Infrastructure

From Wikipedia, the free encyclopedia

Infrastructure is the fundamental facilities and systems serving a country, city, or other area,^[1] including the services and facilities necessary for its economy to function.^[2] Infrastructure is composed of public and private physical improvements such as roads, bridges, tunnels, water supply, sewers, electrical grids, telecommunications (including Internet connectivity and broadband speeds). In general, it has also been defined as "the physical components of interrelated systems providing commodities and services essential to enable, sustain, or enhance societal living conditions."^[3]

San Francisco Ferry Building at night

There are two general types of ways to view infrastructure, hard or soft. Hard infrastructure refers to the physical networks necessary for the functioning of a modern industry.^[4] This includes roads, bridges, railways, etc. Soft infrastructure refers to all the institutions that maintain the economic, health, social, and cultural standards of a country.^[4]This includes educational programs, parks and recreational facilities, law enforcement agencies, and emergency services.

The word infrastructure has been used in English since 1887 and in French since 1875, originally meaning "The installations that form the basis for any operation or system".^[5][6] The word was imported from French, where it means subgrade, the native material underneath a constructed pavement or railway. The word is a combination of the Latin prefix "infra", meaning "below" and many of these constructions are underground, for example, tunnels, water and gas systems, and railways. The army use of the term achieved currency in the United States after the formation of NATO in the 1940s, and by 1970 was adopted by urban planners in its modern civilian sense.^[7]

Classifications

A 1987 US National Research Council panel adopted the term "public works infrastructure", referring to:

"... both specific functional modes – highways, streets, roads, and bridges; mass transit; airports and airways; water supply and water resources; wastewater management; solid-waste treatment and disposal; electric power generation and transmission; telecommunications; and hazardous waste management – and the combined system these modal elements comprise. A comprehension of infrastructure spans not only these public works facilities, but also the operating procedures, management practices, and development policies that interact together with societal demand and the physical world to facilitate the transport of people and goods, provision of water for drinking and a variety of other uses, safe disposal of society's waste products, provision of energy where it is needed, and transmission of information within and between communities."^[8]

The OECD also classifies communications as a part of infrastructure.^[9]

The American Society of Civil Engineers publish a "Infrastructure Report Card" which represents the organizations opinion on the condition of various infrastructure every 2–4 years.^[10] As of 2017 they grade 16 categories, namely Aviation, Bridges, Dams, Drinking Water, Energy, Hazardous Waste, Inland Waterways, Levees, Parks & Recreation, Ports, Rail, Roads, Schools, Solid Waste, Transit and Wastewater.^[10]:4

Personal

A way to embody personal infrastructure is to think of it in term of human capital.^[11] Human capital is defined by the Encyclopedia Britannica as “intangible collective resources possessed by individuals and groups within a given population."^[12] The goal of personal infrastructure is to determine the quality of the economic agents’ values. This results in three major tasks: the task of economic proxies’ in the economic process (teachers, unskilled and qualified labor, etc.); the importance of personal infrastructure for an individual (short and long-term consumption of education); and the social relevance of personal infrastructure.^[11]

Institutional

Institutional infrastructure branches from the term “economic constitution.” According to Gianpiero Torrisi, Institutional infrastructure is the object of economic and legal policy. It compromises the grown and sets norms.^[11] It refers to the degree of actual equal treatment of equal economic data and determines the framework within which economic agents may formulate their own economic plans and carry them out in co-operation with others.

Material

Material infrastructure is defined as “those immobile, non-circulating capital goods that essentially contribute to the production of infrastructure goods and services needed to satisfy basic physical and social requirements of economic agents."^[11] There are two distinct qualities of material infrastructures: 1) Fulfillment of social needs and 2) Mass production. The first characteristic deals with the basic needs of human life. The second characteristic is the non-availability of infrastructure goods and services.^[11]

Economic

According to the business dictionary, economic infrastructure can be defined as “internal facilities of a country that make business activity possible, such as communication, transportation and distribution networks, financial institutions and markets, and energy supply systems.”^[13] Economic infrastructure support productive activities and events. This includes roads, highways, bridges, airports, water distribution networks, sewer systems, irrigation plants, etc.^[11]

Social

Social infrastructure can be broadly defined as the construction and maintenance of facilities that support social services.^[14] Social infrastructures are created to increase social comfort and act on economic activity. These being schools, parks and playgrounds, structures for public safety, waste disposal plants, hospitals, sports area, etc.^[11]

Core

Core assets provide essential services and have monopolistic characteristics.^[15] Investors seeking core infrastructure look for five different characteristics: Income, Low volatility of returns, Diversification, Inflation Protection, and Long-term liability matching.^[15] Core Infrastructure incorporates all the main types of infrastructure. For instance; roads, highways, railways, public transportation, water and gas supply, etc.

