I don’t claim this to be new, the earliest reference I can find is Baez and Bunn [1], though I am sure the idea is older than that.\u00a0\u00a0 The claim is that there is a kind of duality in the Einstein field equations between the Ricci tensor and the energy-momentum tensor. That is one can in essence switch the roles of the Ricci tensor and the energy-momentum tensor in the field equations.\u00a0 I will assume familiarity with the\u00a0 tensors and the field equations.<\/p>\n
Lets see how this works. Recall the Einstein field equations in 4d<\/p>\n
\\(R_{\\mu \\nu}{-} \\frac{1}{2}g_{\\mu \\nu}R\u00a0 + g_{\\mu \\nu} \\Lambda = \\kappa T_{\\mu \\nu} \\),<\/p>\n
here \\(\\kappa = \\frac{8 \\pi \\: G}{c^{4}}\\) is the gravitational constant<\/p>\n
and of course\u00a0 \\(R:= R_{\\lambda}^{\\:\\: \\lambda}\\).<\/p>\n
The field equations imply that<\/p>\n
\\(R_{\\mu}^{\\:\\: \\mu} {-} \\frac{1}{2} g_{\\mu}^{\\:\\: \\mu}R_{\\lambda}^{\\:\\: \\lambda} + g_{\\mu}^{\\:\\: \\mu} \\Lambda = \\kappa T_{\\mu}^{\\:\\: \\mu} \\).<\/p>\n
We assume that we are in 4d thus<\/p>\n
\\(g_{\\mu}^{\\:\\: \\mu} =4\\).<\/p>\n
One could consider other dimensions, but things work out clearer in 4d and anyway this is where classical general relativity is formulated.<\/p>\n
Thus we arrive at<\/p>\n
\\({-}R_{\\mu}^{\\:\\: \\mu} = \\kappa T_{\\mu}^{\\:\\: \\mu} {-} 4 \\Lambda\\).<\/p>\n
Now using this result in the field equations produces<\/p>\n
\\(R_{\\mu \\nu} = \\kappa \\left(\u00a0 T_{\\mu \\nu}{-} \\frac{1}{2}g_{\\mu \\nu}T_{\\lambda}^{\\:\\: \\lambda}\\right) + g_{\\mu\\nu} \\Lambda \\).<\/p>\n
Now divide by the\u00a0 gravitational constant to write the field equations as<\/p>\n
\\(T_{\\mu \\nu} {-} \\frac{1}{2} g_{\\mu \\nu}T_{\\lambda}^{\\:\\: \\lambda} + g_{\\mu \\nu} \\left(\u00a0 \\frac{\\Lambda}{\\kappa}\\right) = \\left(\\frac{1}{\\kappa}\\right)R_{\\mu \\nu} \\).<\/p>\n
Comparing the above with the original form of the field equations we see that we have a kind of duality given by<\/p>\n
\\(R \\rightarrow T\\)<\/p>\n
\\(T \\rightarrow R\\)<\/p>\n
\\(\\Lambda \\rightarrow\u00a0 \\frac{\\Lambda}{\\kappa} \\)<\/p>\n
together with the inversion of the gravitational constant,<\/p>\n
\\(\\kappa \\rightarrow \\kappa^{-1}\\).<\/p>\n
I some sense we have done nothing. Both forms of the Einstein field equations are equally valid and describe exactly the same physics. The difference, as I see it is that the second form, this “dual form”, is better from a geometric perspective.<\/p>\n
In particular the Ricci curvature tensor has a clear geometric origin.\u00a0 Via the \u00a0 Raychaudhuri equation, the Ricci tensor (for a Lorentzian signature metric) measures\u00a0 the degree to which near by test particles will tend to converge or diverge.<\/p>\n
Then one can then paraphrase the Einstein field equations as<\/p>\n
The degree test particles tend to converge or diverge\u00a0 in time is determined by the matter content\u00a0 + the cosmological constant. <\/em><\/p>\n I am not aware of any such nice interpretation of the Einstein tensor.<\/p>\n Another interesting point is that one gets at the vacuum equations very quickly with this “dual form”.\u00a0 Just “turn off” T.<\/p>\n The real question here is “does this duality have a deeper meaning?”. This I really do not know.\u00a0 It would also be interesting to understand if any technical issues can be addressed via this “duality” and how this really helps us understand gravity.<\/p>\n My literature hunts needs to continue…<\/p>\n References<\/strong><\/p>\n [1] John C. Baez & Emory F. Bunn. The Meaning of Einstein’s Equation.\u00a0Amer. Jour. Phys.<\/em>\u00a073<\/strong> (2005), 644-652. Also available as \u00a0arXiv:gr-qc\/0103044.<\/p>\n <\/p>\n Update<\/strong><\/p>\n The alternative form of the field equations is also presented in Wolfgang Rindler’s Essential Relativity, revised second edition, 1977.\u00a0 So the idea is old and I am sure to be found in other books.<\/p>\n","protected":false},"excerpt":{"rendered":" I don’t claim this to be new, the earliest reference I can find is Baez and Bunn [1], though I am sure the idea is older than that.\u00a0\u00a0 The claim is that there is a kind of duality in the Einstein field equations between the Ricci tensor and the energy-momentum tensor. That is one can … Continue reading A duality between the Ricci and energy-momentum tensors<\/span>