# Quantum Gravity

Over the centuries, physics as a science has evolved through a close interplay between experiment and theory. However, every once in a while, new phenomena have been discovered which do not conform to existing scientific paradigms. This gives root to radical new ideas which stand in direct conflict to traditional ways of thinking. The history of physics has long been embroiled with such conflicts of paradigms. Consider the friction between classical and quantum physicists of the early 20th century. This shook the very foundations of physics, thus sowing the seeds for a quantum revolution. It is precisely this tension brewing at the interface of antipodal schools of thought which leads to the emergence of new fundamental principles. The objective of a more fundamental theory is then to resolve disparities in preceding paradigms and thereby operate as one unified framework.

The quest for a fundamental theory of gravity is one such instance where the resolution of conflicting paradigms has led physicists to revolutionary new ideas. That gravity is a universal attractive force between objects is something that has been known since the time of Isaac Newton in 1687. However, it was only with Einstein’s theory of general relativity1 in 1915, that a concrete mathematical theory of classical gravity was formulated. This is the classical paradigm of gravity.

Then, in the early nineteen hundreds, came the quantum revolution, lead by the likes of Schrödinger, Heisenberg and Dirac. Now quantum mechanics is a theory of microscopic particles such as atoms and electrons, and their interactions. These are objects typically characterized as small in size and light in weight. Classical gravity, on the other hand, is a theory of macroscopic bodies, which are typically large in size and heavy in weight. So the question is, how should one describe the physics of objects, which are small in size and yet very heavy due to the huge mass they carry? Black holes2 serve as fitting examples of such extreme density objects in our universe. To explain these and, more fundamentally, to comprehend gravity itself, a complete quantum description of gravity is necessary. It was Einstein’s dream to find a unified description of gravity that reconciled the classical and quantum viewpoints. Though a complete answer to this problem is still in the making, a promising candidate for such a quantum theory of gravity emerges in the form of string theory. In addition to unifying Einstein’s theory to quantum mechanics, string theory seeks to go even further and unify all the forces of nature in such a way that they can be understood through a common set of fundamental principles.

Gravity: Nature’s Enigmatic Force

Cosmic ingredients

Our physical universe comprises matter, radiation and their interactions (some would add ‘information’, but here we take the stance that information is encoded in the interactions). The fundamental building blocks of matter are particles called fermions3 – examples of fermions are quarks4 and electrons. The building blocks of radiation are particles called bosons5 – examples being photons, gluons6. The interactions constitute the four fundamental forces of nature: (1) Electromagnetic force – which holds together the electrons in an atom; (2) Strong nuclear force – which holds together the quarks in a proton; (3) Weak nuclear force7 – this interaction facilitates nuclear fusion reactions in stars like our own sun; and (4) Gravity – the attractive force between massive objects. Interestingly, while electromagnetism and gravity have been known since eons, the two nuclear forces were discovered relatively recently, only in the latter half of the 20th century … nearly after the time of Einstein.

An unlikely union

Before the advent of quantum mechanics and Einstein’s special theory of relativity8 in the 20th century, much of physics was based on Newton’s principles of dynamics. Quantum theory shook the very foundations of the Newtonian paradigm and presented us with a whole new world which behaves very differently at microscopic scales (that of atoms and nuclei), while aggregating to classical laws at large distances (those between billiard balls and planets, say). Moreover, quantum mechanics and special relativity gelled together easily to give rise to what we now call relativistic quantum theories. However, general relativity, which is a theory of classical gravity, has remained largely unaffected by the quantum bandwagon. Reconciliation between the two was expected to deliver what would be the ultimate theory of nature: quantum gravity. Unifying these paradigms of classical gravity with quantum theory is what lies at the heart of present day research in theoretical physics.

So what do these quantum theories tell us about the fundamental forces? From the perspective of these theories, a force between two fermions is mediated by the exchange of a boson. For instance, the repulsion between two electrons is facilitated by the emission and absorption of a photon. Photons are the force carriers of electromagnetism. Similarly, the gluons are carriers of the strong nuclear force between quarks, while the W and Z bosons9 do the same for the weak nuclear force. Moreover, the predictions of quantum theory for each of these three forces conform perfectly to experimental evidence. Put together, the quantum description of these three forces is what is called the standard model of particle physics.

