Mass in general relativity

From Wikipedia, the free encyclopedia

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

The concept of mass in general relativity (GR) is more subtle to define than the concept of mass in special relativity. In fact, general relativity does not offer a single definition of the term mass, but offers several different definitions that are applicable under different circumstances. Under some circumstances, the mass of a system in general relativity may not even be defined.

The reason for this subtlety is that the energy and momentum in the gravitational field cannot be unambiguously localized. So, rigorous definitions of the mass in general relativity are not local, as in classical mechanics or special relativity, but make reference to the asymptotic nature of the spacetime. A well defined notion of the mass exists for asymptotically flat spacetimes and for asymptotically Anti-de Sitter space. However, these definitions must be used with care in other settings.

Defining mass in general relativity: concepts and obstacles

In special relativity, the rest mass of a particle can be defined unambiguously in terms of its energy and momentum as described in the article on mass in special relativity. Generalizing the notion of the energy and momentum to general relativity, however, is subtle. The main reason for this is that that gravitational field itself contributes to the energy and momentum. However, the "gravitational field energy" is not a part of the energy–momentum tensor; instead, what might be identified as the contribution of the gravitational field to a total energy is part of the Einstein tensor on the other side of Einstein's equation (and, as such, a consequence of these equations' non-linearity). While in certain situations it is possible to rewrite the equations so that part of the "gravitational energy" now stands alongside the other source terms in the form of the stress–energy–momentum pseudotensor, this separation is not true for all observers, and there is no general definition for obtaining it.

How, then, does one define a concept as a system's total mass – which is easily defined in classical mechanics? As it turns out, at least for spacetimes which are asymptotically flat (roughly speaking, which represent some isolated gravitating system in otherwise empty and gravity-free infinite space), the ADM 3+1 split leads to a solution: as in the usual Hamiltonian formalism, the time direction used in that split has an associated energy, which can be integrated up to yield a global quantity known as the ADM mass (or, equivalently, ADM energy). Alternatively, there is a possibility to define mass for a spacetime that is stationary, in other words, one that has a time-like Killing vector field (which, as a generating field for time, is canonically conjugate to energy); the result is the so-called Komar mass Although defined in a totally different way, it can be shown to be equivalent to the ADM mass for stationary spacetimes. The Komar integral definition can also be generalized to non-stationary fields for which there is at least an asymptotic time translation symmetry; imposing a certain gauge condition, one can define the Bondi energy at null infinity. In a way, the ADM energy measures all of the energy contained in spacetime, while the Bondi energy excludes those parts carried off by gravitational waves to infinity. Great effort has been expended on proving positivity theorems for the masses just defined, not least because positivity, or at least the existence of a lower limit, has a bearing on the more fundamental question of boundedness from below: if there were no lower limit to the energy, then no isolated system would be absolutely stable; there would always be the possibility of a decay to a state of even lower total energy. Several kinds of proofs that both the ADM mass and the Bondi mass are indeed positive exist; in particular, this means that Minkowski space (for which both are zero) is indeed stable. While the focus here has been on energy, analogue definitions for global momentum exist; given a field of angular Killing vectors and following the Komar technique, one can also define global angular momentum.

The disadvantage of all the definitions mentioned so far is that they are defined only at (null or spatial) infinity; since the 1970s, physicists and mathematicians have worked on the more ambitious endeavor of defining suitable quasi-local quantities, such as the mass of an isolated system defined using only quantities defined within a finite region of space containing that system. However, while there is a variety of proposed definitions such as the Hawking energy, the Geroch energy or Penrose's quasi-local energy–momentum based on twistor methods, the field is still in flux. Eventually, the hope is to use a suitable defined quasi-local mass to give a more precise formulation of the hoop conjecture, prove the so-called Penrose inequality for black holes (relating the black hole's mass to the horizon area) and find a quasi-local version of the laws of black hole mechanics.

Types of mass in general relativity

Komar mass in stationary spacetimes

Main article: Komar mass

A non-technical definition of a stationary spacetime is a spacetime where none of the metric coefficients ${\displaystyle g_{\mu \nu }\,}$ are functions of time. The Schwarzschild metric of a black hole and the Kerr metric of a rotating black hole are common examples of stationary spacetimes.

By definition, a stationary spacetime exhibits time translation symmetry. This is technically called a time-like Killing vector. Because the system has a time translation symmetry, Noether's theorem guarantees that it has a conserved energy. Because a stationary system also has a well defined rest frame in which its momentum can be considered to be zero, defining the energy of the system also defines its mass. In general relativity, this mass is called the Komar mass of the system. Komar mass can only be defined for stationary systems.

Komar mass can also be defined by a flux integral. This is similar to the way that Gauss's law defines the charge enclosed by a surface as the normal electric force multiplied by the area. The flux integral used to define Komar mass is slightly different from that used to define the electric field, however – the normal force is not the actual force, but the "force at infinity". See the main article for more detail.

Of the two definitions, the description of Komar mass in terms of a time translation symmetry provides the deepest insight.

