Principle of locality

From Wikipedia, the free encyclopedia

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

In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. A theory that includes the principle of locality is said to be a "local theory". This is an alternative to the concept of instantaneous, or "non-local" action at a distance. Locality evolved out of the field theories of classical physics. The idea is that for a cause at one point to have an effect at another point, something in the space between those points must mediate the action. To exert an influence, something, such as a wave or particle, must travel through the space between the two points, carrying the influence.

The special theory of relativity limits the maximum speed at which causal influence can travel to the speed of light, ${\displaystyle c}$. Therefore, the principle of locality implies that an event at one point cannot cause a truly simultaneous result at another point. An event at point ${\displaystyle A}$ cannot cause a result at point ${\displaystyle B}$ in a time less than ${\displaystyle T=D/c}$, where ${\displaystyle D}$ is the distance between the points and ${\displaystyle c}$ is the speed of light in vacuum.

The principle of locality plays a critical role in one of the central results of quantum mechanics. In 1935 Albert Einstein, Boris Podolsky, and Nathan Rosen, with their EPR paradox thought experiment, raised the possibility that quantum mechanics might not be a complete theory. They described two systems physically separated after interacting; this pair would be called entangled in modern terminology. They reasoned that without additions, now called hidden variables, quantum mechanics would predict illogical relationships between the physically separated measurements.

In 1964 John Stewart Bell formulated Bell's theorem, an inequality which, if violated in actual experiments, implies that quantum mechanics violates local causality (referred to as local realism in later work), a result now considered equivalent to precluding local hidden variables. Progressive variations on those Bell test experiments have since shown that quantum mechanics broadly violates Bell's inequalities. According to some interpretations of quantum mechanics, this result implies that some quantum effects violate the principle of locality.

Pre-quantum mechanics

Main article: Action at a distance

During the 17th century, Newton's principle of universal gravitation was formulated in terms of "action at a distance", thereby violating the principle of locality. Newton himself considered this violation to be absurd:

It is inconceivable that inanimate Matter should, without the Mediation of something else, which is not material, operate upon, and affect other matter without mutual Contact…That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it. Gravity must be caused by an Agent acting constantly according to certain laws; but whether this Agent be material or immaterial, I have left to the Consideration of my readers.

— Isaac Newton, Letters to Bentley, 1692/3

Coulomb's law of electric forces was initially also formulated as instantaneous action at a distance, but in 1880, James Clerk Maxwell showed that field equations – which obey locality – predict all of the phenomena of electromagnetism. These equations show that electromagnetic forces propagate at the speed of light.

In 1905, Albert Einstein's special theory of relativity postulated that no matter or energy can travel faster than the speed of light, and Einstein thereby sought to reformulate physics in a way that obeyed the principle of locality. He later succeeded in producing an alternative theory of gravitation, general relativity, which obeys the principle of locality.

However, a different challenge to the principle of locality developed subsequently from the theory of quantum mechanics, which Einstein himself had helped to create.

Models for locality

Diagram for locality in quantum mechanics

Simple spacetime diagrams can help clarify the issues related to locality. A way to describe the issues of locality suitable for discussion of quantum mechanics is illustrated in the diagram. A particle is created in one location, then split and measured in two other, spatially separated, locations. The two measurements are named for Alice and Bob. Alice performs measurements (A) and gets a result ${\displaystyle \mathbf{a}}$); Bob performs (${\displaystyle B}$) and gets result ${\displaystyle \mathbf{b}}$. The experiment is repeated many times and the results are compared.

Alice and Bob in spacetime

Alice and Bob in spacetime diagram

A spacetime diagram has a time coordinate going vertical and a space coordinate going horizontal. Alice, in a local region on the left, can affect events only in a cone extending in the future as shown; the finite speed of light prevent her from affecting other areas including Bob's location in this case. Similarly we can use the diagram to reason that Bob's local circumstances cannot be altered by Alice at the same time: all events that cause an effect on Bob are in the cone below his location on the diagram. Dashed lines around Alice show her valid future locations; dashed lines around Bob show events that could have caused his present circumstance. When Alice measures quantum states in her location she gets the results labeled ${\displaystyle \mathbf{a}}$; similarly Bob gets ${\displaystyle \mathbf{b}}$. Models of locality attempt to explain the statistical relationship between these measured values.

