![Olbers' Paradox - All Points](https://upload.wikimedia.org/wikipedia/commons/7/76/Olbers%27_Paradox_-_All_Points.gif)
Olbers's paradox, also known as the dark night sky paradox, is an argument in
astrophysics
Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the h ...
and
physical cosmology
Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...
that says that the
darkness
Darkness, the direct opposite of lightness, is defined as a lack of illumination, an absence of visible light, or a surface that absorbs light, such as black or brown.
Human vision is unable to distinguish colors in conditions of very low lu ...
of the
night sky
The night sky is the nighttime appearance of celestial objects like stars, planets, and the Moon, which are visible in a clear sky between sunset and sunrise, when the Sun is below the horizon.
Natural light sources in a night sky include ...
conflicts with the assumption of an infinite and eternal
static universe
In cosmology, a static universe (also referred to as stationary, infinite, static infinite or static eternal) is a cosmological model in which the universe is both spatially and temporally infinite, and space is neither expanding nor contract ...
. In the hypothetical case that the universe is static,
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
at a large scale, and populated by an infinite number of
star
A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
s, any line of sight from
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
must end at the surface of a star and hence the night sky should be completely illuminated and very bright. This contradicts the observed darkness and non-uniformity of the night.
The darkness of the night sky is one of the pieces of evidence for a dynamic universe, such as the
Big Bang model
The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
. That model explains the observed non-uniformity of brightness by invoking
expansion of the universe
The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion whereby the scale of space itself changes. The universe does not exp ...
, which increases the
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
of
visible light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 te ...
originating from the Big Bang to
microwave
Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ran ...
scale via a process known as ''
redshift
In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in f ...
''. The resulting
microwave radiation background has wavelengths much longer (millimeters instead of nanometers), which appears dark to the naked eye and bright for a radio receiver.
Other explanations for the paradox have been offered, but none have wide acceptance in cosmology. Although he was not the first to describe it, the paradox is popularly named after the German astronomer
Heinrich Wilhelm Olbers
Heinrich Wilhelm Matthias Olbers (; ; 11 October 1758 – 2 March 1840) was a German physician and astronomer.
Life and career
Olbers was born in Arbergen, Germany, today part of Bremen, and studied to be a physician at Göttingen (1777–80 ...
(1758–1840).
History
The first one to address the problem of an infinite number of stars and the resulting heat in the Cosmos was
Cosmas Indicopleustes
Cosmas Indicopleustes ( grc-x-koine, Κοσμᾶς Ἰνδικοπλεύστης, lit=Cosmas who sailed to India; also known as Cosmas the Monk) was a Greek merchant and later hermit from Alexandria of Egypt. He was a 6th-century traveller who ma ...
, a 6th-century Greek monk from
Alexandria
Alexandria ( or ; ar, ٱلْإِسْكَنْدَرِيَّةُ ; grc-gre, Αλεξάνδρεια, Alexándria) is the second largest city in Egypt, and the largest city on the Mediterranean coast. Founded in by Alexander the Great, Alexandria ...
, who states in his ''
Topographia Christiana
The ''Christian Topography'' ( grc, Χριστιανικὴ Τοπογραφία, la, Topographia Christiana) is a 6th-century work, one of the earliest essays in scientific geography written by a Christian author. It originally consisted of fiv ...
'': "The crystal-made sky sustains the heat of the Sun, the moon, and the infinite number of stars; otherwise, it would have been full of fire, and it could melt or set on fire."
Edward Robert Harrison
Edward R. Harrison (8 January 1919 – 29 January 2007)
"Physics Today Obituaries: Edward R. (Ted) Harrison",
William M. Irvine, PhysicsToday.org, 2007-02-23, webpage:
PToday-125
was a British astronomer and cosmologist, noted for his work ...
's ''Darkness at Night: A Riddle of the Universe'' (1987) gives an account of the dark night sky paradox, seen as a problem in the history of science. According to Harrison, the first to conceive of anything like the paradox was
Thomas Digges
Thomas Digges (; c. 1546 – 24 August 1595) was an English mathematician and astronomer. He was the first to expound the Copernican system in English but discarded the notion of a fixed shell of immoveable stars to postulate infinitely many s ...
