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Recently, I came across something I found quite interesting on Wikipedia, which is the addition of the cosmological constant to Newtonian gravity. The Wikipedia page (Alternatives to General Relativity) writes the modified Newton-Poisson equation as follows: $$ \nabla^2 \phi + \frac{1}{2}\Lambda c^2 = 4 \pi\rho G . $$

It does not provide relevant sources for this, which has left me scratching my head. So I am wondering why we add the $\frac{1}{2}\Lambda c^2 $ term instead of, say, just adding $\Lambda$. Is this some sort of limit of general relativity, or that this form is perhaps easier to deal with in some way?

P.S. I have at best limited knowledge of general relativity, and perhaps this is something obvious from the GR viewpoint which I have overlooked.

Jonathan Huang
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Why ${1 \over2}$ and why ${c^2}$: the short answer is, so that you don't scratch your head in despair with splintered glass, when it comes to assigning actual numerical values in actual physical units to $G$ and $\Lambda$ (and yes, to $\pi$ as well).

The long answer is, have a look a bit further down, the same § of the WP article: they chute the GR field eq. as $T^{\mu\nu} = {1 \over {8 \pi G}} (R^{\mu\nu} - {1 \over 2} g^{\mu\nu} R + g^{\mu\nu} \Lambda) $. Which translates, in covariant coordinates, to ${4 \pi G} T_{\mu\nu} = {1 \over 2} R_{\mu\nu} - {1 \over 4} R + {1 \over 2} \Lambda $.
Rings any bell? Well, it says to me: $T_{00}$ reduces to $\rho $, $g_{00}$ to $\phi$ and ${1 \over 2} R_{00} - {1 \over 4} R$ to $ \nabla^2 \phi$ in the Newtonian approximation. That's because, in the GR world, time measures in meters (at a wee bit under $ 1.8 \times 10^{10}$ m per minute); among normal people, $T_{00}\rightarrow {\rho \over {c^2}}$ and ${1 \over 2} R_{00} - {1 \over 4} R \rightarrow {1 \over {c^2}} \nabla^2 \phi$.

That must be why the factor ${1 \over2}$ and why the factor ${c^2}$: so that $G$ and $\Lambda$ denote the two exact same things in two different theories, just like $4$ and $\pi$ do.

François Jurain
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