# Gravitational… confinement?

This is a followup to Ergil's question "Weak isospin confinement?".

According to the Wikipedia article on color confinement:

The current theory is that confinement is due to the force-carrying gluons having color charge [...],

i.e. because the gauge group is non-abelian. But that is equally true for the weak force.

It would also appear to be true for gravity. Gravitons should have energy and momemtum, which is "charge" under the gravitational interaction. (though I have no idea whether the gauge group is non-abelian)

The stress-energy tensor is different from other charges because it is not frame-independent, but maybe it still counts.

In this answer, Johannes gives a formula for the potential for the strong force

$$V(r) = - \frac{4}{3} \frac{\alpha_s(r) \hbar c}{r} + kr$$

It's empirically derived, but the interesting feature is that at short distances the field strength ($$\frac{dV}{dr}$$) is dominated by an inverse-square term, and at longer distances a constant term. (I'm glossing over the unspecified $$\alpha_s(r)$$ function)

Contrary to the question title, I'm not suggesting that the gravitational force would become constant at long distances, maybe just fall off more slowly than it does at shorter distances... similar to Modified Newtonian dynamics.

There would be a large ratio between the "inverse-square cut-off distances" for the strong force and gravity, but then there is a large ratio between their strengths as well.

Is this plausible? Has anyone with the necessary mathematical tools looked into it?

• Gravity is not exactly a gauge theory (though similar to one). And even in ordinary non-Abelian gauge theories, we understand confinement only through toy models and simulations. – ACuriousMind Dec 24 '14 at 1:30
• One can always suggest these kinds of modifications to force laws and experimental data can, at best, set an upper limit on the strength of the deformation on the current scale. From a theoretical perspective it may be satisfying to speculate this way, but it has very little predictive power. – CuriousOne Dec 24 '14 at 1:48
• Confinement does not occur just because the gauge group is non-abelian. Even a standard non-abelian gauge theory can be non-confining if there are many charged fields. For instance, if I only have fermions, the theory is non-confining if $N_f > \frac{11}{2} N_c$ where $N_f$ is the number of fermions and $N_c$ is the rank of the gauge group. – Prahar Mitra Jul 13 '20 at 19:56

## 3 Answers

This is definitely not the case with gravity. It has been shown after many years of observation and evidence gathering, and through the study of galaxies, solar systems, moon systems etc, that the gravitational force obeys the inverse square law at arbitrarily large distances. This is to the best of our knowledge how gravity operates.

• Well, the rotational velocity curves for galaxies based on all detectable matter demonstrably to NOT agree with GR predictions, which is the major driver for the postulation of dark matter. It'd be more accurate to say "We have postulated 25% of matter/energy in the universe as undetectable for reasons as yet unknown in order to bring observations into agreement with General Relativity. – JPattarini Jun 15 '17 at 19:20

There are a number of papers suggesting that "gravitational confinement" can explain MOND phenomenology / replace dark matter. Here is a reference (Implications of Graviton-Graviton Interaction to Dark Matter): https://arxiv.org/abs/0901.4005

The suggestion is motivated by similarities between QCD and gravity lagrangians, and the numerical calculation makes use of lattice QCD methods. While the calculation is very approximate, the result does provide a good fit for galaxy rotation curves (a few examples are shown in the paper). In the QCD analogy, the Tully Fisher relation (the relation between the baryonic galaxy mass and its rotation speed) corresponds to Regge trajectories (relation between the hadron mass and its angular momentum).

Confinement can be roughly interpreted as you need infinite amount of energy to pull things apart. For example, in QCD there is a proverbial 'string' (BTW, this is where superstring is originally coming from) attached to quarks, so that the string related potential energy is $$kr,$$ which tends in infinity if you try to separate quarks by forcing $$r \rightarrow \infty$$.

As for gravity, the Newtonian potential is of course not confining, since $$-\frac{G}{r}$$ tends to zero when $$r \rightarrow \infty$$.

An interesting case is MOND on the galactic level. The MOND potential is sort of (in the deep MOND regime) $$ln(r/r_0),$$ which is marginally confining, because the energy required is only logarithmically divergent.

If you look at the whole universe, the cosmological constant will contribute an energy term like (only in the sense of Newtonian interpretation of the Friedman equation, since energy conservation is a moot point in GR), $$-\Lambda a^2,$$ where $$a$$ of the FLRW metric is the scale parameter of the universe. Therefore, if the cosmological constant $$\Lambda$$ is negative, the universe can be figuratively regarded as in a confined phase since infinite energy will be required for $$a\rightarrow \infty$$. With a negative cosmological constant, the universe will transition from expansion to contraction at some maximal $$a$$ and eventually will collapse on to itself (big crunch).