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Additional resources for A Stubbornly Persistent Illusion: The Essential Scientific Works of Albert Einstein
Those consist in a mix of outer multiplication with contraction. Examples. —From the covariant tensor of the second one rank Amn and the contravariant tensor of the 1st rank Bs we shape through outer multiplication the combined tensor s Dmn ϭ Amn Bs. On contraction with admire to the indices n and s, we receive the covariant four-vector Dm ϭ Dnmn ϭ Amn Bn. This we name the internal manufactured from the tensors Amn and Bs. Analogously we shape from the tensors Amn and Bst, via outer multiplication and double contraction, the internal product Amn Bmn. via outer multiplication and one contraction, we receive from Amn and Bst the combined tensor of the second one rank Dtm ϭ Amn Bnt. This operation will be aptly characterised as a combined one, being “outer” with recognize to the indices m and t, and “inner” with admire to the indices n and s. We now end up a proposition that is frequently important as proof of tensor personality. From what has simply been defined, Amn Bmn is a scalar if Amn and Bst are tensors. yet we can also make the next statement: If AmnBmn is a scalar for any collection of the tensor Bmn, then Amn has tensor personality. For, by means of speculation, for any substitution, A¿st B¿ st ϭ Amn Bmn. yet through an inversion of (9) 0xm 0xn st B¿ . Bmn ϭ 0x¿s 0x¿t This, inserted within the above equation, offers 0xm 0xn aA¿st Ϫ A b B¿ st ϭ zero. 0x¿s 0x¿t mn sixty one THE precept OF R E L AT I V I T Y this may merely be happy for arbitrary values of B¿ st if the bracket vanishes. the outcome then follows by means of equation (11). This rule applies correspondingly to tensors of any rank and personality, and the facts is comparable in all instances. the guideline can also be verified during this shape: If Bm and Cn are any vectors, and if, for all values of those, the internal product Amn B m Cn is a scalar, then Amn is a covariant tensor. This latter proposition additionally holds strong whether simply the extra unique statement is right, that with any selection of the four-vector Bm the interior product Amn B m Bn is a scalar, if additionally it truly is identified that Amn satisfies the situation of symmetry Amn ϭ Anm. For by way of the strategy given above we turn out the tensor personality of 1Amn ϩ A nm 2, and from this the tensor personality of Amn follows because of symmetry. This may also be simply generalized to the case of covariant and contravariant tensors of any rank. ultimately, there follows from what has been proved, this legislations, which can even be generalized for any tensors: If for any number of the fourvector Bn the amounts Amn Bn shape a tensor of the 1st rank, then Amn is a tensor of the second one rank. For, if Cm is any four-vector, then because of the tensor personality of Amn Bn, the interior product Amn BnC m is a scalar for any number of the 2 four-vectors Bn and Cm. From which the proposition follows. § eight. a few points OF the elemental TENSOR gmn The Covariant primary Tensor. —In the invariant expression for the sq. of the linear aspect, ds 2 ϭ g mn dxm dxn, the half performed by way of the dxm is that of a contravariant vector that could be selected at will. because extra, gmn ϭ gnm, it follows from the concerns of the previous paragraph that gmn is a covariant tensor of the second one rank.