On the other hand, according to Einstein’s general theory of relativity, gravity is a property of space-time10. From this point of view, space-time is a dynamic rather than a static entity, whose geometry is responsible for the gravitational attraction between massive bodies. The presence of matter has the effect of distorting the ‘shape’ of the space-time around it. You can create visual analog for this distorting effect by modeling space as a two-dimensional rubber sheet, held stretched. Placing a heavy billiard ball, which could represent a massive object such as a star, at the centre, has the effect of distorting the rubber sheet due to its weight. The distortion is greatest at the location of the ball, and gradually diminishes towards the edges of the sheet. If we were then to make things more interesting by throwing a very small and light pebble onto the sheet, the curvature of the sheet would allow the pebble to orbit around the heavy ball for a while. This models the attraction of a planet around a star as something that results from the distortion of space around it. In fact, this distortion effect exactly accounted for anomalies observed in Mercury’s orbit around the sun and this served as one of the earliest pieces of evidence for Einstein’s theory. Similarly, matter distorts time, but that is much harder to visualize.

Returning to the standard model of the electromagnetic and strong/weak nuclear force, we saw that those theories depicted force as a boson exchange between fermions. But in Einstein’s theory force has something to do with the ‘shape’ of space-time. Therefore, there is apparently no simple way to extend those quantum theories to incorporate gravity. A quantum theory of gravity requires a very new approach which unifies Einstein with the quantum. In a quantum theoretic approach a force is necessarily described in terms of its fundamental building blocks, the bosons that transmit this force. The carriers of gravity are called the gravitons11. However, it turns out that gravitons cannot be found in the spectrum of any of the conventional quantum theories, which work for the other forces. A full description of quantum gravity should reconcile these two notions of force, one as an exchange of gravitons and the other as a manifestation of space-time geometry. String theory is one such attempt to answer these questions. And, remarkably enough, the resolution of some of these paradoxes has stretched our intuitions of space, time and dimensions to their limits.

The World According to String Theory

What is string theory?

The fundamental units of string theory are not particles, but one-dimensional objects called strings. The ends of a string are either joined together or open-ended; the former are called closed strings, the latter open strings. Depending on the energy they possess, strings can vibrate with different frequencies. Analogous to the chords of a musical instrument, a string of a given length and fixed tension has a discrete range of vibrating frequencies (modes or musical harmonics as they are known). The idea now is that each vibrating mode of a string represents a particle of nature. Low energy vibration modes correspond to lighter mass particles, high-energy modes correspond to more massive particles. Remarkably, the spectrum of these vibrations includes the building blocks of matter, radiation and gravity all in one package. Moreover, strings can interact with each other: two closed strings intersect each other at a certain point and open up to form another closed string – a visual analog (although it includes an extra dimension) would be to think of soap bubbles hitting each other and forming another bubble of similar shape. Similarly, open strings interact with other open strings by gluing together at one end to form another open string. This is a consistent quantum theory of interacting strings. The world is then fundamentally comprised of such quantum strings and the different particles we see around us are only manifestations of these strings in different vibration modes. Imagine a ball of a wound up woolen string: that is what a particle would look like. Add some vibration to your woolen ball and there you have a toy model for a specific particle. In this way, vibrating strings can account for the carriers of all four forces of nature, including gravity. At microscopic scales, gravity is indeed described as the exchange of gravitons, while at macroscopic scales string theory reproduces Einstein’s results. Thus, a quantum description of gravity forced upon us a major paradigm shift from the traditional way of thinking in particles to a revolutionary new stringy way of thinking.

Confronting extra dimensions

In string theory, the notion of dimensions takes on a whole new meaning. These can either be macroscopic – the familiar three directions of space and one axis of time; or microscopic. The macroscopic dimensions can be visualized as, say, co-ordinates on a map (at least for the spatial dimensions) that extend outward towards infinity and denote the location of a point. The microscopic dimensions are, instead, rolled up into themselves, like a circle projecting out of every point on a map – only a sufficiently small bug, whose size is smaller than the radius of this circle, will be able to walk into this extra dimension on our map, whereas big bugs only see the map as a grainy surface. String theory predicts six of these extra curled up dimensions12 in our universe and the ‘bugs’ that can probe into these new spaces are the strings themselves. So how small should a string be in order to sense the effects of these microscopic dimensions? A string length is typically of the order of 10-33 cm (this is 1 divided by 10 to the power 33); compare this to the typical diametric size of an atomic nucleus, which is in the order of 10-17 cm (1 divide by 10 raise to 17) – this is 10,000,000,000,000,000 times larger than the length of a string!

