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

Observations of galaxies commonly show ‘jets’ and other ejection phenomenon like these ‘Fermi bubbles’ The Fermi Bubbles are two enormous orbs of gas and cosmic rays that tower over the Milky Way, covering a region roughly as large as the galaxy itself. These giant space bubbles may be fueled by a strong outflow of matter from the center of the Milky Way.

Source: Something Strange Is Happening in the Fermi Bubbles (Space.com)

Even the Big Bang is thought by some to be a ‘Big Bounce’. To explain these things, it seems desirable to have the feature, in a theory of gravity, that the strength of gravity reduces for dense regions of matter.

It would also help avoid infinity problems, such as those predicted by General Relativity in the singularity of a black hole.

But does General Relativity allow such a feature? Can it be interpreted or amended to allow it?

Apparently, mass and energy increase the curvature of space-time (gravity) and so does pressure, so that, for example, at the center of a galaxy, the pressure would add to the strength of gravity and not reduce it.

But using a Newtonian approach, a reduction is predicted and here is the work done so far.

I’m wondering whether General Relativity can be interpreted in such a way as to reach a similar conclusion.

For a universe near critical density, for each mass $$m$$

$$mc^2-\frac{GMm}{R^2} = 0$$

or rearranging to make $$G$$ the subject

$$G=\frac{Rc^2}{M}$$

Small numerical constants omitted, $$M$$ is the mass of the visible universe within radius $$R=\frac{c}{H}$$ and $$H$$ is Hubbles constant.

Why it’s true is similar to the ‘flatness problem’.

If we just accept it, and accept it’s true for any mass, then for a larger mass, or one with a small radius $$r$$, and we include the ‘self gravitational energy’

$$mc^2-\frac{GMm}{R^2} - \frac{Gm^2}{r} = 0$$

it rearranges to

$$G_{effective}=\frac{c^2}{c^2/G + m/r}$$

and a reduction of $$G$$ is predicted for dense matter.

Is there any way that General Relativity can be interpreted to allow a reduction, not an increase, of the strength of gravity for dense regions of matter?

• en.wikipedia.org/wiki/Brans%E2%80%93Dicke_theory Mar 25, 2021 at 11:52
• That's interesting, it allows a variation in G with place and time, do you know if it predicts a reduction in G for dense regions, such as the center of a galaxy? Mar 25, 2021 at 12:07
• @Nihar Karve also is it true that Brans-Dicke theory has apparently been ruled out due to Lunar Laser Ranging showing no reduction of G with time? I thought to combine a reducing G theory with physics.stackexchange.com/q/620794 - in that approach Brans-Dicke isn't ruled out by Lunar Laser Ranging and maybe a cosmology without the drawbacks of concordance cosmology can be found. Mar 25, 2021 at 12:18
• I don't understand what the Fermi Bubbles have to do with the question. Mar 25, 2021 at 14:44
• @JohnHunter pure Brans-Dicke only allows variations in the (dynamical) inverse coupling constant on the order $O(1/\omega)$, and $\omega$ is experimentally greater than $40\,000$, so you probably won't see a huge change in the gravitational coupling (though it is entirely possible that the "long-range" nature/propagation of the $\phi$ field allows such a reduction - but this would require some additional analysis) Mar 29, 2021 at 9:47

Any theory that extends general relativity must reduce to GR in some limit, and must (of course) agree with experiment. Probably the simplest such extension is Brans-Dicke gravity, which corresponds to replacing the gravitational constant by a scalar field on spacetime: $$\frac1G\to\phi(x)$$. Leaving $$\phi$$ as non-dynamical leaves too many degrees of freedom that we need to put in by hand (the values of $$\phi$$ at every point in spacetime) - while this is a well-defined theory, it is not very predictive.

We remedy this by giving a kinetic term for $$\phi$$ in the Lagrangian, which when appropriately dimensionalised, has a constant prefactor $$\omega$$. This field now couples to the Ricci scalar in the Lagrangian, and so has an equation of motion which is determined by the geometry and hence the energy-momentum content. Note that the only free parameter in the theory is $$\omega$$, which must be constrained by experimental bounds. Current observational evidence suggests $$\omega>40\,000$$ - for comparison, general relativity is heuristically reobtained in the $$\omega\to\infty$$ limit.

