Newton--Cartan gravity

Intuition

See first axiomatic Newtonian mechanics#Schuller approach.

In this approach, gravity is no longer a force, but something to be encoded in the geometry (linear connection) of spacetime.
For motion in space, we had the particle's motion given by x:RR3. We need to convert this into the particle's worldline, which we get from the map X:RR4 given by

X(t)=(t,x1(t),x2(t),x3(t)):=(X0(t),X1(t),X2(t),X3(t)).

Assume that

(1)x¨α(t)=gα(x(t))

for α=1,2,3, where g is given by Newton's gravitational law. This is the equation of motion for a particle in the gravitational potential ϕ above.

The question is: can this equation of motion be the geodesic equations of the manifold R4 endowed with a particular linear connection? Or in other word, can spacetime have a "shape" such that it makes that gravity appears as the corresponding "straight lines" of this shape?

Observe that

X˙0(t)=1,X¨0(t)=0.

We can rewrite (1) as

X¨α(t)gα(X(t))=0,

for α=1,2,3. Now, we can multiply by X˙0(t) because it is equal to 1, and so we have:

X¨α(t)gα(X(t))X˙0(t)X˙0(t)=0.

Now combining this with the X¨0(t)=0 equation, we see that we have an geodesic equations:

X¨a+ΓabcX˙bX˙c=0,

where a,b,c=0,1,2,3, with the choice of the Γs:

Γα00=gα,α=1,2,3.

Now, this could just be a coordinate-choice artifact, and so could be transformed away. It turns out that this is not the case, and you can show it by calculating the Riemann curvature tensor components. We have non-vanishing ones:

Rα0β0=βgα.

As this is a tensor, if it is non-vanishing in one chart, it must be non-vanishing in all charts.
Remark:
Given the Riemann tensor at the end of the proof above, we can actually work out the Ricci curvature tensor by setting α=β:

Ric00=αgα.

which, using the Poisson equation, gives:

Ric00=4πGρ.

This is actually one of the so-called Einstein equations:

Ric00=8πGT00,

where T00=ρ/2. T is known as the energy-momentum tensor, which describes the energy and momentum distribution in a region of spacetime.
The interesting thing of this equation is that links the distribution of "matter" with the "curvature" of spacetime, even in Galilean spacetime.
Remark:
Note that the fact that the only non-vanishing Γs have the lower indices both as "time" (i.e., they are 0) tells us that the curvature is taking place in spacetime, not just in space. That is, the Riemann tensor vanishes for all spatial indices Rαβγδ=0 for all α,β,γ,δ=1,2,3.

Tidal Forces

The result above, about not being able to transform away the curvature result, is known as tidal forces. The basic idea is that you can only transform away gravitational fields locally. In other words, the only way you can transform away a gravitational field globally is if it is uniform.

To see why this is the case, imagine being inside a box in space with two balls. Now imagine the box is in a gravitational field, and so is in free fall towards some massive object. We shall ignore the gravitational fields generated by our body and by the balls themselves. If the gravitational field is uniform across the box, everything experiences the same pull and so falls exactly the same. That is, if we put the balls out at our sides, they would appear to just float there, and if there were no windows on our box to see things moving past us, we wouldn't even know we were in a gravitational field. Obviously, someone stationary (w.r.t. the massive object) outside the box would see the balls moving down and would say they are in a gravitational field.

What is going on here is that we have transformed ourselves to a frame of reference (which for this remark is just a chart) which falls with the balls and so we have "removed" the effects of gravity via such a change of chart.