Basic

Basic infrastructure refers to main railways, roads, canals, harbors and docks, the electromagnetic telegraph, drainage, dikes, and land reclamation.^[11] It consist of the more well-known features of infrastructure. The things in the world we come across everyday (buildings, roads, docks, etc).

Complementary

Complementary infrastructure refers to things like light railways, tramways, gas/electricity/water supply, etc.^[11] To complement something, means to bring to perfection or complete it. So, complementary infrastructure deals with the little parts of the engineering world the brings more life. The lights on the sidewalks, the landscaping around buildings, the benches for pedestrians to rest, etc.

Related concepts

The term infrastructure may be confused with the following overlapping or related concepts.

Land improvement and land development are general terms that in some contexts may include infrastructure, but in the context of a discussion of infrastructure would refer only to smaller scale systems or works that are not included in infrastructure, because they are typically limited to a single parcel of land, and are owned and operated by the land owner. For example, an irrigation canal that serves a region or district would be included with infrastructure, but the private irrigation systems on individual land parcels would be considered land improvements, not infrastructure. Service connections to municipal service and public utility networks would also be considered land improvements, not infrastructure.^[16][17]

The term public works includes government-owned and operated infrastructure as well as public buildings, such as schools and court houses. Public works generally refers to physical assets needed to deliver public services. Public services include both infrastructure and services generally provided by government.

Ownership and financing

Infrastructure may be owned and managed by governments or by private companies, such as sole public utility or railway companies. Generally, most roads, major airports and other ports, water distribution systems, and sewage networks are publicly owned, whereas most energy and telecommunications networks are privately owned.^{[citation needed]} Publicly owned infrastructure may be paid for from taxes, tolls, or metered user fees, whereas private infrastructure is generally paid for by metered user fees.^{[citation needed]} Major investment projects are generally financed by the issuance of long-term bonds.^{[citation needed]}

Government-owned and operated infrastructure may be developed and operated in the private sector or in public-private partnerships, in addition to in the public sector. As of 2008 in the United States for example, public spending on infrastructure has varied between 2.3% and 3.6% of GDP since 1950.^[18] Many financial institutions invest in infrastructure.

Types

Engineering and construction

Engineers generally limit the term "infrastructure" to describe fixed assets that are in the form of a large network; in other words, hard infrastructure.^{[citation needed]} Efforts to devise more generic definitions of infrastructures have typically referred to the network aspects of most of the structures, and to the accumulated value of investments in the networks as assets.^{[citation needed]} One such definition from 1998 defined infrastructure as the network of assets "where the system as a whole is intended to be maintained indefinitely at a specified standard of service by the continuing replacement and refurbishment of its components".^[19]

Civil defense and economic development

Civil defense planners and developmental economists generally refer to both hard and soft infrastructure, including public services such as schools and hospitals, emergency services such as police and fire fighting, and basic financial services. The notion of infrastructure-based development combining long-term infrastructure investments by government agencies at central and regional levels with public private partnerships has proven popular among economists in Asia (notably Singapore and China), mainland Europe, and Latin America.

Military

Military infrastructure is the buildings and permanent installations necessary for the support of military forces, whether they are stationed in bases, being deployed or engaged in operations. For example, barracks, headquarters, airfields, communications facilities, stores of military equipment, port installations, and maintenance stations.^[20]

Green

Green infrastructure (or blue-green infrastructure) highlights the importance of the natural environment in decisions about land use planning.^[21][22] In particular there is an emphasis on the "life support" functions provided by a network of natural ecosystems, with an emphasis on interconnectivity to support long-term sustainability. Examples include clean water and healthy soils, as well as the more anthropocentric functions such as recreation and providing shade and shelter in and around towns and cities. The concept can be extended to apply to the management of stormwater runoff at the local level through the use of natural systems, or engineered systems that mimic natural systems, to treat polluted runoff.^[23][24]

Communications

Communications infrastructure is the informal and formal channels of communication, political and social networks, or beliefs held by members of particular groups, as well as information technology, software development tools. Still underlying these more conceptual uses is the idea that infrastructure provides organizing structure and support for the system or organization it serves, whether it is a city, a nation, a corporation, or a collection of people with common interests. Examples include IT infrastructure, research infrastructure, terrorist infrastructure, employment infrastructure and tourism infrastructure.^{[citation needed]}

In the developing world

According to researchers at the Overseas Development Institute, the lack of infrastructure in many developing countries represents one of the most significant limitations to economic growth and achievement of the Millennium Development Goals (MDGs). Infrastructure investments and maintenance can be very expensive, especially in such areas as landlocked, rural and sparsely populated countries in Africa. It has been argued that infrastructure investments contributed to more than half of Africa's improved growth performance between 1990 and 2005, and increased investment is necessary to maintain growth and tackle poverty. The returns to investment in infrastructure are very significant, with on average thirty to forty percent returns for telecommunications (ICT) investments, over forty percent for electricity generation, and eighty percent for roads.^[26]