Action at a distance

Action at a distance

The simplest locality model is no locality: instantaneous action at a distance with no limits for relativity. The locality model for action at a distance is called continuous action. The gray area (a circle here) is a mathematical concept called a "screen". Any path from a location through the screen becomes part of the physical model at that location. The gray ring indicates events from all parts of space and time can affect the probability measured by Alice or Bob. So in the case of continuous action, events at all times and places affect Alice's and Bob's model. This simple model is highly successful for solar planetary dynamics with Newtonian gravity and in electrostatics, cases where relativistic effects are insignificant.

No future-input dependence

No future-input dependence

Many locality models explicitly or implicitly ignore the possible effect of future events. The spacetime diagram at the right shows the effect of such a restriction when combined with continuous action. Inputs from the future (above the dashed line) are no longer considered part of Alice's or Bob's model. Comparing this diagram with the one for continuous action makes it clear that these are not the same locality model. Common sense arguments about the future not affecting the present are reasonable criteria but such assumptions alter the mathematical character of the models.

Bell's local causality

Bell's local causality

John Stewart Bell when discussing his Bell's theorem uses the screening model shown at the right. Events in the common past of Alice and Bob are part of the model used in calculating probabilities for Alice and for Bob as indicated the way the screen absorbs those events. However events at Bob's location during Alice measurement and events in the future are excluded. Bell called this assumption local causality, but with the diagram we can reason about the meaning of the assumption without getting tripped up by other meanings of local combined with other meanings of causal. Dash lines show relativistically valid regions in the past of Alice or Bob. The gray arc is the assumed Bell "screen".

Quantum mechanics

The relative positions of our few, easily distinguishable planets (for example) can be seen directly: understanding and measuring their relative location poses only technical issues. The submicroscopic world on the other hand is known only by measurements that average over many seemingly random ("statistical" or "probabilistic") events and measurements can show either particle-like or wave-like results depending on their design. This world is governed by quantum mechanics. The concepts of locality are more complex and they are described in the language of probability and correlation.

In the 1935 Einstein–Podolsky–Rosen paradox paper (EPR paper), Albert Einstein, Boris Podolsky and Nathan Rosen imagined such an experiment. They observed that quantum mechanics predicts what is now known as quantum entanglement and examined its consequences. In their view, the classical principle of locality implied that "no real change can take place" at Bob's site as a result of whatever measurements Alice was doing. Since quantum mechanics does predict a wavefunction collapse that depends on Alice's choice of measurement, they concluded that this was a form of action-at-distance and that the wavefunction could not be a complete description of reality. Other physicists did not agree: they accepted the quantum wavefunction as complete and questioned the nature of locality and reality assumed in the EPR paper.

In 1964 John Stewart Bell investigated whether it might be possible to fulfill Einstein's goal—to "complete" quantum theory—with local hidden variables to explain the correlations between spatially separated particles as predicted by quantum theory. Bell established a criterion to distinguish between local hidden-variables theory and quantum theory by measuring specific values of correlations between entangled particles. Subsequent experimental tests have shown that some quantum effects do violate Bell's inequalities and cannot be reproduced by a local hidden-variables theory. Bell's theorem depends on careful defined models of locality.

Locality and hidden variables

Main article: Bell's theorem

Bell described local causality in terms of probability needed for analysis of quantum mechanics. Using the notation that ${\displaystyle P(\mathbf{r}\mid g)}$ for the probability of a result ${\displaystyle \mathbf{r}}$ with given state ${\displaystyle g}$, Bell investigated the probability distribution $${\displaystyle P(\mathbf{a}\mathbf{b}\mid AB,\lambda ),}$$ where ${\displaystyle \lambda }$ represents hidden state variables set (locally) when the two particles are initially co-located. If local causality holds, then the probabilities observed by Alice and by Bob should be only coupled by the hidden variables, and we can show that $${\displaystyle P(\mathbf{a}\mathbf{b}\mid AB,\lambda )=P(\mathbf{a}\mid A,\lambda )P(\mathbf{b}\mid B,\lambda ).}$$ Bell proved that a consequence of this factorization are limits on the correlations observed by Alice and Bob known as Bell inequalities. Since quantum mechanics predicts correlations stronger than this limit, locally set hidden variables cannot be added to "complete" quantum theory as desired by the EPR paper.

Numerous experiments specifically designed to probe the issues of locality confirm the predictions of quantum mechanics; these include experiments where the two measurement locations are more than a kilometer apart. The 2022 Nobel Prize in Physics was awarded to Alain Aspect, John Clauser and Anton Zeilinger, in part "for experiments with entangled photons, establishing the violation of Bell inequalities". The specific aspect of quantum theory that leads to these correlations is termed quantum entanglement, and versions of Bell's scenario are now used to verify entanglement experimentally.

