Level 1: Spacetime — \(0\text{s}\)
Fundamental quanta
Size: \(10^{-35} \text{m}\)
Initial state: Big Bang monistic substance
Production process: Gravity decoupling
Meta-stable equilibrium: GUT Quantum fields
Stable future: Vacuum with macro-fluctuations
The Universe
Temperature: \(10^{32} \text{K}\)
Initial state: Big Bang monistic substance
Production process: Compactification
Meta-stable equilibrium: Curved 4D universe
Stable future: Frozen / Ripped universe
Level 1 describes structures that appeared at the same time as our universe. We’ve talked a lot about other worlds and the possibility of the multiverse in previous sections. To be clear, when talking about "the universe" here, I refer to the entity that originated in the Big Bang (which we know occurred thanks to the Cosmic Microwave Background surrounding us). It might well be part of a larger structure (such as a world undergoing eternal inflation, with Big Bangs simmering here and there), but I’ll stick to the conventional universe.
It is tempting to picture the Big Bang as the explosive growth of a tiny "egg" at an unimaginable pace. But remember that the universe might well be infinite. In that case, it hasn’t been growing in size following the Big Bang. Expansion just means that the density decreases everywhere: an infinite fabric is getting stretched more and more — but it was already infinite, and stays infinite. That is why there is no size information under \(1\Omega\) but rather a temperature metric.
To study what happened at or right after the Big Bang requires what Gamow called "\(G\,\hbar c\) physics".
The physics cube
\(G\), \(\hbar\) and \(c\) are respectively the gravitational constant, the speed of light and the Planck constant.
In classical mechanics, one can ignore gravitation (\(G=0\)), light is assumed to move infinitely fast (\(c=\infty\)) and energy transfers are taken to be completely continuous (no quanta: \(\hbar=0\)). For each constant, we have the choice to "switch it on" or not, resulting in the "physics cube".
Newton’s Universal Gravitation (UG) is introduced.
Einstein’s special relativity (SR) reconciles classical physics with a finite \(c\).
Planck, Einstein and Bohr consolidate early quantum mechanics (QM).
Einstein’s general relativity (GR) elegantly expands SR to explain gravity.
Dirac’s quantum field theory (QFT) incorporates QM with SR.
Non-relativistic quantum gravity (QG) has never been seriously attempted.
The holy Grail of physics, supposedly the key to a "theory of everything" (TOE).
The current dichotomy between the Standard Model of quantum physics and General Relativity is tenable because each theory has its own domain. At macroscopic scales, quantum effects are negligible, so \(h=0\) (the semi-classical limit) is a good approximation. At microscopic scales (molecular or below), quantum effects become significant but it is gravitation that is unimportant: the mass of the systems under consideration is too small, or to put it another way, spacetime curvature looks locally flat over the tiny patch that we’re observing.
However, those remarks are true only in the context of "normal physics", that is at the levels of energy and at the scales that we can find in outer space, on earth, or in a star’s core. In extreme conditions, both gravitational and quantum effects become essential, hence the quest for quantum gravity as the only type of model that can teach us about the early universe.
Yet quantum gravity is not just as a needlessly precise version of our current physics, nor one that is only valid in specific environments such as black holes: if we choose to focus on a very small patch of spacetime, then by virtue of Heisenberg’s uncertainty principle, quantum fluctuations will be enormous enough to bring us back to energy levels comparable to the very early universe.
Planck’s natural units
In fact, \(G\, \hbar c\) also yields the system of Planck units which define the scales at which quantum gravity effects dominate.
\[\begin{aligned} &\text{Planck length} && l_P= \sqrt{\frac{\hbar G}{c^3}} && 1.6 \times 10^{-35} \text{m}\\ &\text{Planck time} && t_P= \sqrt{\frac{\hbar G}{c^5}} && 5.4 \times 10^{-44} \text{kg}\\ &\text{Planck mass} && l_P= \sqrt{\frac{\hbar c}{G}} && 2.2 \times 10^{-8} \text{kg}\\ &\text{Planck energy} && l_P= \sqrt{\frac{\hbar c^5}{G}} && 1.2 \times 10^{19} \text{GeV}\\ &\text{Planck temperature} && l_P= \sqrt{\frac{\hbar c^5}{Gk_B^2}} && 1.4 \times 10^{32} \text{K}\\ \end{aligned}\]
These are indeed the characteristic values of physical metrics right after the Big Bang. And as noted above, since Heisenberg’s relations teach us that \(\Delta E \Delta t \geq \hbar\), then zooming on the Planck scales should introduce quantum fluctuations with amplitudes comparable to the Planck energy, and virtual particles as heavy as the Planck mass popping up...
