I've recently been looking at the way in which pressure terms contribute to the Komar expression for the total energy in GR and I've been trying to understand the Newtonian equivalent.(adsbygoogle = window.adsbygoogle || []).push({});

I found something which surprised me, and I'm wondering if it's well-known:

If you take a couple of masses and take an arbitrary plane between them perpendicular to the line between them, then in Newtonian theory there should be some sort of "pull" going on distributed across that plane in space, consisting of a negative pressure which when integrated over the whole area gives the force between the masses.

It turns out that there's a trivial model which gives the right result. Basically, it's the difference between the square of the total field and the squares of the separate fields of the two masses, but expressed as a separate product for each component (as in the pressure terms in the diagonal of the stress-energy tensor) rather than just the sum of those three terms.

That is, if the gravitational field from the first mass on its own would beg_{1}and that from the second would beg_{2}, and the total field isg, we have the following expression for the pressure in each direction:

T_{ii}= (1/4 \pi G) ( g^{2}- g_{1}^{2}- g_{2}^{2}) = 1/(4 \pi G) ( 2 g_{1}.g_{2})

Geometrically, what this says is that you take the component of the field on each side of the plane in a direction perpendicular to the plane (using the same direction for both components) and multiply them together (and then multiply by 2).

Is this a known result?

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# Tension in space in Newtonian gravity

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