A Brief Overview of Loop Quantum Gravity

Quantum gravity is one of the hardest problems in Physics. The force of gravity is one of the four known fundamental forces of nature. Theoretical physicists have successfully discovered mathematical models that explain and unify the first three fundamental forces, electromagnetic force, strong nuclear force, and weak nuclear force into one single framework called Quantum Field Theory. But there is a big challenge when we try to incorporate the force of gravity into Quantum Field Theory. For eight decades, physicists have been struggling to find a way to successfully incorporate gravity into Quantum Field Theory but up till today, there is no success in all the attempts that have been made. This article focuses on briefly overviewing one novel approach taken by many groups of theoretical physicists called Loop Quantum Gravity.

A Short Account of the Past Scientific Literature

The discovery of the gravitational force is attributed to Sir Issac Newton. It was perhaps an accidental event that one day Sir Issac was sitting under the shade of a big apple tree and an apple fell on this head. A shocked Sir Issac then asked himself what could be the reason behind the apple falling on his head. He started a deep investigation of the matter and finally ended up with the famous Newton’s Law Of Gravitation. His law of gravitation was very successful to the extent that he could prove on theoretical grounds why the planets in our solar system go around the sun in elliptic orbits. He successfully proved all the three laws of planetary motion which were discovered by Kepler after carrying out ten years of astronomical observations of planetary motion.

But as time progressed and detailed astronomical observations were carried out, it was found that the orbital of the planet Mercury shows precession, an effect which could not be deduced from Newton’s law of gravitation. Many attempts were made to explain the above effect by hypothesizing the presence of some celestial object that causes the orbit of Mercury to precise but none of them were very successful.

Now Albert Einstein came into the picture with his revolutionary Special Theory Of Relativity. Albert Einstein was not satisfied with the Newtonian description of the gravitational force. From the special theory of relativity, he was convinced that no information in the universe can travel faster than the speed of light and a force can be perceived as information given from one body to another body to change its present configuration. How is it that sun exerts gravitation attraction on the planets located a very large distance apart in no time? Also, he was looking to generalise his special theory of relativity and incorporate accelerating frames of reference into a final framework called General Theory Of Relativity. After publishing his paper on the special theory of relativity, he devoted the next ten years of his life to find answers to his above queries. In 1915, he managed to discover ten equations that collectively describe the geometry of space-time influenced by the matter that is present in it. His philosophy can be described with the following phrase, “Matter tells space-time how to curve and space-time tells matter how to move.” Einstein’s General Theory of Relativity was astonishingly successful in explaining the long unresolved problem of precession of Mercury’s orbit around the sun. Following that Einstein himself predicted the existence of travelling oscillating vibrations on the fabric of space-time called Gravitational Waves which has been recently detected by project LIGO in the year 2015. Another completely strange and unexpected deduction from Einstein’s General Theory of Relativity was the existence of Black Holes. It was Karl Schwarzschild, a German astrophysicist who surprisingly solved the complex Einstein’s field equations and predicted the existence of Black Holes, a region in spacetime with intensely strong gravitational field. Since then, General Relativity has been a topic of great interest among physicists.

Parallel with the development of Theory of Relativity, another theory was under development, the Quantum Theory. Max Planck was the first one to introduce the idea of quanta or quantum in the literature of physics. Quantum means discrete. Using the concept of quantum, he could account for the phenomenon of black-body radiation shown by black-bodies. Later Albert Einstein went further and predicted that light is a stream of quanta or particles called the photon. Neils Bohr proposed a new model of the atom in which he introduced the idea of quantised angular momentum leading to discrete circular orbits for the motion of the electrons around the nucleus and surprisingly it could account for the absorption and emission spectrum of hydrogen atoms. This model of the atom was developed further and it was discovered that systems at the atomic level tend to have discrete or quantised values of physical quantities. Louis Victor de Broglie hypothesised the wave-like behaviour of subatomic particles and it was proved experimentally by Clinton Davisson and Lester Germer that particles apart from the photon, like electron does show wave-like property. Erwin Schrodinger derived an equation for the hypothetical wave property of matter. But the solutions of the equation did not have any meaningful interpretation, though from the equation many observed properties of atomic systems could be naturally derived. It was after Werner Heisenberg proposed his uncertainty relation for subatomic systems that a probabilistic interpretation for the solutions of the Schrodinger’s equation was realised. A new field called Quantum Mechanics eventually started following the above-mentioned discoveries. Quantum Mechanics and Theory of Relativity are the two pillars of Modern Physics. The former is used for studying phenomena in the microcosm and the latter is used for studying phenomena in the macrocosm. Attempts to unify Quantum Mechanics and Special Theory Of Relativity have been successful leading to Quantum Field Theory but it is very difficult to unify General Relativity with Quantum Mechanics.