The Planck epoch: \(0\) to \(10^{-43} \text{s}\)
The period betwen the Big Bang and one Planck time is known as the Planck epoch. Energy levels throughout the period are unknown (it’s not even clear if that concept would make sense), but at the end of that period, temperature is still at an unfathomable \(10^{32} \text{K}\).
Entropy is also very low (estimated around \(10^{88} k_B\)), which is quite reassuring since it gives the universe room to grow before becoming fully homogeneous (and therefore time for complexity to develop). At the same time, something might seem counter-intuitive. Didn’t we say that highly homogeneous states had high entropy, and isn’t the early universe expected to be largely homogeneous? In fact, the universe did not have any structure or asymmetries indeed, but the combination of an extremely strong interaction and of immense quantum (therefore, random) fluctuations actually yields a state of very low entropy (relative to the energy).
If I had to summarily depict the universe in the Planck epoch, I would say: one object, one interaction. Indeed, the kind of distinctions that seem fundamental in physics: between spacetime and its content, between interactions and the particles they act upon, between radiation and matter, make no sense at the Planck energy level. There’s only one object, the universe, and no clear way to distinguish anything more fundamental.
Unified forces
The root cause is the unification of all forces at such energies. To fully unpack what that means, let’s refresh our memories a tad...
There are four fundamental forces in the world today (interactions) that explain all physical phenomena.
The strong nuclear force binds quarks together into protons, neutrons and other hadrons. These structures neutralize the strong force but the residual is strong enough to induce a powerful attraction that "glues" protons and neutrons together, ensuring that atomic nuclei can exist. It doesn’t behave like forces we are more used to, such as electromagnetism or gravitation: between quarks the strong force actually grows with distance, ensuring that apart from super-high energy environments, quarks have no way of being free.
The electromagnetic force is more familiar: it is the source of all the electric, magnetic, and optical phenomena (including, more generally, all EM waves). The attraction or repulsion between electrically charged bodies quickly decreases with distance.
The weak nuclear force is mostly evident at our scale in the form of beta radioactive decay. Unlike the other three, it is characterized by a short range: long-distance interactions are always possible but vanishingly unlikely beyond microscopic scale.
The gravitational force is a well-known attractive force between all entities that have mass or energy (which covers every physical object). It has an infinite range. General relativity interprets it as a manifestation of spacetime curvature.
None of the objects we’ve just mentioned (protons, neutrons, electrons...) exist at the Planck epoch. But we want to give a sense of what the forces do, and how they could be unified during the Planck epoch... First, what does that even concretely mean? Unification has always been a powerful driving force in the history of science, taking seemingly disparate theories and finding they’re entirely reducible to a more fundamental one. That leads to ontological parsimony, epistemological efficiency, and most of all to great beauty. Physics offers textbook examples of unification:
Universal gravitation: Johannes Kepler’s laws of planetary movement and Galileo Galilei’s mathematical representation of gravity were both landmark achievements of the seventeenth century, but must have seemed to contemporaries as very disjointed in their scope. What could be in common between an apple falling to the ground and a planet orbiting the Sun? Isaac Newton’s concept of gravitation elegantly explains both by integrating a new gravitational force with his laws of dynamics. The apple that I throw and which falls on the ground is as much in orbit as the moon, only given its low speed, it is an unstable orbit that ends in a crash. And the moon, in a stable orbit, is constantly "falling on the earth", and missing.
Electromagnetism: Electricity and magnetism are amongst the oldest physical phenomena to have been discussed by "natural philosophers". Similarities had always been observed between the two, despite one major difference. Electrically charged objects can exist independently from each other, exerting an attractive or repulsive force (unlike gravitation, electrical interactions act on charges that can be positive or negative). But magnetic objects found in nature are always dipoles, endowed with a North and a South pole (like the needle of a compass, which obviously inspired this terminology). Coulomb’s law and later Gauss’s formulation aimed to describe these two forces in isolation.
But important connections were evidenced between them. Ampère, Biot and Savart’s work formalized the earlier observation that an electrical current generates a magnetic force; in fact a circular current would create a field that approximates an actual magnet, suggesting that magnets’ properties have their source in micro-currents. Lastly, Michael Faraday discovered induction of an electrical current by a varying magnetic field.
