Gravitational binding energy

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https://en.wikipedia.org/wiki/Gravitational_binding_energy

Galaxy clusters are the largest known gravitationally bound structures in the universe.

The gravitational binding energy of a system is the minimum energy which must be added to it in order for the system to cease being in a gravitationally bound state. A gravitationally bound system has a lower (i.e., more negative) gravitational potential energy than the sum of the energies of its parts when these are completely separated—this is what keeps the system aggregated in accordance with the minimum total potential energy principle.

The gravitational binding energy can be conceptually different within the theories of newtonian gravity and Albert Einstein's theory of gravity called General Relativity. In newtonian gravity, the binding energy can be considered to be the linear sum of the interactions between all pairs of microscopic components of the system, while in General Relativity, this is only approximately true if the gravitational fields are all weak. When stronger fields are present within a system, the binding energy is a nonlinear property of the entire system, and it cannot be conceptually attributed among the elements of the system. In this case the binding energy can be considered to be the (negative) difference between the ADM mass of the system, as it is manifest in its gravitational interaction with other distant systems, and the sum of the energies of all the atoms and other elementary particles of the system if disassembled.

For a spherical body of uniform density, the gravitational binding energy U is given in newtonian gravity by the formula $${\displaystyle U=-\frac{3G{M}^2}{5R}}$$ where G is the gravitational constant, M is the mass of the sphere, and R is its radius.

Assuming that the Earth is a sphere of uniform density (which it is not, but is close enough to get an order-of-magnitude estimate) with M = 5.97×10²⁴ kg and r = 6.37×10⁶ m, then U = 2.24×10³² J. This is roughly equal to one week of the Sun's total energy output. It is 37.5 MJ/kg, 60% of the absolute value of the potential energy per kilogram at the surface.

The actual depth-dependence of density, inferred from seismic travel times (see Adams–Williamson equation), is given in the Preliminary Reference Earth Model (PREM). Using this, the real gravitational binding energy of Earth can be calculated numerically as U = 2.49×10³² J.

According to the virial theorem, the gravitational binding energy of a star is about two times its internal thermal energy in order for hydrostatic equilibrium to be maintained. As the gas in a star becomes more relativistic, the gravitational binding energy required for hydrostatic equilibrium approaches zero and the star becomes unstable (highly sensitive to perturbations), which may lead to a supernova in the case of a high-mass star due to strong radiation pressure or to a black hole in the case of a neutron star.

Derivation within Newtonian gravity for a uniform sphere

The gravitational binding energy of a sphere with radius ${\displaystyle R}$ is found by imagining that it is pulled apart by successively moving spherical shells to infinity, the outermost first, and finding the total energy needed for that.

Assuming a constant density ${\displaystyle \rho }$, the masses of a shell and the sphere inside it are: $${\displaystyle m_{\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{l}\mathrm{l}}=4\pi r^2\rho dr}$$ and $${\displaystyle m_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}}=\frac{4}{3}\pi r^3\rho }$$

The required energy for a shell is the negative of the gravitational potential energy: $${\displaystyle dU=-G\frac{m_{\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{l}\mathrm{l}}{m}_{\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}}}{r}}$$

Integrating over all shells yields: $${\displaystyle U=-G{\int }_0^R\frac{\left(4\pi r^2\rho \right)\left(\frac{4}{3}\pi r^3\rho \right)}{r}dr=-G\frac{16}{3}{\pi }^2{\rho }^2{\int }_0^R{r}^{4}dr=-G\frac{16}{15}{\pi }^2{\rho }^2{R}^5}$$

Since ${\displaystyle \rho }$ is simply equal to the mass of the whole divided by its volume for objects with uniform density, therefore

$${\displaystyle \rho =\frac{M}{\frac{4}{3}\pi R^3}}$$

And finally, plugging this into our result leads to $${\displaystyle U=-G\frac{16}{15}{\pi }^2{R}^5{\left(\frac{M}{\frac{4}{3}\pi R^3}\right)}^2=-\frac{3G{M}^2}{5R}}$$