Regional differences

The demand for infrastructure, both by consumers and by companies is much higher than the amount invested.^[26] There are severe constraints on the supply side of the provision of infrastructure in Asia.^[27] The infrastructure financing gap between what is invested in Asia-Pacific (around US$48 billion) and what is needed (US$228 billion) is around US$180 billion every year.^[26]

In Latin America, three percent of GDP (around US$71 billion) would need to be invested in infrastructure in order to satisfy demand, yet in 2005, for example, only around two percent was invested leaving a financing gap of approximately US$24 billion.^[26]

In Africa, in order to reach the seven percent annual growth calculated to be required to meet the MDGs by 2015 would require infrastructure investments of about fifteen percent of GDP, or around US$93 billion a year. In fragile states, over thirty-seven percent of GDP would be required.^[26]

Sources of funding

The source of financing varies significantly across sectors. Some sectors are dominated by government spending, others by overseas development aid (ODA), and yet others by private investors.^[26] In California, infrastructure financing districts are established by local governments to pay for physical facilities and services within a specified area by using property tax increases.^[28] In order to facilitate investment of the private sector in developing countries' infrastructure markets, it is necessary to design risk-allocation mechanisms more carefully, given the higher risks of their markets.^[29]

In Sub-Saharan Africa, governments spend around US$9.4 billion out of a total of US$24.9 billion. In irrigation, governments represent almost all spending. In transport and energy a majority of investment is government spending. In ICT and water supply and sanitation, the private sector represents the majority of capital expenditure. Overall, between them aid, the private sector, and non-OECD financiers exceed government spending. The private sector spending alone equals state capital expenditure, though the majority is focused on ICT infrastructure investments. External financing increased in the 2000s (decade) and in Africa alone external infrastructure investments increased from US$7 billion in 2002 to US$27 billion in 2009. China, in particular, has emerged as an important investor.^[26]

Exact solutions in general relativity

From Wikipedia, the free encyclopedia

In general relativity, an exact solution is a Lorentzian manifold^{[clarification needed]} equipped with tensor fields modeling states of ordinary matter, such as a fluid, or classical nongravitational fields such as the electromagnetic field.

Background and definition

These tensor fields should obey any relevant physical laws (for example, any electromagnetic field must satisfy Maxwell's equations). Following a standard recipe^[which?] which is widely used in mathematical physics, these tensor fields should also give rise to specific contributions to the stress–energy tensor ${\displaystyle T^{\alpha \beta }}$.^[1] (A field is described by a Lagrangian, varying with respect to the field should give the field equations and varying with respect to the metric should give the stress-energy contribution due to the field.)

Finally, when all the contributions to the stress–energy tensor are added up, the result must be a solution of the Einstein field equations (written here in geometrized units, where speed of light c = Gravitational constant G = 1)

${\displaystyle G^{\alpha \beta }=8\pi \,T^{\alpha \beta }.}$

In the above field equations, ${\displaystyle G^{\alpha \beta }}$ is the Einstein tensor, computed uniquely from the metric tensor which is part of the definition of a Lorentzian manifold. Since giving the Einstein tensor does not fully determine the Riemann tensor, but leaves the Weyl tensor unspecified (see the Ricci decomposition), the Einstein equation may be considered a kind of compatibility condition: the spacetime geometry must be consistent with the amount and motion of any matter or nongravitational fields, in the sense that the immediate presence "here and now" of nongravitational energy–momentum causes a proportional amount of Ricci curvature "here and now". Moreover, taking covariant derivatives of the field equations and applying the Bianchi identities, it is found that a suitably varying amount/motion of nongravitational energy–momentum can cause ripples in curvature to propagate as gravitational radiation, even across vacuum regions, which contain no matter or nongravitational fields.

Difficulties with the definition

Any Lorentzian manifold is a solution of the Einstein field equation for some right hand side. This is illustrated by the following procedure:

• take any Lorentzian manifold, compute its Einstein tensor ${\displaystyle G^{\alpha \beta }}$, which is a purely mathematical operation
• divide by ${\displaystyle 8\pi }$
• declare the resulting symmetric second rank tensor field to be the stress–energy tensor ${\displaystyle T^{\alpha \beta }}$.

This shows that there are two complementary ways to use general relativity:

• One can fix the form of the stress–energy tensor (from some physical reasons, say) and study the solutions of the Einstein equations with such right hand side (for example, if the stress–energy tensor is chosen to be that of the perfect fluid, a spherically symmetric solution can serve as a stellar model)
• Alternatively, one can fix some geometrical properties of a spacetime and look for a matter source that could provide these properties. This is what cosmologists have done since the 2000s: they assume that the Universe is homogeneous, isotropic, and accelerating and try to realize what matter (called dark energy) can support such a structure.