ADM and Bondi masses in asymptotically flat space-times

If a system containing gravitational sources is surrounded by an infinite vacuum region, the geometry of the space-time will tend to approach the flat Minkowski geometry of special relativity at infinity. Such space-times are known as "asymptotically flat" space-times.

For systems in which space-time is asymptotically flat, the ADM and Bondi energy, momentum, and mass can be defined. In terms of Noether's theorem, the ADM energy, momentum, and mass are defined by the asymptotic symmetries at spatial infinity, and the Bondi energy, momentum, and mass are defined by the asymptotic symmetries at null infinity. Note that mass is computed as the length of the energy–momentum four-vector, which can be thought of as the energy and momentum of the system "at infinity".

The ADM energy is defined through the following flux integral at infinity. If a spacetime is asymptotically flat this means that near "infinity" the metric tends to that of flat space. The asymptotic deviations of the metric away from flat space can be parametrized by

${\displaystyle g_{\mu \nu }=\eta _{\mu \nu }+h_{\mu \nu }}$

where ${\displaystyle \eta _{\mu \nu }}$ is the flat space metric. The ADM energy is then given by an integral over a surface, ${\displaystyle S}$ at infinity

${\displaystyle P^{0}={1 \over 16\pi G}\int \left(\partial ^{k}h_{jk}-\partial ^{j}h_{kk}\right)d^{2}S_{j},}$

where ${\displaystyle S_{j}}$ is the outward-pointing normal to ${\displaystyle S}$. The Einstein summation convention is assumed for repeated indices but the sum over k and j only runs over the spatial directions. The use of ordinary derivatives instead of covariant derivatives in the formula above is justified because of the assumption that the asymptotic geometry is flat.

Some intuition for the formula above can be obtained as follows. Imagine that that we take the surface, S, to be a spherical surface so that the normal points radially outwards. At large distances from the source of the energy, r, the tensor ${\displaystyle h_{ij}}$ is expected to fall off as ${\displaystyle r^{-1}}$ and the derivative with respect to r converts this into ${\displaystyle r^{-2}}$ The area of the sphere at large radius also grows precisely as ${\displaystyle r^{2}}$ and therefore one obtains a finite value for the energy.

It is also possible to obtain expressions for the momentum in asymptotically flat spacetime. To obtain such an expression one defines

${\displaystyle H^{\mu \alpha \nu \beta }=-{\bar {h}}^{\mu \nu }\eta ^{\alpha \beta }-\eta ^{\mu \nu }{\bar {h}}^{\alpha \beta }+{\bar {h}}^{\alpha \nu }\eta ^{\mu \beta }+{\bar {h}}^{\mu \beta }\eta ^{\alpha \nu }}$

where

${\displaystyle {\bar {h}}_{\mu \nu }=h_{\mu \nu }-{1 \over 2}\eta _{\mu \nu }h_{\alpha }^{\alpha }}$

Then the momentum is obtained by a flux integral in the asymptotically flat region

${\displaystyle P^{\mu }={1 \over 16\pi G}\int \partial _{\alpha }H^{\mu \alpha 0j}d^{2}S_{j}}$

Note that the expression for ${\displaystyle P^{0}}$ obtained from the formula above coincides with the expression for the ADM energy given above as can easily be checked using the explicit expression for H.

The Newtonian limit for nearly flat space-times

In the Newtonian limit, for quasi-static systems in nearly flat space-times, one can approximate the total energy of the system by adding together the non-gravitational components of the energy of the system and then subtracting the Newtonian gravitational binding energy.

Translating the above statement into the language of general relativity, we say that a system in nearly flat space-time has a total non-gravitational energy E and momentum P given by:

${\displaystyle E=\int _{v}T_{00}dV\qquad P^{i}=\int _{V}T_{0i}dV}$

When the components of the momentum vector of the system are zero, i.e. Pⁱ = 0, the approximate mass of the system is just (E+E_binding)/c², E_binding being a negative number representing the Newtonian gravitational self-binding energy.

Hence when one assumes that the system is quasi-static, one assumes that there is no significant energy present in the form of "gravitational waves". When one assumes that the system is in "nearly-flat" space-time, one assumes that the metric coefficients are essentially Minkowskian within acceptable experimental error.

The formulas for the total energy and momentum can be seen to arise naturally in this limit as follows. In the linearized limit, the equations of general relativity can be written in the form

${\displaystyle \partial _{\alpha }\partial _{\beta }H^{\mu \alpha \nu \beta }=16\pi GT^{\mu \nu }}$

In this limit, the total energy-momentum of the system is simply given by integrating the stress-tensor on a spacelike slice.

${\displaystyle P^{\mu }=\int T^{\mu 0}d^{3}x}$

But using the equations of motion, one can also write this as

${\displaystyle P^{\mu }={1 \over 16\pi G}\int \partial _{\alpha }\partial _{\beta }H^{\mu \alpha 0\beta }d^{3}x={1 \over 16\pi G}\int \partial _{\alpha }\partial _{j}H^{\mu \alpha 0j}d^{3}x}$

where the sum over j runs only over the spatial directions and the second equality uses the fact that ${\displaystyle H^{\mu \alpha \nu \beta }}$ is anti-symmetric in ${\displaystyle \nu }$ and ${\displaystyle \beta }$. Finally, one uses the Gauss law to convert the integral of a divergence over the spatial slice into an integral over a Gaussian sphere

${\displaystyle {1 \over 16\pi G}\int \partial _{\alpha }\partial _{j}H^{\mu \alpha 0j}d^{3}x={1 \over 16\pi G}\int \partial _{\alpha }H^{\mu \alpha 0j}d^{2}S_{j}}$

which coincides precisely with the formula for the total momentum given above.