Gravitational binding energy

${\displaystyle U=-\frac{3G{M}^2}{5R}}$

Negative mass component

Two bodies, placed at the distance R from each other and reciprocally not moving, exert a gravitational force on a third body slightly smaller when R is small. This can be seen as a negative mass component of the system, equal, for uniformly spherical solutions, to: $${\displaystyle M_{\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g}}=-\frac{3G{M}^2}{5R{c}^2}}$$

For example, the fact that Earth is a gravitationally-bound sphere of its current size costs 2.49421×10¹⁵ kg of mass (roughly one fourth the mass of Phobos – see above for the same value in Joules), and if its atoms were sparse over an arbitrarily large volume the Earth would weigh its current mass plus 2.49421×10¹⁵ kg kilograms (and its gravitational pull over a third body would be accordingly stronger).

It can be easily demonstrated that this negative component can never exceed the positive component of a system. A negative binding energy greater than the mass of the system itself would indeed require that the radius of the system be smaller than: $${\displaystyle R\le \frac{3GM}{5c^2}}$$ which is smaller than $\frac{3}{10}$ its Schwarzschild radius: $${\displaystyle R\le \frac{3}{10}{r}_{\mathrm{s}}}$$ and therefore never visible to an external observer. However this is only a Newtonian approximation and in relativistic conditions other factors must be taken into account as well.

Non-uniform spheres

Planets and stars have radial density gradients from their lower density surfaces to their much denser compressed cores. Degenerate matter objects (white dwarfs; neutron star pulsars) have radial density gradients plus relativistic corrections.

Neutron star relativistic equations of state include a graph of radius vs. mass for various models. The most likely radii for a given neutron star mass are bracketed by models AP4 (smallest radius) and MS2 (largest radius). BE is the ratio of gravitational binding energy mass equivalent to observed neutron star gravitational mass of M with radius R, $${\displaystyle BE=\frac{0.60\beta }{1-\frac{\beta }{2}}}$$ $${\displaystyle \beta =\frac{GM}{R{c}^2}.}$$

Given current values

• ${\displaystyle G=6.6743\times {10}^{-11}{\mathrm{m}}^3\cdot \mathrm{k}{\mathrm{g}}^{-1}\cdot {\mathrm{s}}^{-2}}$
• ${\displaystyle c^2=8.98755\times {10}^{16}{\mathrm{m}}^2\cdot {\mathrm{s}}^{-2}}$
• ${\displaystyle M_{\odot }=1.98844\times {10}^{30}\mathrm{k}\mathrm{g}}$

and the star mass M expressed relative to the solar mass, $${\displaystyle M_x=\frac{M}{M_{\odot }},}$$

then the relativistic fractional binding energy of a neutron star is

$${\displaystyle BE=\frac{885.975M_x}{R-738.313M_x}}$$

BCS theory

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/BCS_theory

A commemorative plaque placed in the Bardeen Engineering Quad at the University of Illinois at Urbana-Champaign. It commemorates the Theory of Superconductivity developed here by John Bardeen and his students, for which they won a Nobel Prize for Physics in 1972.

In physics, the Bardeen–Cooper–Schrieffer (BCS) theory (named after John Bardeen, Leon Cooper, and John Robert Schrieffer) is the first microscopic theory of superconductivity since Heike Kamerlingh Onnes's 1911 discovery. The theory describes superconductivity as a microscopic effect caused by a condensation of Cooper pairs. The theory is also used in nuclear physics to describe the pairing interaction between nucleons in an atomic nucleus.

It was proposed by Bardeen, Cooper, and Schrieffer in 1957; they received the Nobel Prize in Physics for this theory in 1972.