Within the first approach the alleged stress–energy tensor must arise in the standard way from a "reasonable" matter distribution or nongravitational field. In practice, this notion is pretty clear, especially if we restrict the admissible nongravitational fields to the only one known in 1916, the electromagnetic field. But ideally we would like to have some mathematical characterization that states some purely mathematical test which we can apply to any putative "stress–energy tensor", which passes everything which might arise from a "reasonable" physical scenario, and rejects everything else. Unfortunately, no such characterization is known. Instead, we have crude tests known as the energy conditions, which are similar to placing restrictions on the eigenvalues and eigenvectors of a linear operator. But these conditions, it seems, can satisfy no-one. On the one hand, they are far too permissive: they would admit "solutions" which almost no-one believes are physically reasonable. On the other, they may be far too restrictive: the most popular energy conditions are apparently violated by the Casimir effect.

Einstein also recognized another element of the definition of an exact solution: it should be a Lorentzian manifold (meeting additional criteria), i.e. a smooth manifold. But in working with general relativity, it turns out to be very useful to admit solutions which are not everywhere smooth; examples include many solutions created by matching a perfect fluid interior solution to a vacuum exterior solution, and impulsive plane waves. Once again, the creative tension between elegance and convenience, respectively, has proven difficult to resolve satisfactorily.

In addition to such local objections, we have the far more challenging problem that there are very many exact solutions which are locally unobjectionable, but globally exhibit causally suspect features such as closed timelike curves or structures with points of separation ("trouser worlds"). Some of the best known exact solutions, in fact, have globally a strange character.

Types of exact solution

Many well-known exact solutions belong to one of several types, depending upon the intended physical interpretation of the stress–energy tensor:

• Vacuum solutions: ${\displaystyle T^{\alpha \beta }=0}$; these describe regions in which no matter or nongravitational fields are present,
• Electrovacuum solutions: ${\displaystyle T^{\alpha \beta }}$ must arise entirely from an electromagnetic field which solves the source-free Maxwell equations on the given curved Lorentzian manifold; this means that the only source for the gravitational field is the field energy (and momentum) of the electromagnetic field,
• Null dust solutions: ${\displaystyle T^{\alpha \beta }}$ must correspond to a stress–energy tensor which can be interpreted as arising from incoherent electromagnetic radiation, without necessarily solving the Maxwell field equations on the given Lorentzian manifold,
• Fluid solutions: ${\displaystyle T^{\alpha \beta }}$ must arise entirely from the stress–energy tensor of a fluid (often taken to be a perfect fluid); the only source for the gravitational field is the energy, momentum, and stress (pressure and shear stress) of the matter comprising the fluid.

In addition to such well established phenomena as fluids or electromagnetic waves, one can contemplate models in which the gravitational field is produced entirely by the field energy of various exotic hypothetical fields:

• Scalar field solutions: ${\displaystyle T^{\alpha \beta }}$ must arise entirely from a scalar field (often a massless scalar field); these can arise in classical field theory treatments of meson beams, or as quintessence,
• Lambdavacuum solutions (not a standard term, but a standard concept for which no name yet exists): ${\displaystyle T^{\alpha \beta }}$ arises entirely from a nonzero cosmological constant.

One possibility which has received little attention (perhaps because the mathematics is so challenging) is the problem of modeling an elastic solid. Presently, it seems that no exact solutions for this specific type are known.

Below we have sketched a classification by physical interpretation. This is probably more useful for most readers than the Segre classification of the possible algebraic symmetries of the Ricci tensor, but for completeness we note the following facts:

• nonnull electrovacuums have Segre type ${\displaystyle \{\,(1,1)(11)\}}$ and isotropy group SO(1,1) x SO(2),
• null electrovacuums and null dusts have Segre type ${\displaystyle \{\,(2,11)\}}$ and isotropy group E(2),
• perfect fluids have Segre type ${\displaystyle \{\,1,(111)\}}$ and isotropy group SO(3),
• Lambdavacuums have Segre type ${\displaystyle \{\,(1,111)\}}$ and isotropy group SO(1,3).

The remaining Segre types have no particular physical interpretation and most of them cannot correspond to any known type of contribution to the stress–energy tensor.

Examples

Noteworthy examples of vacuum solutions, electrovacuum solutions, and so forth, are listed in specialized articles (see below). These solutions contain at most one contribution to the energy–momentum tensor, due to a specific kind of matter or field. However, there are some notable exact solutions which contain two or three contributions, including:

• NUT-Kerr–Newman–de Sitter solution contains contributions from an electromagnetic field and a positive vacuum energy, as well as a kind of vacuum perturbation of the Kerr vacuum which is specified by the so-called NUT parameter,
• Gödel dust contains contributions from a pressureless perfect fluid (dust) and from a positive vacuum energy.

Some hypothetical possibilities which don't fit into our rough classification^{[clarification needed]} are:

• certain wormhole metrics^[which?],
• Alcubierre metric.
• "Time machines", i.e. initially nice spacetimes^{[clarification needed]} in which at some stage of evolution closed causal curves appear^{[clarification needed]}.