History

In 1918, David Hilbert wrote about the difficulty in assigning an energy to a "field" and "the failure of the energy theorem" in a correspondence with Klein. In this letter, Hilbert conjectured that this failure is a characteristic feature of the general theory, and that instead of "proper energy theorems" one had 'improper energy theorems'.

This conjecture was soon proved to be correct by one of Hilbert's close associates, Emmy Noether. Noether's theorem applies to any system which can be described by an action principle. Noether's theorem associates conserved energies with time-translation symmetries. When the time-translation symmetry is a finite parameter continuous group, such as the Poincaré group, Noether's theorem defines a scalar conserved energy for the system in question. However, when the symmetry is an infinite parameter continuous group, the existence of a conserved energy is not guaranteed. In a similar manner, Noether's theorem associates conserved momenta with space-translations, when the symmetry group of the translations is finite-dimensional. Because General Relativity is a diffeomorphism invariant theory, it has an infinite continuous group of symmetries rather than a finite-parameter group of symmetries, and hence has the wrong group structure to guarantee a conserved energy. Noether's theorem has been extremely influential in inspiring and unifying various ideas of mass, system energy, and system momentum in General Relativity.

As an example of the application of Noether's theorem is the example of stationary space-times and their associated Komar mass.(Komar 1959). While general space-times lack a finite-parameter time-translation symmetry, stationary space-times have such a symmetry, known as a Killing vector. Noether's theorem proves that such stationary space-times must have an associated conserved energy. This conserved energy defines a conserved mass, the Komar mass.

ADM mass was introduced (Arnowitt et al., 1960) from an initial-value formulation of general relativity. It was later reformulated in terms of the group of asymptotic symmetries at spatial infinity, the SPI group, by various authors. (Held, 1980). This reformulation did much to clarify the theory, including explaining why ADM momentum and ADM energy transforms as a 4-vector (Held, 1980). Note that the SPI group is actually infinite-dimensional. The existence of conserved quantities is because the SPI group of "super-translations" has a preferred 4-parameter subgroup of "pure" translations, which, by Noether's theorem, generates a conserved 4-parameter energy–momentum. The norm of this 4-parameter energy–momentum is the ADM mass.

The Bondi mass was introduced (Bondi, 1962) in a paper that studied the loss of mass of physical systems via gravitational radiation. The Bondi mass is also associated with a group of asymptotic symmetries, the BMS group at null infinity. Like the SPI group at spatial infinity, the BMS group at null infinity is infinite-dimensional, and it also has a preferred 4-parameter subgroup of "pure" translations.

Another approach to the problem of energy in General Relativity is the use of pseudotensors such as the Landau–Lifshitz pseudotensor.(Landau and Lifshitz, 1962). Pseudotensors are not gauge invariant – because of this, they only give consistent gauge-independent answers for the total energy when additional constraints (such as asymptotic flatness) are met. The gauge dependence of pseudotensors also prevents any gauge-independent definition of the local energy density, as every different gauge choice results in a different local energy density.

Relational database

From Wikipedia, the free encyclopedia

A relational database is a (most commonly digital) database based on the relational model of data, as proposed by E. F. Codd in 1970. A system used to maintain relational databases is a relational database management system (RDBMS). Many relational database systems are equipped with the option of using the SQL (Structured Query Language) for querying and maintaining the database.

History

The term "relational database" was first defined by E. F. Codd at IBM in 1970. Codd introduced the term in his research paper "A Relational Model of Data for Large Shared Data Banks". In this paper and later papers, he defined what he meant by "relational". One well-known definition of what constitutes a relational database system is composed of Codd's 12 rules. However, no commercial implementations of the relational model conform to all of Codd's rules, so the term has gradually come to describe a broader class of database systems, which at a minimum:

1. Present the data to the user as relations (a presentation in tabular form, i.e. as a collection of tables with each table consisting of a set of rows and columns);
2. Provide relational operators to manipulate the data in tabular form.

In 1974, IBM began developing System R, a research project to develop a prototype RDBMS. The first system sold as an RDBMS was Multics Relational Data Store (June 1976). Oracle was released in 1979 by Relational Software, now Oracle Corporation. Ingres and IBM BS12 followed. Other examples of an RDBMS include IBM Db2, SAP Sybase ASE, and Informix. In 1984, the first RDBMS for Macintosh began being developed, code-named Silver Surfer, and was released in 1987 as 4th Dimension and known today as 4D.