In the new "Einstein equations" relating $$R_{\mu\nu}$$ and $$T_{\mu\nu}$$, you can interpret $$\phi$$ and its derivatives either as constituting a "generalised mass" or a "generalised spacetime curvature" - either way, in the Newtonian limit, it is manifest that this is equivalent to a changing gravitational coupling (even though the Einstein equations themselves have become rather more complicated). This means that regions where $$\phi$$ is large have a low gravitational strength and vice versa.

Here's the equation of motion for $$\phi(x)$$:

$$\square\phi=\frac{8\pi}{2\omega+3}T_\alpha^\alpha$$

which is a wave equation for massless $$\phi$$, sourced by the trace of the energy-momentum tensor. This means that $$\phi$$-waves propagate at the speed of light away from regions where $$T_\alpha^\alpha>0$$.

With the reasonable additional assumption that $$\phi$$ approaches a constant value $$\tilde\phi\sim\frac1G$$ asymptotically, we can expand around this value in powers of $$\frac{1}{\omega}$$. Provided that the trace of the energy-momentum tensor does not vanish identically, this means that $$\phi\sim\tilde\phi+O(1/\omega)$$ and

$$R_{\mu\nu}-\frac12 Rg_{\mu\nu}=\frac{8\pi}{\phi}T_{\mu\nu}+O(1/\omega)$$

If you imagine a single $$\phi$$-wave propagating through space, regions where $$\phi(x)$$ has a crest have below-average gravity while regions in a trough have above-average gravity.

Since, as mentioned, $$\omega>40\,000$$, we would expect only very small variations in the gravitational strength. However, regions dominated by electromagnetic energy-momentum have $$T_\alpha^\alpha=0$$ and so allow $$\phi$$-waves to travel through them unimpeded, as in a vacuum (since the equation of motion for $$\phi$$ reduces to the free wave equation) - this is what is alluded to in their "long-range nature". So due to their long-range nature and luminal travel, it is possible that many such waves, generated from various sources and different spacetime locations superpose to reduce $$G$$ in a region (bearing in mind that the paths of these fluctuations are themselves affected by the spacetime geometry, and so lens naturally into a dense region).

This certainly isn't going to resolve the singularity problem (indeed, if any theory that agrees with experiment could, you would see a much larger uptake of it), but it does provide a mechanism to reduce the gravitational strength in regions of high density. Pertinent to your cosmological case is the possibility that $$\omega$$ itself is not a constant, and has stabilised at a high value over time - thus allowing $$\phi$$-waves of greater magnitude, at that epoch creating regions of highly varying gravitational strength.

• Thankyou, wikipedia says in BD theory that there is a scalar field, $\phi$ , which has the physical effect of changing the effective gravitational constant from place to place. Do you know how $\phi$ gets the different values, i.e. what makes it vary from place to place? en.wikipedia.org/wiki/Brans%E2%80%93Dicke_theory Mar 29, 2021 at 18:28
• Also,Faraoni ( arxiv.org/abs/gr-qc/9902083 ) has argued that BR might not tend to GR for traceless matter, when $\omega$ tend to infinity, and apparently the electromagnetic stress tensor is trace free, thus for a region dominated by electro-magnetic radiation is it possible that BD can provide the reduction of gravity described? Mar 29, 2021 at 19:52
• @JohnHunter $\phi$ evolves according to the equation in the answer - there's no way to put in the values by hand. The $\omega\to\infty$ limit is a bit of a red herring since its not really relevant to your question, but the point about an EM-dominated region is a good one: the $\phi$-waves here behave essentially as they would in a vacuum, allowing fluctuations in gravity to become long-range Mar 30, 2021 at 7:50
• Thankyou for the answer and comments, you should have the bonus as it's the best answer. If it isn't ticked yet, that's because I was hoping for a definite reason why the gravity is likely to reduce at galactic centres. Personally it seems that the effect should happen and would be good for understanding ejection phenomenon and improving our cosmological models. Happy Easter Apr 3, 2021 at 12:33
• @JohnHunter Happy Easter! Apr 3, 2021 at 12:36