, who was also the first to expound the
Copernican system in English and also postulated an infinite universe with infinitely many stars.
Kepler
Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws o ...
also posed the problem in 1610, and the paradox took its mature form in the 19th-century work of
Halley Halley may refer to:
Science
* Halley's Comet, officially designated 1P/Halley, a comet that becomes visible from Earth every 75-76 years
* Halley (lunar crater), a lunar crater named after Edmond Halley
* Halley (Martian crater), a Martian cra ...
and
Cheseaux.
The paradox is commonly attributed to the
German
German(s) may refer to:
* Germany (of or related to)
** Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...
amateur
astronomer
An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, natural satellite, moons, comets and galaxy, g ...
Heinrich Wilhelm Olbers
Heinrich Wilhelm Matthias Olbers (; ; 11 October 1758 – 2 March 1840) was a German physician and astronomer.
Life and career
Olbers was born in Arbergen, Germany, today part of Bremen, and studied to be a physician at Göttingen (1777–80 ...
, who described it in 1823, but Harrison shows convincingly that Olbers was far from the first to pose the problem, nor was his thinking about it particularly valuable. Harrison argues that the first to set out a satisfactory resolution of the paradox was
Lord Kelvin
William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, Mathematical physics, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy (Glasgow), Professor of Natural Philoso ...
, in a little known 1901 paper, and that
Edgar Allan Poe
Edgar Allan Poe (; Edgar Poe; January 19, 1809 – October 7, 1849) was an American writer, poet, editor, and literary critic. Poe is best known for his poetry and short stories, particularly his tales of mystery and the macabre. He is wide ...
's essay ''
Eureka
Eureka (often abbreviated as E!, or Σ!) is an intergovernmental organisation for research and development funding and coordination. Eureka is an open platform for international cooperation in innovation. Organisations and companies applying th ...
'' (1848) curiously anticipated some qualitative aspects of Kelvin's argument:
The paradox
The paradox is that a static, infinitely old universe with an infinite number of stars distributed in an infinitely large space would be bright rather than dark.
![Olbers' Paradox](https://upload.wikimedia.org/wikipedia/commons/c/cd/Olbers%27_Paradox.svg)
To show this, we divide the universe into a series of concentric shells, 1 light year thick. A certain number of stars will be in the shell 1,000,000,000 to 1,000,000,001 light years away. If the universe is homogeneous at a large scale, then there would be four times as many stars in a second shell, which is between 2,000,000,000 and 2,000,000,001 light years away. However, the second shell is twice as far away, so each star in it would appear one quarter as bright as the stars in the first shell. Thus the total light received from the second shell is the same as the total light received from the first shell.
Thus each shell of a given thickness will produce the same net amount of light regardless of how far away it is. That is, the light of each shell adds to the total amount. Thus the more shells, the more light; and with infinitely many shells, there would be a bright night sky.
While dark clouds could obstruct the light, these clouds would heat up, until they were as hot as the stars, and then radiate the same amount of light.
Kepler saw this as an argument for a finite
observable universe
The observable universe is a ball-shaped region of the universe comprising all matter that can be observed from Earth or its space-based telescopes and exploratory probes at the present time, because the electromagnetic radiation from these obj ...
, or at least for a finite number of stars. In
general relativity theory
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. G ...
, it is still possible for the paradox to hold in a finite universe: though the sky would not be infinitely bright, every point in the sky would still be like the surface of a star.
Explanation
The poet
Edgar Allan Poe
Edgar Allan Poe (; Edgar Poe; January 19, 1809 – October 7, 1849) was an American writer, poet, editor, and literary critic. Poe is best known for his poetry and short stories, particularly his tales of mystery and the macabre. He is wide ...
suggested that the finite size of the observable universe resolves the apparent paradox. More specifically, because the universe is
finitely old and the speed of light is finite, only finitely many stars can be observed from Earth (although the whole universe can be infinite in space). The density of stars within this finite volume is sufficiently low that any line of sight from Earth is unlikely to reach a star.