The first systems that were relatively faithful implementations of the relational model were from:

• University of Michigan – Micro DBMS (1969)
• Massachusetts Institute of Technology (1971)
• IBM UK Scientific Centre at Peterlee – IS1 (1970–72) and its successor, PRTV (1973–79)

The most common definition of an RDBMS is a product that presents a view of data as a collection of rows and columns, even if it is not based strictly upon relational theory. By this definition, RDBMS products typically implement some but not all of Codd's 12 rules.

A second school of thought argues that if a database does not implement all of Codd's rules (or the current understanding on the relational model, as expressed by Christopher J. Date, Hugh Darwen and others), it is not relational. This view, shared by many theorists and other strict adherents to Codd's principles, would disqualify most DBMSs as not relational. For clarification, they often refer to some RDBMSs as truly-relational database management systems (TRDBMS), naming others pseudo-relational database management systems (PRDBMS).

As of 2009, most commercial relational DBMSs employ SQL as their query language.

Alternative query languages have been proposed and implemented, notably the pre-1996 implementation of Ingres QUEL.

Relational model

Main article: Relational model

A relational model organizes data into one or more tables (or "relations") of columns and rows, with a unique key identifying each row. Rows are also called records or tuples. Columns are also called attributes. Generally, each table/relation represents one "entity type" (such as customer or product). The rows represent instances of that type of entity (such as "Lee" or "chair") and the columns representing values attributed to that instance (such as address or price).

For example, each row of a class table corresponds to a class, and a class corresponds to multiple students, so the relationship between the class table and the student table is "one to many"

Keys

Each row in a table has its own unique key. Rows in a table can be linked to rows in other tables by adding a column for the unique key of the linked row (such columns are known as foreign keys). Codd showed that data relationships of arbitrary complexity can be represented by a simple set of concepts.

Part of this processing involves consistently being able to select or modify one and only one row in a table. Therefore, most physical implementations have a unique primary key (PK) for each row in a table. When a new row is written to the table, a new unique value for the primary key is generated; this is the key that the system uses primarily for accessing the table. System performance is optimized for PKs. Other, more natural keys may also be identified and defined as alternate keys (AK). Often several columns are needed to form an AK (this is one reason why a single integer column is usually made the PK). Both PKs and AKs have the ability to uniquely identify a row within a table. Additional technology may be applied to ensure a unique ID across the world, a globally unique identifier, when there are broader system requirements.

The primary keys within a database are used to define the relationships among the tables. When a PK migrates to another table, it becomes a foreign key in the other table. When each cell can contain only one value and the PK migrates into a regular entity table, this design pattern can represent either a one-to-one or one-to-many relationship. Most relational database designs resolve many-to-many relationships by creating an additional table that contains the PKs from both of the other entity tables – the relationship becomes an entity; the resolution table is then named appropriately and the two FKs are combined to form a PK. The migration of PKs to other tables is the second major reason why system-assigned integers are used normally as PKs; there is usually neither efficiency nor clarity in migrating a bunch of other types of columns.

Relationships

Relationships are a logical connection between different tables, established on the basis of interaction among these tables.

Transactions

In order for a database management system (DBMS) to operate efficiently and accurately, it must use ACID transactions.

Stored procedures

Part of the programming within a RDBMS is accomplished using stored procedures (SPs). Often procedures can be used to greatly reduce the amount of information transferred within and outside of a system. For increased security, the system design may grant access to only the stored procedures and not directly to the tables. Fundamental stored procedures contain the logic needed to insert new and update existing data. More complex procedures may be written to implement additional rules and logic related to processing or selecting the data.

Terminology

Relational database terminology

The relational database was first defined in June 1970 by Edgar Codd, of IBM's San Jose Research Laboratory.^[1] Codd's view of what qualifies as an RDBMS is summarized in Codd's 12 rules. A relational database has become the predominant type of database. Other models besides the relational model include the hierarchical database model and the network model.

The table below summarizes some of the most important relational database terms and the corresponding SQL term:

SQL term	Relational database term	Description
Row	Tuple or record	A data set representing a single item
Column	Attribute or field	A labeled element of a tuple, e.g. "Address" or "Date of birth"
Table	Relation or Base relvar	A set of tuples sharing the same attributes; a set of columns and rows
View or result set	Derived relvar	Any set of tuples; a data report from the RDBMS in response to a query

Relations or tables

Main articles: Relation (database) and Table (database)

In a relational database, a relation is a set of tuples that have the same attributes. A tuple usually represents an object and information about that object. Objects are typically physical objects or concepts. A relation is usually described as a table, which is organized into rows and columns. All the data referenced by an attribute are in the same domain and conform to the same constraints.

The relational model specifies that the tuples of a relation have no specific order and that the tuples, in turn, impose no order on the attributes. Applications access data by specifying queries, which use operations such as select to identify tuples, project to identify attributes, and join to combine relations. Relations can be modified using the insert, delete, and update operators. New tuples can supply explicit values or be derived from a query. Similarly, queries identify tuples for updating or deleting.

Tuples by definition are unique. If the tuple contains a candidate or primary key then obviously it is unique; however, a primary key need not be defined for a row or record to be a tuple. The definition of a tuple requires that it be unique, but does not require a primary key to be defined. Because a tuple is unique, its attributes by definition constitute a superkey.