\begin{document} 
\begin{tikzpicture}[scale=1.4][domain=0:4] 
\draw[thick] (0,0) -- (4,0) -- (4,3) -- (0,3) -- (0,0); \draw[thick] (1,1.5) circle [radius=0.3cm]; \draw[thick] (3,1.5) circle [radius=0.3cm]; \draw[ultra thick, ->, blue] (0.5,3.5) -- (0.5,-0.5); \draw[ultra thick, ->, blue] (1,3.5) -- (1,-0.5); \draw[ultra thick, ->, blue] (1.5,3.5) -- (1.5,-0.5); \draw[ultra thick, ->, blue] (2,3.5) -- (2,-0.5); \draw[ultra thick, ->, blue] (2.5,3.5) -- (2.5,-0.5); \draw[ultra thick, ->, blue] (3,3.5) -- (3,-0.5); \draw[ultra thick, ->, blue] (3.5,3.5) -- (3.5,-0.5); \draw[thick] (6,0) -- (10,0) -- (10,3) -- (6,3) -- (6,0); \draw[thick] (7,1.5) circle [radius=0.3cm]; \draw[thick] (9,1.5) circle [radius=0.3cm]; \node at (8,-0.5) {\large{Gravitational effect transformed away}};
\end{tikzpicture} 
\end{document}

Now imagine we do the same thing, but the gravitational field is not uniform but comes radially from some spherical object. Again everything still falls at the same rate, but now the ball to our left will be pulled slightly to the right and the ball to our right will be pulled slightly left. To us inside the box, then, the balls slowly move towards each other. This is not an effect that we can remove by going to another frame of reference, and so it represents something physical. This is "real" gravity.

\begin{document} 
\begin{tikzpicture}[scale=1.4][domain=0:4] 
\draw[thick] (0,0) -- (4,0) -- (4,3) -- (0,3) -- (0,0); \draw[thick] (1,1.5) circle [radius=0.3cm]; \draw[thick] (3,1.5) circle [radius=0.3cm]; \draw[ultra thick, ->, blue] (-0.5,3.5) -- (1,-0.25); \draw[ultra thick, ->, blue] (0.5,3.5) -- (1.5,-0.5); \draw[ultra thick, ->, blue] (1.25,3.5) -- (1.75,-0.25); \draw[ultra thick, ->, blue] (2,3.5) -- (2,-0.5); \draw[ultra thick, ->, blue] (2.75,3.5) -- (2.25,-0.25); \draw[ultra thick, ->, blue] (3.5,3.5) -- (2.5,-0.5); \draw[ultra thick, ->, blue] (4.5,3.5) -- (3,-0.25); \draw[thick] (6,0) -- (10,0) -- (10,3) -- (6,3) -- (6,0); \draw[ultra thick, ->, blue] (7,1.5) -- (7.75,1.5); \draw[ultra thick, ->, blue] (9,1.5) -- (8.25,1.5); \draw[thick, fill=white] (7,1.5) circle [radius=0.3cm]; \draw[thick, fill=white] (9,1.5) circle [radius=0.3cm]; \node at (8,-0.5) {\large{Tidal force}};
\end{tikzpicture} 
\end{document}

The inability to remove this effect by a change of chart is what we refer to as a tidal force. From this, we see that when we feel gravity pulling us, it's actually the inhomogeneous nature of the gravity we feel; it pulls our feet harder than it pulls our head and pushes our arms towards each other.

Formally

Definition (Newtonian Spacetime)
A Newtonian spacetime is a quintuple of structures (M,O,A,,t), where (M,O,A) is a 4-dimensional smooth manifold, and t:MR is a smooth function called the absolute time, which satisfies:

  1. (dt)p0 for all pM — there is a concept of absolute space (defined below).
  2. dt=0 everywhere — absolute time flows uniformly.
  3. is torsion-free.
    The connection is determined by a mass distribution ρ by means of the "Einstein equation":
Ric00=8πGρ/2.

Related relativistic spacetime.

Definition (Absolute Space)
Let (M,O,A,,t) be a Newtonian spacetime. Absolute space at time τ is the set:

Sτ:={pM|t(p)=τ}.

It follows that:

M=τSτ.

Note that it is only once we introduce the absolute time function that we can think of splitting spacetime into space and time. Before that, it was just a 4-dimensional manifold.

