There might be a relation to the Cosmic Influx Theory (CIT). According to CIT, gravitational acceleration is not caused by the curvature of spacetime, as in general relativity, but by an influx of energy or particles (possibly "neutrino-like" particles) streaming into massive objects. This influx can be seen as analogous to the flow of heat or matter in other physical systems.

In this context, the energy gradient that drives gravitational acceleration can be compared to how gradients in temperature or concentration drive flows in Fourier's Law of heat conduction or Fick's Law of diffusion, respectively. CIT proposes that the flow of energy or matter (the influx) causes a gravitational potential gradient, leading to the observed effects of gravity, such as objects being "pulled" towards massive bodies. Thus, gravitational acceleration in CIT is a result of an external energy influx, similar to how heat or matter moves due to their respective gradients.

In summary, CIT does allow the addition of a gravity flow to these examples, suggesting that gravity itself is driven by an influx flow, akin to how energy, heat, or matter follows gradients in other systems.

If you want to read more about the Cosmic Influx Theory, for a start please read: Introducing the Cosmic Influx Theory (CIT) https://zenodo.org/records/12770177

There is one simple example that can throw light on this matter. An object rotating round earth at a constant velocity and altitude is said to be under centripetal acceleration given by v^2/r per unit mass, where v is velocity and r is its distance from the centre. This acceleration can also be considered as a continuous free fall towards earth and as such the circular path is a geodesic by definition according to GR. We can also view the case as that of a constant energy v^2 placed at a fixed distance r from the centre. In this case the gradient of energy is the energy difference divided by distance r. Since energy at the centre is zero, the negative gradient of the energy is -v^2/r which is acceleration opposite to the outward direction of r – that is towards the centre, which is the centripetal force as we determined earlier.

So, it appears then that finding the free fall path aimed at in GR is the same as finding the path of equal energy distribution that would lead to zero acceleration. More usefully, the bending of space time to produce zero acceleration motion amounts to finding equal energy density path that allows a mass to move at a constant speed.

See also this link:

“ A simple geodesic equation for gravity, electromagnetism and all sources of energy” Advanced Studies in Theoretical Physics, Vol. 18, 2024, no. 4, 195-202 doi: 10.12988/astp.2024.92137

https://www.m-hikari.com/astp/astp2024/astp1-4-2024/92137.html