Quantum Gravity

1. Theories of everything

In the 20th century, people realised that just four types of force - electromagnetism, the strong and weak nuclear forces, and gravity - are responsible for all the forces that we experience in the universe.

Electromagnetism and the strong and weak nuclear forces are now understood using quantum field theories. Quantum mechanics shows that at high energies, electromagnetism and the weak force combine to form a single force, known as the electroweak force[1][2][3].

It's generally expected that at even higher energies, the strong force may combine with the electroweak force, although this has not yet been proven. A theory that shows that the electroweak and strong force combine into a single force at high energies is known as a grand unified theory (GUT)[4].

A theory which shows that gravity combines with this other force at even higher energies, so that all the forces can be described by a single theory, is sometimes called a theory of everything (TOE), although it will almost certainly not explain everything[5].

Diagram showing that all forces may have been unified until about 10<sup>-43</sup> seconds after the big bang, when the force of gravity separated. The strong force separated about 10<sup>-35</sup> seconds after the big bang, leaving the electroweak force, and the weak force separated 10<sup></sup> seconds after the big bang, leaving the electromagnetic force.

The unification of forces. Image credit: Helen Klus/CC-NC-SA.

1.1 Gravity and quantum mechanics

Although the electroweak force has not yet been shown to unify with the strong force, it's thought to be even more difficult to unify the rest of the forces with gravity. This is because we have not yet combined gravity with quantum mechanics, as we have done with the other forces[6a].

A theory of quantum gravity is needed to describe things that are very small but also very heavy, like black holes or the early universe. Yet gravity is described by general relativity, which seems fundamentally incompatible with quantum mechanics. This is because in general relativity all physical qualities have definite values, whereas in quantum mechanics they do not. This is shown in Heisenberg's uncertainty principle[7].

Heisenberg's uncertainty principle shows that it is a physical law of the universe that the more precisely you measure the position of something at any one time, the more its momentum - which depends on its mass and velocity - can fluctuate.

A similar relationship holds for energy and time, so that something can gain a higher energy than is classically possible for a short period of time. This can be observed in quantum tunnelling, which sometimes allows particles to 'borrow' enough energy to travel through an otherwise impenetrable barrier.

Classical theories of the electromagnetic field were once considered incompatible with quantum mechanics, and this was overcome by quantising the field. This means associating the quantum state of the electromagnetic field with the quantum state of charged particles[8].

It would seem then, that we should be able to associate the quantum state of the gravitational field with the quantum state of particles with mass. Yet when scientists tried to do this, the answers made no sense, both mathematically and conceptually[6b].

If Heisenberg's uncertainty principle were applied to spacetime itself, then we would expect it to also be continually fluctuating, with no definite values. This would mean that on very small scales, spacetime is not 'smooth', as is implied by general relativity, but is in fact 'foaming' with energy continually coming into, and out of, existence.

The smaller the scale, the higher the energy. Special relativity shows that energy is related to mass[9], and so this allows for very massive objects - such as microscopic black holes - to come into, and out of, existence in otherwise empty space.

General relativity shows that mass curves spacetime[10], and so these objects should drastically affect the shape of empty space, causing it to collapse in on itself. We know that 'empty' space is full of virtual particles coming into and out of existence due to quantum field theory, but we do not observe spacetime ripping itself apart, and so this idea must be wrong.

Another problem with the idea of quantising spacetime is that quantum mechanics and general relativity both have different, and incompatible, definitions of time:

  • In general relativity, time is a component of spacetime, which is shaped by mass. The 'shape' of spacetime can change, it does not provide a fixed background, and so time is relative.

  • In quantum mechanics, time is a fixed background in which things take place, and does not change. Time is absolute.

Quantum theory can predict what will happen at an 'instant', but in general relativity there are no real 'instants', so it is unclear how things can change in time.

So far, no one has been able to prove how the force of gravity works on a quantum scale, although there are many different ideas[6c]. Two of the most popular approaches to quantum gravity are loop quantum gravity and string theory.

2. Loop quantum gravity

Indian physicist Abhay Ashtekar devised loop quantum gravity in 1986[11], and Italian physicist Carlo Rovelli and American physicist Lee Smolin developed the theory in 1988[12].

Loop quantum gravity attempts to quantise spacetime, so that there’s a minimum observable distance. It addresses the problem of combining two incompatible definitions of time by fixing the time before spacetime is quantised, so that only space is really quantised.