History

Rapid progress in the understanding of superconductivity gained momentum in the mid-1950s. It began with the 1948 paper, "On the Problem of the Molecular Theory of Superconductivity", where Fritz London proposed that the phenomenological London equations may be consequences of the coherence of a quantum state. In 1953, Brian Pippard, motivated by penetration experiments, proposed that this would modify the London equations via a new scale parameter called the coherence length. John Bardeen then argued in the 1955 paper, "Theory of the Meissner Effect in Superconductors", that such a modification naturally occurs in a theory with an energy gap. The key ingredient was Leon Cooper's calculation of the bound states of electrons subject to an attractive force in his 1956 paper, "Bound Electron Pairs in a Degenerate Fermi Gas".

In 1957 Bardeen and Cooper assembled these ingredients and constructed such a theory, the BCS theory, with Robert Schrieffer. The theory was first published in April 1957 in the letter, "Microscopic theory of superconductivity". The demonstration that the phase transition is second order, that it reproduces the Meissner effect and the calculations of specific heats and penetration depths appeared in the December 1957 article, "Theory of superconductivity". They received the Nobel Prize in Physics in 1972 for this theory.

In 1986, high-temperature superconductivity was discovered in La-Ba-Cu-O, at temperatures up to 30 K. Following experiments determined more materials with transition temperatures up to about 130 K, considerably above the previous limit of about 30 K. It is experimentally very well known that the transition temperature strongly depends on pressure. In general, it is believed that BCS theory alone cannot explain this phenomenon and that other effects are in play. These effects are still not yet fully understood; it is possible that they even control superconductivity at low temperatures for some materials.

Overview

At sufficiently low temperatures, electrons near the Fermi surface become unstable against the formation of Cooper pairs. Cooper showed such binding will occur in the presence of an attractive potential, no matter how weak. In conventional superconductors, an attraction is generally attributed to an electron-lattice interaction. The BCS theory, however, requires only that the potential be attractive, regardless of its origin. In the BCS framework, superconductivity is a macroscopic effect which results from the condensation of Cooper pairs. These have some bosonic properties, and bosons, at sufficiently low temperature, can form a large Bose–Einstein condensate. Superconductivity was simultaneously explained by Nikolay Bogolyubov, by means of the Bogoliubov transformations.

In many superconductors, the attractive interaction between electrons (necessary for pairing) is brought about indirectly by the interaction between the electrons and the vibrating crystal lattice (the phonons). Roughly speaking the picture is the following:

An electron moving through a conductor will attract nearby positive charges in the lattice. This deformation of the lattice causes another electron, with opposite spin, to move into the region of higher positive charge density. The two electrons then become correlated. Because there are a lot of such electron pairs in a superconductor, these pairs overlap very strongly and form a highly collective condensate. In this "condensed" state, the breaking of one pair will change the energy of the entire condensate - not just a single electron, or a single pair. Thus, the energy required to break any single pair is related to the energy required to break all of the pairs (or more than just two electrons). Because the pairing increases this energy barrier, kicks from oscillating atoms in the conductor (which are small at sufficiently low temperatures) are not enough to affect the condensate as a whole, or any individual "member pair" within the condensate. Thus the electrons stay paired together and resist all kicks, and the electron flow as a whole (the current through the superconductor) will not experience resistance. Thus, the collective behavior of the condensate is a crucial ingredient necessary for superconductivity.

Details

BCS theory starts from the assumption that there is some attraction between electrons, which can overcome the Coulomb repulsion. In most materials (in low temperature superconductors), this attraction is brought about indirectly by the coupling of electrons to the crystal lattice (as explained above). However, the results of BCS theory do not depend on the origin of the attractive interaction. For instance, Cooper pairs have been observed in ultracold gases of fermions where a homogeneous magnetic field has been tuned to their Feshbach resonance. The original results of BCS (discussed below) described an s-wave superconducting state, which is the rule among low-temperature superconductors but is not realized in many unconventional superconductors such as the d-wave high-temperature superconductors.

Extensions of BCS theory exist to describe these other cases, although they are insufficient to completely describe the observed features of high-temperature superconductivity.

