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.Orodruin said: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.
A weak gravitational field is a region in space where the effects of gravity are relatively small compared to other forces. This is typically seen in situations where the masses involved are small and/or the distances between them are large.
The Einstein field equations are a set of equations developed by Albert Einstein as part of his theory of general relativity. They describe the relationship between the curvature of space-time and the distribution of matter and energy within it.
The Einstein field equations can be solved for a weak gravitational field by using the linearized version of the equations. This involves making certain approximations and simplifications to the equations to make them easier to solve.
Solving the Einstein field equations for a weak gravitational field has many practical applications, such as predicting the behavior of objects in orbit around a planet or star, understanding the gravitational lensing effect, and developing accurate models of the universe.
Yes, there are limitations to using the linearized version of the Einstein field equations. This version only applies to weak gravitational fields and cannot accurately describe extreme gravitational fields, such as those near black holes.