In loop quantum gravity, space is discrete rather than continuous, in a similar way to how the electromagnetic force is discrete because it's composed of photons. In loop quantum gravity, spacetime is composed of finite loops, called spin networks.

The smallest observable length is on the same scale as a spin network, and so anything that happens below this length has no observable effect on the rest of the universe.

Loop quantum gravity is not a potential TOE because it doesn't try to combine gravity with the other three fundamental forces.

3. String theories

String theory began with equations developed by Italian physicist Gabriele Veneziano in order to describe the strong nuclear force in 1968[13]. This theory has since been surpassed by quantum chromodynamics (QCD).

In 1970, American physicists Yoichiro Nambu[14] and Leonard Susskind[15], and Danish physicist Holger Nielsen[16] independently discovered that Veneziano's theory is a quantum theory of relativistic vibrating strings, a type of string theory[17].

The initial idea behind string theory was that elementary particles are not dimensionless 'points', but instead exist in one dimension. They have a length but no width, and are known as strings.

The one-dimensional strings of string theory can vibrate in a similar way to three-dimensional strings, like the strings on a guitar. Whereas we experience the vibration of three-dimensional strings as music, the vibrations of one-dimensional strings give rise to what we experience as the fundamental properties of elementary particles, such as mass, charge, and spin. Different types of vibrations give rise to different types of particles.

The vibrations of guitar strings differ, depending on how tightly the strings are stretched, where a tighter string produces a higher pitch. The vibrations of one-dimensional strings also differ depending on their tension. Although the strings in string theory are unlike guitar strings because they do not necessarily have to be tied down to have tension. One-dimensional strings can be open, like a line, or closed, like a circle.

In 1974, French physicist Joel Scherk and American physicist John Schwarz showed that the graviton - the particle thought to be associated with the force of gravity - naturally arises from string theory; one of the particles states of a closed string has its exact properties[18]. Japanese physicist Tamiaki Yoneya came to the same conclusions independently[19][20], and string theory was reassessed as a quantum theory of gravity[21].

The strings in string theory are around the same length as the spin networks of loop quantum gravity. Unlike loop quantum gravity, however, string theory does not treat spacetime as a quantum mechanical entity, and so does not face the problem of combining two different concepts of time.

Strings are not part of spacetime, but exist within it, and so are probably not truly fundamental.

3.1 Bosonic string theories and extra dimensions

The first string theory only applied to bosons, these include all the elementary particles that transmit forces but none of the elementary particles that constitute matter, which are fermions.

In the late 1960s and early 1970s, four types of bosonic string theories were developed. They differed depending on whether the theory just involved closed strings, or whether it included strings that were open and closed, and on whether or not the strings were orientable (i.e. whether or not you can tell which way a string is wound). All of these theories predicted the existence of extra dimensions, having 26 dimensions in total, where 25 are spatial and 1 is temporal[22].

We clearly don't experience more than three spatial dimensions in everyday life. We can specify our location, for example, with three coordinates, our latitude, longitude, and height above the surface of the Earth. We cannot visualise a fourth spatial dimension, yet the existence of 'hidden' spatial dimensions had already been suggested before string theory was developed.

In 1921, German physicist Theodor Kaluza showed that the force of electromagnetism could be combined with the force of gravity if there were four spatial dimensions, instead of three[23].

In 1926, Swedish physicist Oskar Klein showed that a fourth dimension may exist in our universe but be unobservable because it is 'curled up', so that it appears to be extremely small[24][25].

General relativity shows that dimensions can be curved, since spacetime can be curved, and it's easy to imagine a finite curved coordinate system. The surface area of the Earth, for example, is two-dimensional, but it's curved so that if you travelled around it you could end up back where you started.

It's also relatively easy to imagine a dimension could be too small to be noticed. A wire, for example, may be so long and thin that it looks like a line from a distance, but something smaller and closer, like an ant walking across it, will notice its second and third dimensions.

Klein suggested that the fourth spatial dimension is similar to this. It's like a tiny circle, and you can travel around it and get back to where you started without noticing. It is said to be 'compacted'.

In ordinary life, you do not need to specify your location on the fourth spatial dimension in order to tell someone where you are because your position in this dimension is inconsequential. You could be anywhere on it and still appear in the same place.

Even though we don't notice this extra dimension, it can still affect spacetime, and so when string theory was developed, the extra dimensions it predicted were explained by Kaluza and Klein's idea that they are too small to be noticed.

