Taming infinity with loops

How loop quantum gravity could replace the absurd state of infinite density, the big bang with which, according to Einstein’s relativity, the universe began

An article by Martin Bojowald

In the conventional big bang models based on general relativity, our universe began in a patently absurd physical state: With the big bang singularity, at a time when all the space and matter we see around us today was compressed to a single point of infinitely high density. Such infinities are a sure sign for “pathological” physics; a sure sign that Einstein’s equations which, in the big bang models, govern the evolution of the cosmos, lose their meaning directly at the big bang.

For decades, there has been hope that a theory unifying general relativity with the concepts of quantum theory (in short, a theory of quantum gravity) would solve the problem of these infinities. These hopes have recently taken on a more concrete shape, in the framework of so-called loop quantum gravity and its applications to cosmology.

Quanta of space and time

In this theory, there is a kind of quantization of the structure of space and time. In Einstein’s theory, space is a continuum: every region of space can be subdivided further. In loop quantum gravity, matters are different: there, every length is a multiple of an elementary length called the Planck length. Two points in space can have a distance of one, two, three or more Planck length, but they cannot be one and a half, or two-thirds of a Planck length apart. Length is not arbitrarily divisible – there is an ultimate subdivision, that into the minimal units of length.

The Planck length is extremely small – around 10^-35 metres, and hence far beyond the microscopic distances that can be explored with our most powerful “microscopes”: the world’s largest particle accelerators. That’s why the subdivision of length into fixed minimal building blocks – its “quantization” – is irrelevant for everyday life, and the same goes for most of physics. However, according to the big bang models, roughly 14 billion years ago, the whole of the observable universe was confined to a space a few Planck lengths in diameter – and at that point in time, close to the big bang, the quantum structure of space and time becomes important indeed.

Avoiding infinite density

As it turns out, the quantization of space as predicted by loop quantum gravity promises to fulfill physicists’ hopes of a cosmology that is free of infinities. In ordinary physics, these infinities are inescapable: there, the average density in a region of space is equal to the mass contained in that region, divided by the region’s volume. If that region contracts to zero volume, the density becomes infinite. In loop quantum gravity, the relation between density and volume is more complicated, as far as very small volumes are concerned. The following image shows the density of a fixed amount of energy, confined to smaller and smaller volumes, as predicted by loop quantum gravity:

density loop quantum gravity

Volume is plotted on the horizontal axis, density on the vertical axis. The red, solid line shows the evolution of density in classical physics, for instance in general relativity. The voilet dots show the density values predicted by loop quantum gravity for matter that is, to the far right, confined to 30 times the elementary volume, then to a mere 29, and so on, until, on the left, space has the volume of a single elementary cube – Planck length cubed – and, finally, space contracts to volume zero. The crucial feature: as space shrinks, density can take on large values, but it does not become infinite, even when space shrinks to a point.

In this way, the ugly big bang singularity with its infinite density can be eliminated. Instead, in cosmological models based on loop quantum gravity, there is history before the big bang. The evolution of the universe is governed by its energy density. The non-classical dependence of such densities on volume corresponds to a repulsive force at very small scales. This effect clearly shows the difference between quantum gravity and ordinary general relativity – in relativity, gravity is always attractive. Thanks to this repulsive force, the quantized Einstein equations can describe physics at the big bang, and beyond – a clear advantage over general relativity. (More informations about this modified evolution can be found in the spotlight topic Avoiding the big bang.)

Further Information

Colophon

Martin Bojowald

is a physics professor at Pennsylvania State University with a research focus on cosmological applications of loop quantum gravity.

