Newton's shell theorem has two corollaries:
The gravitational attraction of a spherically symmetric body acts as if all its mass were concentrated at the center, and
The gravitational acceleration inside the cavity of a spherically symmetric body vanishes.
Consider a spaceship floating freely in space. In a homogeneous universe, the combined attraction from all matter should cancel out, and the spaceship should stay motionless. Nevertheless, I'm free to divide the attraction into several parts originating from different parts of the universe: In the figure below, I've divided the universe into a red sphere centered on some arbitrary point (×) with my spaceship located at the edge of the sphere, plus infinitely many shells centered on the same point.
By corollary #1, the gravitational attraction of the red sphere equals that of all its mass centered at the point ×. By corollary #2, the combined acceleration of the spaceship from all mass in the green shell vanishes. The same can be said for the blue shell, the orange shell, and so on ad infinitum.
Hence my spaceship should start accelerating toward ×. By choosing the sphere large enough, I should be able to make it accelerate arbitrarily fast, and by choosing the location of × I can make it accelerate in any direction.
Of course this doesn't work, but why?
My best guess is that, even in an infinite universe, you can't keep adding spheres because you'll exit the observable universe, in which case there's no way to feel the gravity in part of the shell so that it's no longer symmetric. Perhaps also the expansion of the universe matters. But see the last two points below.
A few more things to consider:
The mass of the red sphere increases with the chosen radius $r$ as $r^3$, while the acceleration it generates is proportional to $r^{-2}$; hence the acceleration increases linearly with the chosen $r$.
Our universe — the "Universe" — has an average density of some $10^{-29}\,\mathrm{g}\,\mathrm{cm}^{-3}$. Hence if I set $r$ equal to the radius of the observable Universe (46.3 billion light-years), the acceleration is a minuscule $10^{-7}\,\mathrm{cm}\,\mathrm{s}^{-2}$. If that bothers you, choose another universe where $\rho$ is ten orders of magnitude higher.
Our Universe is not really homogeneous, but on large enough scales ($\gtrsim$ half a billion light-years) it seems it is. Still, the acceleration of the spaceship will be dominated by nearby sources. If that bothers you, choose a sufficiently homogeneous universe.
On the scales we're considering, the Universe is not governed by Newtonian dynamics, but by general relativity. If that bothers you, use Birkhoff's theorem instead — I think the issue is the same.
If the issue is really that the size of the observable universe matters, then my intuition tells me that I can just choose an arbitrarily old universe where the asymmetric contribution from the most distant shells is arbitrarily small.
If the issue is that the universe expands (so that gravity from the far side of a shell is somehow weakened, or "redshifted"), then my intuition tells me that I can just choose a sufficiently static universe.
Answer
Updated 07.11
We can chose the model to discuss the problem and so let us chose:
Model: Newtonian mechanics/Newtonian gravity, with the Universe filled with uniformly dense matter, interacting only gravitationally (in cosmology this called “dust matter”), and at the initial time of our spaceship journey all this matter is at rest.
Hence my spaceship should start accelerating toward ×. By choosing the sphere large enough, I should be able to make it accelerate arbitrarily fast, and by choosing the location of × I can make it accelerate in any direction.
Absolutely!
Of course this doesn't work, but why?.
It does work. If we assume that initially the spaceship was at rest together with the whole universe it will reach the point × in time needed for the ship to fall into a point mass equal to the mass of the pink sphere.
The problem is that by that time all of the pink sphere also falls toward that same point as well, as do all other colored spheres and the rest of the universe also. If our astronaut checks its distance to the point × before the spaceship falls into it she would notice that this distance has decreased, but at the same time is she checks her surroundings she would notice that the spaceship is surrounded by precisely the same matter particles that when the journey started only they are closer to each other and to the spaceship. This distance contraction is simply a Newtonian version of Big Crunch event.
If the universe is filled with matter interacting only gravitationally and we assume that the density of matter will stay uniform throughout the universe, then the only conclusion would be that such universe is not static. It has either (Newtonian version of) Big Bang in its past or Big Crunch in its future (or in our model, since we chose initial moment as a turning point from expansion to contraction, it has both).
It may seem that the whole Universe falling toward our chosen point × is an absurdity, since we have chosen this point arbitrarily. But in this situation there is no paradox, the acceleration of all matter toward this point is due to the fact that in our setup there is no “absolute space”, no set of outside stationary inertial observers which could give us absolute accelerations, instead we can only choose a reference point × (or rather specify an observer located at this point and at rest with respect to surrounding matter) and calculate relative accelerations toward this point.
