As part of research trying to develop a cosmological model that can be static (scale-factor), but in dynamic equilibrium - a way is being considered whereby dense collapsing regions of matter 'bounce' or form jets such as those in Active Galactic Nuclei.

The background thinking and a few relevant links are at the bottom, but basically, the reasoning is this. The singularity in a black hole seems unphysical, a state of infinite density - and something should happen to prevent it from forming. An example of an infinite quantity hinting at different physics is the ultraviolet catastrophe.

Black holes are said to form due to the pressure, resisting the collapse, adding to the gravity. A kind of positive feedback for positive pressure and gravity. High gravity requires high pressure to resist the collapse, but the pressure adds to the gravity attracting the matter more strongly requiring higher pressure etc...

The density of active gravitational mass according to General Relativity is

$$\rho +\frac{3p}{c^2}\tag 1$$

and this expression shows that positive pressure adds to the active gravitational mass.

The question is: Can there be a situation whereby negative pressure causes a decrease in gravity, that allows matter to escape from a dense region, causing more negative pressure etc... this time positive feedback for negative pressure?

Are there any models like that, presumably for galactic nuclei, that have already been developed?

enter image description here

In diagram 1. A dense region of matter (blue circles), is resisting collapse due to pressure (green arrows). The pressure is positive at the outside circle but at the middle circle, since the pressure is acting inwards and outwards, some inward pressure arrows have been put on.

Diagram 2. The region of matter inside the inner circle of diagram 1. has now escaped, due to the high pressure and possibly some asymmetry or disturbance such as might occur if a star collided with a galactic nucleus. The red arrows show the direction of the escaping matter.

Now we are left with the green arrows on diagram 2 i.e. negative pressure and less outward positive pressure at that circle. According to formula 1) the active gravitational mass of matter within that circle is reduced, but there is still the high pressure that would speed up the matter, which would then be ejected also in the direction of the red arrows.

So in this way could a negative pressure - positive feedback cycle occur in a galactic nucleus? When enough matter has been ejected, the pressure would become negligible and the situation would then revert to the usual situation, where positive pressure is resisting the gravitational collapse.

Links to other work/questions I have done on this:

  1. Does General Relativity allow a reduction in the strength of gravity?

  2. Cosmology - an expansion of all length scales

  3. John Hunter, A New Solution of the Friedman Equations, https://vixra.org/abs/2006.0209

  • $\begingroup$ This doesn't answer the specific question you asked, but if the motive is finding a mechanism that prevents singularities, then why restrict the scope to clasical physics? Classical general relativity is an approximation, and the singularities it predicts are expected to be mere artifacts of the approximation. Even in a semiclassical model, where gravity is still treated classically and everything else is quantum, quantum effects can already change the picture substantially (see arXiv:1912.06047). $\endgroup$ Jun 19, 2021 at 13:33
  • $\begingroup$ Yes, there seems to be a problem with singularities - true, but also the motivation is to complete a cosmological model that is static (in scale-factor) but with a redshift. That's link 2) at the bottom of the question - it predicts a matter density between 0.25 and 0.33. Being apparently static, but with a redshift, it's also required that collapsing matter 'bounce' and this may include the Big Bang itself. Early attempts were made in link 1). This question is to do with finding a way in which a static universe (with a redshift) can be in dynamic equilibrium. $\endgroup$ Jun 19, 2021 at 14:39

1 Answer 1


There seems to be confusion about pressure here. The diagrams portray pressure as a vector, but it is a component of a second-order tensor -- or in the case of isotropic fluids, simply a scalar, as you are denoting $p$. There is no distinction between "outward" and "inward" pressure; they are the same thing.

Moreover, all familiar forms of matter, including massive particles as well as massless ones (radiation), have nonnegative pressure. (The main exception is "dark energy".)

The removal of matter from a region of space, leaving vacuum, results in zero pressure there, not negative pressure. Likewise, matter that is adjacent to vacuum develops lower pressure as a result, but not negative pressure.

To be clear: Vacuum doesn't suck (well, except perhaps a tiny amount due to cosmological dark energy). The intuition that it does is based on Earth conditions where the baseline is ambient atmospheric pressure. But the $p$ in the gravitational formulas is absolute pressure.

  • $\begingroup$ Would the pressure outside the 'missing region' mean that matter expanded inwards to fill it and can that be thought of as negative pressure - in the sense that expansion does work and so the energy of the region between the middle and outer circle of diagram 1 is reduced, and it's gravitational mass reduced? $\endgroup$ Jun 25, 2021 at 13:48

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