I am reading the new Discovery Institute Press book, The Big Bang Revolutionaries, written by Jean-Pierre Luminet and endorsed by three physics Nobel Prize winners. My own 2015 Discovery Institute Press book, In the Beginning and Other Essays on Intelligent Design, draws its name from a chapter on the Big Bang and its philosophical/theological implications.

Since very few people understand Einstein’s General Theory of Relativity (GR), for most of us the Big Bang theory, which is based on the GR, seems very mysterious and counterintuitive. Even though I am a mathematician and am able to understand Einstein’s special theory of relativity and its mathematics, I have to confess that the general theory is beyond my grasp. 

The history in Luminet’s new book is very interesting, but it is a bit frustrating that I, like most everyone else, have to trust the scientists who do understand GR when reading about the Big Bang. Fortunately, it is possible to develop a mathematical model for the expanding universe that involves only classical gravitational theory, and I included a section in my 2015 book that does this. It still requires a bit of mathematics, but only some calculus: no understanding of GR is required!

In the new Luminet book, Georges Lemaȋtre is quoted (p. 131):

To study in detail the variation of the radius of space, it is necessary to appeal to the equations of general relativity. It is possible, however, to illustrate the result of relativistic computation by elementary considerations involving the laws of classical mechanics. This is possible because the laws of relativity are reduced as a limit to the laws of Newton, when they are applied to an infinitely small volume.

These equations account for the dynamics of the universe; they accustom us to thinking of the radius of the universe as a physical quantity, able to vary. The manner in which these equations have been obtained must not be regarded as rigorous demonstration. A demonstration which is not open to criticism can be deduced only from the general equations of relativity.

This is a nice summary of the significance, and lack of significance, of the model in my Section 6.5, reproduced below with some additional calculations added in a Postscript at the end. The classical model given here is obviously not to be trusted blindly when r(t) is very small, but it nevertheless “correctly” (in agreement with the standard conclusions of GR without a speculative cosmological constant) predicts that if the density of the universe is less than some critical density (which means, according to GR, that space has negative “curvature”) the universe will continue to expand forever, while if it is greater than this critical density (which has by the way the same value as derived by GR), it will eventually stop expanding and begin to contract. And it correctly “predicts” that in either case, there was a time in the finite past when r=0 and r’=∞, which is very suggestive of a “Big Bang”!

6.5 A Model for the Expanding Universe

If I dig a hole to the center of the Earth and excavate a small chamber there, I will be able to float about weightlessly in my chamber, because the gravitational attraction of the Earth on my body is equally strong in all directions. What if I only tunnel halfway to the center? As you might expect, I will weigh less than I did on the surface, but I will not be completely weightless; there is still a net force toward the center of the Earth. It can be shown by summing up the gravitational forces exerted on my body by all the molecules in the Earth (thanks to integral calculus, this is not as difficult as it sounds), that the net force exerted by that portion of the Earth which is closer to the surface than I am is zero, so that only the portion closer to the center than I contributes to my weight. If I could hollow out the inner core of the Earth, the entire hollow core would be a giant weightless chamber and I could float about in it, because at any point within the hollow core the tug of gravity would be the same in every direction. Thus the Earth’s gravitational attraction on my body can be calculated by throwing away the outer shell and pretending that I am standing on the surface of a smaller planet, half the diameter of the Earth.

Now let us replace the Earth in our story by the entire universe, and let us take our planet to be the center of the universe. (Copernicus showed us that the Earth is not the center of the universe, but the cosmological principle and the theory of relativity tell us that it is as good a center as any!) Then consider a certain galaxy (A) whose distance from the Earth is given as a function of time by R(t) and let us calculate the “weight” of that galaxy — that is, the gravitational force with which the rest of the matter in the universe pulls it toward the center (us). By the same reasoning as used above, we conclude that we can ignore all matter further away from the center than A and calculate the force pulling A toward the center as the force of gravity between A and a sphere of matter whose center is the Earth and whose radius is R. According to Newtonian gravitational theory, this force is GMm/R(t)², where G is the universal gravitation constant, m is the mass of A, and M is the mass of the above-described sphere, on whose surface A rests. M is just the density ρ(t) times the volume of this sphere, 4/3 πR(t)³.  As the universe expands or contracts, this sphere will expand or contract proportionally, so the quantity of matter in the sphere will remain constant. Then we may use the current (t=0) values for ρ(t) and R(t) and write M=4/3 πρ₀R₀³_.

This gives, for the gravitational force on A:

          GMm/R(t)² = 4/3 πρ₀R₀³Gm/R(t)²

By Newton’s second law, then, the acceleration (R’’) of A is equal to this force divided by the mass (m) of A:

          R’’(t) = -4/3 πρ₀R₀³G/R(t)²

where the negative sign is used because the acceleration is negative, that is, gravity decelerates (slows down) the expansion.

