In our universe, due to reasons usually attributed to the 3D geometry of it, the forces of nature seem to mostly be $\propto \frac{1}{r^2}$, inversely proportional to the distance of the source of the force squared.
However, what would it look like if we lived in a universe where common forces, such as the gravitational force, or the electrostatic force, were instead $\propto \frac{1}{r} -$ simply inversely proportional to the distance from the source of the force?
For example, if the gravitational force obeyed an inverse law ($f_\text{g} \propto \frac{1}{r}$) rather than an inverse square law ($f_\text{g} \propto \frac{1}{r^2} ,$) the amount of work needed to separate two masses $m$ and $M$ to distances infinitely far away from one another would be infinite. $$ \int_r^\infty \frac{GMm}{r} ~=~ \left. GMm \ln(r) \right|_r^\infty ~=~ \infty $$
Similarly, if the electrostatic force obeyed an inverse law instead of an inverse square law, the work a negative charge $Q$ would do on a positive charge $q$ as it attracted it from infinitely far away would be infinite.
We could no longer define “infinitely far away” as a place of zero potential energy, because as $Q$ and $q$ approach being infinitely apart, the work they would do on each other as they were brought closer to a distance $r$ from one another would be infinite.
What would be the physical consequences of this, of living in a world with an inverse law instead of an inverse square law?
I can think of one right away: there would be no such thing as an escape velocity. No matter how great the velocity with which you launch a rocket from a planet’s surface into empty (let’s assume space is empty) space, the gravitational force of the planet would eventually cause the rocket to fall back towards the planet.
What other consequences would there be? What would such a universe look like? Could such a universe even exist, in 2 OR 3 dimensions?
