It should be clear that General Relativity describes gravitation in terms of curvature of spacetime and reduces to Special Theory of Relativity for Local Inertial Frame (LIF). However, it is important to explicitly check that the description reduces to the Newtonian treatment when we select the correct boundary conditions.
These conditions, referred to as the Newtonian limit, are applicable to physical systems exhibiting:
- objects moving relatively slowly (compared to the speed of light).
- weak gravitional field.
- gravitational field does not change with time, ie it is static.
Mathematically, this leads to the following approximations:
- v <<c
- Without gravitation, spacetime possesses the Minkowski metric ημν. Weak gravitational fields only cause small curvatures of spacetime. Therefore, we assume that coordinates exist, such that the metric takes the following form gμν = ημν + hμν with | hμν|<<1
- the time-component (ie the 0th component of the particle's vector) dominates the other (spatial) components. For i=1,2,3 we have dxi<<dt
So far, we have used the Newtonian limit approximation in two use cases:
- for retrieving the Newtonian gravitational law of motion a = - ∇Φ (acceleration = - gradient of gravitational field) from the geodesic equation - see Geodesic equation in the Newtonian Limit
- for deducing the exact value of the proportional constant K=-8πG/c4 in the Einstein's field equation from the equivalent Poisson's differential equation for the potential in terms of the matter density ρ and Newton's gravitationnal constant G ∇2Φ = 4πGρ - see Einstein Field Equations for the full demonstration.
Newton gravity as the curvature of the time
Curvature in time is nothing more than the gravitational redshift: time advances at different rates in different places, so time is curved.
And gravitational redshift is enough to ensure that free-falling bodies follow their Newtonian trajectories.
All of Newtonian gravitation is simply the curvature of time.
Now it is time to raise an interesting question: what could be a curved spacetime picture for Newtonian gravitation? Put it in other words, how do we reconcile the parabolic movement of projectiles as observed on Earth (weak gravitational field) with Einstein's spacetime curvature vision?
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