Citation

Cite this article as:
Martin Bojowald, “Taming infinity with loops” in: Einstein Online Band 04 (2010), 01-1024

Definitioner

time

It is a fact of life that not all events in our universe happen concurrently - instead, there is a certain order. Defining a time coordinate or defining time, the way physicists do it, is to define a prescription to associate with each event a number so as to reflect that order - if event B happens after event A, then the number associated with B should be larger than that associated with A. The first step of this definition is to construct a clock: Choose a simple process that repeats regularly. (What is "regular"? Luckily, in our universe, all elementary processes such as a swinging pendulum, the oscillations of atoms or of electronic circuits lead to the same concept of regularity.) As a second step, install a counter: A mechanism that, with every repetition of the chosen process, raises the count by one. With this definition, one can at least assign a time (the numerical value of the counter) to events happening at location of the clock. For events at different locations, an additional definition is necessary: One needs to define simultaneity. After all, the statement that some far-away event A happens at 12 o'clock is the same as saying that event A and "our clock counter shows 12:00:00" are simultaneous. The how and why of defining simultaneity - a centre-piece of Einstein's special theory of relativity - are described in the spotlight topic Defining "now". With all these preparations, physicists can, in principle, assign a time coordinate value ("a time") to any possible events, and describes how fast or how slow processes happen, compared to that time coordinate.

space

In a strict sense: Space as we know it from everyday life: the totality of all locations in which objects can sit, with three dimensions. In a more general sense used by mathematicians, all kinds of sets of points are spaces - a line for instance, which has but a single dimension, or a two-dimensional surface, but also higher-dimensional spaces. Also, in such more general spaces, geometry can be different from the standard Euclidean geometry taught in high schools - such spaces can be curved.

singularity

Irregular boundary of spacetime in general relativity - region where spacetime simply comes to an end. Often, such boundaries are associated with spacetime curvature growing beyond all bounds and becoming infinitely large - so-called curvature singularities (notably Ricci singularities or Weyl singularities - but there are exceptions (for instance a conic singularities). According to general relativity, a singularity exists inside every black hole, and the starting point of any universe described by a big bang model is a singularity as well. The occurrence of singularities is a failure of general relativity - and a strong indication that the theory is incomplete. Instead, one could describe the earliest universe and the interior of black holes using a theory of quantum gravity. More information about singularities can be found in the spotlight texts Spacetime singularities and Of singularities and breadmaking.

quantum theory

The framework for formulating the physical laws that govern the world at microscopic length-scales - the physics of the micro-world, for instance of atoms, atomic nuclei or elementary particles, but also the physics of ultra-precise measurements such as those made by gravitational wave detectors.

The laws of quantum theory are fundamentally different from our everyday experience and from those of classical physics.

The first unusual feature is that, in many cases, quantum theory merely allows statements about probabilities. For instance, in classical physics, one can assign to every particle, at every point in time, a location and a velocity. Whosoever can measure those quantities precisely can, in principle, predict where the particle in question can be found at every point in the future. In quantum theory, all one can assign to a system of particles is an abstract quantum state from which can be derived no precise predictions, but merely the probabilities for detecting a specific particle at a given time in a given place. Whether or not one will really find the particle at that location is governed by chance.

The second unusual feature is a fundamental restriction placed on the exactness of certain measurements (Heisenberg uncertainty relation). For instance, the more precise one measures a particle's location, the less definite any statements one can make about its velocity.

The third feature is how quantum theory came by its name: A number of physical quantities in nature come in little packets, in quanta. For instance, according to quantum theory, electromagnetic radiation is made of tiny quanta of energy called photons.