Some doubt has been cast^{[according to whom?]} upon whether sufficient quantity of exotic matter needed for wormholes and Alcubierre bubbles can exist.^[2] Later, however, these doubts were shown^[3] to be mostly groundless. The third of these examples, in particular, is an instructive example of the procedure mentioned above for turning any Lorentzian manifold into a "solution". It is along this way that Hawking succeeded in proving^[4] that time machines of a certain type (those with a "compactly generated Cauchy horizon") cannot appear without exotic matter. Such spacetimes are also a good illustration of the fact that unless a spacetime is especially nice ("globally hyperbolic") the Einstein equations do not determine its evolution uniquely. Any spacetime may evolve into a time machine, but it never has to do so.^[5]

Constructing solutions

The Einstein field equations are a system of coupled, nonlinear partial differential equations. In general, this makes them hard to solve^[why?]. Nonetheless, several effective techniques for obtaining exact solutions have been established.

The simplest involves imposing symmetry conditions on the metric tensor, such as stationarity (symmetry under time translation) or axisymmetry (symmetry under rotation about some symmetry axis). With sufficiently clever assumptions of this sort, it is often possible to reduce the Einstein field equation to a much simpler system of equations, even a single partial differential equation (as happens in the case of stationary axisymmetric vacuum solutions, which are characterized by the Ernst equation) or a system of ordinary differential equations (as happens in the case of the Schwarzschild vacuum).

This naive approach usually works best if one uses a frame field rather than a coordinate basis.

A related idea involves imposing algebraic symmetry conditions on the Weyl tensor, Ricci tensor, or Riemann tensor. These are often stated in terms of the Petrov classification of the possible symmetries of the Weyl tensor, or the Segre classification of the possible symmetries of the Ricci tensor. As will be apparent from the discussion above, such Ansätze often do have some physical content, although this might not be apparent from their mathematical form.

This second kind of symmetry approach has often been used with the Newman–Penrose formalism, which uses spinorial quantities for more efficient bookkeeping.

Even after such symmetry reductions, the reduced system of equations is often difficult to solve. For example, the Ernst equation is a nonlinear partial differential equation somewhat resembling the nonlinear Schrödinger equation (NLS).

But recall that the conformal group on Minkowski spacetime is the symmetry group of the Maxwell equations. Recall too that solutions of the heat equation can be found by assuming a scaling Ansatz. These notions are merely special cases of Sophus Lie's notion of the point symmetry of a differential equation (or system of equations), and as Lie showed, this can provide an avenue of attack upon any differential equation which has a nontrivial symmetry group. Indeed, both the Ernst equation and the NLS have nontrivial symmetry groups, and some solutions can be found by taking advantage of their symmetries. These symmetry groups are often infinite dimensional, but this is not always a useful feature.

Emmy Noether showed that a slight but profound generalization of Lie's notion of symmetry can result in an even more powerful method of attack. This turns out to be closely related to the discovery that some equations, which are said to be completely integrable, enjoy an infinite sequence of conservation laws. Quite remarkably, both the Ernst equation (which arises several ways in the studies of exact solutions) and the NLS turn out to be completely integrable. They are therefore susceptible to solution by techniques resembling the inverse scattering transform which was originally developed to solve the Korteweg-de Vries (KdV) equation, a nonlinear partial differential equation which arises in the theory of solitons, and which is also completely integrable. Unfortunately, the solutions obtained by these methods are often not as nice as one would like. For example, in a manner analogous to the way that one obtains a multiple soliton solution of the KdV from the single soliton solution (which can be found from Lie's notion of point symmetry), one can obtain a multiple Kerr object solution, but unfortunately, this has some features which make it physically implausible.^[6]

There are also various transformations (see Belinski-Zakharov transform) which can transform (for example) a vacuum solution found by other means into a new vacuum solution, or into an electrovacuum solution, or a fluid solution. These are analogous to the Bäcklund transformations known from the theory of certain partial differential equations, including some famous examples of soliton equations. This is no coincidence, since this phenomenon is also related to the notions of Noether and Lie regarding symmetry. Unfortunately, even when applied to a "well understood", globally admissible solution, these transformations often yield a solution which is poorly understood and their general interpretation is still unknown.

Existence of solutions

Given the difficulty of constructing explicit small families of solutions, much less presenting something like a "general" solution to the Einstein field equation, or even a "general" solution to the vacuum field equation, a very reasonable approach is to try to find qualitative properties which hold for all solutions, or at least for all vacuum solutions. One of the most basic questions one can ask is: do solutions exist, and if so, how many?

To get started, we should adopt a suitable initial value formulation of the field equation, which gives two new systems of equations, one giving a constraint on the initial data, and the other giving a procedure for evolving this initial data into a solution. Then, one can prove that solutions exist at least locally, using ideas not terribly dissimilar from those encountered in studying other differential equations.

