**Warm-up: Laws of nature have their limit**

Newton’s law of motion (*F=ma*) is OK as long as you don’t go “too fast”. What does “too fast” mean here? It means slower than the speed of light. So as long as you want to compute what happens to your car, this plane or this train, no problem, Newton is good. High speed trains in Spain run a best at 0.000028% of the speed of light. So yes, it’s slower.

But when you play with things approaching the speed of light, Newton progressively gets it wrong. In the equation “*F=ma*”, Special Relativity multiplies the mass “*m*” by a term which is 1 for small velocities, but progressively goes to infinity as you approach *c* (“*c*” is the symbol for the speed of light). No wonder its changes the math.

We know this thanks to Einstein’s theory of Special Relativity, and to the zillions of experiments which proved it right since 1905. In plasma physics for example, my field of research, there’s no way you can understand nowadays intense laser-plasma experiments, without Special Relativity.

Newton’s law of gravity also has allergies. It’s when the gravitational field becomes “too strong”. It wants it “low”. Here, “low” means you’re much farther from the central mass than its “Schwarzschild radius”. Important detail: this Schwarzschild radius is proportional to the central mass. It grows with that mass.

This radius is usually so small that it fits way inside the mass itself. For example, the Schwarzschild radius of the Sun is only 3 km. But if you’re close enough to the Sun, like Mercury, you can detect tiny, tiny, deviations from Newton’s law of gravitation. Einstein’s “General Relativity” (GR) solved this.

The bottom line? Every physical theory we know has its limits. Newton doesn’t like too fast a motion, or too close to the Sun. Maxwell’s equations also have their limits, etc. And GR Relativity, does it have limits? Yes. A little Thought experiment shows it. Here it goes.

**General Relativity has its limit ***also*

*also*

Take a mass *M* with the electric charge of a proton. Put an electron around. Quantum Mechanics (QM) tells the electron will settle from the mass *M* at a distance equal to the so-called “Bohr radius”. Then it can jump between energy levels and all that, but the Bohr radius is the typical distance it will orbit from the central mass *M*.

The Bohr radius does not depend on the central mass *M*. Only on its electric charge.

Now, increase the central mass. A lot. If *M* is an everyday proton, its Schwarzschild radius is so incredibly smaller than the Bohr radius that no one cares about gravity. But since the Schwarzschild radius grows with *M* and the Bohr radius does *not*, the Schwarzschild radius* must* eventually reach the Bohr radius, for a large enough *M* (see figure below).

Clearly, squeezing such a mass inside its Bohr or Schwarzschild radius implies an immense density. But in principle, it’s inescapable. As you push up density, both radii will eventually merge.

*For a large enough central mass M, the Schwarzschild radius must catch up with the Bohr radius. **The left figure is absolutely not to scale.*

Now, for such a huge mass, what do QM and GR say? The Bohr radius was computed forgetting about gravity. And the Schwarzschild was computed forgetting about QM. But now that both radii are equal, we can no longer be so forgetful. How should we then modify QM and GR to describe the motion of the electrons?

No one knows.

GR is no longer valid when QM effects must be accounted for, and vice versa. Like Newton’s law of motion is no longer valid when you approach the speed of light. Like Newton’s law of gravity is no longer valid when you’re too close to the Sun.

But while Einstein found how to extend Newton if you go to fast, or you’re too close to your sun, we still don’t know how to extend GR when QM has a word.

In case you’re familiar with the double slit experiment in QM, here’s another thought experiment which shows the same: GR has its limit, which we could write “GR+QM=?”.

Now we can talk about the Big Bang.

**What does it have to do with the Big Bang?**

I won’t elaborate too much on the Big Bang here. Let me just remind that the idea was born with the observation of the expansion of the universe. That some have tried to interpret these observations otherwise, without success. And that such observations are perfectly in line with GR.

Since the universe is expanding today, we just have to rewind the movie to find any 2 points of it must have been closer to each other in the past. That’s precisely what GR tells us. In addition, GR provides the mathematical description of how this distance changed with time (see the famous “FLRW metric”).

It just happens that at time t=0, the distance between any 2 points goes to 0. This is called the “Big Bang singularity”. So the question comes: is it physical? In other words, can we trust GR it *really happened*, like we trust GR when our GPS tells we’re or there?

No. The reason for this is simple. As we approach t=0, the universe squeezes matter in an ever-decreasing volume. So the density goes up, together with the temperature. Not only they go up, but they mathematically go up to infinity at the singularity. Now, if the density goes to infinity, sooner or later you *must* reach a point where our thought experiment describe above applies. You must reach a point where GR and QM have to work together. Where GR can no longer ignore QM effects. That is, when GR fails.

There’s no way around. The FLRW metric fails before it reaches t=0. We cannot trust it down to t=0. GR alone cannot tell whether the BB singularity really happened, or not. It becomes blind before.

**Any way out?**

To resolve the singularity, that is, to know what really happened instead of these mathematical infinites at t=0, we need to marry GR with QM. Many bright people have been working on this for decades (Einstein included), so far without success, probably in part because it is extremely difficult to make experiments or observations that could help discriminate between various options.

The good old days when you could test GR with Mercury’s orbit and QM in your kitchen, are gone. It took billions of dollars to test de Standard Model and to find the Higgs boson at CERN, and still, the accelerator they used for that is way too small to test any proposal of GR/QM unification.

So there are candidates out there, like String Theory or Loop Quantum Gravity, all of them untested, be it through observation or experiment. Interestingly, both String theory and Loop Quantum Gravity could give a “Big Bounce”. Other kinds of “bounce cosmology” are also proposed. Stephen Hawking also had a proposal, this one with a real beginning. But again, nothing certain, for nothing successfully tested like GR or QM.

One last word about the “BGV theorem“, frequently cited in this context. It is classical, which means it does *not* account for QM. Just GR here. Its conclusions are therefore very useful, but we know they do *not* describe the real world (it’s always useful to know the behavior of a model, even if you know it does not describe the real world). That’s what Sean Carroll tries to explain to William Lane Craig around min 58 of this debate. He even shows the picture below, where Alan Guth, the “G” of BGV, tells he thinks the universe may be past eternal but that basically, **we don’t know**.

We may close further this case with this text from Avi Loeb, who teaches cosmology at Harvard, and Paul Steinhard who does the same thing at Princeton. Toward the end, we find this sentence,

“Although most cosmologists assume a bang, there is currently no evidence—zero—to say whether the event that occurred 13.7 billion years ago was a bang or a bounce”

Finally, those who worry about entropy can rest assure that Alan Guth, Avi Loeb, Neil Turok or Paul Steinhard, to name a few, also know about it and for example, read this.

The current scientific answer to the question “did the universe had a beginning?” is therefore simple, and for simple reasons.

Current science simply **doesn’t know**.