﻿What does it mean for “spacetime” to be flat or curved? It means that the riemannian tensor field is not vanishing.


Newtonian spacetime is curved. What does that mean?


In the traditional formulation of Newtonian space, it's flat. But in this case we desire to reformulate in a way that the force of gravity is manifested as curvature.


This is not general relativity, only Newtonian spacetime.


Basically, gravity should not be a force. It should rather be encoded in the curvature of spacetime. That’s how the argument will go. Although all this ends up being a choice computationally apparently.


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Separate: what are Newton's three laws:
1. Inertia. A body without a force will continue along it’s path.
2. F=ma
3. Every action has an equal and opposite reaction


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Ok back to the formulation of Newtonian spacetime with gravity manifesting itself as curvature.


So Newton’s first two laws, but expressed in more familiar language, say that:
1. A body without a force traverses in a uniformly straight line
2. Deviations off this straight line are caused by a force, reduced by a reciprocal of the body’s mass.


Now looking at these formulations, you realize that 1) is a specific case of 2) and so looks redundant. But this is in fact because we are reading 1) wrongly. We should read the first law as if it’s saying a statement about the underlying geometry. Given some particles without a force on them, measure what path they take, and that path is officially the straightest line. So the first law is a measurement prescription for geometry.


Now another thing you notice about 1) is that in the standard formulation, unless you live in a universe with only one particle, then the law is useless nominally because gravity exists on every pair of particles, regardless of how far away they are. But thankfully if we encode gravity in terms of curvature, then we don’t need to worry about this.


So the reason 1) is not a redundancy of 2) is because 1) says what a straight line is, and 2) explains the deviation from that straight line caused by a force, reduced by a reciprocal of the body’s mass.
Laplace was the first to ask the question of whether it’s possible or not to encode gravity into the curvature of space.


The answer for laplace is that if you don’t include time, it’s a resounding no.


Why is it a resounding no? Well, let's try it out.