To get some idea of "how many" solutions we might optimistically expect, we can appeal to Einstein's constraint counting method. A typical conclusion from this style of argument is that a generic vacuum solution to the Einstein field equation can be specified by giving four arbitrary functions of three variables and six arbitrary functions of two variables. These functions specify initial data, from which a unique vacuum solution can be evolved. (In contrast, the Ernst vacuums, the family of all stationary axisymmetric vacuum solutions, are specified by giving just two functions of two variables, which are not even arbitrary, but must satisfy a system of two coupled nonlinear partial differential equations. This may give some idea of how just tiny a typical "large" family of exact solutions really is, in the grand scheme of things.)

However, this crude analysis falls far short of the much more difficult question of global existence of solutions. The global existence results which are known so far turn out to involve another idea.

Global stability theorems

We can imagine "disturbing" the gravitational field outside some isolated massive object by "sending in some radiation from infinity". We can ask: what happens as the incoming radiation interacts with the ambient field? In the approach of classical perturbation theory, we can start with Minkowski vacuum (or another very simple solution, such as the de Sitter lambda vacuum), introduce very small metric perturbations, and retain only terms up to some order in a suitable perturbation expansion—somewhat like evaluating a kind of Taylor series for the geometry of our spacetime. This approach is essentially the idea behind the post-Newtonian approximations used in constructing models of a gravitating system such as a binary pulsar. However, perturbation expansions are generally not reliable for questions of long-term existence and stability, in the case of nonlinear equations.

The full field equation is highly nonlinear, so we really want to prove that the Minkowski vacuum is stable under small perturbations which are treated using the fully nonlinear field equation. This requires the introduction of many new ideas. The desired result, sometimes expressed by the slogan that the Minkowski vacuum is nonlinearly stable, was finally proven by Demetrios Christodoulou and Sergiu Klainerman only in 1993. Analogous results are known for lambdavac perturbations of the de Sitter lambdavacuum (Helmut Friedrich) and for electrovacuum perturbations of the Minkowski vacuum (Nina Zipser).

The positive energy theorem

Another issue we might worry about is whether the net mass-energy of an isolated concentration of positive mass-energy density (and momentum) always yields a well-defined (and non-negative) net mass. This result, known as the positive energy theorem was finally proven by Richard Schoen and Shing-Tung Yau in 1979, who made an additional technical assumption about the nature of the stress–energy tensor. The original proof is very difficult; Edward Witten soon presented a much shorter "physicist's proof", which has been justified by mathematicians—using further very difficult arguments. Roger Penrose and others have also offered alternative arguments for variants of the original positive energy theorem.

Magical thinking

From Wikipedia, the free encyclopedia

Magical thinking is a term used in anthropology and psychology, denoting the fallacious attribution of causal relationships between actions and events, with subtle differences in meaning between the two fields. In anthropology, it denotes the attribution of causality between entities grouped with one another (coincidence) or similar to one another. In psychology, the entities between which a causal relation has to be posited are more strictly delineated; here it denotes the belief that one's thoughts by themselves can bring about effects in the world or that thinking something corresponds with doing it.^[1] In both cases, the belief can cause a person to experience fear, seemingly not rationally justifiable to an observer outside the belief system of performing certain acts or having certain thoughts because of an assumed correlation between doing so and threatening calamities.

Anthropology

In religion, folk religion, and superstitious beliefs, the posited causality is between religious ritual, prayer, sacrifice, or the observance of a taboo, and an expected benefit or recompense. The use of a lucky item or ritual, for example, is assumed to increase the probability that one will perform at a level so that one can achieve a desired goal or outcome.^[2]

Researchers have identified two possible principles as the formal causes of the attribution of false causal relationships:

• the temporal contiguity of two events
• "associative thinking", the association of entities based upon their semblance to one another

Prominent Victorian theorists identified associative thinking (a common feature of practitioners of magic) as a characteristic form of irrationality. As with all forms of magical thinking, association-based and similarities-based notions of causality are not always said to be the practice of magic by a magician. For example, the doctrine of signatures held that similarities between plant parts and body parts indicated their efficacy in treating diseases of those body parts, and was a part of Western medicine during the Middle Ages. This association-based thinking is a vivid example of the general human application of the representativeness heuristic.^[3]

Edward Burnett Tylor coined the term "associative thinking", characterizing it as pre-logical,^{[4][not in citation given]} in which the "magician's folly" is in mistaking an imagined connection with a real one. The magician believes that thematically linked items can influence one another by virtue of their similarity.^[5] For example, in E. E. Evans-Pritchard's account, members of the Azande tribe^[6] believe that rubbing crocodile teeth on banana plants can invoke a fruitful crop. Because crocodile teeth are curved (like bananas) and grow back if they fall out, the Azande observe this similarity and want to impart this capacity of regeneration to their bananas. To them, the rubbing constitutes a means of transference.