Base and derived relations

Main articles: Relvar and View (database)

All data are stored and accessed via relations. Relations that store data are called "base relations", and in implementations are called "tables". Other relations do not store data, but are computed by applying relational operations to other relations. These relations are sometimes called "derived relations". In implementations these are called "views" or "queries". Derived relations are convenient in that they act as a single relation, even though they may grab information from several relations. Also, derived relations can be used as an abstraction layer.

Domain

Main article: Data domain

A domain describes the set of possible values for a given attribute, and can be considered a constraint on the value of the attribute. Mathematically, attaching a domain to an attribute means that any value for the attribute must be an element of the specified set. The character string "ABC", for instance, is not in the integer domain, but the integer value 123 is. Another example of domain describes the possible values for the field "CoinFace" as ("Heads","Tails"). So, the field "CoinFace" will not accept input values like (0,1) or (H,T).

Constraints

Constraints are often used to make it possible to further restrict the domain of an attribute. For instance, a constraint can restrict a given integer attribute to values between 1 and 10. Constraints provide one method of implementing business rules in the database and support subsequent data use within the application layer. SQL implements constraint functionality in the form of check constraints. Constraints restrict the data that can be stored in relations. These are usually defined using expressions that result in a boolean value, indicating whether or not the data satisfies the constraint. Constraints can apply to single attributes, to a tuple (restricting combinations of attributes) or to an entire relation. Since every attribute has an associated domain, there are constraints (domain constraints). The two principal rules for the relational model are known as entity integrity and referential integrity.

Primary key

Main articles: Unique key and Primary key

Every relation/table has a primary key, this being a consequence of a relation being a set. A primary key uniquely specifies a tuple within a table. While natural attributes (attributes used to describe the data being entered) are sometimes good primary keys, surrogate keys are often used instead. A surrogate key is an artificial attribute assigned to an object which uniquely identifies it (for instance, in a table of information about students at a school they might all be assigned a student ID in order to differentiate them). The surrogate key has no intrinsic (inherent) meaning, but rather is useful through its ability to uniquely identify a tuple. Another common occurrence, especially in regard to N:M cardinality is the composite key. A composite key is a key made up of two or more attributes within a table that (together) uniquely identify a record.

Foreign key

Main article: Foreign key

Foreign key refers to a field in a relational table that matches the primary key column of another table. It relates the two keys. Foreign keys need not have unique values in the referencing relation. A foreign key can be used to cross-reference tables, and it effectively uses the values of attributes in the referenced relation to restrict the domain of one or more attributes in the referencing relation. The concept is described formally as: "For all tuples in the referencing relation projected over the referencing attributes, there must exist a tuple in the referenced relation projected over those same attributes such that the values in each of the referencing attributes match the corresponding values in the referenced attributes."

Stored procedures

Main article: Stored procedure

A stored procedure is executable code that is associated with, and generally stored in, the database. Stored procedures usually collect and customize common operations, like inserting a tuple into a relation, gathering statistical information about usage patterns, or encapsulating complex business logic and calculations. Frequently they are used as an application programming interface (API) for security or simplicity. Implementations of stored procedures on SQL RDBMS's often allow developers to take advantage of procedural extensions (often vendor-specific) to the standard declarative SQL syntax. Stored procedures are not part of the relational database model, but all commercial implementations include them.

Index

Main article: Index (database)

An index is one way of providing quicker access to data. Indices can be created on any combination of attributes on a relation. Queries that filter using those attributes can find matching tuples directly using the index (similar to Hash table lookup), without having to check each tuple in turn. This is analogous to using the index of a book to go directly to the page on which the information you are looking for is found, so that you do not have to read the entire book to find what you are looking for. Relational databases typically supply multiple indexing techniques, each of which is optimal for some combination of data distribution, relation size, and typical access pattern. Indices are usually implemented via B+ trees, R-trees, and bitmaps. Indices are usually not considered part of the database, as they are considered an implementation detail, though indices are usually maintained by the same group that maintains the other parts of the database. The use of efficient indexes on both primary and foreign keys can dramatically improve query performance. This is because B-tree indexes result in query times proportional to log(n) where n is the number of rows in a table and hash indexes result in constant time queries (no size dependency as long as the relevant part of the index fits into memory).

Relational operations

Main article: Relational algebra

Queries made against the relational database, and the derived relvars in the database are expressed in a relational calculus or a relational algebra. In his original relational algebra, Codd introduced eight relational operators in two groups of four operators each. The first four operators were based on the traditional mathematical set operations:

• The union operator (υ) combines the tuples of two relations and removes all duplicate tuples from the result. The relational union operator is equivalent to the SQL UNION operator.
• The intersection operator (∩) produces the set of tuples that two relations share in common. Intersection is implemented in SQL in the form of the INTERSECT operator.
• The set difference operator (-) acts on two relations and produces the set of tuples from the first relation that do not exist in the second relation. Difference is implemented in SQL in the form of the EXCEPT or MINUS operator.
• The cartesian product (X) of two relations is a join that is not restricted by any criteria, resulting in every tuple of the first relation being matched with every tuple of the second relation. The cartesian product is implemented in SQL as the Cross join operator.