Unification of Quantum Mechanics and General Relativity

Quantum Mechanics and General Relativity are two ends of the natural world we know and there are many contradictory ideas in between these two theories. Quantum Mechanics has a probabilistic interpretation whereas General Relativity has a causal interpretation. Previous experience has shown the scientists that using the established theories to reduce General Relativity to the microcosmic domain and Quantum Mechanics to the macrocosmic domain does not give us acceptable results. Another sharp contradiction between Quantum Mechanics and General Relativity is that the former allows non-local interactions between the components of a physical system (“spooky action at distance” in Albert Einstein’s words) which occurs instantaneously irrespective of distance, whereas General Relativity allows only local interactions and no information in the framework of General Relativity can travel faster than the speed of light. General Relativity also predicts that electrons in an atom should emit very weak gravitational waves due to their accelerated motion, lose energy and collapse into the nucleus whereas Quantum Mechanics has proved atoms to be very stable. Solutions of Einsteins’ field equation for black holes predicts the existence of a point called singularity where space-time curvature tends to be infinity. It was also believed that information gets lost once it crosses the event horizon of the black hole.

Probably a quantum mechanical description of gravity can resolve the above issues. Thus, world-wide theoretical physicists have set forward to develop a quantum mechanical description of gravity.

What is Loop Quantum Gravity?

Loop Quantum Gravity is an approach to develop a quantum mechanical description of gravity as described by General Relativity. According to General Relativity, gravity is not a force but a property of spacetime. Loop Quantum Gravity attempts to formulate a geometric description of gravity at the Planck scale ( 10−35m) respecting all the core principles of Quantum Mechanics. All quantum mechanical system possess quantised values of energy, momentum and respects the principle of uncertainty. Loop Quantum Gravity claims a discrete or granular structure of spacetime.

Just like atoms in crystal organise into a lattice framework of certain geometry, spacetime also prefers to adopt a lattice-like framework. Each discrete unit of spacetime is called a loop and hence the name of the theory is Loop Quantum Gravity. Quantisation of spacetime implies the existence of a smallest length of space (smallest area of space, smallest volume of space and smallest interval of time). General Relativity is formulated using a class of higher-dimensional geometry called Riemann Geometry and Loop Quantum Gravity is formulated with a similar geometric model but including quantisations -Quantum Geometry.

The development of Loop Quantum Gravity started with an Indian theoretical physicist Abhay Ashtekar who formulated a theory which partially unified Quantum Mechanics and General Relativity on theoretical grounds. Inspired by Ashtekar’s work, Lee Smolin and Ted Jacobson found that an equation named Wheeler-DeWitt equation as an excellent candidate for quantum gravity. Wheeler-deWitt’s equation admitted solutions which are loops just as we have discussed above. Later Carlo Rovelli and Lee Smolin formulated Loop Quantum Gravity as a background-independent, non-perturbative theory of quantum gravity. Background independence is a property of a theory which does not assume a prior geometry of its physical environment but deduces them by the dynamical equations of the theory. General Relativity is a background-independent theory as Einstein’s field equations gives the space-time manifold depending on the given conditions of the physical system. Loop Quantum Gravity also proceeds with the same spirit. In theoretical physics, a non-perturbative solution is one that cannot be described in terms of perturbations about some simple background, such as empty space.

We next discuss the geometries in Loop Quantum Gravity.

Spin- Networks and Spin- Foam

Spacetime geometry in Loop Quantum Gravity is described using spin-networks. We will discuss this idea in short. Mathematicians often represent three-dimensional geometric objects in one-dimensional labelled graphs. These representations of an object are called dual representations. A graph is a one-dimensional geometric structure composed of nodes interconnected to each other by lines labelled with some values to highlight some attribute of that line. Carefully consider the example of a cube above. Observe that there is a zero-dimensional central point from which six one-dimensional lines/appendages originate and intersect the six different faces of the cube. The dual one-dimensional graph of this cube is given above on the right with 1 as an arbitrary label on each line.

Spin-Networks are graphs like the one on the side which depict the spin-state of spacetime at the Planck scale for a given time. The name spin is derived from the spin-states of elementary particles in Particle Physics.

Now consider a spin-network of space-time as given on the side. We will just reverse the way we approached the previous object. Each node is representing a central point in space surrounding which all the lines represent unique faces forming a complex structure around that central point. Each complex structure has the property that the faces intersect in one-dimensional lines, forming a long connected mesh-like structure representing space-time.