BCS is able to give an approximation for the quantum-mechanical many-body state of the system of (attractively interacting) electrons inside the metal. This state is now known as the BCS state. In the normal state of a metal, electrons move independently, whereas in the BCS state, they are bound into Cooper pairs by the attractive interaction. The BCS formalism is based on the reduced potential for the electrons' attraction. Within this potential, a variational ansatz for the wave function is proposed. This ansatz was later shown to be exact in the dense limit of pairs. Note that the continuous crossover between the dilute and dense regimes of attracting pairs of fermions is still an open problem, which now attracts a lot of attention within the field of ultracold gases.

Underlying evidence

The hyperphysics website pages at Georgia State University summarize some key background to BCS theory as follows:

• Evidence of a band gap at the Fermi level (described as "a key piece in the puzzle")

the existence of a critical temperature and critical magnetic field implied a band gap, and suggested a phase transition, but single electrons are forbidden from condensing to the same energy level by the Pauli exclusion principle. The site comments that "a drastic change in conductivity demanded a drastic change in electron behavior". Conceivably, pairs of electrons might perhaps act like bosons instead, which are bound by different condensate rules and do not have the same limitation.

• Isotope effect on the critical temperature, suggesting lattice interactions

The Debye frequency of phonons in a lattice is proportional to the inverse of the square root of the mass of lattice ions. It was shown that the superconducting transition temperature of mercury indeed showed the same dependence, by substituting the most abundant natural mercury isotope, ²⁰²Hg, with a different isotope, ¹⁹⁸Hg.

• An exponential rise in heat capacity near the critical temperature for some superconductors

An exponential increase in heat capacity near the critical temperature also suggests an energy bandgap for the superconducting material. As superconducting vanadium is warmed toward its critical temperature, its heat capacity increases greatly in a very few degrees; this suggests an energy gap being bridged by thermal energy.

• The lessening of the measured energy gap towards the critical temperature

This suggests a type of situation where some kind of binding energy exists but it is gradually weakened as the temperature increases toward the critical temperature. A binding energy suggests two or more particles or other entities that are bound together in the superconducting state. This helped to support the idea of bound particles – specifically electron pairs – and together with the above helped to paint a general picture of paired electrons and their lattice interactions.

Implications

BCS derived several important theoretical predictions that are independent of the details of the interaction, since the quantitative predictions mentioned below hold for any sufficiently weak attraction between the electrons and this last condition is fulfilled for many low temperature superconductors - the so-called weak-coupling case. These have been confirmed in numerous experiments:

• The electrons are bound into Cooper pairs, and these pairs are correlated due to the Pauli exclusion principle for the electrons, from which they are constructed. Therefore, in order to break a pair, one has to change energies of all other pairs. This means there is an energy gap for single-particle excitation, unlike in the normal metal (where the state of an electron can be changed by adding an arbitrarily small amount of energy). This energy gap is highest at low temperatures but vanishes at the transition temperature when superconductivity ceases to exist. The BCS theory gives an expression that shows how the gap grows with the strength of the attractive interaction and the (normal phase) single particle density of states at the Fermi level. Furthermore, it describes how the density of states is changed on entering the superconducting state, where there are no electronic states any more at the Fermi level. The energy gap is most directly observed in tunneling experiments and in reflection of microwaves from superconductors.
• BCS theory predicts the dependence of the value of the energy gap Δ at temperature T on the critical temperature T_c. The ratio between the value of the energy gap at zero temperature and the value of the superconducting transition temperature (expressed in energy units) takes the universal value $${\displaystyle \mathrm{\Delta }(T=0)=1.764k_{\mathrm{B}}{T}_{\mathrm{c}},}$$ independent of material. Near the critical temperature the relation asymptotes to $${\displaystyle \mathrm{\Delta }(T\to T_{\mathrm{c}})\approx 3.06k_{\mathrm{B}}{T}_{\mathrm{c}}\sqrt{1-(T/T_{\mathrm{c}})}}$$ which is of the form suggested the previous year by M. J. Buckingham based on the fact that the superconducting phase transition is second order, that the superconducting phase has a mass gap and on Blevins, Gordy and Fairbank's experimental results the previous year on the absorption of millimeter waves by superconducting tin.
• Due to the energy gap, the specific heat of the superconductor is suppressed strongly (exponentially) at low temperatures, there being no thermal excitations left. However, before reaching the transition temperature, the specific heat of the superconductor becomes even higher than that of the normal conductor (measured immediately above the transition) and the ratio of these two values is found to be universally given by 2.5.
• BCS theory correctly predicts the Meissner effect, i.e. the expulsion of a magnetic field from the superconductor and the variation of the penetration depth (the extent of the screening currents flowing below the metal's surface) with temperature.
• It also describes the variation of the critical magnetic field (above which the superconductor can no longer expel the field but becomes normal conducting) with temperature. BCS theory relates the value of the critical field at zero temperature to the value of the transition temperature and the density of states at the Fermi level.
• In its simplest form, BCS gives the superconducting transition temperature T_c in terms of the electron-phonon coupling potential V and the Debye cutoff energy E_D: $${\displaystyle k_{\mathrm{B}}{T}_{\mathrm{c}}=1.134E_{\mathrm{D}}{e}^{-1/N(0)V},}$$ where N(0) is the electronic density of states at the Fermi level. For more details, see Cooper pairs.
• The BCS theory reproduces the isotope effect, which is the experimental observation that for a given superconducting material, the critical temperature is inversely proportional to the square-root of the mass of the isotope used in the material. The isotope effect was reported by two groups on 24 March 1950, who discovered it independently working with different mercury isotopes, although a few days before publication they learned of each other's results at the ONR conference in Atlanta. The two groups are Emanuel Maxwell, and C. A. Reynolds, B. Serin, W. H. Wright, and L. B. Nesbitt. The choice of isotope ordinarily has little effect on the electrical properties of a material, but does affect the frequency of lattice vibrations. This effect suggests that superconductivity is related to vibrations of the lattice. This is incorporated into BCS theory, where lattice vibrations yield the binding energy of electrons in a Cooper pair.
• Little–Parks experiment - One of the first indications to the importance of the Cooper-pairing principle.

Gravitational energy

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https://en.wikipedia.org/wiki/Gravitational_energy

Image depicting Earth's gravitational field. Objects accelerate towards the Earth, thus losing their gravitational energy and transforming it into kinetic energy.

Gravitational energy or gravitational potential energy is the potential energy a massive object has due to its position in a gravitational field. It is the mechanical work done by the gravitational force to bring the mass from a chosen reference point (often an "infinite distance" from the mass generating the field) to some other point in the field, which is equal to the change in the kinetic energies of the objects as they fall towards each other. Gravitational potential energy increases when two objects are brought further apart and is converted to kinetic energy as they are allowed to fall towards each other.

Formulation

For two pairwise interacting point particles, the gravitational potential energy ${\displaystyle U}$ is the work done by the gravitational force in bringing the masses together: $${\displaystyle U=-W_g={\mathbf{\text{-F}}}_g\cdot \mathbf{\text{r}},}$$ where $\mathbf{\text{r}}$ is the displacement vector between the two particles and $\cdot$ denotes the scalar product. Since the gravitational force is always parallel to the axis joining the particles, this simplifies to:

$${\displaystyle U=-\frac{GMm}{|\mathbf{\text{r}}|},}$$

where ${\displaystyle M}$ and ${\displaystyle m}$ are the masses of the two particles and ${\displaystyle G}$ is the gravitational constant.

Close to the Earth's surface, the gravitational field is approximately constant, and the gravitational potential energy of an object reduces to $${\displaystyle U=m(\mathbf{\text{g}}\cdot \mathbf{\text{r}})=m|\mathbf{\text{g}}||\mathbf{\text{r}}|=mh|\mathbf{\text{g}}|,}$$ where ${\displaystyle m}$ is the object's mass, $\mathbf{\text{g}}=\frac{G{M}_{\oplus }\widehat{\mathbf{\text{r}}}}{|{\mathbf{\text{r}}}_{\oplus }^2|}$ is the gravity of Earth, and ${\displaystyle h}$ is the height of the object's center of mass above a chosen reference level.

