I Weak Gravitational Field: Solving Einstein Field Eqs

[deRoy]

What do I have to do if I want the EFE's to approximate a weak gravitational field, where for example, an inversely proportional to the cube ( $1 / r^3$ ) of the distance law between the masses applies?

 

[Orodruin]

What should go as $1/r^3$? This is not really a question of GR, it is a question of the solutions to the Laplace equation, which depends on the number of dimensions. For example, to get a potential that goes as $\sim 1/r^3$ you would need to add two additional spatial dimensions.

 

[deRoy]

What should go as $1/r^3$? This is not really a question of GR, it is a question of the solutions to the Laplace equation, which depends on the number of dimensions. For example, to get a potential that goes as $\sim 1/r^3$ you would need to add two additional spatial dimensions.

Yes, that's exactly what I mean, I want to begin with a non-Newtonian law of universal gravitation let's say: $F = GMm/r^3$ then find a curved spacetime (5 + 1 dimensions) to approximate this law in weak gravitational fields.

Can I do the same for 3+1 dimensions?

 

[Orodruin]

In 5+1 dimensions the potential goes as $1/r^3$. In 3+1 dimensions the potential goes as $1/r$.