However, the
Big Bang theory
The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
seems to introduce a new problem: it states that the sky was much brighter in the past, especially at the end of the
recombination era, when it first became transparent. All points of the local sky at that era were comparable in brightness to the surface of the Sun, due to the high temperature of the universe in that
era
An era is a span of time defined for the purposes of chronology or historiography, as in the regnal eras in the history of a given monarchy, a calendar era used for a given calendar, or the geological eras defined for the history of Earth.
Comp ...
; and most light rays will originate not from a star but the relic of the Big Bang.
This problem is addressed by the fact that the Big Bang theory also involves the
expansion of space
The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion whereby the scale of space itself changes. The universe does not exp ...
, which can cause the energy of emitted light to be reduced via
redshift
In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in f ...
. More specifically, the extremely energetic radiation from the
Big Bang
The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
has been redshifted to microwave wavelengths (1100 times the length of its original wavelength) as a result of the cosmic expansion, and thus forms the
cosmic microwave background radiation
In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...
. This explains the relatively low light densities and energy levels present in most of our sky today despite the assumed bright nature of the Big Bang. The redshift also affects light from distant stars and
quasar
A quasar is an extremely Luminosity, luminous active galactic nucleus (AGN). It is pronounced , and sometimes known as a quasi-stellar object, abbreviated QSO. This emission from a galaxy nucleus is powered by a supermassive black hole with a m ...
s, but this diminution is minor, since
the most distant galaxies and quasars have redshifts of only around 5 to 8.6.
Other factors
Steady state
The redshift hypothesised in the Big Bang model would by itself explain the darkness of the night sky even if the universe were infinitely old. In the
Steady state theory
In cosmology, the steady-state model, or steady state theory is an alternative to the Big Bang theory of evolution of the universe. In the steady-state model, the density of matter in the expanding universe remains unchanged due to a continuous ...
the universe is infinitely old and uniform in time as well as space. There is no Big Bang in this model, but there are stars and quasars at arbitrarily great distances. The
expansion of the universe
The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion whereby the scale of space itself changes. The universe does not exp ...
causes the light from these distant stars and quasars to redshift, so that the total light flux from the sky remains finite. Thus the observed radiation density (the sky brightness of
extragalactic background light The diffuse extragalactic background light (EBL) is all the accumulated radiation in the universe due to star formation processes, plus a contribution from active galactic nuclei (AGNs). This radiation covers almost all wavelengths of the electroma ...
) can be independent of finiteness of the universe. Mathematically, the total electromagnetic energy density (radiation energy density) in
thermodynamic equilibrium
Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermod ...
from
Planck's law
In physics, Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature , when there is no net flow of matter or energy between the body and its environment.
At ...
is
e.g. for temperature 2.7 K it is 40 fJ/m
3 ... 4.5×10
−31 kg/m
3 and for visible temperature 6000 K we get 1 J/m
3 ... 1.1×10
−17 kg/m
3. But the total radiation emitted by a star (or other cosmic object) is at most equal to the total
nuclear binding energy
Nuclear binding energy in experimental physics is the minimum energy that is required to disassemble the atomic nucleus, nucleus of an atom into its constituent protons and neutrons, known collectively as nucleons. The binding energy for stable n ...
of
isotope
Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), and that differ in nucleon numbers (mass numbers) ...
s in the star. For the density of the
observable universe
The observable universe is a ball-shaped region of the universe comprising all matter that can be observed from Earth or its space-based telescopes and exploratory probes at the present time, because the electromagnetic radiation from these obj ...
of about 4.6×10
−28 kg/m
3 and given the known
abundance of the chemical elements
The abundance of the chemical elements is a measure of the Type–token distinction#Occurrences, occurrence of the chemical elements relative to all other elements in a given environment. Abundance is measured in one of three ways: by the mass frac ...