James Clerk Maxwell, in a work of physical and mathematical acumen that can only be compared to Newton and Einstein’s achievements, re-framed all these laws as part of a simple theorization of electromagnetism as a whole. The objects under consideration are not charged bodies; that Newtonian style of physics was replaced by a focus on fields: Maxwell’s theory is a dynamics of the electromagnetic field. From these equations in a vacuum, Maxwell landed on a wave equation, whose solutions has a propagation speed remarkably close to the speed of light. Maxwell had proved that light was in fact not an irreducible entity, but an electromagnetic wave, governed by the laws of electromagnetism. All the conclusions of optical theories could then be folded into Maxwell’s.Statistical mechanics: The development of thermodynamics during the nineteenth century had turned the old calorics (science of heat) into a rigorous theory of gases, where pressure, temperature, entropy played the same fundamental role as mass or velocity in classical mechanics. The newly minted concept of energy played the role of a bridge between these domains. Boltzmann’s incredible insight was not only to espouse the right side in the controversy over atomism (thereby positing that a gas is made up of molecules in an excited state), but also to run away with the idea, developing the appropriate statistical model, and showing that all thermodynamic quantities were just statistical moments (average, volatility...) of the molecules’ mechanical properties (mass, momentum...). Boltzmann’s work was the first step in unifying thermodynamics with classical mechanics.
Based on this triumphant history of unification in physics, it may come as no surprise that similar successes bloomed in the twentieth century. But the unification of fundamental interactions that quantum physics has achieved is different: it is both conceptual and historical.
Newton, Maxwell and Boltzmann proved a connection between phenomena that were thought to be grounded in different forces, but actually never were. The acceleration of a falling apple relates to the orbit of the moon in a way that’s always been the same, since there is no qualitative difference, only circumstantial ones (the earth is big and I cannot throw the apple fast enough). They are two facets of the same interaction.
The modern unification of the fundamental forces does not attempt to correct the record on our understanding of the present: we are right in considering those four interactions as distinct since they do have very different strengths. The idea is that:
Their relative strength varies with energy levels: they are closer to each other at higher energy.
Since the energy density of the universe has decreased over time, so have the fundamental forces diverged from each other.
Past the energy level where two or more forces have reached an equal strength, that relationship remains stable: the forces can be considered to be unified.
All of this ought to be described by a single theory, which would describe both the high-energy, unified state of affairs, and the low-energy environment we’re experiencing today.
This is quite remarkable. Unification is not just a metascientific process indicating progress in our understanding of the world. It is both that and a time machine showing us what actually was. We’re not just unifying theories, we also describing the very physical, very concrete fact that if we turn the temperature high enough, these forces will be unified again!
Coming back to our main throughline: what marks the end of the vanishingly short Planck epoch is that gravity separates from the electronuclear force (the unified composite of the strong, electromagnetic and weak interactions). The intensity of gravitational interactions, which was equivalent to that of electronuclear interactions, starts to undergo a relative decline.
What does it mean for an interaction to be more or less intense than another? Let’s put it simply: if you drop an iron nail on the floor, a small magnet should be enough to have it "fly" towards you. The attractive force from that small magnet has handily beaten the attraction from the whole planet Earth... Today, electromagnetism is about \(10^{37}\) stronger than gravity. The only reason that gravity seems so mighty, creating galaxies, solar systems, keeping our feet stuck to our planet, is that most matter in the universe is electrically neutral, being either a plasma, or made of atoms. When a stronger force gets neutralized that way, and only then, a weaker one can dominate.
The strong nuclear force is even more intense, 100 times more than electromagnetism. The mere existence of atoms is striking evidence of that: how else could positive protons be so tightly bound within a nucleus otherwise? Electromagnetic forces are desperately trying to separate them, but they stubbornly stay together, so much so that it takes a tremendous energy to break nuclei... And we’re only talking about the very frail residual of the strong nuclear force: the one that acts between nucleons, and not between their quark components! The weak nuclear force is a specific case: despite its name, it is not the weakest. It’s a million times weaker than electromagnetism, but still \(10^{31}\) stronger than gravitation. However, its concrete effects are alleviated by its tiny range.