Terminology

Main article: Quantum nonlocality

See also: Bell's theorem § Interpretations, and Reality § Realism and locality in physics

Bell's mathematical results, when compared to experimental data, eliminate local hidden-variable mathematical quantum theories. But the interpretation of the math with respect to the physical world remains under debate. Bell described the assumptions behind his work as "local causality", shortened to "locality"; later authors referred to the assumptions as local realism. These different names do not alter the mathematical assumptions.

A review of papers using this phrase suggests that a common (classical) physics definition of realism is

the assumption that measurement outcomes are well defined prior to and independent of the measurements.

This definition includes classical concepts like "well-defined", which conflicts with quantum superposition, and "prior to ... measurements", which implies (metaphysical) preexistence of properties. Specifically, the term local realism in the context of Bell's theorem cannot be viewed as a kind of "realism" involving locality other than the kind implied by the Bell screening assumption. This conflict between common ideas of realism and quantum mechanics requires careful analysis whenever local realism is discussed. Adding a "locality" modifier, that the results of two spatially well-separated measurements cannot causally affect each other, does not make the combination relate to Bell's proof; the only interpretation that Bell assumed was the one he called local causality.  Consequently, Bell's theorem does not restrict the possibility of nonlocal variables as well as theories based on retrocausality or superdeterminism.

Because of the probabilistic nature of wave function collapse, this apparent violation of locality in quantum mechanics cannot be used to transmit information faster than light, in accordance to the no communication theorem. Asher Peres distinguishes between weak and strong nonlocality, the latter referring to the theories that allow faster-than-light communication. Under these terms, quantum mechanics would allow weakly nonlocal correlations but not strong nonlocality.

Relativistic quantum mechanics

One of the main principles of quantum field theory is the principle of locality. The field operators and the Lagrangian density describing the dynamics of the fields are local, in the sense that interactions are not described by action-at-a-distance. This condition can be achieved by avoiding terms in the Lagrangian that are products of two fields that depend on distant coordinates. Specifically, in relativistic quantum field theory, to enforce the principles of locality and causality the following condition is required: if there are two observables, each localized within two distinct spacetime regions which happen to be at a spacelike separation from each other, the observables must commute. This condition is sometimes imposed as one of the axioms of relativistic quantum field theory.

Action at a distance

From Wikipedia, the free encyclopedia

Action at a distance is the concept in physics that an object's motion can be affected by another object without the two being in physical contact; that is, it is the concept of the non-local interaction of objects that are separated in space. Coulomb's law and Newton's law of universal gravitation are based on action at a distance.

Historically, action at a distance was the earliest scientific model for gravity and electricity and it continues to be useful in many practical cases. In the 19th and 20th centuries, field models arose to explain these phenomena with more precision. The discovery of electrons and of special relativity led to new action at a distance models providing alternative to field theories. Under our modern understanding, the four fundamental interactions (gravity, electromagnetism, the strong interaction and the weak interaction) in all of physics are not described by action at a distance.

Categories of action

In the study of mechanics, action at a distance is one of three fundamental actions on matter that cause motion. The other two are direct impact (elastic or inelastic collisions) and actions in a continuous medium as in fluid mechanics or solid mechanics.  Historically, physical explanations for particular phenomena have moved between these three categories over time as new models were developed.

Action-at-a-distance and actions in a continuous medium may be easily distinguished when the medium dynamics are visible, like waves in water or in an elastic solid. In the case of electricity or gravity, no medium is required. In the nineteenth century, criteria like the effect of actions on intervening matter, the observation of a time delay, the apparent storage of energy, or even the possibility of a plausible mechanical model for action transmission were all accepted as evidence against action at a distance. Aether theories were alternative proposals to replace apparent action-at-a-distance in gravity and electromagnetism, in terms of continuous action inside an (invisible) medium called "aether".

Direct impact of macroscopic objects seems visually distinguishable from action at a distance. If however the objects are constructed of atoms, and the volume of those atoms is not defined and atoms interact by electric and magnetic forces, the distinction is less clear.

Roles

The concept of action at a distance acts in multiple roles in physics and it can co-exist with other models according to the needs of each physical problem.

