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Section 2: The Origins of Quantum Gravity

In the early 20th century, physicists succeeded in explaining a wide range of phenomena on length scales ranging from the size of an atom (roughly 10-8 centimeters) to the size of the currently visible universe (roughly 1028 centimeters). They accomplished this by using two different frameworks for physical law: quantum mechanics and the general theory of relativity.

Quantum effects meet general relativity somewhere near the center of this length scale.

Figure 3: Quantum effects meet general relativity somewhere near the center of this length scale.

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Built on Albert Einstein's use of Max Planck's postulate that light comes in discrete packets called "photons" to explain the photoelectric effect and Niels Bohr's application of similar quantum ideas to explain why atoms remain stable, quantum mechanics quickly gained a firm mathematical footing. ("Quickly" in this context, means over a period of 25 years). Its early successes dealt with systems in which a few elementary particles interacted with each other over short distance scales, of an order the size of an atom. The quantum rules were first developed to explain the mysterious behavior of matter at those distances. The end result of the quantum revolution was the realization that in the quantum world—as opposed to a classical world in which individual particles follow definite classical trajectories—positions, momenta, and other attributes of particles are controlled by a wave function that gives probabilities for different classical behaviors to occur. In daily life, the probabilities strongly reflect the classical behavior we intuitively expect; but at the tiny distances of atomic physics, the quantum rules can behave in surprising and counterintuitive ways. These are described in detail in Units 5 and 6.

In roughly the same time period, but for completely different reasons, an equally profound shift in our understanding of classical gravity occurred. One of the protagonists was again Einstein, who realized that Newton's theory of gravity was incompatible with his special theory of relativity. In Newton's theory, the attractive gravitational force between two bodies involves action at a distance. The two bodies attract each other instantaneously, without any time delay that depends on their distance from one another. The special theory of relativity, by contrast, would require a time lapse of at least the travel time of light between the two bodies. This and similar considerations led Einstein to unify Newtonian gravity with special relativity in his general relativity theory.

Einstein proposed his theory in 1915. Shortly afterward, in the late 1920s and early 1930s, theorists found that one of the simplest solutions of Einstein's theory, called the Friedmann-Lemaître-Robertson-Walker cosmology after the people who worked out the solution, can accommodate the basic cosmological data that characterize our visible universe. As we will see in Unit 11, this is an approximately flat or Euclidean geometry, with distant galaxies receding at a rate that gives us the expansion rate of the universe. This indicates that Einstein's theory seems to hold sway at distance scales of up to 1028 centimeters.

In the years following the discoveries of quantum mechanics and special relativity, theorists worked hard to put the laws of electrodynamics and other known forces (eventually including the strong and weak nuclear forces) into a fully quantum mechanical framework. The quantum field theory they developed describes, in quantum language, the interactions of fields that we learned about in Unit 2.

The theoretical problem of quantizing gravity

In a complete and coherent theory of physics, one would like to place gravity into a quantum framework. This is not motivated by practical necessity. After all, gravity is vastly weaker than the other forces when it acts between two elementary objects. It plays a significant role in our view of the macroscopic world only because all objects have positive mass, while most objects consist of both positive and negative electric charges, and so become electromagnetically neutral. Thus, the aggregate mass of a large body like the Earth becomes quite noticeable, while its electromagnetic field plays only a small role in everyday life.

The rules of quantum gravity must predict the probability of different collision fragments forming at the LHC, such as the miniature black hole simulated here.

Figure 4: The rules of quantum gravity must predict the probability of different collision fragments forming at the LHC, such as the miniature black hole simulated here.

Source: © ATLAS Experiment, CERN. More info

But while the problem of quantizing gravity has no obvious practical application, it is inescapable at the theoretical level. When we smash two particles together at some particular energy in an accelerator like the Large Hadron Collider (LHC), we should at the very least expect our theory to give us quantum mechanical probabilities for the nature of the resulting collision fragments.

Gravity has a fundamental length scale—the unique quantity with dimensions of length that one can make out of Planck's constant, Newton's universal gravitational constant, G, and the speed of light, c. The Planck length is 1.61 x 10-35 meters, 1020 times smaller than the nucleus of an atom. A related constant is the Planck mass (which, of course, also determines an energy scale); is around 10-5 grams, which is equivalent to ~ 1019 giga-electron volts (GeV). These scales give an indication of when quantum gravity is important, and how big the quanta of quantum gravity might be. They also illustrate how in particle physics, energy, mass, and 1/length are often considered interchangeable, since we can convert between these units by simply multiplying by the right combination of fundamental constants. math icon See the math

Since gravity has a built-in energy scale, MPlanck, we can ask what happens as we approach the Planckian energy for scattering. Simple approaches to quantum gravity predict that the probability of any given outcome when two energetic particles collide with each other grows with the energy, E, of the collision at a rate controlled by the dimensionless ratio (E/MPlanck)2. This presents a serious problem: At some energy close to the Planck scale, one finds that the sum of the probabilities for final states of the collision is greater than 100%. This contradiction means that brute-force approaches to quantizing gravity are failing at sufficiently high energy.

We should emphasize that this is not yet an experimentally measured problem. The highest energy accelerator in the world today, the LHC, is designed to achieve center-of-mass collision energies of roughly 10 tera-electron volts (TeV)—15 orders of magnitude below the energies at which we strongly suspect that quantum gravity presents a problem.

General relativity is consistent with all the cosmological data that characterizes our visible universe.

Figure 5: General relativity is consistent with all the cosmological data that characterizes our visible universe.

Source: © NASA, WMAP Science Team. More info

On the other hand, this does tell us that somewhere before we achieve collisions at this energy scale (or at distance scales comparable to 10-32 centimeters), the rules of gravitational physics will fundamentally change. And because gravity in Einstein's general relativity is a theory of spacetime geometry, this also implies that our notions of classical geometry will undergo some fundamental shift. Such a shift in our understanding of spacetime geometry could help us resolve puzzles associated with early universe cosmology, such as the initial cosmological singularity that precedes the Big Bang in all cosmological solutions of general relativity.

These exciting prospects of novel gravitational phenomena have generated a great deal of activity among theoretical physicists, who have searched long and hard for consistent modifications of Einstein's theory that avoid the catastrophic problems in high-energy scattering and that yield new geometrical principles at sufficiently short distances. As we will see, the best ideas about gravity at short distances also offer tantalizing hints about structures that may underlie the modern theory of elementary particles and Big Bang cosmology.