Examples for quantum theories are quantum mechanics and relativistic quantum field theories such as Quantum electrodynamics or other parts of the standard model of particle physics.

quantum gravity

Theory based both on the effects, concepts and laws of quantum theory and on those of general relativity. To date, no complete such theory exists; the best-known candidate theories are string theory and loop quantum gravity. Some information on the question of quantum gravity can be found in the chapter Relativity and the quantum of Elementary Einstein, starting with the page Border regions of gravity. Selected aspects of quantum gravity are described in the category Relativity and the quantum of our Spotlights on relativity.

quantization (quantize)

First of all, quantization is the process describing the transition from a classical theory to the corresponding quantum theory. For instance, if you quantize classical electrodynamics, you will end up with its quantum version, quantum electrodynamics. On the other hand, to be quantized means for a physical quantity to be divided into little building blocks or packets. For instance, in quantum theory, the energy of light is quantized: a given quantity of light consists of a finite number of energy packets called photons.

point

Elementary "building block" of geometrical entities such as surfaces or more general spaces. For instance, a surface is the set of all its points, of all possible locations on the surface, and all geometrical objects in that surface are defined by the points that belong to them - for instance, a line on the surface is the set of (infinitely many) points.

Planck length

Natural length that can be obtained by combining the fundamental natural constants that govern spacetime, the strength of gravity and the quantum world: the gravitational constant, Planck's constant and the speed of light. It amounts to roughly 1.6·10^-35 meters [see exponential notation]. At such length scales, both the effects of quantum theory and those of general relativity should become important, in short: whatever concerns such scales can only be described adequately using a theory of quantum gravity. Compare Planck time, Planck energy, Planck mass.

matter

In general relativity: All contents of spacetime that contribute to its curvature: particles, dust, gases, fluids, electromagnetic and other fields. In particle physics: All elementary particles with half-integer spin, such as electrons and quarks, as well as their composites such as protons and neutrons, in contrast with force particles.

mass

In classical physics, mass plays a triple role. First of all, it is a measure for how easy it is to influence the motion of a body. Imagine that you're drifting in emtpy space. Drifting by are an elephant and a mouse, and you give each of them a push of equal strength. The fact that the mouse abruptly changes its path, while the elephant's course is as good as unaltered, is a sure sign that the mass (or, in the language of physics, the inertia or inertial mass) of the elephant is much greater than that of the mouse. Secondly, mass is a measure of how many atoms there are in a body, and of what type they are. All atoms of one and the same type have the same mass, and adding up all those tiny component masses, the total mass of the body results. Thirdly, in Newton's theory of gravity, mass determines how strongly a body attracts other bodies via the gravitational force, and how strongly these bodies attract it (in this sense, mass is the charge associated with the gravitational force). In special relativity, one can also define a mass that is a measure for a bodies resistance to changing its motion. However, the value of this relativistic mass depends on the relative motion of the body and the observer. The relativistic mass is the "m" in Einstein's famous E=mc² (cf. equivalence of mass and energy). The relativistic mass has a minimum for an observer that is at rest relative to the body in question. This value is the so-called rest mass of the body, and when particle physicists talk of mass, this is usually what they mean. Just as in classical physics, the rest mass is a kind of measure for how much matter the body is made up of - with one caveat: For composite bodies, the energies associated with the forces holding the body together contribute to the total mass, as well (another consequence of the equivalence of mass and energy). In general relativity, mass still plays a role as a source of gravity; however, it has been joined by physical quantities such as energy, momentum and pressure.

loop quantum gravity

Candidate for a theory of quantum gravity. Loop quantum gravity is an attempt to apply the concepts and laws of quantum theory directly to the geometry that is at the heart of general relativity. A brief description can be found on the page Loop quantum gravity in the chapter Relativity and the quantum of Elementary Einstein.

line

Geometric object with a single dimension. A line can either be an independent one-dimensional space (in the abstract mathematical space where a space does not need to have three dimensions), or it can be embedded into a more general space, like a line drawn onto a piece of paper (i.e. a surface).

gravity

See gravitation

gravity (gravitation)

In classical physics: An action-at-a-distance force by which all bodies that possess mass attract each other (see Newtonian theory of gravity), synonym: gravitational force. In Einstein's general theory of relativity: The fact that matter that possesses mass, energy, pressure or similar properties distorts spacetime, and that this distortion in turn influences whatever matter might be present. An introduction to the basic ideas of general relativity is provided by the section General relativity of Elementary Einstein. More information about the nature of gravity in general relativity can be found in the spotlight text Gravity: From weightlessness to curvature.