One role is as a summary of physical phenomena, independent of any understanding of the cause of such an action. For example, astronomical tables of planetary positions can be compactly summarized using Newton's law of universal gravitation, which assumes the planets interact without contact or an intervening medium. As a summary of data, the concept does not need to be evaluated as a plausible physical model.

Action at a distance also acts as a model explaining physical phenomena even in the presence of other models. Again in the case of gravity, hypothesizing an instantaneous force between masses allows the return time of comets to be predicted as well as predicting the existence of previously unknown planets, like Neptune. These triumphs of physics predated the alternative more accurate model for gravity based on general relativity by many decades.

Introductory physics textbooks discuss central forces, like gravity, by models based on action-at-distance without discussing the cause of such forces or issues with it until the topics of relativity and fields are discussed. For example, see The Feynman Lectures on Physics on gravity.

History

Early inquiries into motion

Main articles: Scientific Revolution and History of gravitational theory

Action-at-a-distance as a physical concept requires identifying objects, distances, and their motion. In antiquity, ideas about the natural world were not organized in these terms. Objects in motion were modeled as living beings. Around 1600, the scientific method began to take root. René Descartes held a more fundamental view, developing ideas of matter and action independent of theology. Galileo Galilei wrote about experimental measurements of falling and rolling objects. Johannes Kepler's laws of planetary motion summarized Tycho Brahe's astronomical observations. Many experiments with electrical and magnetic materials led to new ideas about forces. These efforts set the stage for Newton's work on forces and gravity.

Newtonian gravity

Main article: History of gravitational theory

In 1687 Isaac Newton published his Principia which combined his laws of motion with a new mathematical analysis able to reproduce Kepler's empirical results. His explanation was in the form of a law of universal gravitation: any two bodies are attracted by a force proportional to their mass and inversely proportional to the square of the distance between them. Thus the motions of planets were predicted by assuming forces working over great distances.

This mathematical expression of the force did not imply a cause. Newton considered action-at-a-distance to be an inadequate model for gravity.^[6] Newton, in his words, considered action at a distance to be:

so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it.

— Isaac Newton, Letters to Bentley, 1692/3

Metaphysical scientists of the early 1700s strongly objected to the unexplained action-at-a-distance in Newton's theory. Gottfried Wilhelm Leibniz complained that the mechanism of gravity was "invisible, intangible, and not mechanical". Moreover, initial comparisons with astronomical data were not favorable. As mathematical techniques improved throughout the 1700s, the theory showed increasing success, predicting the date of the return of Halley's comet and aiding the discovery of planet Neptune in 1846. These successes and the increasingly empirical focus of science towards the 19th century led to acceptance of Newton's theory of gravity despite distaste for action-at-a-distance.

Electrical action at a distance

Main article: History of electromagnetic theory

Jean-Antoine Nollet reproducing Stephan Gray's “electric boy” experiment, in which a boy hanging from insulating silk ropes is given an electric charge. A group are gathered around. A woman is encouraged to bend forward and poke the boy's nose, to get an electric shock.

Electrical and magnetic phenomena also began to be explored systematically in the early 1600s. In William Gilbert's early theory of "electric effluvia," a kind of electric atmosphere, he rules out action-at-a-distance on the grounds that "no action can be performed by matter save by contact". However subsequent experiments, especially those by Stephen Gray showed electrical effects over distance. Gray developed an experiment call the "electric boy" demonstrating electric transfer without direct contact. Franz Aepinus was the first to show, in 1759, that a theory of action at a distance for electricity provides a simpler replacement for the electric effluvia theory. Despite this success, Aepinus himself considered the nature of the forces to be unexplained: he did "not approve of the doctrine which assumes the possibility of action at a distance", setting the stage for a shift to theories based on aether.

By 1785 Charles-Augustin de Coulomb showed that two electric charges at rest experience a force inversely proportional to the square of the distance between them, a result now called Coulomb's law. The striking similarity to gravity strengthened the case for action at a distance, at least as a mathematical model.

As mathematical methods improved, especially through the work of Pierre-Simon Laplace, Joseph-Louis Lagrange, and Siméon Denis Poisson, more sophisticated mathematical methods began to influence the thinking of scientists. The concept of potential energy applied to small test particles led to the concept of a scalar field, a mathematical model representing the forces throughout space. While this mathematical model is not a mechanical medium, the mental picture of such a field resembles a medium.

