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The Trace in the Footnote |
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An article appearing on the Scientific American web site on the subject of the cosmological constant included the surprising comment that "it is a little-known fact that Einstein had noted the possibility of such an extension to the field equations in his original exposition of 1916." This claim has been repeated subsequently in other venues, so it may be worthwhile to point out that the claim is incorrect. It refers to the footnote in Section 14 of Einstein’s paper “The Foundation of the General Theory of Relativity”. In this paper Einstein used the symbol Bμν to denote the Ricci tensor, defined as the contraction of the full curvature tensor (see Section 12 of the paper), but the most common English translation used Gμν (not to be confused with the Einstein tensor for which the symbol Gμν later became conventional). For consistency with modern convention, we will denote the Ricci tensor here as Rμν. Einstein derives the vacuum field equations Rμν = 0 in Section 14, and then says |
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It must be pointed out that there is only a minimum of arbitrariness in the choice of these equations. For besides Rμν there is no tensor of the second rank which is formed from the gμν and its derivatives, contains no derivatives higher than second, and is linear in these derivatives.* |
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The asterisk points to the footnote at the bottom of the page, which qualifies this remark: |
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Properly speaking, this can be affirmed only of the tensor Rμν + λ gμν gαβ Rαβ where λ is a constant. If, however, we set this tensor = 0, we come back again to the equation Rμν = 0. |
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The fact that Einstein chose the symbol λ for the constant seems to have misled some people into thinking that he is referring here to the cosmological constant, which later came to be associated with the symbol λ. However, this footnote actually has nothing to do with the cosmological constant. Recall that the curvature scalar R is given by the contraction of the Ricci tensor, i.e., we have R = gαβ Rαβ. Thus Einstein’s footnote is just referring to the fact that the stated conditions are satisfied not just by Rμν but by Rμν + λgμνR for any constant λ. The extra term is called the “trace” term, because R is the trace of Rμν. He then goes on to note that it doesn’t really matter, for the vacuum field equations, whether we include the trace term or not, because Rμν + λgμνR = 0 if and only if Rμν = 0. |
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Later in the paper he goes on to consider the field equations in the presence of mass-energy represented by the tensor Tμν (which vanishes in vacuum). One might think the full field equations would be given by setting Rμν proportional to Tμν, but this leads to a problem, because (local) conservation of mass-energy implies that the covariant derivative of Tμν must vanish, whereas the covariant derivative of Rμν does not vanish. However, as Einstein shows, the covariant derivative of Rμν + λgμνR does vanish identically, provided we set λ = –1/2. Thus we arrive at the final form of the field equations |
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Noting that R = 8πT, this is equivalent to equation 53 of Einstein’s paper. Needless to say, this isn’t an “extension of the field equations”, it’s just the basic field equations in the presence of mass-energy. Einstein’s use of the symbol B for the Ricci tensor, later translated as G, may contribute to the confusion for modern readers, since today G is commonly used to refer to the “Einstein tensor”, i.e., the entire left side of the above equation. Also, writing the field equations in terms of the trace T instead of R contributes to the confusion, because it encourages the mistaken impression that G (standing alone on the left side) represents the full Einstein tensor. |
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As is well known, the actual cosmological term was not introduced by Einstein until the following year, in his 1917 paper “Cosmological Considerations on the General Theory of Relativity”. Equation 13a of that paper, converted to modern notation conventions, and using the trace R instead of T, is |
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The footnote in the 1916 paper referred to the second term on the left side (the trace term), whereas the cosmological term introduced in 1917 is the third term on the left side of this equation. These are completely different terms, with completely different physical meanings, and the constant denoted by λ in the 1916 footnote has the value –1/2, completely unrelated to the cosmological constant, for which Einstein also happened to use the symbol λ in the 1917 paper. Of course, the cosmological term is a fairly obvious formal possibility, since the covariant derivative of the metric tensor identically vanishes, but Einstein was not looking for ambiguity and new indeterminate constants, so he didn’t recognize (or at least didn’t acknowledge) this term in his 1916 paper. |
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Needless to say, like the cosmological term, the trace term has been a subject of controversy. For the trace term, the controversy has been largely historical, focusing on the question of whether Einstein arrived at the necessity of that term on his own, or got it from Hilbert, or Hilbert got it from Einstein, or both found it independently. It’s interesting that the first appearance of this term in Einstein’s 1916 paper is in a footnote, amending a statement in the main text that claims the Ricci tensor is the only possible tensor meeting the stated conditions. One might speculate that, at some stage, Einstein thought it was the unique tensor, and then added the footnote after realizing that the Ricci tensor plus the trace term still meets all the conditions. |
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In any case, the trace term was the last piece of the puzzle that finally completed the basic field equations (just as the “displacement current” term completed Maxwell’s equations). Einstein would surely have been relieved to note that it does not affect the vacuum solution, so his predictions for light deflection and orbital precession were undisturbed. But, again, this term had nothing to do with the cosmological term, which first appeared only in the following year. |
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