The diagram on side roughly depicts the above elaboration of spin- networks. Observe the spin-value on each of the lines in the spin-network graph. These numbers are abstract but in some sense, they represent the amount of geometry or area associated with the faces represented by those lines. Quantisation rules are also applicable to the areas of these faces and hence the spin-values of the faces are discrete in nature. This mesh structure of space-time is very dynamic in nature with continuous appearance and disappearance of those faces. This is called Quantum Evolution of spacetime and the resulting chaotic form of spacetime is called spin foam. Matter is allowed to move along those faces of spacetime. This is a non-mathematical idea of quantum loops.

Applications of Loop Quantum Gravity

Loop Quantum Gravity has been applied to many different scenarios where our present theories of physics seems to break down. Black hole singularity and the Big Bang singularity are the two major places where this theory has been used and it has given some probable insight into the nature of our universe and peculiar objects like black holes. Below we discuss some of them in short.

Planck Stars and Black Holes

Black holes are one of the most peculiar objects in our universe. Anyone who reads about it for the first time definitely asks the question, “What is inside a black hole?”. General Relativity yields a point of singularity inside a black hole where spacetime curvature tends to infinity, but this result is not very convincing.

Black holes with their enormously strong gravitational field eat up all matter that passes nearby it’s event horizon but where does all this matter go as they try to reach the singularity? Energy cannot be destroyed. There is one such problem known as the black hole information loss paradox. Calculations suggest that physical information could permanently disappear in a black hole, allowing many physical states to devolve into the same state. This is controversial because it violates the core precept of modern physics or the Copenhagen interpretation in Quantum Mechanics—that in principle the value of a wave function of a physical system at one point in time should determine its value at any other time. To resolve these contradiction, Loop Quantum Gravity proposes the presence of an astrophysical object in the core of a black hole called the Planck Star.

Planck Star is an extremely small star of Planck dimensions and has high energy density. Planck Star is formed due to the strong repulsive forces operating at the Planck scale when a massive star collapses into a black hole. These repulsive forces originate from the quantum nature of spacetime. Planck star formation prevents the formation of a singularity inside a black hole which otherwise will contradict the quantum nature of spacetime itself. Planck star has sufficient room for storing all the information that is consumed by a black hole and thus we solve the black hole information loss paradox.

Astrophysicists are well acquainted with the stellar evolution cycle. In particular, it is known that certain massive stars collapse into black holes in the last stage of their lives. Herein we present a continuation of the process with Black Hole Evolution. Stephen Hawking has proved that black hole continuously emits radiations from the event horizon called the Hawking radiation. Following the above property, after some finite amount of time, black holes should get evaporated and all its information should be lost but we have shown above that Planck Star resolves the issue. So as the event horizon of the black hole collapses to the centre of the black hole, physicists Abhay Ashtekar, Javier Olmendo and Parampreet Singh have applied loop quantum gravity to the centre of black holes. They discarded the previously accepted notion of singularity. Their calculation predicts that space-time is curved very strongly near the centre of the black hole. The result is that space-time continues into a region in the future that has the structure of a white hole. A white hole is like a black hole in reverse, meaning that unlike a black hole, which pulls matter in, a white hole shoots matter out. Though the existence of white holes could be derived from General Relativity it was still a matter of debate. Now, that the theory of loop quantum gravity manages to do it, it is an indication that this theory of black hole- white hole conversion, has ripened enough to tackle real-world situations.

Loop Quantum Cosmology

One of the central quests of the field of Cosmology is to understand how the universe evolved and what existed before the beginning of our universe. Big Bang hypothesis is the most widely accepted hypothesis in the scientific community. All that supports for Big Bang hypothesis has come up from empirical evidence. Unfortunately, physicists do not have a clear physical model of Big Bang. General Relativity breaks down at the point of Big Bang singularity because it does not consider the quantum effects operating at such a level. Thus, one of the motivations for a quantum theory of gravity is to understand the Big Bang singularity.

Loop Quantum Cosmology provides a hypothetical model of our universe by resolving the issue of Big Bang singularity with the help of Loop Quantum Gravity. Big Bang is now replaced by two events Big Bounce and Big Crunch. This model is a cyclic model of the universe where the universe is depicted to go through eternal cycles of expansion and contraction. When one cycle is coming to an end and the universe is contracting, it contracts up to the limiting energy density called Planck energy density with dimensions up to the Planck limits imposed by the quantum nature of space-time. This is called the Big Crunch. Strong repulsive forces are operative at the Planck scale which forces all the energy to bounce out from the highly contracted state and a new cycle of the universe is started. This is called the Big Bounce. Quantum fluctuations occur during the transition from one universe to another allowing the cycles to be distinct from one another.

Naturally one may ask a question that how did this cyclic process start? At present we do not have any answer to the above question.