Sir James Frazer (1854-1941) elaborated upon Tylor's principle by dividing magic into the categories of sympathetic and contagious magic. The latter is based upon the law of contagion or contact, in which two things that were once connected retain this link and have the ability to affect their supposedly related objects, such as harming a person by harming a lock of his hair. Sympathetic magic and homeopathy operate upon the premise that "like affects like", or that one can impart characteristics of one object to a similar object. Frazer believed that some individuals think the entire world functions according to these mimetic, or homeopathic, principles.^[7]

In How Natives Think (1925), Lucien Lévy-Bruhl describes a similar notion of mystical, "collective representations". He too sees magical thinking as fundamentally different from a Western style of thought. He asserts that in these representations, "primitive" people's "mental activity is too little differentiated for it to be possible to consider ideas or images of objects by themselves apart from the emotions and passions which evoke those ideas or are evoked by them".^[8] Lévy-Bruhl explains that natives commit the post hoc, ergo propter hoc fallacy, in which people observe that x is followed by y, and conclude that x has caused y.^[9] He believes that this fallacy is institutionalized in native culture and is committed regularly and repeatedly.

Despite the view that magic is less than rational and entails an inferior concept of causality, in The Savage Mind (1966), Claude Lévi-Strauss suggested that magical procedures are relatively effective in exerting control over the environment. This outlook has generated alternative theories of magical thinking, such as the symbolic and psychological approaches, and softened the contrast between "educated" and "primitive" thinking: "Magical thinking is no less characteristic of our own mundane intellectual activity than it is of Zande curing practices."^{[10][n 1]}

Other forms

Bronisław Malinowski's Magic, Science and Religion (1954) discusses another type of magical thinking, in which words and sounds are thought to have the ability to directly affect the world.^[11] This type of wish fulfillment thinking can result in the avoidance of talking about certain subjects ("speak of the devil and he'll appear"), the use of euphemisms instead of certain words, or the belief that to know the "true name" of something gives one power over it, or that certain chants, prayers, or mystical phrases will bring about physical changes in the world. More generally, it is magical thinking to take a symbol to be its referent or an analogy to represent an identity.

Sigmund Freud believed that magical thinking was produced by cognitive developmental factors. He described practitioners of magic as projecting their mental states onto the world around them, similar to a common phase in child development.^[12] From toddlerhood to early school age, children will often link the outside world with their internal consciousness, e.g. "It is raining because I am sad."

Symbolic approach to magic

Another theory of magical thinking is the symbolic approach. Leading thinkers of this category, including Stanley J. Tambiah, believe that magic is meant to be expressive, rather than instrumental. As opposed to the direct, mimetic thinking of Frazer, Tambiah asserts that magic utilizes abstract analogies to express a desired state, along the lines of metonymy or metaphor.^[13]

An important question raised by this interpretation is how mere symbols could exert material effects. One possible answer lies in John L. Austin's concept of "performativity," in which the act of saying something makes it true, such as in an inaugural or marital rite.^[14] Other theories propose that magic is effective because symbols are able to affect internal psycho-physical states. They claim that the act of expressing a certain anxiety or desire can be reparative in itself.^[15]

Psychological functions of magic

A healing ritual (the laying on of hands)

Some scholars believe that magic is effective psychologically. They cite the placebo effect and psychosomatic disease as prime examples of how our mental functions exert power over our bodies.^[16] Similarly, Robin Horton suggests that engaging in magical practices surrounding healing can relieve anxiety, which could have a significant positive physical effect. In the absence of advanced health care, such effects would play a relatively major role, thereby helping to explain the persistence and popularity of such practices.^[17][18]

According to theories of anxiety relief and control, people turn to magical beliefs when there exists a sense of uncertainty and potential danger and few logical or scientific responses to such danger. Magic is used to restore a sense of control over circumstance. In support of this theory, research indicates that superstitious behavior is invoked more often in high stress situations, especially by people with a greater desire for control.^[19][20]

Another potential reason for the persistence of magic rituals is that the rituals prompt their own use by creating a feeling of insecurity and then proposing themselves as precautions.^[21] Boyer and Liénard propose that in obsessive-compulsive rituals — a possible clinical model for certain forms of magical thinking — focus shifts to the lowest level of gestures, resulting in goal demotion. For example, an obsessive-compulsive cleaning ritual may overemphasize the order, direction, and number of wipes used to clean the surface. The goal becomes less important than the actions used to achieve the goal, with the implication that magic rituals can persist without efficacy because the intent is lost within the act.^[21] Alternatively, some cases of harmless "rituals" may have positive effects in bolstering intent, as may be the case with certain pre-game exercises in sports.^[22]

Phenomenological approach

Ariel Glucklich tries to understand magic from a subjective perspective, attempting to comprehend magic on a phenomenological, experientially based level. Glucklich seeks to describe the attitude that magical practitioners feel which he calls "magical consciousness" or the "magical experience." He explains that it is based upon "the awareness of the interrelatedness of all things in the world by means of simple but refined sense perception."^[23]

Another phenomenological model is that of Gilbert Lewis, who argues that "habit is unthinking." He believes that those practicing magic do not think of an explanatory theory behind their actions any more than the average person tries to grasp the pharmaceutical workings of aspirin.^[24] When the average person takes an aspirin, he does not know how the medicine chemically functions. He takes the pill with the premise that there is proof of efficacy. Similarly, many who avail themselves of magic do so without feeling the need to understand a causal theory behind it.

