D O C U M E N T 9 0 O C T O B E R 1 9 2 5 1 0 5 We want to introduce the designations (11) in (10) as well. Additionally, we set [5] [this new relation is actually superfluous, because a comparison with (7) shows simply that ] and regard as the components of a three-dimen- sional vector a. Then one obtains out of (10) two equation triples (the one with the other with ), (to abbreviate ). Instead of (10) one can write and the two equation triples become In summary: and that means, If one defines as you have done, then these equations be- come and 2Γμμ μ gμμ----------- ∂gμμ ∂xμ – aμ = αμ 2ϕμ –= a1, a2, a3 μ 2, = ν 3 = μ 3, = ν 1 = μ 1, = ν 2 = μ 1 2 2, , , = v 4= Δ ∂2 ∂x ------- - ∂2 ∂y2 ------- - ∂2 ∂z2 ------- - + + = ∂xν ∂aμ ∂xμ ∂aν² – ¢ Δ ∂2 ∂t2¹ ------ -– © § · ψνμ ∂aμ ∂xν -------- - ∂aν ∂xμ -------- –= 15) ( Δ ∂2 ∂t2¹ ------· -– © § Ex ∂a2 ∂z -------- ∂a3 ∂y -------- –= Δ ∂2 ∂t2¹ ------· -– © § Ey ∂a3 ∂x -------- ∂a1 ∂z -------- ,–= Δ ∂2 ∂t2¹ ------· -– © § Ez ∂a1 ∂y -------- ∂a2 ∂x -------- –= ¿ ° ° ¾ ° ° ½ 16) ( Δ ∂2 ∂t2¹ ------· -– © § E rota –= ……(17) Δ ∂2 ∂t2¹ ------· -– © § Hx ∂a1 ∂t -------- ∂a4 ∂x -------- –= Δ ∂2 ∂t2¹ ------· -– © § Hy ∂a2 ∂t -------- ∂a4 ∂y -------- –= Δ ∂2 ∂t2¹ ------ -– © § · Hz ∂a3 ∂t -------- ∂a4 ∂z -------- –= ¿ ° ° ¾ ° ° ½ 18) ( Δ ∂2 ∂t2¹ ------· -– © § H a · grad a4 –= ……(19) aμ 0 d. h. ϕμ 0= ( ), = Δ ∂2 ∂t2¹ ------ -– © § · E 0= ………(20) Δ ∂2 ∂t2¹ ------ -– © § · H 0= ………(21)

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1 0 6 D O C U M E N T 9 0 O C T O B E R 1 9 2 5 It follows from the first in relation with (13)[6] and out of (21) in connection with (14)[6] The last two equations are in good agreement with the second group of Maxwell’s formulas All of this agrees with what you found and I also notice that if one defines , it follows from (7) that and from (6) and (5). This means that if , equation (3) holds for all the values of the indices α, μ, ν. It now suggests itself that if vector is not set equal to zero one could perhaps obtain a link to the second group of Maxwell equations as they read in the presence of charges, namely: , It goes without saying that , will be related to . Nev- ertheless, it is not as simple as that on the contrary, it can be shown that the men- tioned link is scarcely possible. In order to see this, it suffices to look at the elec- trostatic case (e.g., of an electron at rest). Then (17) changes into . Because , it follows that . If we hold that then follows, which is not valid in general, of course. grad div E ∂ ∂t ---- rot H E) · – ( + 0. = rot (rot H E) · 0 =– rot H E · = div E 0. = ϕα 0= Γαα α 1 2 --gαα---------αxα∂ - ∂g = Γμα μ 1 2 -- - gμμ© ∂gμα ∂xμ ----------- - ∂gμμ ∂xα ----------- ∂ -----------¹·αμxμg∂ -–+ § = ϕα 0= ϕα rot H E · ρv += div E ρ = a1, a2, a3, a4 ρvx, ρvy, ρvz, ρ ΔE rot a –= div rot a 0= Δ div E 0= div E ρ ,= Δρ 0=

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Volume 15: The Berlin Years: Writings & Correspondence, June 1925-May 1927 (English Translation Supplement) resources

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1 0 4 D O C U M E N T 9 0 O C T O B E R 1 9 2 5 In a postscript I shall indicate the steps in the calculation perhaps Mr. Grommer[4] will be kind enough to check them. By the way, through direct substi- tution I have convinced myself that the presented values in connection with (2) sat- isfy your basic conditions (5). It only remains to be seen what follows out of your conditions (4) . They can be replaced by and It turns out that the unknown ’s do not occur in (8) and that the equation contains only the symmetrical components of the ’s. With (8) we are dealing with the field equations of gravitation, about which I have nothing to say. I there- fore restrict myself to (9). Let μ, ν, κ, λ be the four numbers 1, 2, 3, 4 in any sequence. Then (9) becomes These equations, together with (2), should determine the electromagnetic field. If one puts then the four equations (2) read, if one writes x, y, z, t for the coordinates x, y, z, t: (i.e., ) and (d. h. ) This is the first half of Maxwell’s equations. Rμν 0= 1 2 -- - Rμν Rνμ) + ( 0= ……(8) 1 2 --( - Rμν Rνμ) – 0= ……(9) Γαα α gαβ ∂Γμμ μ ∂xν ------------ ∂Γνν ν ∂xμ ----------- -– 1 2 -- - gμμ∂xμ∂xν ∂2gμμ ----------------- – 1 2 -- - gνν∂xμ∂xν ∂2gνν ----------------- 1 2 -- - gμμ---------------νμ2xμ∂ ∂2ψ + + 1 2 -- - gνν-------------- ∂2ψνμ ∂xν 2 - 1 2 -- - gκκ-------------- ∂2ψνμ ∂xκ 2 - 1 2 -- - gλλ-------------- ∂2ψνμ- ∂xλ2 + + + 0 = 10) ( ψ23 Ex, ψ31 Ey, ψ12 Ez = = = ψ41 Hx, ψ42 Hy, ψ43 Hz = = = ¿ ¾ ½ ……(11) x1, x2, x3, x4, ∂Ez ∂y -------- ∂Ey ∂z -------- -– ∂Hx ∂t --------- ,–= ∂Ex ∂z -------- - ∂Ez ∂x -------- – ∂Hy ∂t --------- ,–= ∂Ey ∂x -------- - ∂Ex ∂y -------- -– ∂Hz ∂t --------- –= 12) ( rotE H · –= ……(13) ∂Hx ∂x --------- ∂Hy ∂y --------- ∂Hz ∂z --------- + + 0 = divH 0= 14) (

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