, the corresponding maximal radiation energy density of 9.2×10
−31 kg/m
3, i.e. temperature 3.2 K (matching the value observed for the optical radiation temperature by
Arthur Eddington
Sir Arthur Stanley Eddington (28 December 1882 – 22 November 1944) was an English astronomer, physicist, and mathematician. He was also a philosopher of science and a populariser of science. The Eddington limit, the natural limit to the lumin ...
). This is close to the summed energy density of the
cosmic microwave background
In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...
(CMB) and the
cosmic neutrino background
The cosmic neutrino background (CNB or CB) is the universe's background particle radiation composed of neutrinos. They are sometimes known as relic neutrinos.
The CB is a relic of the Big Bang; while the cosmic microwave background radiation (CM ...
. The Big Bang hypothesis predicts that the
cosmic background radiation
Cosmic background radiation is electromagnetic radiation from the Big Bang. The origin of this radiation depends on the region of the spectrum that is observed. One component is the cosmic microwave background. This component is redshifted pho ...
(CBR) should have the same energy density as the binding energy density of
the primordial helium, which is much greater than the binding energy density of the non-primordial elements; so it gives almost the same result. However, the steady-state model does not predict the angular distribution of the microwave background temperature accurately (as the standard ΛCDM paradigm does). Nevertheless, the modified gravitation theories (without metric expansion of the universe) cannot be ruled out by CMB and
BAO observations.
Finite age of stars
Stars have a finite age and a finite power, thereby implying that each star has a finite impact on a sky's light field density.
Edgar Allan Poe
Edgar Allan Poe (; Edgar Poe; January 19, 1809 – October 7, 1849) was an American writer, poet, editor, and literary critic. Poe is best known for his poetry and short stories, particularly his tales of mystery and the macabre. He is wide ...
suggested that this idea could provide a resolution to Olbers's paradox; a related theory was also proposed by
Jean-Philippe de Cheseaux
Jean-Philippe Loys de Cheseaux (; 4 May 1718 – 30 November 1751) was a Swiss astronomer.
Biography
Loys de Cheseaux was born on 4 May 1718 in Lausanne, Vaud, to Paul-Etienne Loys de Cheseaux, a banneret, and Estienne-Judith de Crousaz. His ...
. However, stars are continually being born as well as dying. As long as the density of stars throughout the universe remains constant, regardless of whether the universe itself has a finite or infinite age, there would be infinitely many other stars in the same angular direction, with an infinite total impact. So the finite age of the stars does not explain the paradox.
Brightness
Suppose that the universe were not expanding, and always had the same stellar density; then the temperature of the universe would continually increase as the stars put out more radiation. Eventually, it would reach 3000 K (corresponding to a typical photon energy of 0.3
eV and so a frequency of 7.5×10
13 Hz), and the photons would begin to be absorbed by the hydrogen plasma filling most of the universe, rendering outer space opaque. This maximal radiation density corresponds to about eV/m
3 = , which is much greater than the observed value of .
So the sky is about five hundred billion times darker than it would be if the universe was neither expanding nor too young to have reached equilibrium yet. However, recent observations increasing the lower bound on the number of galaxies suggest UV absorption by hydrogen and reemission in near-IR (not visible) wavelengths also plays a role.
Fractal star distribution
A different resolution, which does not rely on the Big Bang theory, was first proposed by
Carl Charlier
Carl Vilhelm Ludwig Charlier (1 April 1862 – 4 November 1934) was a Swedish astronomer. His parents were Emmerich Emanuel and Aurora Kristina (née Hollstein) Charlier.
Career
Charlier was born in Östersund. He received his Ph.D. from ...
in 1908 and later rediscovered by
Benoît Mandelbrot
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
in 1974. They both postulated that if the stars in the universe were distributed in a hierarchical
fractal cosmology
In physical cosmology, fractal cosmology is a set of minority cosmological theories which state that the distribution of matter in the Universe, or the structure of the universe itself, is a fractal across a wide range of scales (see also: multif ...
(e.g., similar to
Cantor dust
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.
...