Idiomatic difference

Robin Horton maintains that the difference between the thinking of Western and of non-Western peoples is predominantly "idiomatic." He asserts that the members of both cultures use the same practical common-sense, and that both science and magic are ways beyond basic logic by which people formulate theories to explain whatever occurs. However, non-Western cultures use the idiom of magic and have community spiritual figures, and therefore non-Westerners turn to magical practices or to a specialist in that idiom. Horton sees the same logic and common-sense in all cultures, but notes that their contrasting ontological idioms lead to cultural practices which seem illogical to observers whose own culture has correspondingly contrasting norms. He explains, "[T]he layman's grounds for accepting the models propounded by the scientist are often no different from the young African villager's ground for accepting the models propounded by one of his elders."^[25]

Along similar lines, Michael F. Brown argues that the Aguaruna of Peru see magic as a type of technology, no more supernatural than their physical tools. Brown says that the Aguaruna utilize magic in an empirical manner; for example, they discard any magical stones which they have found to be ineffective. To Brown—as to Horton—magical and scientific thinking differ merely in idiom.^[26]

These theories blur the boundaries between magic, science, and religion, and focus on the similarities in magical, technical, and spiritual practices. Brown even ironically writes that he is tempted to disclaim the existence of 'magic.'^[27]

Substantive difference

One theory of substantive difference is that of the open versus closed society. Horton describes this as one of the key dissimilarities between traditional thought and Western science. He suggests that the scientific worldview is distinguished from a magical one by the scientific method and by skepticism, requiring the falsifiability of any scientific hypothesis. He notes that for native peoples "there is no developed awareness of alternatives to the established body of theoretical texts."^[28] He notes that all further differences between traditional and Western thought can be understood as a result of this factor. Because there are no alternatives in societies based on magical thought, a theory does not need to be objectively judged to be valid.

In children

Magical thinking is most prominent in children between ages 2 and 7. During this age, children strongly believe that their personal thoughts have a direct effect on the rest of the world. Therefore, if they experience something tragic that they do not understand, e.g. a death, their minds will create a reason to feel responsible. Jean Piaget, a developmental psychologist, came up with a theory of four developmental stages. Children between ages 2 and 7 would be classified under his preoperational stage of development. During this stage children are still developing their use of logical thinking. A child's thinking is dominated by perceptions of physical features, meaning that if the child is told that a family pet has gone away, then the child will have difficulty comprehending the transformation of the dog not being around anymore. Magical thinking would be evident here, since the child may believe that the family pet being gone is just temporary. Their young minds in this stage do not understand the finality of death and magical thinking may bridge the gap.

Grieving children

Children who evidence magical thinking often feel that they are responsible for an event or events occurring or are capable of reversing an event simply by thinking about it and wishing for a change.^[29] Make-believe and fantasy are an integral part of life at this age and are often used to explain the inexplicable.^[30][31]

According to Piaget, children within this age group are often "egocentric," believing that what they feel and experience is the same as everyone else's feelings and experiences.^[32] Also at this age, there is often a lack of ability to understand that there may be other explanations for events outside of the realm of things they have already comprehended. What happens outside their understanding needs to be explained using what they already know, because of an inability to fully comprehend abstract concepts.^[32]

Magical thinking is found particularly in children's explanations of experiences about death, whether the death of a family member or pet, or their own illness or impending death. These experiences are often new for a young child, who at that point has no experience to give understanding of the ramifications of the event.^[33] A child may feel that they are responsible for what has happened, simply because they were upset with the person who died, or perhaps played with the pet too roughly. There may also be the idea that if the child wishes it hard enough, or performs just the right act, the person or pet may choose to come back, and not be dead any longer.^[34] When considering their own illness or impending death, some children may feel that they are being punished for doing something wrong, or not doing something they should have, and therefore have become ill.^[35] If a child's ideas about an event are incorrect because of their magical thinking, there is a possibility that the conclusions the child makes could result in long-term beliefs and behaviours that create difficulty for the child as they mature.^[36]

Related terms

• "Quasi-magical thinking" describes "cases in which people act as if they erroneously believe that their action influences the outcome, even though they do not really hold that belief."^[37] People may realize that a superstitious intuition is logically false, but act as if it were true because they do not exert an effort to correct the intuition.^[38]