Fields as an alternative

Main article: History of field theory

Glazed frame, containing "Delineation of Lines of Magnetic Force by Iron filings" prepared by Michael Faraday

Michael Faraday was the first who suggested that action at a distance was inadequate as an account of electric and magnetic forces, even in the form of a (mathematical) potential field. Faraday, an empirical experimentalist, cited three reasons in support of some medium transmitting electrical force: 1) electrostatic induction across an insulator depends on the nature of the insulator, 2) cutting a charged insulator causes opposite charges to appear on each half, and 3) electric discharge sparks are curved at an insulator. From these reasons he concluded that the particles of an insulator must be polarized, with each particle contributing to continuous action. He also experimented with magnets, demonstrating lines of force made visible by iron filings. However, in both cases his field-like model depends on particles that interact through an action-at-a-distance: his mechanical field-like model has no more fundamental physical cause than the long-range central field model.

Faraday's observations, as well as others, led James Clerk Maxwell to a breakthrough formulation in 1865, a set of equations that combined electricity and magnetism, both static and dynamic, and which included electromagnetic radiation – light. Maxwell started with elaborate mechanical models but ultimately produced a purely mathematical treatment using dynamical vector fields. The sense that these fields must be set to vibrate to propagate light set off a search of a medium of propagation; the medium was called the luminiferous aether or the aether.

In 1873 Maxwell addressed action at a distance explicitly. He reviews Faraday's lines of force, carefully pointing out that Faraday himself did not provide a mechanical model of these lines in terms of a medium. Nevertheless the many properties of these lines of force imply these "lines must not be regarded as mere mathematical abstractions". Faraday himself viewed these lines of force as a model, a "valuable aid" to the experimentalist, a means to suggest further experiments.

In distinguishing between different kinds of action Faraday suggested three criteria: 1) do additional material objects alter the action?, 2) does the action take time, and 3) does it depend upon the receiving end? For electricity, Faraday knew that all three criteria were met for electric action, but gravity was thought to only meet the third one. After Maxwell's time a fourth criteria, the transmission of energy, was added, thought to also apply to electricity but not gravity. With the advent of new theories of gravity, the modern account would give gravity all of the criteria except dependence on additional objects.

Fields fade into spacetime

Main article: Luminiferous ether

The success of Maxwell's field equations led to numerous efforts in the later decades of the 19th century to represent electrical, magnetic, and gravitational fields, primarily with mechanical models. No model emerged that explained the existing phenomena. In particular no good model for stellar aberration, the shift in the position of stars with the Earth's relative velocity. The best models required the ether to be stationary while the Earth moved, but experimental efforts to measure the effect of Earth's motion through the aether found no effect.

In 1892 Hendrik Lorentz proposed a modified aether based on the emerging microscopic molecular model rather than the strictly macroscopic continuous theory of Maxwell. Lorentz investigated the mutual interaction of a moving solitary electrons within a stationary aether. He rederived Maxwell's equations in this way but, critically, in the process he changed them to represent the wave in the coordinates moving electrons. He showed that the wave equations had the same form if they were transformed using a particular scaling factor, $${\displaystyle \gamma =\frac{1}{\sqrt{1-(u^2/c^2)}}.}$$ where ${\displaystyle u}$ is the velocity of the moving electrons and ${\displaystyle c}$ is the speed of light. Lorentz noted that if this factor were applied as a length contraction to moving matter in a stationary ether, it would eliminate any effect of motion through the ether, in agreement with experiment.

In 1899, Henri Poincaré questioned the existence of an aether, showing that the principle of relativity prohibits the absolute motion assumed by proponents of the aether model. He named the transformation used by Lorentz the Lorentz transformation but interpreted it as a transformation between two inertial frames with relative velocity ${\displaystyle u}$. This transformation makes the electromagnetic equations look the same in every uniformly moving inertial frame. Then, in 1905, Albert Einstein demonstrated that the principle of relativity, applied to the simultaneity of time and the constant speed of light, precisely predicts the Lorentz transformation. This theory of special relativity quickly became the modern concept of spacetime.

Thus the aether model, initially so very different from action at a distance, slowly changed to resemble simple empty space.

In 1905, Poincaré proposed gravitational waves, emanating from a body and propagating at the speed of light, as being required by the Lorentz transformations and suggested that, in analogy to an accelerating electrical charge producing electromagnetic waves, accelerated masses in a relativistic field theory of gravity should produce gravitational waves. However, until 1915 gravity stood apart as a force still described by action-at-a-distance. In that year, Einstein showed that a field theory of spacetime, general relativity, consistent with relativity can explain gravity. New effects resulting from this theory were dramatic for cosmology but minor for planetary motion and physics on Earth. Einstein himself noted Newton's "enormous practical success".