4. Superstring theories

Bosonic string theories are all unstable, and were soon surpassed by five stable theories, which could describe all of the elementary particles, both fermions and bosons. These are known as Type I, Type IIA, Type IIB, E8xE8 heterotic, and SO(32) heterotic theories, and they were developed in the 1970s and 1980s. These string theories only predict 10 dimensions, 9 of which are spatial. They all involve something known as supersymmetry and so are known as superstring theories.

Supersymmetry was independently discovered in 1971 by physicists Yuri Gol'fand and Evgeny Likhtman while working at the USSR Academy of Sciences[26], American physicist Pierre Ramond[27], and American physicist John Schwarz and French physicist André Neveu[28].

Supersymmetry relates fermions to bosons by showing that at high energies there is a fermionic partner for every boson and vice versa. These are known as 'superpartners' and it is thought that the superpartners of some bosons, known as neutralinos, might constitute dark matter. There is no evidence of superpartners yet, although physicists at CERN have been searching since 2010[29].

In the 1980s, it was shown that string theories can be made supersymmetric. This idea was developed by Schwarz and British physicist Michael Green in 1981[30]. Superstring theories were found to not just be theories of one-dimensional objects; they allow for objects with zero to nine dimensions, called p-branes.

Strings are p-branes of one dimension, also known as one-branes. Branes are able to interact and form D-branes. These occur when the ends of open p-branes are attached to another static p-brane. Theories of D-branes were independently developed in 1989 by Czech physicist Petr Hořava[31], and physicists Joe Polchinski, Jin Dai, and Rob Leigh[32], working at the University of Texas.

Diagram of branes. A D0-brane is a circle, a D1-brane is a line, and a D2-brane is a plane.

Different types of D-branes. Image credit: Helen Klus/CC-NC-SA.

4.1 Calabi-Yau space

Different explanations for how extra dimensions can remain unobservable were developed using superstring theories. Kaluza and Klein showed that extra dimensions could exist in our universe if they were compacted and, in the 1980s, it was shown that the 6 extra spatial dimensions of superstring theories could be compacted together in something known as a Calabi-Yau manifold.

Illustration of a Calabi-Yau manifold.

Depiction of a Calabi-Yau manifold. Image credit: Jbourjai/Public domain.

The term 'Calabi-Yau space' was first used in a paper by British mathematician Philip Candelas and American physicists Gary Horowitz, Andrew Strominger, and Edward Witten, in 1985[33]. It was named after American mathematician Eugenio Calabi who had first developed the idea in 1954[34], and American mathematician Shing-Tung Yau who had extended Calabi's work in 1976[35].

The main problem with this idea is that there are thousands of ways that extra dimensions could be compacted and no way to know which is correct.

One way to prove there are compacted extra dimensions is to see how certain forces act at these scales. In the three-dimensional universe we usually experience, the force of gravity between two objects that have mass, and the electromagnetic force between two objects that have charge, are both inversely proportional to the square of the distance between the objects.

The fact that the number is squared is related to the fact that there are three spatial dimensions to consider. If we were able to look at how these forces behave on a scale so small that compacted dimensions need to be considered, then they should decrease more rapidly. For a string theory with 9 spatial dimensions, these forces should be inversely proportional to distance to the power of 8, rather than 2.

Evidence of forces decreasing more rapidly at smaller scales might one day be found using particle accelerators at extremely high energies, or from cosmic rays, although it is not yet known if this will be possible. This is because it is not known how small these extra dimensions really are, and the smaller they are, the more energy is required for us to find them.

4.2 Brane worlds

The fact that there's no way to predict the correct way to compact dimensions in Calabi-Yau space led to a different option being proposed in the 1990s, that of brane worlds[36][37][38][39][40].

In brane world scenarios, at least one of the extra spatial dimensions posited by superstring theories is not compacted. Non-compacted spatial dimensions must be extremely large, possibly infinite. They make up the 'bulk', and the universe we experience is a three-dimensional D-brane, or brane world, a sub-space of the bulk.

This is similar to the idea that if you were alive inside of a computer screen, then you would only be able to move in two dimensions, despite knowing you really exist within a larger, higher-dimensional universe.

The open strings, which make up most of the particles and forces we experience, are attached to the brane world, and so cannot be used to probe the bulk, but the closed strings, which carry gravitational forces, and determine the shape of spacetime, are not. This means that we may be able to feel the effects of massive objects outside of our brane world.

