Tension in space in Newtonian gravity

[Jonathan Scott]

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.

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 be g₁ and that from the second would be g₂, and the total field is g, we have the following expression for the pressure in each direction:

T_ii = (1/4 \pi G) ( g² - g₁² - g₂² ) = 1/(4 \pi G) ( 2 g₁.g₂ )

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?

 

[Jonathan Scott]

Hmmm. Obviously not THAT well known.

I think it must be closely related to the way in which electromagnetic energy is assumed to be distributed in space, in that the "gravitational tension" expression above is very close to being the gravitational equivalent of the electrostatic energy density:

$$\frac{1}{2} \varepsilon_0 E^2$$

A few more points about this relationship:

1. The factor of 1/2 for the electromagnetic case means that the integral of this tension expression over a plane is only half of the total force in that case. (In electromagnetic theory, this factor is required to avoid counting the potential energy of each pair of charges twice). It seems odd that it should be a half of the force.

2. The sign of this "pressure" behaves in an odd way, in that between two like charges it behaves in the same way as between two gravitational masses, showing a negative change in the pressure, and between two unlike charges it gives a positive change in pressure. (The term for the square of the field of each particle is also positive, even though an electric charge would repel itself but a gravitational source mass attracts itself).

3. Mathematically, there is a standard identity which says that the electrostatic potential energy is equal and opposite to the integral of the field energy in this form over a sufficiently large volume V of space:

$$\frac{\varepsilon_0}{2} \int_V \phi \, \nabla^2 \phi + (\nabla \phi)^2 \, dV \\ = \frac{\varepsilon_0}{2} \int_V \nabla (\phi \, \nabla \phi) \, dV \\ = \frac{\varepsilon_0}{2} \int_S \phi \, \mathbf{n}.(\nabla \phi) \, dS$$

where the final expression becomes zero when the surface S of the volume is sufficiently far from sources so $\phi$ tends to zero.

However, this doesn't seem very useful when extended to gravity (with or without the factor of 1/2), because in that case it asserts that the total energy in a volume is unchanged by gravitational interactions. (There's a sense in which this is true, in that gravitational interactions convert energy between potential and kinetic, and it is only when some of the kinetic energy is removed from the system that the total energy changes, but it's not clear how this helps).

I'm interested in this area because I'm trying to understand whether it is possible to find a way to extend the GR Komar energy pressure term to provide a representation of the total energy in simple dynamic but stable situations such as masses going round one another in a circular orbit.

 

[Jonathan Scott]

While I'm continuing to talk to myself, I'll just mention that I've now found a very interesting paper which is directly relevant to this area:

arXiv:gr-qc/9605011
CONSISTENTLY IMPLEMENTING THE FIELDS SELF-ENERGY IN NEWTONIAN GRAVITY by Domenico Giulini

 

[Naty1]

I recently tried to read a little on Komar mass...Wikipedia has a description but as usual they launch into rather heavy duty mathematics...it might be useful to you however...

 

[Jonathan Scott]

I recently tried to read a little on Komar mass...Wikipedia has a description but as usual they launch into rather heavy duty mathematics...it might be useful to you however...

Thanks, but I already know about that; Pervect, a previous mentor at PF, pointed me to that write-up a couple of years ago (and I think Pervect may have actually written it).