The remaining operators proposed by Codd involve special operations specific to relational databases:

• The selection, or restriction, operation (σ) retrieves tuples from a relation, limiting the results to only those that meet a specific criterion, i.e. a subset in terms of set theory. The SQL equivalent of selection is the SELECT query statement with a WHERE clause.
• The projection operation (π) extracts only the specified attributes from a tuple or set of tuples.
• The join operation defined for relational databases is often referred to as a natural join (⋈). In this type of join, two relations are connected by their common attributes. MySQL's approximation of a natural join is the Inner join operator. In SQL, an INNER JOIN prevents a cartesian product from occurring when there are two tables in a query. For each table added to a SQL Query, one additional INNER JOIN is added to prevent a cartesian product. Thus, for N tables in an SQL query, there must be N−1 INNER JOINS to prevent a cartesian product.
• The relational division (÷) operation is a slightly more complex operation and essentially involves using the tuples of one relation (the dividend) to partition a second relation (the divisor). The relational division operator is effectively the opposite of the cartesian product operator (hence the name).

Other operators have been introduced or proposed since Codd's introduction of the original eight including relational comparison operators and extensions that offer support for nesting and hierarchical data, among others.

Normalization

Main article: Database normalization

Normalization was first proposed by Codd as an integral part of the relational model. It encompasses a set of procedures designed to eliminate non-simple domains (non-atomic values) and the redundancy (duplication) of data, which in turn prevents data manipulation anomalies and loss of data integrity. The most common forms of normalization applied to databases are called the normal forms.

RDBMS

The general structure of a relational database

 

See also: Database § Database management system

Connolly and Begg define Database Management System (DBMS) as a "software system that enables users to define, create, maintain and control access to the database". RDBMS is an extension of that acronym that is sometimes used when the underlying database is relational.

An alternative definition for a relational database management system is a database management system (DBMS) based on the relational model. Most databases in widespread use today are based on this model.

RDBMSs have been a common option for the storage of information in databases used for financial records, manufacturing and logistical information, personnel data, and other applications since the 1980s. Relational databases have often replaced legacy hierarchical databases and network databases, because RDBMS were easier to implement and administer. Nonetheless, relational stored data received continued, unsuccessful challenges by object database management systems in the 1980s and 1990s, (which were introduced in an attempt to address the so-called object–relational impedance mismatch between relational databases and object-oriented application programs), as well as by XML database management systems in the 1990s. However, due to the expanse of technologies, such as horizontal scaling of computer clusters, NoSQL databases have recently become popular as an alternative to RDBMS databases.

Distributed relational databases

Distributed Relational Database Architecture (DRDA) was designed by a workgroup within IBM in the period 1988 to 1994. DRDA enables network connected relational databases to cooperate to fulfill SQL requests. The messages, protocols, and structural components of DRDA are defined by the Distributed Data Management Architecture.

Market share

According to DB-Engines, in April 2022, the most widely used systems were:

1. Oracle Database
2. MySQL
3. Microsoft SQL Server
4. PostgreSQL (free software)
5. IBM Db2
6. Microsoft Access
7. SQLite (free software)
8. MariaDB (free software)
9. Snowflake
10. Microsoft Azure SQL Database
11. Apache Hive (free software)
12. Teradata Vantage

According to research company Gartner, in 2011, the five leading proprietary software relational database vendors by revenue were Oracle (48.8%), IBM (20.2%), Microsoft (17.0%), SAP including Sybase (4.6%), and Teradata (3.7%).

Carbon fee and dividend

From Wikipedia, the free encyclopedia

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

 

Concept of a carbon fee and dividend

 

A coal power plant in Germany. Fee and dividend will make fossil fuels – coal, oil, and gas – less competitive as a fuel than other options.

A carbon fee and dividend or climate income is a system to reduce greenhouse gas emissions and address climate change. The system imposes a carbon tax on the sale of fossil fuels, and then distributes the revenue of this tax over the entire population (equally, on a per-person basis) as a monthly income or regular payment.

Since the adoption of the system in Canada and Switzerland, it has gained increased interest worldwide as a cross-sector and socially just approach to reducing emissions and tackling climate change.

Designed to maintain or improve economic vitality while speeding the transition to a sustainable energy economy, carbon fee and dividend has been proposed as an alternative to emission reduction mechanisms such as complex regulatory approaches, cap and trade or a straightforward carbon tax. While there is general agreement among scientists and economists on the need for a carbon tax, economists are generally neutral on specific uses for the revenue, though there tends to be more support than opposition for returning the revenue as a dividend to taxpayers.