Ours may not be the only brane world in the bulk. There may even be an infinite amount of brane worlds, and interactions with other brane worlds could affect our own. The big bang, for example, could have occurred from a collision between brane worlds, which caused our own to expand.

There's currently no observational evidence for the brane world scenario, although some models could be tested by future experiments using CERN's Large Hadron Collider (LHC).

5. M-theory and duality transformations

The number of superstring theories was reduced in the 1990s, when it was shown that different superstring theories are connected by duality transformations. This means that one superstring theory can be transformed into another; they are two manifestations of a single theory[41a][42a].

Duality transformations are not unique to string theory. Given a room that is 10 metres wide, for example, the two sentences 'the ball is 1 metre from the edge of the wall on the right hand side' and 'the ball is 9 metres from the edge of the wall on the left hand side' are equivalent and related by duality transformations.

There are two types of dualities in string theory, known as T-duality and S-duality.

Diagram showing how the different types of superstring theories are related. The Type IIA and Type IIB, and SO(32) heterotic and E8xE8 heterotic superstring theories are related by T-duality. The SO(32) heterotic and Type I superstring theories are related by S-duality. This also relates the Type IIB superstring theory with itself. The E8xE8 heterotic and Type IIA superstring theories are related by S-duality under M-theory.

Types of superstring theories. Image credit: modified by Helen Klus, original image by Alex Dunkel/ShotmanMaslo/Polytope24/CC-SA.

5.1 T-duality

T-duality relates theories where one of the extra spatial dimensions is very small, with theories where this extra spatial dimension is very large.

The Type IIA and Type IIB superstring theories are related by T-duality, and so are thought to be two manifestations of a single theory. The same is true for the SO(32) heterotic and E8xE8 heterotic superstring theories. These are all theories that only involve closed strings.

Closed strings interact by splitting and joining, and, if a dimension is small enough, they can wrap around it. The number of times they wrap around is known as the winding number, which is quantised. If the dimension is too large for them to wrap around, then they will contract to their smallest size, acting similarly to a point particle with a quantised momentum.

Mathematically, theories with relatively large dimensions, high momentums, and low winding numbers, are equivalent to theories with relatively small dimensions, low momentums, and large winding numbers.

The winding number in one theory is identical to the momentum in another, and so large and small distance scales also appear to be mathematically equivalent, even though they appear very different to us.

This could have implications for the three spatial dimensions that we experience. If it were proven that the universe is not infinite, and we could, theoretically, travel around it and get back to where we started, then T-duality shows that we could also think of it as being extremely small.

What we think of as momentum could be strings wrapped around the whole universe.

5.2 S-duality

S-duality relates the SO(32) heterotic and Type I superstring theories, and relates the Type IIB superstring theory with itself. The SO(32) heterotic and Type IIB superstring theories only involve closed strings, but the Type I theory works for both open and closed strings.

S-duality relates coupling constants. Coupling constants show how strong an interaction is. The coupling constant for the gravitational force, G, shows how strong the gravitational force is. If it were twice as large then the force of gravity would be twice as strong.

The coupling constant in string theory is related to the probability that a closed string will break apart, or that an open or closed string will join with other strings. S-duality shows that a theory where strings hardly ever break apart, or join together, can be mathematically equivalent to a theory where they break and join easily.

5.3 M-theory

Dualities between superstring theories show that what we think of as different theories are actually the same, and so it's possible that all five superstring theories are related. They may be five manifestations of a single theory, known as M-theory.

M-theory relates Type IIA and E8xE8 superstring theories so that all the superstring theories are related, and was developed in the mid-1990s, following papers by Witten[41b] and British physicists Chris Hull and Paul Townsend[42b].

M-theory requires an extra spatial dimension, making it an 11-dimensional theory, where 10 dimensions are spatial and one is temporal.

5.4 Could there be more than one time dimension?

All of the extra dimensions predicted by string theories are spatial, although it may also be possible for universes, or brane worlds, to exist with more than one temporal dimension.

Spatial dimensions give us the options for directions to move in space. We can turn around, for example, so objects that were once in front of us are now behind us. An extra temporal dimension could give us this same freedom to move in time, so that the future can become the past, from our perspective. Although it's difficult to imagine what this would really mean.

In 1997, Swedish-American cosmologist Max Tegmark showed that any possible brane world with a different number of visible spatial or temporal dimensions to our own would be uninhabitable[43].