Definition (Future Directed / Spatial / Past Directed)
A vector XTpM is called:

  1. Future directed if dt(X)>0.
  2. Spatial if dt(X)=0.
  3. Past directed if dt(X)<0.
\begin{document} 
\begin{tikzpicture}[scale=1.4][domain=0:4]         \draw[thick, rotate around={-25:(0,0)}, xscale=1.5, yshift=-1.5cm, xshift=0.5cm, fill = gray!40, opacity = 0.8] (0,0) .. controls (0.8,0.5) and (1.2,0.5) .. (3.5,1) .. controls (4,1.5) and (4,3) .. (4.5,4.5) .. controls (3.2,4) and (3.7,4) .. (1,3.5) .. controls (0.5,3) and (0.5,1.5) .. (0,0);
        \draw[thick, rotate around={-25:(0,0)}, xscale=1.5, yshift=-1.5cm, xshift=0.5cm] (0,0) .. controls (0.8,0.5) and (1.2,0.5) .. (3.5,1) .. controls (4,1.5) and (4,3) .. (4.5,4.5) .. controls (3.2,4) and (3.7,4) .. (1,3.5) .. controls (0.5,3) and (0.5,1.5) .. (0,0);
        \draw[ultra thick, ->, red, rotate around={180:(4,0.5)}] (4,0.5) -- (4,2.5);
        \draw[thick, rotate around={-25:(0,0)}, xscale=1.5, fill = gray!40, opacity = 0.8] (0,0) .. controls (0.8,0.5) and (1.2,0.5) .. (3.5,1) .. controls (4,1.5) and (4,3) .. (4.5,4.5) .. controls (3.2,4) and (3.7,4) .. (1,3.5) .. controls (0.5,3) and (0.5,1.5) .. (0,0);
        \draw[thick, rotate around={-25:(0,0)}, xscale=1.5] (0,0) .. controls (0.8,0.5) and (1.2,0.5) .. (3.5,1) .. controls (4,1.5) and (4,3) .. (4.5,4.5) .. controls (3.2,4) and (3.7,4) .. (1,3.5) .. controls (0.5,3) and (0.5,1.5) .. (0,0);
        \draw[ultra thick, ->, blue, rotate around={10:(4,0.5)}] (4,0.5) -- (4,2.5);
        \draw[ultra thick, ->, green, rotate around={-105:(4,0.5)}] (4,0.5) -- (4,2.5);
        \draw[fill=black] (4,0.5) circle [radius=0.05cm];
        %
        \node at (8.5,1.25) {\large{$S_{\tau_2}$}};
        \node at (8.5,-0.5) {\large{$S_{\tau_1}$}};
        %
        \draw[ultra thick, ->, blue, rotate around={10:(12.5,0.5)}] (12.5,0.5) -- (12.5,2);
        \draw[ultra thick, ->, red, rotate around={-10:(12.5,0.5)}] (12.5,0.5) -- (12.5,-1);
        \draw[ultra thick, ->, green] (12.5,0.5) -- (14.5,0.5);
        \draw[thick] (10,0.5) -- (15,0.5);
        \draw[thick] (10,-1.5) -- (15,-1.5);
        %
        \node at (12.2,2.3) {\textcolor{blue}{Future directed}};
        \node at (14,0.8) {\textcolor{green}{Spatial}};
        \node at (12.5,-1.2) {\textcolor{red}{Past directed}};
\end{tikzpicture} 
\end{document}

Newton's laws can be restated as:
(I) The worldline of a particle under the influence of no force (gravity is not one here) is a future directed autoparallel. That is vγvγ=0 and dt(vγ)>0 everywhere.

(II) The acceleration along a worldline is

aγ:=vγvγ=Fm,

where the force, F, is a spatial vector field, dt(F)=0, and where m is the mass of the particle. Keep an eye: maybe here there is a problem, since aγ is a vector but forces are usually 1-forms (the differential of a potential function). Probably we should admit the existence of a spatial metric.

Worked example

For the moment, what I have is the problem of the evolution of a matter density under Newtonian gravity, without this geometric approach. But I think it is a question of language translation.