Length scales, energy scales and unification of forces

Now let us get a feeling for the interplay between length and energy scales in order to understand at what point the effects of string theory become relevant and how that leads to a subsequent unification of forces. In physics, the scale of length is inversely proportional to that of energy, meaning that shorter distance interactions occur at higher energies and vice-versa. So the energy involved in billiard ball collisions is much less than that involved in nuclear collisions of the type achieved in nuclear reactors. By the same logic, energies of dynamical processes that directly involve string interactions are of the order of cosmic events such as cataclysmic stellar explosions or the big bang itself. The shortest distance scales that present-day technology can probe are processes that occur inside protons and neutrons (quark interactions) and the energy required to probe such interactions lies in the order of giga (109 – that is 10 to the power 9) electron volt units (denoted as GeV13). The most massive/heaviest fundamental particle observed in a lab is the ‘Top Quark’ carrying a mass that is the energy equivalent of 174 GeV. Collision experiments in particle accelerators have to be performed at such high energies in order to observe these massive particles and their short distance interactions. In comparison, creating a light particle like an electron only requires 0.0005 GeV of energy. The hotly anticipated accelerator in Geneva, the LHC12 will, on the other hand, be able to achieve energies of up to 7000 GeV!

The question, then, is what the energy scale is at which string theory interactions can be probed by an experimenter? This is called the Planck Scale and it stands at 1019 GeV. Unfortunately, this is far beyond the reach of current laboratory technology. But these are precisely the energy scales relevant to the processes that occurred during the early history of the universe, just after the big bang, and string theory provides the appropriate theoretical framework to answer some of those questions. But, at a fundamental level, there is something very significant about the Planck scale. It is crucial for understanding the possible unification of forces. Moreover, this is the energy scale at which the quantum nature of gravity becomes manifest. The reason is, quite simply, that the strength of the fundamental forces is not the same at every energy scale. It does in fact vary as we probe physical processes at different energies. For example, in experiments involving billiard balls or even atomic processes, the force of gravity is much weaker than the weak nuclear force, which in turn is weaker than the electromagnetic force between charged bodies, and that itself is weaker than the strong nuclear force. Moving up to about 102 GeV, the regime of sub-nuclear interactions, this gap begins to shrink. Though gravity is still the weakest and the strong nuclear force still the strongest, electromagnetism is now of equal strength to the weak nuclear force. In fact, at this scale, the latter two forces unify into a single force called the electroweak force. Tuning this further up to 1014 GeV, one enters the domain of so-called grand unification theories, wherein the electroweak and strong nuclear forces are expected to be of equal strength and unify. And finally, at the Planck scale and beyond, gravity is as strong a force as the others and its effects have to be considered on a par with the other forces. This is what physicists mean by the unification of all fundamental interactions and string theory stands out as a promising candidate for that endeavor.

The challenge ahead

Though string theory has enjoyed a limited to fair amount of success in explaining some of the puzzles in cosmology and black hole physics, the challenges are far from surmounted and, as things stand right now, it is fair to say that there are more questions on the table than we have answers to. Moreover, in the last two decades, string theory itself has been constantly evolving, with increasingly refined machinery entering the game. It has now been understood that the theory not only comprises one dimensional strings, but also a restricted class of higher dimensional membranes (called D-branes in stringy jargon), which can themselves emit and absorb strings. The theory as such is still very much a work in progress.

Most of what string theory can currently say holds for physics at very high energies like the Planck scale. Experimental signatures for phenomena at these scales are currently sparse due to the huge divide between technology and theory, the latter being far ahead, and that unfortunately hinders many verifications of theoretical predictions. The gap between tested physics at energies up to 102 GeV and Planck physics15 at 1019 GeV is pretty wide. There is absolutely no reason to believe that there is no new interesting physics out there yet to be discovered in that intermediate domain. In one sense, if string theory is a fundamental theory of matter, energy and their interactions, it ought to bridge this divide too – in the same way that quantum mechanics, as a more fundamental theory than Newtonian mechanics, not only explains collisions between atoms, but also shows how classical dynamics, of say billiard balls, emerge as averages over quantum states (at least in principle). Despite its remarkable elegance and heavy mathematical machinery, bridging this gap still remains a daunting task for string theory.16

1. General relativity: In 1915, Einstein generalized special relativity to include accelerated motion in space-time. Starting from this, the theory was able to explain classical gravity through the geometric properties of space-time. One of the predictions of this theory, which was later verified, is that gravity even attracts light!