The idea that interactions’ intensity has varied over time may seem strange. After all, aren’t we touching to the most fundamental aspects of the physical world? That stems in fact from the deep nature of quantum field theories. They don’t represent forces the way Newton did, as two particles acting on each other, or like Maxwell did, as field acting on a charge. In the quantum world, what we at our scale perceive as an attractive or a repulsive force is the result of exchanging a virtual boson (a force-mediating particle). For instance two electrons exchange a photon (the mediator for the electromagnetic interaction). But that’s not all: one of the main teachings of quantum field theory is that the vacuum, far from being empty, is constantly disturbed by virtual particle/antiparticle pairs popping up. The more energy around, the more such events. The quantum field of a charge particle carries energy, and that particle is then "dressed" by a could of virtual pairs. An electron is never just an electron then: it is constantly surrounded by myriads of electron-positron pairs.
All those virtual particles intervene in the calculations of concrete, measurable values, and ultimately of the coupling constant (the measure of interaction strength) that we observe today. But now, it should be clearer how that is not really a constant: at very high energy levels, the production of virtual particles is bound to change according to quantum-relativistic effect; therefore, the interaction’s coupling "constant" is equally bound to vary.
Quantum gravity and spacetime compactification
As we turn back time from today to the most distant past (or equivalently, "turn up the heat" from a cool summer night to the fiery Big Bang furnace), we should then see the weak interaction become unified with electromagnetism, then the resulting electroweak force wed the strong force to make the electronuclear interaction, which would then become one with gravitation.
Conceptually, it is now understood. Mathematically, it also means that we should have a unique formalism that could encompass general relativity and quantum field theories at lower energies. We don’t have that yet, although the work done to outline the superstring landscape is giving us precious hints, which we’ll go on to describe soon. But most of all, we should understand concretely why the Planck epoch reality was so different and monistic.
In classical physics, things are pretty clear: space and time are the background onto which physics happen, not an actor of physics itself. In modern physics, things are a bit more complicated: spacetime is definitely an actor in general relativity, which is at its core a law of dynamics for spacetime itself. But quantum field theories ignore that, using the flat spacetime of special relativity as a fixed background for their physics. Attemps at doing quantum physics in curved spacetime keep that wall up: you can try to solve equations at the quantum scale inside a star, or near a black hole, under severely stressed conditions of spacetime curvature, but that still stays fixed at over the timeframes under consideration.
At the Planck scale, this distinction has no meaning. If you insisted to talk about spacetime and particles (despite my warnings!), you could say that particles’ interactions warp spacetime, while spacetime distorsions themselves are quantum fields interacting with the others and between themselves... But that is encouraging confusion. Mathematically there was indeed a topological entity commonly called "spacetime" in the literature, but which doesn’t resemble ours, neither in its dimensionality (11 dimensions is a fairly consensual, though still speculative value) nor its properties. It cannot be the "neutral background" where quantum physics takes place that we commonly picture.
However, the key event at the end of the Planck epoch, besides the divergence of gravity (and of course it is not unrelated) is the spontaneous compactification of the extra dimensions. A quick historical note may be warranted, lest all of this appears as a fanciful deus ex machina.
As I’ve stated a couple times I think, the theoretical combination of general relativity is a massive challenge. We know that now, but it didn’t have to be a priori. Maxwell’s electromagnetic theory was quantized in the form of QED (Quantum Electrodynamics); it was a long and complex endeavor for sure, but in the end a success, and QED has made some of the most accurate predictions in science to have been experimentally confirmed. Why not try and do the same?
General relativity is entirely governed by Einstein’s field equation: \[R_{\mu \nu} - \frac{1}{2} R = T_{\mu\nu}\] The left-hand side term depends curvature of spacetime, and the right-hand side is the stress-energy tensor. Together with the usual spacetime symmetries (the Poincaré group), from which we deduce all the conservation laws (of energy, momentum, etc.), that’s the whole theory: Curvature = Energy.
In a modern field-theoretic context, we prefer to start from the lagrangian density, from which the field equations can be derived via the Euler-Lagrange equations anyway. In general relativity that is simply: \[\mathcal{L}_{GR} = \sqrt{-g}R\]
What QFT then goes on to quantize a lagrangian by turning each variable into a quantum field. Then, in principle, the resulting term could be added to any other lagrangian, say the one for the free electron, processed through the machinery, to get some verifiable prediction at the end. For QED that is far from being an easy process. The calculations involve lots of seemingly infinite term, which end up cancelling each other — technically, we say the QED is renormalizable. However a basic attempt to quantize general relativity doesn’t work that way. All the "equipment" is there: where QED identified Maxwell’s electromagnetic potential with a mediating particle: the photon, quantum gravity has the graviton. But in general relativity, the force field is a \((0,2)\) tensor field while in electromagnetism, it is a vector (or \((0,1)\) tensor) field. That makes this basic quantization non-renormalizable.