)—the average density of any region diminishes as the region considered increases—it would not be necessary to rely on the Big Bang theory to explain Olbers's paradox. This model would not rule out a Big Bang, but would allow for a dark sky even if the Big Bang had not occurred.
Mathematically, the light received from stars as a function of star distance in a hypothetical fractal cosmos is
where:
* ''r''
0 = the distance of the nearest star, ''r''
0 > 0;
* ''r'' = the variable measuring distance from the Earth;
* ''L''(''r'') = average
luminosity
Luminosity is an absolute measure of radiated electromagnetic power (light), the radiant power emitted by a light-emitting object over time. In astronomy, luminosity is the total amount of electromagnetic energy emitted per unit of time by a st ...
per star at distance ''r'';
* ''N''(''r'') = number of stars at distance ''r''.
The function of luminosity from a given distance ''L''(''r'')''N''(''r'') determines whether the light received is finite or infinite. For any luminosity from a given distance proportional to ''r''
''a'',
is infinite for ''a'' ≥ −1 but finite for ''a'' < −1. So if is proportional to ''r''
−2, then for
to be finite, must be proportional to ''r''
''b'', where ''b'' < 1. For ''b'' = 1, the numbers of stars at a given radius is proportional to that radius. When integrated over the radius, this implies that for ''b'' = 1, the ''total'' number of stars is proportional to ''r''
2. This would correspond to a
fractal dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...
of 2. Thus the fractal dimension of the universe would need to be less than 2 for this explanation to work.
This explanation is not widely accepted among cosmologists, since the evidence suggests that the
fractal dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...
of the universe is at least 2.
Moreover, the majority of cosmologists accept the
cosmological principle
In modern physical cosmology, the cosmological principle is the notion that the spatial distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throu ...
, which assumes that matter at the scale of billions of light years is distributed
isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
ally. Contrarily, fractal cosmology requires
anisotropic
Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
matter distribution at the largest scales.
See also
*
Heat death paradox
The heat death paradox, also known as thermodynamic paradox, Clausius' paradox and Kelvin’s paradox, is a ''reductio ad absurdum'' argument that uses thermodynamics to show the impossibility of an infinitely old universe. It was formulated in Feb ...
*
List of paradoxes
This list includes well known paradoxes, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. This list collects only scenarios that have been called a paradox by at least one source and have their ...
*
Horizon problem
The horizon problem (also known as the homogeneity problem) is a cosmological fine-tuning problem within the Big Bang model of the universe. It arises due to the difficulty in explaining the observed homogeneity of causally disconnected region ...
References
Further reading
*
Edward Robert Harrison
Edward R. Harrison (8 January 1919 – 29 January 2007)
"Physics Today Obituaries: Edward R. (Ted) Harrison",
William M. Irvine, PhysicsToday.org, 2007-02-23, webpage:
PToday-125
was a British astronomer and cosmologist, noted for his work ...
(1987). ''Darkness at Night: A Riddle of the Universe'', Harvard University Press.
*
Edward Robert Harrison
Edward R. Harrison (8 January 1919 – 29 January 2007)
"Physics Today Obituaries: Edward R. (Ted) Harrison",
William M. Irvine, PhysicsToday.org, 2007-02-23, webpage:
PToday-125
was a British astronomer and cosmologist, noted for his work ...
(2000). ''Cosmology: The Science of the Universe'', 2nd ed. Cambridge University Press. Chapter 24.
*
* Zamarovský, Peter (2013). ''Why is it Dark at Night? Story of Dark Night Sky Paradox''. AuthorHouseUK. ISBN 978-1491878804.
External links
Relativity FAQ about Olbers's paradox*
Why is the sky dark?physics.org page about Olbers's paradox
Why is it dark at night?A 60-second animation from the
Perimeter Institute
Perimeter Institute for Theoretical Physics (PI, Perimeter, PITP) is an independent research centre in foundational theoretical physics located in Waterloo, Ontario, Canada. It was founded in 1999. The institute's founding and major benefactor i ...
exploring the question with Alice and Bob in Wonderland
{{Authority control
Physical cosmology
Physical paradoxes
Unsolved problems in astronomy