2. Black holes: Remnants of a massive stellar collapse, whose gravity is so strong that anything which comes within a certain critical radius of a black hole is sucked into it forever – even light rays. That is why they are called ‘black holes’.

3. Fermions: The class of particles that constitute all forms of matter. Quarks and electrons are typical examples.

4. Quarks: Particles that make up protons and neutrons in atomic nuclei.

5. Bosons: The class of particles that constitute all forms of radiation. Photons and gluons come under this category.

6. Gluons: Carriers of the strong nuclear interaction between quarks.

7. Strong/Weak nuclear force: Two types of nuclear forces exist within atomic nuclei. Both enable short-range interactions within constituents of the nucleus. The stronger one is mediated by gluons, the weaker of the two, by W and Z bosons. The mechanism that holds protons and neutrons together in the nucleus is governed by the strong force, while the mechanism that is responsible for radioactive beta-decay is governed by the weak force.

8. Special relativity: Einstein’s 1905 theory stating that all uniform motion (of particles) and measurements thereof are relative with respect to the observer. This had far-reaching implications for the laws of Newtonian mechanics and shattered the notion that space and time are absolute entities.

9. W and Z bosons: Both these particles are carriers of the weak nuclear force.

10. Space-time dimensions: In general relativity, length, breadth and height are the three dimensions of space. Time is the fourth dimension. Combined together this gives the ‘fabric’ of space-time.

11. Gravitons: Carriers of the gravitational force.

12. These 10 dimensions exist in the formulation of ‘Superstring theory’, as it is also called. A generalization of this theory goes by the name ‘M-theory’, which lives in 11 dimensional space-time. Currently, very little is known about M-theory.

13. GeV: Giga electron volts = 109 electron volts. An electron volt is a unit of energy used in particle physics. For example, if the entire mass of an electron were converted into energy, it would be 500,000 electron volts of energy.

14. LHC: The Large Hadron Collider is a particle colliding machine at the European laboratory for nuclear research (aka CERN) in Geneva. When fully operational, it will be well poised to discover new physics beyond what current particle physics models tell us, and at higher energy scales than what we are able to probe today.

15. Planck physics: Physical phenomena relevant at Planck scale energies, that is at 1019 GeV. The behavior of physical interactions is drastically different as we probe processes at different energy scales. The Planck scale is the regime of quantum gravity and the unification of forces.

16. Let us also acknowledge other independent approaches to quantum gravity; the two most prominent among them being Loop Quantum Gravity and the theory of Causal Dynamical Triangulations. Owing to the author’s limited knowledge there is very little to discuss in relation to these.**Recommended literature***Basic reading:*

Close, F.E., *The Cosmic Onion: Quarks and the Nature of the Universe*, Melville, 1986.

Gilmore, R., *Alice in Quantumland: An Allegory of Quantum Physics*, New York, 1995.

Greene, B.R., *The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory*, New York, 1999.

Hawking, S.W., *A Brief History of Time: From the Big Bang to Black Holes*, London, 1988.

NOVA, *The Elegant Universe*, 2003,http://www.pbs.org/wgbh/nova/elegant/program.html (4 November 2009).*Advanced reading:*

Halzen, F. and Martin, A.D., *Quarks and Leptons: An Introductory Course in Modern Particle Physics*, Hoboken, 1984.

Misner, C.W., J.A. Wheeler and K.S. Thorne, *Gravitation*, San Francisco, 1970.

Peskin, M.E. and D.V. Schroeder, *An Introduction to Quantum Field Theory*, Boulder, 1995.

Polchinski, J., *String Theory, Vol. 1 : An Introduction to the Bosonic String*, Cambridge, 1998.

Polchinski, J., *String Theory, Vol. 2 : Superstring Theory and Beyond*, Cambridge, 1998.