Once that was ascertained, many different candidates for quantum gravity were proposed. One interesting observation was that gravity arose naturally in the context of locally supersymmetric theories. Supersymmetry is an extension of the familiar symmetries of quantum field theories. It builds a bridge between bosons (force-mediating particles) and fermions (particles subject to force). It was discovered that imposing supersymmetry as a local gauge symmetry of the lagrangian, a procedure that was introduced to solve a range of independent issues, made gravity a necessary addition. That model, called supergravity, still wasn’t renormalizable, however work continued on various models including higher dimensions (\(D>4\)).
It might seem fanciful to do so. After all, the world as we observe it is definitely four-dimensional (and more precisely, (3,1)-D: three spatial dimensions, one time dimension). That said, the idea of compact, and therefore invisible dimensions had been around for a while. Theodor Kaluza, only a few years after Einstein publicized general relativity, developed his own version of a unified field theory, tentatively unifying gravity and electromagnetism. The theory was a 5D version of general relativity but with a catch: the extra spatial dimension didn’t have the same geometry: it was curled up and microscopic, so that a circular condition could be imposed, with a very small radius (around what we now call the Planck length). This both explains why that dimension is undetectable to us and negligible in terms of our regular experience, but it also is the cause, mathematically, for the appearance of Maxwell-style electromagnetic terms in the lagrangian! Kaluza’s theory was not verified, buy the compactification idea stayed.
The largest possible "realistic" supergravity theory (that is, with only one time dimension and one graviton) has 11 dimensions. It is to this day a promising avenue for quantum gravity in the context of superstring theory. In short, string theory is an approach to field theory that considers quantum fields to be describing tiny strings, open or closed instead of points. The extra dimension involved is not visible in our 4D world, so compact dimensions are a normal feature of string theory. What was a fringe projects suddenly became mainstream in the 1980s when supersymmetry applied to strings ("superstring") was shown to be anomaly-free and perturbatively finite for certain dimensions and certain gauge symmetries, including some that could include all the Standard Model symmetries.
The 1990s brought a second "superstring revolution". By that time, several superstring theories existed that were thought of as rivals. In comes "M-theory", whose main architect was Edward Witten. I should be clear that M-theory is not a full-blown theory in the same sense as Einstein’s, or QED. It has no fundamental equations, and makes no testable predictions. All I can say is that it is still, thirty years on, a promising field of research, that it has proved to be a fertile ground for mathematical discoveries, and that whatever happens, it does give us insights about the ultimate reality.
M-theory is a hypothetical 11-dimensional theory of branes. Out of the five rival superstring theories, it admits two as 10D-compactified approximations, and connects the other three via dualities. Even better, its "low-energy" limit is 11D supergravity! Branes are simply generalizations of strings to N dimensions: for a 2D brane, imagine a sheet. A strong argument in favor of superstring theories in general, and hopefully, transitively, of M-theory, is the compactification solution it’s developed. Extra spatial dimensions don’t end up curled over a tiny cylinder like in the Kaluza model, but on a much more complicated Calabi-Yau or \(G_2\) manifold. But that compactification not only yields the correct result, compactifying 7 dimensions, leaving 4 large ones as required: it also should determine the nature of the residual 4D physics... which should encompass everything we know about general relativity and the standard model.
We have been delving into speculative territories, but this is how I would sum up the various steps during the Planck epoch, should this speculation, or more likely some heavily amended version, become reality:
At the beginning, the universe is a single object, a single substance, taking the form of an eleven-dimensional brane field.
Around the Planck scale, the universe’s energy density is "low enough" that 11D supergravity is a good approximation, with individual branes appearing as distinct entities over an 11D spacetime.
Above the Planck scale, characteristic lengths become superior to the string coupling constant and spontaneous compactification kicks in: 7 dimensions now appear as compact, 4 as "large". Brane- or string-related effects become negligible and everything behaves as per an 11D, (7-4)-compactified supergravity theory that includes a supersymmetric grand-unified QFT.