Newtonian mechanics

In classical mechanics, two or more masses always have a gravitational potential. Conservation of energy requires that this gravitational field energy is always negative, so that it is zero when the objects are infinitely far apart. The gravitational potential energy is the potential energy an object has because it is within a gravitational field.

The magnitude of the force between a point mass, ${\displaystyle M}$, and another point mass, ${\displaystyle m}$, is given by Newton's law of gravitation: $${\displaystyle F=\frac{GMm}{r^2}}$$

To get the total work done by the gravitational force in bringing point mass ${\displaystyle m}$ from infinity to final distance ${\displaystyle R}$ (for example, the radius of Earth) from point mass $M$, the force is integrated with respect to displacement: $${\displaystyle W={\int }_{\mathrm{\infty }}^R\frac{GMm}{r^2}dr=-{\frac{GMm}{r}|}_{\mathrm{\infty }}^R}$$

Because $\underset{r\to \mathrm{\infty }}{lim}\frac{1}{r}=0$, the total work done on the object can be written as:

Gravitational Potential Energy

${\displaystyle U=-\frac{GMm}{R}}$

In the common situation where a much smaller mass ${\displaystyle m}$ is moving near the surface of a much larger object with mass ${\displaystyle M}$, the gravitational field is nearly constant and so the expression for gravitational energy can be considerably simplified. The change in potential energy moving from the surface (a distance ${\displaystyle R}$ from the center) to a height ${\displaystyle h}$ above the surface is $${\displaystyle \begin{array}{rl}\mathrm{\Delta }U& =\frac{GMm}{R}-\frac{GMm}{R+h}\\ & =\frac{GMm}{R}\left(1-\frac{1}{1+h/R}\right).\end{array}}$$ If ${\displaystyle h/R}$ is small, as it must be close to the surface where ${\displaystyle g}$ is constant, then this expression can be simplified using the binomial approximation $${\displaystyle \frac{1}{1+h/R}\approx 1-\frac{h}{R}}$$ to $${\displaystyle \begin{array}{rl}\mathrm{\Delta }U& \approx \frac{GMm}{R}\left[1-\left(1-\frac{h}{R}\right)\right]\\ \mathrm{\Delta }U& \approx \frac{GMmh}{R^2}\\ \mathrm{\Delta }U& \approx m\left(\frac{GM}{R^2}\right)h.\end{array}}$$ As the gravitational field is ${\displaystyle g=GM/R^2}$, this reduces to $${\displaystyle \mathrm{\Delta }U\approx mgh.}$$ Taking ${\displaystyle U=0}$ at the surface (instead of at infinity), the familiar expression for gravitational potential energy emerges: $${\displaystyle U=mgh.}$$

General relativity

Main article: Mass in general relativity

A 2 dimensional depiction of curved geodesics ("world lines"). According to general relativity, mass distorts spacetime and gravity is a natural consequence of Newton's First Law. Mass tells spacetime how to bend, and spacetime tells mass how to move.

In general relativity gravitational energy is extremely complex, and there is no single agreed upon definition of the concept. It is sometimes modelled via the Landau–Lifshitz pseudotensor that allows retention for the energy–momentum conservation laws of classical mechanics. Addition of the matter stress–energy tensor to the Landau–Lifshitz pseudotensor results in a combined matter plus gravitational energy pseudotensor that has a vanishing 4-divergence in all frames—ensuring the conservation law. Some people object to this derivation on the grounds that pseudotensors are inappropriate in general relativity, but the divergence of the combined matter plus gravitational energy pseudotensor is a tensor.

