˆ 1 + ˆ (58) 1 1. ˆ + Because index doesn t exist at all in the second term of the riht side of (57) I can chane dummy index of (57) to dummy index. nd we can handle (58) accordin to Definision 7. The possible rewrite by or of 3 is 3 (59) x x x 3 3 3 x x (60) 3 3 3 3 (61) (6) accordin to Definision 4 Definision 6. Because two covariant Vector of the same index exists in one term I don t handle (59). Because two contravariant Vector of the same index exists in one term I don t handle (61). Because four contravariant Vector of the same index exists in one term I don t handle (6). Therefore we must rewrite (53) to 3 1 1 = + 3 + ( ) + + 3 1 1 3 x 5 1 1 1 + 1 1 1 x x x x x x 1 1 1 + + x x x x x (63) 958
3 = + 3 + x x x x x x x x x ( ) ( ) 1 1 + + 3 1 1 3 x x 5 1 1 1 + 1 1 1 x x x + x 1 1 1 + 1 1 3 = + 3 + + + 3 1 1 3 x 5 1 1 1 + x x x x x 1 1 1 x x x x 1 1 1 + + x x x x by usin Definision 4 Definision 6 with considerin (60). I et ( ) = ( ) 3 3 = 3 3 = x x x x x in consideration of establishment of ˆ = ˆ from (64) (65) here. I et in consideration of (1) for (66). Therefore I et ( ) (64) (65) (66) = 0 (67) 3 = (68) x x x 959
( ) 3 = (69) x x x ( ) 3 = (70) x x x from (63) (64) (65) in consideration of (1) (67). nd we can rewrite (68) (69) (70) by usin Definision 4 Definision 6 for 3 = =. (71) Because the second third term of the riht side of (68) (69) (70) does not exist here we may adopt (68) (69) (70) and (71) description form of which. Similarly we must rewrite (54) to ( ) 1 1 3 = + 3 + ( ) 3 1 + + 1 3 5 1 1 1 + x x x x x x 1 1 1 x x x x x x 1 1 1 + + x x x x x 3 1 1 = + + 3 ( ) 3 1 + + x x x x x 1 3 5 1 1 1 + x x x x x x 1 1 1 x x x 1 1 1 + + 3 1 1 = + + 3 3 1 + + x x x x x (7) (73) 960
1 3 x x x 5 1 1 1 + x x x x x x 1 1 1 x 1 1 1 + + x x x x (74) by usin Definision 4 Definision 6 with considerin (60). Because (7) includes here I don t handle (7). I et ( ) = ( ) 3 3 = 3 3 = (75) in consideration of establishment of ˆ = ˆ from (73) (74) here. I et = 0 (76) in consideration of (1) for (75). Therefore I et (69) (70) from (73) (74) in consideration of (1) (76). End Proof Proposition 5 If x x x x x = x x = x is established = 0 is established. Proof: I et from λε λε = (77) = if all coordinate systems x x x x λ satisfies Definision. nd I et = 0 (78) from (77) Proposition Proposition 4. End Proof Because (78) is established I decide not to handle the fifth-order covariant derivative of the contravariant Vector. Proposition 6 If x x x x x = x x = x is established = sin x is established. Proof: I et 3 1 3 1 = 1 1 1 = 3 3 = 1 1 1 = (79) 961
from (71) if a dimensional number is. I et from (79). nd I et from (80). I et from (81). I et 3 1 3 1 1 1 dx= x d dx = x d 1 1 1 3 3 1 1 dx= x d dx = x d 1 1 1 1 1 1 = x = x 1 1 1 = x = x 1 1 = x = x = from (8) Definision 5. nd I et 1 1 1 1 = = 1 1 = = 1 1 from (83). I et 1 1 1 = sin x = sin x = sin x = sin x 1 (80) (81) (8) (83) (84) (85) from (84). nd I et from (85). = sin x (86) End Proof 3. Discussion bout Proposition 1 In () we can handle as Tensor similarly. Furthermore = 0 is established. I do not handle the derived function of a hiher order because derived function is already 0. bout Proposition 3 In (49) we can handle bout Proposition 4 3 In (71) we can handle Furthermore = S is established. as Tensor similarly. as Tensor similarly. 96
bout Proposition 5 In (78) = 0 is established. I do not handle the derived function of a hiher order because derived function is already 0. bout Proposition 6 3 If = 0 is established in (71) can t have a wave-like property. 3 However has a wave-like property if 0 is established in (71). These remind me of the matter wave in the quantum theory. References [1] Einstein. (1916) nnalen der Physik 354 769-8. https://doi.or/10.100/andp.1916354070 [] Ichidayama K. (017) Journal of Modern Physics 8. [3] Dirac P..M. (1975) General Theory of Relativity. John Wiley and Sons Inc. [4] Fleisch D. (01) Student s Guide to Vectors and Tensors. Cambride University Press. Submit or recommend next manuscript to SCIRP and we will provide best service for you: cceptin pre-submission inquiries throuh Email Facebook LinkedIn Twitter etc. wide selection of journals (inclusive of 9 subjects more than 00 journals) Providin 4-hour hih-quality service User-friendly online submission system Fair and swift peer-review system Efficient typesettin and proofreadin procedure Display of the result of downloads and visits as well as the number of cited articles Maximum dissemination of your research work Submit your manuscript at: http://papersubmission.scirp.or/ Or contact jmp@scirp.or 963