Modern action at a distance

Main article: Wheeler–Feynman absorber theory

In the early decades of the 20th century, Karl Schwarzschild, Hugo Tetrode, and Adriaan Fokker independently developed non-instantaneous models for action at a distance consistent with special relativity. In 1949 John Archibald Wheeler and Richard Feynman built on these models to develop a new field-free theory of electromagnetism. While Maxwell's field equations are generally successful, the Lorentz model of a moving electron interacting with the field encounters mathematical difficulties: the self-energy of the moving point charge within the field is infinite. The Wheeler–Feynman absorber theory of electromagnetism avoids the self-energy issue. They interpret Abraham–Lorentz force, the apparent force resisting electron acceleration, as a real force returning from all the other existing charges in the universe.

The Wheeler–Feynman theory has inspired new thinking about the arrow of time and about the nature of quantum non-locality. The theory has implications for cosmology; it has been extended to quantum mechanics. A similar approach has been applied to develop an alternative theory of gravity consistent with general relativity. John G. Cramer has extended the Wheeler–Feynman ideas to create the transactional interpretation of quantum mechanics.

"Spooky action at a distance"

Albert Einstein wrote to Max Born about issues in quantum mechanics in 1947 and used a phrase translated as "spooky action at a distance", and in 1964, John Stewart Bell proved that quantum mechanics predicted stronger statistical correlations in the outcomes of certain far-apart measurements than any local theory possibly could. The phrase has been picked up and used as a description for the cause of small non-classical correlations between physically separated measurement of entangled quantum states. The correlations are predicted by quantum mechanics (the Bell theorem) and verified by experiments (the Bell test). Rather than a postulate like Newton's gravitational force, this use of "action-at-a-distance" concerns observed correlations which cannot be explained with localized particle-based models. Describing these correlations as "action-at-a-distance" requires assuming that particles became entangled and then traveled to distant locations, an assumption that is not required by quantum mechanics.

Force in quantum field theory

Main article: Force carriers

 

Quantum field theory does not need action at a distance. At the most fundamental level, only four forces are needed. Each force is described as resulting from the exchange of specific bosons. Two are short range: the strong interaction mediated by mesons and the weak interaction mediated by the weak boson; two are long range: electromagnetism mediated by the photon and gravity hypothesized to be mediated by the graviton. However, the entire concept of force is of secondary concern in advanced modern particle physics. Energy forms the basis of physical models and the word action has shifted away from implying a force to a specific technical meaning, an integral over the difference between potential energy and kinetic energy.

Introduction to the mathematics of general relativity

From Wikipedia, the free encyclopedia

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

The mathematics of general relativity is complicated. In Newton's theories of motion, an object's length and the rate at which time passes remain constant while the object accelerates, meaning that many problems in Newtonian mechanics may be solved by algebra alone. In relativity, however, an object's length and the rate at which time passes both change appreciably as the object's speed approaches the speed of light, meaning that more variables and more complicated mathematics are required to calculate the object's motion. As a result, relativity requires the use of concepts such as vectors, tensors, pseudotensors and curvilinear coordinates.

For an introduction based on the example of particles following circular orbits about a large mass, nonrelativistic and relativistic treatments are given in, respectively, Newtonian motivations for general relativity and Theoretical motivation for general relativity.

Vectors and tensors

Main articles: Euclidean vector and Tensor

Vectors

Illustration of a typical vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric vector or spatial vector, or – as here – simply a vector) is a geometric object that has both a magnitude (or length) and direction. A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "one who carries". The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from A to B. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity.

Tensors

Stress is a second-order tensor that represents the response of a material to force applied at an angle. The two directions of the tensor represent the "normal" (at right angles to the surface) force, and "shear" (parallel to the surface) force.

A tensor extends the concept of a vector to additional directions. A scalar, that is, a simple number without a direction, would be shown on a graph as a point, a zero-dimensional object. A vector, which has a magnitude and direction, would appear on a graph as a line, which is a one-dimensional object. A vector is a first-order tensor, since it holds one direction. A second-order tensor has two magnitudes and two directions, and would appear on a graph as two lines similar to the hands of a clock. The "order" of a tensor is the number of directions contained within, which is separate from the dimensions of the individual directions. A second-order tensor in two dimensions might be represented mathematically by a 2-by-2 matrix, and in three dimensions by a 3-by-3 matrix, but in both cases the matrix is "square" for a second-order tensor. A third-order tensor has three magnitudes and directions, and would be represented by a cube of numbers, 3-by-3-by-3 for directions in three dimensions, and so on.