The initial conditions for this differential equation are obtained by noting that at t=0 (now) we have R(0) = R₀ and R’(0) = HR₀, since the rate at which any galaxy is receding from us is supposed to be approximately H (the current Hubble constant) times its distance from us.

If r(t) is defined to be R(t)/R₀, r can be interpreted as the size of the universe normalized to make the current size equal to 1. Then the differential equation and initial conditions simplify to:

 (6.1)      r’’ = -4/3 πGρ₀/r² ;  r(0) = 1;  r’(0) = H

Now there is an objection which the reader may raise to the way in which (6.1) was derived. It may be argued that there is no net gravitational force on either the Earth or A, because in either case the pull of the rest of the matter in the universe is equally strong in all directions—either can be considered the center of the universe. The answer is that the ultimate justification for (6.1) comes from the general theory of relativity. However, if we look at any sphere of small (cosmologically speaking!) radius, the general theory of relativity allows us to ignore the gravitational effects of material outside that sphere, and to use Newtonian gravitational theory inside the sphere. And we will get equation (6.1) using classical ideas if we take the universe to be a sphere of arbitrary radius, with center at any particular point — Earth, galaxy A, or a neutral third party.

Using techniques found in any elementary differential equations text we find that (6.1) implies:

 (6.2)      r’ = [(8π/3) Gρ₀/r + C]^½                   

where C is found, by applying the initial conditions, to be

           C = H² – 8/3 πGρ₀

Now if C is positive, that is, if ρ₀ < 3H²/(8πG) ≡ ρ_c, then it is clear from (6.2) that the universe will continue expanding forever, since r’ will always be positive. On the other hand, if ρ₀ > ρ_c, C will be negative and there will be a value of r which will make r’=0, so that when the normalized size of the universe reaches that value of r, it will stop expanding and begin to contract. This contraction would presumably continue until the universe ends in a “big squeeze.”

The differential equation (6.2) can be further solved for r(t) using standard differential equations techniques, but the resulting solution is rather complicated to write out. However, for the case ρ₀ = ρ_c, which is thought to be reasonably close to correct, C=0, and (6.2) reduces to r’=H/√r, which can be easily solved (remembering that r(0)=1) to give r(t) = [3/2 Ht + 1]^⅔ . In this case, we can see that r = 0 and r’ = ∞ when t = -⅔H^-1, which places the Big Bang about 10 billion years ago. For other values of ρ₀, the solution is more complicated, but still predicts that r= 0 in the finite past, less than H^-1 ≅ 15 billion years ago. Some estimates of ρ₀ are around 0.01 ρ_c; that would make the age of the universe about 0.98 H^-1.

Postscript, Added May 2024

Since I mentioned above that equation (6.2) can be further solved, let’s do that now.

If we take ρ₀ = q ρ_c, the differential equation (6.2) simplifies to:

(6.3)      r’ = H [q/r + 1-q]^½            

Since it is believed by many that the density of the universe is subcritical (which means the “curvature” of space is negative) let’s assume q < 1. If q > 1 (6.3) can be similarly solved, but the solution is significantly different-looking. The case q = 1 is much easier and was already solved above.

In the case q < 1 we can integrate (6.3) to give (w ≡ 1-q to simplify the formulas), after applying the initial condition r(0) = 1:

   wHt = -1 + [r(q + rw)]^½  –  q/w^½ log { [(rw)^½ + (rw+q)^½] / [w^½ + 1] }

This is very difficult to solve explicitly for r(t). Fortunately, this is not necessary now, because to determine the time that r = 0 and r’ = ∞ we simply set r = 0 and get:

   wHt₀ = -1 –  q/w^½ log{ q^½ / [w^½ + 1] }

If we try q = 0.01 we find t₀ = -0.98 H^-1 as reported at the end of Section 6.5. As q → 0, t₀ approaches, naturally, t₀ = -H^-1.

Second Postscript

Reading the fascinating history of the Big Bang in Luminet’s new book made me think about design in the universe, and not only for the obvious reasons ­— the implications of a beginning of the universe.  It also brought vividly to mind a statement I had made toward the end of the Epilogue of my book:

If God designed this world as a tourist resort where man could rest in comfort and ease, it is certainly a dismal failure. But I believe, with Savage, that man was created for greater things. That is why, I believe, this world presents us with such an inexhaustible array of puzzles in mathematics, physics, astronomy, biology and philosophy to challenge and entertain us, and provides us with so many opportunities for creativity and achievement in music, literature, art, athletics, business, technology and other pursuits; and why there are always new worlds to discover, from the mountains and jungles of South America and the flora and fauna of Africa, to the era of dinosaurs and the surface of Mars, and the astonishing world of microbiology.