The uniform infinite Newtonian universe (the standard solution to the Seeliger paradox)

When Newton contemplated what the universe is like, he considered three possibilities:

• Aristotle's idea: the universe is finite 
• Stoicism's idea: the universe is a finite island of stuff floating in an infinite vacuum.
• Epicurus's idea: the universe is infinite and uniformly filled with stuffs.

He settled on the infinite uniform universe.

Now he faced a question: what is the gravitational force felt by a test mass in this infinite uniform universe? He thought that, by spherical symmetry, it has to be zero, because if the force isn't zero, it would break the spherical symmetry.

Now, a quick thought would show that there is a serious problem. It's known that a uniform, infinite slab attracts a mass with the same force, no matter how far.

So cut the universe into a infinite series of equally thick slabs, on the left and right of the test mass. We see that to calculate the total force on the test mass, we must do this infinite sum:
$$F + F - F + F - F ...$$

which is indeterminate.

In short, these three conditions are incompatible:

• The universe is infinite and uniform
• The universe is Newtonian (Newton's three laws of motion, plus Newtonian gravity)
• The Newtonian laws of physics should fix a unique solution.

This was what Seeliger argued in 1895. Since then, many solutions have been proffered, by giving up some of the conditions. One interesting solution is by Milne and McCrea in 1934, see Newtonian Cosmology (1955). It gives up the uniqueness, but shows that there are some very interesting solutions: an expanding and contracting universe with finite lifespan and a "center point" (despite being infinite and uniform), something that Newton could have discovered!

Let the universe be infinite, filled with mass with density $\rho(t)$, which is a function of time. 

Let the universe have a special point, call it "the center of the universe". Note that we are breaking the symmetry here: the mass distribution is everywhere the same, but we are choosing a point and declare it special.

Let the mass in the universe move radially towards the center of the universe. That is, consider a particular tiny chunk of mass at distance $r(t)$ away from the center of the universe, then the chunk of mass moves only along the ray from the center to the chunk.

So, the motion of the entire universe is described completely by a function $r(r_0, t)$: the radial distance of a chunk of mass at time $t$, if the chunk is at $r_0$ when $t=0$,

Now, we can write down two equations that would give a consistent solution to our Newtonian universe:

$$\frac{d}{dt}(\rho(t) r(r_0, t)^3) = 0 \quad \text{conservation of mass}$$

$$\frac{d^2}{dt^2}r(r_0, t) = -\frac 4 3 \pi r(r_0, t)^3 \rho(t) G \quad \text{acceleration due to force of gravity}$$

Note that, we have not solved the indeterminacy of the force of gravity. We have simply selected by fiat a solution. We simply show that a solution exists, even if it is not unique.

This system is solvable. Indeed, it has a whole family of solutions:

Let $\rho_0$ be a  constant positive density. Let $K(t)$ be a unitless, positive function satisfying 

$$K^2 \frac{d^2}{dt^2} K = -\frac 4 3 \pi G \rho_0$$

Then we have solution

$$\rho = \rho_0/K^3, \quad \frac{d}{dt} r(r_0, t) = \frac{K'}{K}r(r_0, t), \quad r(r_0, 0) = r_0$$

It's easy to see that (draw the $K(t)$ graph) the function $K(t)$ is convex downwards, and bounded above by a parabola. Thus, $K(t)$ must reach $0$ at two points in time $t_0, t_1$, and so we have $\rho = \infty$ at $t_0$, then dropping down to a minimal value at $(t_0 + t_1)/2$, then rising up to $\infty$ again at $t_1$. That is, this universe blows up from a point, then contracts down to that point again.

Newton would like that: God created the world, and it would end with a final Judgment Day.

To read more, check The cosmological woes of Newtonian gravitation theory (Norton 1999).

Norton is a philosopher and historian of physics. His style is very analytic. I took a tour of his website and found some cool ideas:

The castle in the sky (aka "pancakes all the way down, but each pancake is thinner")

If you put a castle on a layer of rocks, then put that layer on another rock, etc, making each layer thinner and thinner, you get a system of static, rigid bodies such that each layer is static, because it is supported by the layer below.

But what about the theorem that a system of objects should have their total momentum changed by external forces? The center of mass theorem.

That theorem is derived only for finite systems. For an infinite system, somehow, the center of mass theorem fails by a failure of convergence!

The Burning Fuse Model of Time: The advancing point of fire is the now. It consumes the future, the unburnt fuse, which is converted into the past, the scattered nothingness of ashes. The future must be real, because it can become the present. The past cannot become the present, therefore it doesn't have to be real in any sense.

How Did Einstein Think?: Einstein didn't know, and suggested that introspection won't work:

If you want to find out anything from the theoretical physicist about the methods they use, I advise you to stick closely to one principle: don't listen to their words, fix your attention on their deeds.

By examining Einstein's work, Norton concluded that Einstein thought by:

• Hard work (no surprises here)
• Flashes of insight braided with systematic examination of possibilities (especially when he was looking for the equations for general relativity, by checking formulas against a wishlist)
• Focus on particular paradoxes instead of erudition and masses of data
• Thought experiments
• Physical insight instead of math (opposite of Dirac)
• Algebraic reasoning instead of geometric reasoning (this one seemed unhelpful to him!)

Norton also claimed that Einstein changed his thinking style after a bad experience:

Einstein himself recognized this special facility for physical thinking when he wrote his Autobiographical Notes (see right), reporting that this physical insight and intuition outstripped his mathematical instincts. 

There is, however, a unexpected ending to the story that also surfaces in these remarks. Einstein's general theory of relativity derived essentially from physical thinking. How could he devise a theory of gravity that frees us from our dependence on any special coordinate systems? How could acceleration be deprived of its absoluteness in such a theory? 

However the theory grew so complicated and difficult that Einstein's physical instincts failed him at the crucial moment. In 1913 he published a defective version of the theory. After nearly three years of hesitation and doubt, he was able to repair the theory by writing down the mathematically simplest equations. They were ones he'd nearly adopted three years before. He had then thought them to be inadmissible for physical reasons. The diagnosis of his earlier error was clear. He had mistakenly trusted his physical insight over mathematical simplicity.  

The experience triggered a profound change in Einstein's thinking. He now became convinced that the path to a deeper understanding of the physical world lay in mathematics.

This could explain why he didn't manage to do anything nearly as cool in quantum mechanics. Or maybe not.

We might lament that Einstein's work took this pathway. What might have happened had he continued to follow his older methods? We cannot know. However I suspect that not much would have emerged. Einstein's physical instincts were the ones needed to develop relativity theory and his other successes. 

When the focus of research moved to quantum theory, a different sort of instinct seemed to be required. That was embodied by the Danish physicist Niels Bohr. He had a characteristic tolerance and even delight in contradiction. That characteristic enabled Bohr to theorize successfully in the bewildering and uncertain quantum domain and in a way that Einstein's physical sensibilities found repugnant. Einstein's role changed to that of a senior sage, warning the new generations of the dangers of the path they had chosen.