Applications

Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has both a magnitude and direction, such as velocity, the magnitude of which is speed. For example, the velocity 5 meters per second upward could be represented by the vector (0, 5) (in 2 dimensions with the positive y axis as 'up'). Another quantity represented by a vector is force, since it has a magnitude and direction. Vectors also describe many other physical quantities, such as displacement, acceleration, momentum, and angular momentum. Other physical vectors, such as the electric and magnetic field, are represented as a system of vectors at each point of a physical space; that is, a vector field.

Tensors also have extensive applications in physics:

• Electromagnetic tensor (or Faraday's tensor) in electromagnetism
• Finite deformation tensors for describing deformations and strain tensor for strain in continuum mechanics
• Permittivity and electric susceptibility are tensors in anisotropic media
• Stress–energy tensor in general relativity, used to represent momentum fluxes
• Spherical tensor operators are the eigenfunctions of the quantum angular momentum operator in spherical coordinates
• Diffusion tensors, the basis of diffusion tensor imaging, represent rates of diffusion in biologic environments

Dimensions

In general relativity, four-dimensional vectors, or four-vectors, are required. These four dimensions are length, height, width and time. A "point" in this context would be an event, as it has both a location and a time. Similar to vectors, tensors in relativity require four dimensions. One example is the Riemann curvature tensor.

Coordinate transformation

• 
  A vector v, is shown with two coordinate grids, e_x and e_r. In space, there is no clear coordinate grid to use. This means that the coordinate system changes based on the location and orientation of the observer. Observer e_x and e_r in this image are facing different directions.
• 
  Here we see that e_x and e_r see the vector differently. The direction of the vector is the same. But to e_x, the vector is moving to its left. To e_r, the vector is moving to its right.

In physics, as well as mathematics, a vector is often identified with a tuple, or list of numbers, which depend on a coordinate system or reference frame. If the coordinates are transformed, such as by rotation or stretching the coordinate system, the components of the vector also transform. The vector itself does not change, but the reference frame does. This means that the components of the vector have to change to compensate.

The vector is called covariant or contravariant depending on how the transformation of the vector's components is related to the transformation of coordinates.

• Contravariant vectors have units of distance (such as a displacement) or distance times some other unit (such as velocity or acceleration) and transform in the opposite way as the coordinate system. For example, in changing units from meters to millimeters the coordinate units get smaller, but the numbers in a vector become larger: 1 m becomes 1000 mm.
• Covariant vectors, on the other hand, have units of one-over-distance (as in a gradient) and transform in the same way as the coordinate system. For example, in changing from meters to millimeters, the coordinate units become smaller and the number measuring a gradient will also become smaller: 1 Kelvin per m becomes 0.001 Kelvin per mm.

In Einstein notation, contravariant vectors and components of tensors are shown with superscripts, e.g. xⁱ, and covariant vectors and components of tensors with subscripts, e.g. x_i. Indices are "raised" or "lowered" by multiplication by an appropriate matrix, often the identity matrix.

Coordinate transformation is important because relativity states that there is not one reference point (or perspective) in the universe that is more favored than another. On earth, we use dimensions like north, east, and elevation, which are used throughout the entire planet. There is no such system for space. Without a clear reference grid, it becomes more accurate to describe the four dimensions as towards/away, left/right, up/down and past/future. As an example event, assume that Earth is a motionless object, and consider the signing of the Declaration of Independence. To a modern observer on Mount Rainier looking east, the event is ahead, to the right, below, and in the past. However, to an observer in medieval England looking north, the event is behind, to the left, neither up nor down, and in the future. The event itself has not changed: the location of the observer has.

Oblique axes

An oblique coordinate system is one in which the axes are not necessarily orthogonal to each other; that is, they meet at angles other than right angles. When using coordinate transformations as described above, the new coordinate system will often appear to have oblique axes compared to the old system.

Nontensors

See also: Pseudotensor

A nontensor is a tensor-like quantity that behaves like a tensor in the raising and lowering of indices, but that does not transform like a tensor under a coordinate transformation. For example, Christoffel symbols cannot be tensors themselves if the coordinates do not change in a linear way.