Recall, that the first principle of Newtonian mechanics states that every particle continues in its state of rest or uniform motion in a straight line unless it is acted upon by some exterior force. For an isolated system, for example collection of gravitating objects of finite total mass we could (at least in principle) place an observer at rest so far away that it could be considered an inertial object. This would allow us to define a reference frame with respect to which we would measure accelerations. But in our Newtonian cosmology matter is filling the whole Universe, there is no observer on which gravity is not acting, so there is no set of reference frames defined by observers “at infinity” only observers inside the matter concentrations that are affected by the gravitational forces.
While there is no absolute accelerations, the relative positions ($\mathbf{d}_{AB}(t)= \mathbf{x}_A(t)-\mathbf{x}_B(t)$ between objects $A$ and $B$ comoving with the matter of the universe) do have a meaning independent of the choice of reference point. This relative positions, relative velocities ($\dot{\mathbf{d}}_{AB}$), relative accelerations, etc. constitute the set of unambiguously defined quantities measurable within our universe.
then my intuition tells me that I can just choose a sufficiently static universe.
This intuition is wrong, if there is a gravitational force that would accelerate your spaceship toward ×, then it would also be acting on a nearby matter (call them dust particles or planets or stars) producing the same acceleration, so all of the universe would be falling toward ×.
Note on Newtonian cosmology it may seems that Newtonian theory of gravitation is ill suited to handle homogeneous spatially infinite distributions of matter. But one can try to separate the physics of the situation from the deficiencies of particular formalism and possibly to overcome them. As a motivation we could note that over large, cosmological distances our universe to a high degree of accuracy could be considered spatially flat, and the velocities of most massive objects relative to each other and to the frame of CMB are very small compared with the speed of light, meaning that Newtonian approximation may be appropriate. While we do know that general relativity provides a better description for the gravitation, Newtonian gravity is computationally and conceptually much simpler. This seems to suggest that it is worthwhile to “fix” whatever problems one encounters while attempting to formalize cosmological solutions of Newtonian gravity.
The most natural approach is to “geometrize” Newtonian gravity and instead of “force” consider it a part of geometry, dynamical connection representing gravity and inertia. This is done within the framework of Newton–Cartan theory.
As a more detailed reference, with an emphasis on cosmology, see this paper (knowledge of general relativity is required):
Newton–Cartan theory underscores conceptual similarities between Newtonian gravity and general relativity, with Galilei group replacing the Lorentz group of GR. The general approach is coordinate-free and is closely related to the machinery of general relativity, but a specific choice of local Galilei coordinates would produce the usual equations for acceleration ($\mathop{\mathrm{div}} \mathbf{g} = - 4\pi \rho$), with gravitational acceleration now being part of Newtonian connection. Homogeneous and isotropic cosmological solutions are a straightforward lifts of FLRW cosmologies.
While equations are the same, we may already answer some conceptual questions.
Since gravitational acceleration is part of the connection, there is no reason to expect it to be an “absolute” object, there would be gauge transformations that would alter it. We can have multiple charts on which we define the physics with the normally defined transition maps between.
We can have a closed FRW cosmology, the “space” does not has to be a Euclidean space, it could be torus $T_3$ (field equations require that locally the space is flat). Since the spatial volume of a closed universe varies, and tend to zero as the universe approaches the Big Crunch, this asserts that not just matter but space itself collapses during the Big Crunch (to answer one of the comments).
It is quite simple to include the cosmological constant / dark energy thus making the models more realistic.
Note on answer by user105620: If we formulate a regularization procedure by introducing a window function $W(\epsilon,x_0)$ that would make potential well behaved. This provides us with an another way to “fix” problems of our cosmological model. The acceleration of our spaceship computed with this regularization is indeed dependent on the choice of $x_0$ in the limit $\epsilon\to 0$, which is the consequence of the same freedom in choosing the reference point ×. But he/she just should not have stopped there. Divergences requiring the use of regulators and ambiguities remaining after regularization are quite normal features in developing physical models. The next step would be identifying the physically meaningful quantities and checking that those are independent on the regulator artifacts. In our case neither potential $\Phi$ nor gravitational acceleration $\mathbf{g}$ are directly observable in this model. Relative positions, relative velocities and relative accelerations are observable and those are turning to be independent of the regulator parameter $x_0$.
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