Non-overlapping magisteria

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Non-overlapping_magisteria

Non-overlapping magisteria (NOMA) is the view, advocated by paleontologist Stephen Jay Gould, that science and religion each represent different areas of inquiry, fact vs. values, so there is a difference between the "nets" over which they have "a legitimate magisterium, or domain of teaching authority", and the two domains do not overlap. He suggests, with examples, that "NOMA enjoys strong and fully explicit support, even from the primary cultural stereotypes of hard-line traditionalism" and that it is "a sound position of general consensus, established by long struggle among people of goodwill in both magisteria." Some have criticized the idea or suggested limitations to it, and there continues to be disagreement over where the boundaries between the two magisteria should be.

Gould's separate magisteria

In a 1997 essay "Non-overlapping Magisteria" for Natural History magazine, and later in his book Rocks of Ages (1999), Gould put forward what he described as "a blessedly simple and entirely conventional resolution to ... the supposed conflict between science and religion", from his puzzlement over the need and reception of the 1996 address of Pope John Paul II to the Pontifical Academy of Sciences "Truth Cannot Contradict Truth". He draws the term magisterium from Pope Pius XII's encyclical, Humani generis (1950), and defines it as "a domain where one form of teaching holds the appropriate tools for meaningful discourse and resolution", and describes the NOMA principle as "Science tries to document the factual character of the natural world, and to develop theories that coordinate and explain these facts. Religion, on the other hand, operates in the equally important, but utterly different, realm of human purposes, meanings, and values—subjects that the factual domain of science might illuminate, but can never resolve." "These two magisteria do not overlap, nor do they encompass all inquiry (consider, for example, the magisterium of art and the meaning of beauty)."

Gould emphasized the legitimacy of each field of endeavor only within its appropriate area of inquiry: "NOMA also cuts both ways. If religion can no longer dictate the nature of factual conclusions residing properly within the magisterium of science, then scientists cannot claim higher insight into moral truth from any superior knowledge of the world's empirical constitution." In the chapter "NOMA Defined and Defended" Gould gave examples of the types of questions appropriate to each area of inquiry, on the topic of "our relationship with other living creatures": "Do humans look so much like apes because we share a recent common ancestor or because creation followed a linear order, with apes representing the step just below us?" represents an inquiry concerning fact, while "Under what conditions (if ever) do we have a right to drive other species to extinction by elimination of their habitats? Do we violate any moral codes when we use genetic technology to place a gene from one creature into the genome of another species?" represent questions in the domain of values. He went on to present "an outline of historical reasons for the existence of conflict, where none should exist".

In a speech before the American Institute of Biological Sciences, Gould stressed the diplomatic reasons for adopting NOMA as well, stating that "the reason why we support that position is that it happens to be right, logically. But we should also be aware that it is very practical as well if we want to prevail." Gould argued that if indeed the polling data was correct—and that 80–90% of Americans believe in a supreme being, and such a belief is misunderstood to be at odds with evolution—then "we have to keep stressing that religion is a different matter, and science is not in any sense opposed to it", otherwise "we're not going to get very far". He did not, however, consider this diplomatic aspect to be paramount, writing in 1997: "NOMA represents a principled position on moral and intellectual grounds, not a mere diplomatic stance."

In 1997 he had elaborated on this position by describing his role as a scientist with respect to NOMA:

Religion is too important to too many people for any dismissal or denigration of the comfort still sought by many folks from theology. I may, for example, privately suspect that papal insistence on divine infusion of the soul represents a sop to our fears, a device for maintaining a belief in human superiority within an evolutionary world offering no privileged position to any creature. But I also know that souls represent a subject outside the magisterium of science. My world cannot prove or disprove such a notion, and the concept of souls cannot threaten or impact my domain. Moreover, while I cannot personally accept the Catholic view of souls, I surely honor the metaphorical value of such a concept both for grounding moral discussion and for expressing what we most value about human potentiality: our decency, care, and all the ethical and intellectual struggles that the evolution of consciousness imposed upon us.