Life would not be able to evolve in a universe with more, or less, than one time dimension, as there would be no way to predict the future.

In a universe with more than three visible spatial dimensions, there may be no stable structures, and with less, the force of gravity may be too weak for life to exist.

Plot showing the number of spatial dimensions against temporal dimensions. If there are zero spatial or zero temporal dimensions, then the results are unpredictable. They are also unpredictable if there is more than one spatial dimension and more than one temporal dimension. If there is one spatial dimension and more than three time dimensions, then the universe will be unstable. The same is true is there is one temporal dimension, and more than three spatial dimensions. If there are three temporal dimensions and one spatial dimension, then only tachyons will exist, and if there are one to two temporal dimensions, and one spatial dimension, or vice versa, then the universe will be too simple for life to form. This leaves one temporal dimension, and three spatial dimensions as the only possibility.

Image credit: Max Tegmark/CC-SA.

Around this time, American physicist Itzhak Bars showed that the mathematics needed to describe a universe with one observable time dimension, and three observable spatial dimensions, is the same as that needed to describe a universe with two observable temporal dimensions and four observable spatial dimensions[44][45].

Our universe may just appear to have one temporal and three spatial dimensions at low energies, and so we do not observe effects that would otherwise lead to the future becoming unpredictable.

When Bars' theory is applied to M-theory there are predicted to be 13 dimensions in total, although this has not yet been proven.

6. Black holes and the holographic universe

Superstring theories were applied to black hole thermodynamics in the 1990s, leading to the discovery of the holographic principle[46a][47a][48a].

Black holes were shown to be thermodynamic objects by Israeli-American physicist Jacob Bekenstein[49] and British physicist Stephen Hawking[50][51] in the 1970s.

Bekenstein and Hawking showed that black holes appear to follow analogous laws to the laws of thermodynamics, where the temperature of a system in thermal equilibrium is analogous to the surface gravity of a black hole at the event horizon, and the entropy of a system is analogous to the surface area.

Entropy is a measure of how disordered a microscopically random system is. This is equivalent to how much information is needed to describe it. There is more disorder, and therefore more entropy, in a glass of water, for example, than in a glass that is half water and half ice.

The second law of thermodynamics states that in a closed system, things get more disordered, increasing in entropy, over time[52].

Black holes were classically thought to have zero entropy. Bekenstein showed that this could not be the case, however, because this would violate the second law of thermodynamics.

If nothing can escape a black hole, then the entropy of anything that falls in it will be lost, the system would decrease in entropy over time. Bekenstein showed that this means black holes must have entropy, and that a black hole's entropy must be proportional to the surface area of its event horizon.

Hawking showed that if black holes are thermodynamic objects, then they must emit particles, known as Hawking radiation. Hawking radiation occurs when the particle-antiparticles pairs that continuously come into and out of existence in a vacuum are created at the event horizon of a black hole. One half of these pairs disappears into the black hole before it can annihilate the other. The other half escapes the black hole, and the mass of the black hole decreases by the same amount as the mass of the particle.

Objects that cross the event horizon of black holes become unobservable and, if black holes decay, then information about them could eventually be lost forever. This violates the idea that information about a physical system at one time can determine its state in the future, and is known as the black hole information paradox[53][54].

The black hole information paradox may be resolved with the holographic principle. In 1993, Dutch physicist Gerard 't Hooft showed that when things fall into a black hole, they deform the surface of the event horizon[46b]. This affects the Hawking radiation that is produced, and so information is not lost forever as the black hole decays.

Inspired by the idea that the entropy of a black hole is proportional to its surface area, rather than its volume, 't Hooft showed that there cannot be more information stored in a three-dimensional space than is needed to store it in a two-dimensional space. Susskind later applied 't Hooft's idea to superstring theories[47b].

Susskind identified black holes with specific string states, and showed that oscillations on the surface of a black hole's event horizon give a complete physical description of both incoming and outgoing matter.

In 1996, American physicist Andrew Strominger and Iranian-American physicist Cumrun Vafa confirmed Bekenstein and Hawking's entropy model using superstring theories[48b].

The conclusions of the holographic principle can be extended to show that the entire observable universe may really have two, rather than three, visible spatial dimensions, with information 'painted' on the surface.

This could mean that the third dimension that we experience is like the third dimension created by an ordinary hologram, although this has not yet been proven.

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