general relativity (general theory of relativity)

Albert Einstein's theory of gravity; a generalization of his special theory of relativity. For information about the concepts and applications of this theory, we recommend the chapter general relativity of our introductory section Elementary Einstein. Further information about many different aspects of general relativity and its applications can be found in our section Spotlights on relativity.

free

In the context of relativity theory, a particle (object, observer...) that is not acted upon by any force except gravity is said to be free or, a bit more specific, to be in free fall. Free test particles play an important role in understanding the structure of general relativity.

force

In mechanics: Influence acting on a body, trying to accelerate it.

More generally: All influences by which elementary or other particles can interact; in this sense, force and interaction are synonymous. In the standard model of particle physics, there are three elementary forces: electromagnetism, the weak (nuclear) force and the strong (nuclear) force, while there is no quantum description of the fourth fundamental interaction, gravity.

focus

In optics: A focus, also referred to as an image point, is the point where incoming, parallel light rays meet after traveling through a lens. In geometry: For an ellipse all points on the curve are located around two focal points, such that the sum of the distances to each focal point is equal for all points.

energy

Physical quantity with the special property that, in physical processes, energy is neither destroyed nor created, simply transformed from one form of energy to another. Some of the different forms of energy that are defined separately are kinetic energy, thermal energy and the energy carried by electromagnetic radiation. Processes that transform one form of energy into another take place in all machines we use in everyday life, from the engine of a subway train (electrical energy into kinetic energy of the train) to an electric blanket (electrical energy into thermal energy). One important consequence of special relativity is that energy and mass are completely equivalent - two different ways to define what is, on closer inspection, one and the same physical quantity. See the keyword equivalence between mass and energy.

density

In a stricter sense synonymous with "mass density": The average density of matter in a region of space is the total mass of all matter contained in that region, divided by the region's volume. More generally, density can refer to other physical quantities as well. The energy density, for instance, is the total sum of energy localized in a region divided by that region's volume.

cosmology

That branch of physics and astronomy dealing with the structure and development of the universe as a whole. At the core of modern cosmology are the big bang models based on Einstein's general theory of relativity. Their basic features are reviewed in the chapter Cosmology of Elementary Einstein. In order to describe the very early universe, it will be necessary to take the effects of quantum gravity into account - this gives rise to models of quantum cosmology.

continuum (continuous)

Space as we are used to thinking about it is a continuum or, equivalently, continuous space: Between every two points, there always exists an infinity of other points, and every volume can be divided into smaller and smaller parts without ever reaching a limit.

classical

In physics, the word is used with two meanings. First of all, it denotes physical models or theories that take into account neither the effects of Einstein's theories of relativity nor those of quantum physics, for example classical mechanics. However, it is also used to denote models or theories that are not formulated in the framework of quantum physics; in that sense, general relativity is an example for a classical theory.

big bang

The big bang models are the foundation of modern cosmology. Firmly grounded in Einstein's theory of general relativity, they describe a universe that began in a very hot initial state and has expanded (and cooled down) ever since. They make precise predictions about nucleosynthesis in the early universe, the existence and properties of the cosmic background radiation, and the distribution of distant galaxies in the cosmos, which have been confirmed by astronomical observation. The word "big bang" has two different meanings. In a strict sense, the big bang is a spacetime singularity, a state of infinite density - the initial state the big bang models predict for our universe. In a more general sense, the term is applied to the earliest cosmic eras, in which the universe was exceedingly hot and dense. Further information about these two meanings and why it is important to distinguish between them can be found in the spotlight text A tale of two big bangs. The basic features of the big bang models are reviewed in the chapter Cosmology of Elementary Einstein. Selected aspects of cosmology are described in the category Cosmology of our Spotlights on relativity.