Ciarán Benson sees a tendency to re-negotiate the borders between the "human sciences and the natural sciences", as in Wilhelm Dilthey's 1883 claim for the distinction between Geisteswissenschaften (humanities) and Naturwissenschaften (science). The astrophysicist Arnold O. Benz proposes that the boundary between the two magisteria is in the different ways they perceive reality: objective measurements in science, participatory experience in religion. The two planes of perceptions differ, but meet each other, for example, in amazement and in ethics.

National Academy of Sciences

Also in 1999, the National Academy of Sciences adopted a similar stance. Its publication Science and Creationism stated that "Scientists, like many others, are touched with awe at the order and complexity of nature. Indeed, many scientists are deeply religious. But science and religion occupy two separate realms of human experience. Demanding that they be combined detracts from the glory of each."

Humani generis

Gould wrote that he was inspired to consider non-overlapping magisteria after being driven to examine the 1950 encyclical Humani generis, in which Pope Pius XII permits Catholics to entertain the hypothesis of evolution for the human body so long as they accept the divine infusion of the soul. Gould cited the following paragraph:

The Teaching Authority of the Church does not forbid that, in conformity with the present state of human sciences and sacred theology, research and discussions, on the part of men experienced in both fields, take place with regard to the doctrine of evolution, in as far as it inquires into the origin of the human body as coming from pre-existent and living matter—for the Catholic faith obliges us to hold that souls are immediately created by God.

Reception

Richard Dawkins has criticized Gould's position on the grounds that religion is not divorced from scientific matters or the material world. He writes, "it is completely unrealistic to claim, as Gould and many others do, that religion keeps itself away from science's turf, restricting itself to morals and values. A universe with a supernatural presence would be a fundamentally and qualitatively different kind of universe from one without. The difference is, inescapably, a scientific difference. Religions make existence claims, and this means scientific claims."

Dawkins also argues that a religion free of divine intervention would be far different from any existing ones, and certainly different from the Abrahamic religions. Moreover, he claims that religions would be only too happy to accept scientific claims that supported their views. For example, if DNA evidence proved that Jesus had no earthly father, Dawkins claims that the argument of non-overlapping magisteria would be quickly dropped.

The theologian Friedrich Wilhelm Graf has been sympathetic to the approach, but claims it for the theological side—Graf assumes that e.g. creationism may be interpreted as a reaction of religious communities on the Verweltanschaulichung (i.e. interpretation as a worldview) of (natural) science in social Darwinism. That said, attempts to compete with religion by natural science may generate a backlash that is detrimental to both sides.

Ciarán Benson, a secular humanist, defends the spiritual as a category against both. He assumes that while Gould claims for NOMA (non-overlapping magisteria of science, morality and religion), and Richard Dawkins for, verbally, "a brand of SM (bondage of the others by the scientific magisterium)", Benson preferred OM (overlapping magisteria), especially in the case of art and religion.

Francis Collins criticized what he saw as the limits of NOMA, arguing that science, religion, and other spheres have "partially overlapped" while agreeing with Gould that morals, spirituality and ethics cannot be determined from naturalistic interpretation. This exceeds the greatest interconnection allowed by Gould in his original 1997 essay "Nonoverlapping Magisteria" in which he writes:

Each ... subject has a legitimate magisterium, or domain of teaching authority ... This resolution might remain all neat and clean if the nonoverlapping magisteria (NOMA) of science and religion were separated by an extensive no man's land. But, in fact, the two magisteria bump right up against each other, interdigitating in wondrously complex ways along their joint border. Many of our deepest questions call upon aspects of both for different parts of a full answer—and the sorting of legitimate domains can become quite complex and difficult.

Matt Ridley notes that religion does more than talk about ultimate meanings and morals, and science is not proscribed from talking about the above either. After all, morals involve human behavior, an observable phenomenon, and science is the study of observable phenomena. Ridley notes that there is substantial scientific evidence on evolutionary origins of ethics and morality.

Sam Harris has heavily criticized this concept in his book The Moral Landscape. Sam notes that "Meaning, values, morality and the good life must relate to facts about the well-being of conscious creatures – and, in our case, must lawfully depend upon events in the world and upon states of the human brain."