Structure

The basic structure of carbon fee and dividend is as follows:

1. A fee is levied on fuels at their point of origin into the economy, such as the well, mine, or port of entry. The fee is based upon the carbon content of a given fuel, with a commonly-proposed starting point being $10–16 /t of carbon that would be emitted once the fuel is burned.
2. The fee is progressively increased, providing a steady, predictable price signal and incentivizing early transition to low-carbon energy sources and products.
3. A border tax adjustment is levied on imports from nations that lack their own equivalent fee on carbon. For example, if the United States legislated a carbon fee-and-dividend system, China would face the choice of paying carbon fees to the United States or creating its own internal carbon pricing system. This would leverage American economic power to incentivize carbon pricing around the world. 
4. Some or all of the fee is returned to households as an energy dividend. Returning 100% of net fees results in a revenue-neutral carbon fee-and-dividend system; this revenue neutrality often appeals to conservatives, such as former Secretary of State George Shultz, who want to reduce emissions without increasing the size and funding of the federal government.

In order to maximize effectiveness, the amount of the fee would be regulated based on the scientific assessments from both economic and climate science in order to balance the size and speed of fee progression.

Advantages

A climate income has several notable advantages over other emission reduction mechanisms:

• Social justice and acceptability. While there is broad scientific consensus that a carbon tax is the most powerful way to reduce emissions, such a tax necessarily increases prices and the cost of living. By handing out the revenue of this tax as a universal climate income, the price rise is largely compensated. It has been calculated that in total, low and middle incomes would go up under a system of climate income.
• Market based and cross-sector. Unlike complex regulatory approaches, a fossil fuel fee allows market forces to reduce emissions in the most efficient and cost effective way.
• Cross-sector. There is a broad range of sources of carbon emissions. Regulatory approaches and emissions trading often address only one or a couple of sectors. A truly universal fossil fuel fee addresses all these sectors at once. Moreover, through a universal price on CO₂-equivalent emissions, the fee can cover other greenhouse gases (such as methane and nitrous oxide) or emission sectors (industry, agriculture) as well.
• Compatible. The mechanism is compatible with other measures and regulations imposed by the government, such as investments in education, research and infrastructure.
• Revenue neutral. A climate income would not increase the budget of the government, or utilise the imposed carbon fee as a means to balance the government deficit.
• Carbon fee and dividend should avoid fuel protests that have occurred in many places.

Studies

Energy Modeling Forum study 2012

In late 2012 the Energy Modeling Forum (EMF), coordinated by Stanford University, released its EMF 29 study titled "The role of border carbon adjustment in unilateral climate policy". It is well understood that unilateral climate policy can lead to emissions leakage. As one example, trade-exposed emissions-intensive industries may simply relocate to regions with laxer climate protection. A border carbon adjustment (BCA) program can help counter this and related effects. Under such a policy, tariffs are levied on the carbon embodied in imported goods from unregulated trading partners while the original climate protection payments for exported goods are rebated. The study finds that the BCA programs evaluated:

• can reduce emissions leakage
• yield modest gains in global economic efficiency
• shift substantial costs from abating OECD counties to non-abating non-OECD countries

In light of these findings, the study recommends care when designing and implementing BCA programs. Moreover, the regressive impact of shifting part of the abatement burden southward conflicts with the UNFCCC principle of common but differentiated responsibility and respective capabilities, which explicitly acknowledges that developing countries have less ability to shoulder climate protection measures.

Regional Economic Models study 2014

A 2014 economic impact analysis by Regional Economic Models, Incorporated (REMI) concluded that a carbon fee that began at US$10 per ton and increased by US$10 per year, with all net revenue returned to households as an energy dividend, would carry substantial environmental, health, and economic benefits:^[

• CO₂ emissions in the United States would decrease to 50% of 1990 levels in the first 20 years.
• Over the same timespan, reductions in airborne pollution that accompanies CO₂ emissions would result in 230,000 fewer premature deaths.
• Regular dividend payments would stimulate the U.S. economy, leading to the creation of 2.8 million jobs over baseline during the program's first two decades.
• The stimulative effect was also found to positively affect national GDP, adding $70–85 billion/year for a cumulative 20-year increase of $1.375 trillion over baseline (the approximate equivalent of adding an additional year of growth during that span).

International Institute for Applied Systems Analysis study 2016

A 2016 working paper from the International Institute for Applied Systems Analysis (IIASA) looked more narrowly at the impact of a proposed carbon fee and dividend on American households during the first year. Due to the shorter window analyzed (which did not allow for considerations of changes to personal energy use under the policy) the paper found a smaller percentage of households benefiting from carbon fee and dividend than the REMI report summarized above (53% versus approximately two-thirds in the REMI report). It also found that an additional 19% of households suffered a loss of less than 0.2% of annual income, an amount that might be experienced as effectively "breaking even" by households in the upper income quintiles most likely affected.

Implementation

Revenue recycling in real-world carbon tax schemes

As of 2021, Switzerland and Canada were the only two countries with implemented fee and dividend policies. In Switzerland, the carbon tax refund is returned to citizens as a discount on their health insurance; an annual notification of the benefit appears on their health insurance forms. Canada began rolling out a federal fee and dividend scheme from 2019 (though not to the provinces that already had a province level scheme: British Columbia and Alberta). The Canadian federal carbon tax rebate is made via a credit given to one adult in each household (though based on total household size, including children).

