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
Assume that
for
The question is: can this equation of motion be the geodesic equations of the manifold
Observe that
We can rewrite
for
Now combining this with the
where
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:
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
which, using the Poisson equation, gives:
This is actually one of the so-called Einstein equations:
where
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
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
for all — there is a concept of absolute space (defined below). everywhere — absolute time flows uniformly. is torsion-free.
The connection is determined by a mass distributionby means of the "Einstein equation":
Related relativistic spacetime.
Definition (Absolute Space)
Let
It follows that:
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
- Future directed if
. - Spatial if
. - Past directed if
.
\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
(II) The acceleration along a worldline is
where the force,
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.