The British Columbia carbon tax could be considered as "fee and dividend", although there are some differences. Rather than entirely or mostly being returned as a dividend to households, 73% of the carbon tax is used to reduce corporate and small business taxes. Unlike most governments, British Columbia's electricity portfolio largely consists of hydroelectric power and their energy costs, even with the tax, are lower than most countries.

Country	Region	Year started	Price of CO2	Per year progression	Repayment
Canada	British Columbia	2008	40 CAD per ton CO2 from April 2019	5 CAD per year till 50 CAD in April 2021	40% for citizens in 2017
Switzerland		2008	96 CHF per ton CO2 in 2018	12 CHF in 2008 24 CHF in 2009 36 CHF in 2010 60 CHF in 2014 84 CHF in 2016	67% for citizens and companies

Political support

United States

Carbon fee and dividend is the preferred climate solution of Citizens' Climate Lobby (CCL). Citizens' Climate Lobby argues that a fee-and-dividend policy will be easier to adopt and adjust than relatively complicated cap-and-trade or regulatory approaches, enabling a smooth, economically-positive transition to a low-carbon energy economy. James Hansen, Director of the NASA Goddard Institute for Space Studies has frequently promoted awareness of carbon fee and dividend through his writings and frequent public appearances, as well as his position at Columbia University.

A Carbon Dividends plan has been proposed by the Climate Leadership Council, which counts among its members 27 Nobel laureates, 15 Fortune 100 companies, all four past chairs of the Federal Reserve, and over 3000 US economists. Among those supporting the Climate Leadership Council's Carbon Dividends Plan are Greg Mankiw, Larry Summers, James Baker, Henry Paulson, Ted Halstead, and Ray Dalio. It claims to be the most popular, equitable and pro-growth climate solution.

Inspired by the market-friendly structure of carbon fee and dividend, Republican Congressman Bob Inglis introduced H.R. 2380 (the 'Raise Wages, Cut Carbon Act of 2009') in the U.S. House of Representatives on May 13, 2009. Concerned about energy infrastructure as an issue of national security, he supports Fee and Dividend as a reliable means of reducing dependence on foreign oil.

Another bill partly inspired by the Fee and Dividend structure was introduced by Democratic Congressman John B. Larson on July 16, 2015. H.R. 3104, or the "America's Energy Security Trust Fund Act of 2015" includes a steadily rising price on carbon but uses some revenue for job retraining, and returns the remainder of revenue via a payroll tax cut rather than direct dividend payments.

On September 1, 2016, the California Assembly Joint Resolution 43, "Williams. Greenhouse gases: climate change", was filed, having passed both houses. The measure urges the United States Congress to enact a tax on carbon-based fossil fuels. The proposal is revenue-neutral, with all money collected going to the bottom 2⁄3 of American households. It may have difficulty passing in Congress because it would be considered a tax, but if households were to receive an equal share in the form of a dividend then the legislation should properly class as a carbon fee. Thus California's recommendation for national legislation is perhaps close to being acceptable to Congress.

A bipartisan carbon fee and dividend bill, the Energy Innovation and Carbon Dividend Act, was introduced into United States House of Representatives during the second session of the 115th Congress. After the bill died at the end of the session, it was reintroduced in the first session of the 116th Congress on January 24, 2019. The lead sponsor is Democrat Ted Deutch and it is cosponsored by Republican Francis Rooney. The bill would levy a $15 fee per ton of carbon dioxide equivalent which would increase by $10 each year, with all revenue being returned to households.

A similar bill, the Climate Action Rebate Act, was introduced on July 25, 2019, into the Senate by Democrats Chris Coons and Dianne Feinstein and into the House of Representatives by Democrat Jimmy Panetta. This bill's carbon fee would also start at $15 per ton of CO₂-equivalent, but it would increase by $15 each year. The revenue would be split between dividends, infrastructure, research and development, and transition assistance.

Several 2020 presidential candidates have publicly shared their support of the fee and dividend policy, including Bernie Sanders, Pete Buttigieg, Andrew Yang, and John Delaney.

European Union

In the European Union a petition (addressed to the European Commission) was started on May 6, 2019, with the request to introduce a Climate Income in the EU. The petition is a registered European Citizens' Initiative, so if it reaches 1 million signatures, the topic will be placed on the agenda of the European Commission, and will be considered to form a legislative proposal.

Australia

An Australian version was proposed by Professors Richard Holden and Rosalind Dixon at the University of New South Wales (UNSW) and launched by Member for Wentworth Professor Kerryn Phelps AM MP. Surveys conducted by UNSW showed that the proposal would receive 73% support.

Opposition

There are objections on the way the tax revenue is used. Emeritus professor of management Henry Jacoby, formerly of the Massachusetts Institute of Technology, reviewed some of the more common concerns in a Guardian article in January 2021, particularly the stigma of taxation's perceived unpopularity. Some opponents are concerned with governments possibly not returning the revenue to people. A 2021 study looking at the only two countries with implemented carbon dividends – Canada and Switzerland – found that the news of the funds raised being returned to the public had little impact on the carbon taxes unpopularity, and that among Canadian conservatives it may even have increased opposition.