In general relativity, one cannot describe the energy and momentum of the gravitational field by an energy–momentum tensor. Instead, one introduces objects that behave as tensors only with respect to restricted coordinate transformations. Strictly speaking, such objects are not tensors at all. A famous example of such a pseudotensor is the Landau–Lifshitz pseudotensor.

Curvilinear coordinates and curved spacetime

High-precision test of general relativity by the Cassini space probe (artist's impression): radio signals sent between the Earth and the probe (green wave) are delayed by the warping of space and time (blue lines) due to the Sun's mass. That is, the Sun's mass causes the regular grid coordinate system (in blue) to distort and have curvature. The radio wave then follows this curvature and moves toward the Sun.

Curvilinear coordinates are coordinates in which the angles between axes can change from point to point. This means that rather than having a grid of straight lines, the grid instead has curvature.

A good example of this is the surface of the Earth. While maps frequently portray north, south, east and west as a simple square grid, that is not in fact the case. Instead, the longitude lines running north and south are curved and meet at the north pole. This is because the Earth is not flat, but instead round.

In general relativity, energy and mass have curvature effects on the four dimensions of the universe (= spacetime). This curvature gives rise to the gravitational force. A common analogy is placing a heavy object on a stretched out rubber sheet, causing the sheet to bend downward. This curves the coordinate system around the object, much like an object in the universe curves the coordinate system it sits in. The mathematics here are conceptually more complex than on Earth, as it results in four dimensions of curved coordinates instead of three as used to describe a curved 2D surface.

Parallel transport

Main article: Parallel transport

Example: Parallel displacement along a circle of a three-dimensional ball embedded in two dimensions. The circle of radius r is embedded in a two-dimensional space characterized by the coordinates z¹ and z². The circle itself is characterized by coordinates y¹ and y² in the two-dimensional space. The circle itself is one-dimensional and can be characterized by its arc length x. The coordinate y is related to the coordinate x through the relation y¹ = r cos ⁠x/r⁠ and y² = r sin ⁠x/r⁠. This gives ⁠∂y¹/∂x⁠ = −sin ⁠x/r⁠ and ⁠∂y²/∂x⁠ = cos ⁠x/r⁠ In this case the metric is a scalar and is given by g = cos² ⁠x/r⁠ + sin² ⁠x/r⁠ = 1. The interval is then ds² = g dx² = dx². The interval is just equal to the arc length as expected.

The interval in a high-dimensional space

In a Euclidean space, the separation between two points is measured by the distance between the two points. The distance is purely spatial, and is always positive. In spacetime, the separation between two events is measured by the invariant interval between the two events, which takes into account not only the spatial separation between the events, but also their separation in time. The interval, s², between two events is defined as:

${\displaystyle s^2=\mathrm{\Delta }{r}^2-c^2\mathrm{\Delta }{t}^2}$     (spacetime interval),

where c is the speed of light, and Δr and Δt denote differences of the space and time coordinates, respectively, between the events. The choice of signs for s² above follows the space-like convention (−+++). A notation like Δr² means (Δr)². The reason s² is called the interval and not s is that s² can be positive, zero or negative.

Spacetime intervals may be classified into three distinct types, based on whether the temporal separation (c²Δt²) or the spatial separation (Δr²) of the two events is greater: time-like, light-like or space-like.

Certain types of world lines are called geodesics of the spacetime – straight lines in the case of flat Minkowski spacetime and their closest equivalent in the curved spacetime of general relativity. In the case of purely time-like paths, geodesics are (locally) the paths of greatest separation (spacetime interval) as measured along the path between two events, whereas in Euclidean space and Riemannian manifolds, geodesics are paths of shortest distance between two points. The concept of geodesics becomes central in general relativity, since geodesic motion may be thought of as "pure motion" (inertial motion) in spacetime, that is, free from any external influences.

The covariant derivative

Main article: Covariant derivative

The covariant derivative is a generalization of the directional derivative from vector calculus. As with the directional derivative, the covariant derivative is a rule, which takes as its inputs: (1) a vector, u, (along which the derivative is taken) defined at a point P, and (2) a vector field, v, defined in a neighborhood of P. The output is a vector, also at the point P. The primary difference from the usual directional derivative is that the covariant derivative must, in a certain precise sense, be independent of the manner in which it is expressed in a coordinate system.

Parallel transport

Given the covariant derivative, one can define the parallel transport of a vector v at a point P along a curve γ starting at P. For each point x of γ, the parallel transport of v at x will be a function of x, and can be written as v(x), where v(0) = v. The function v is determined by the requirement that the covariant derivative of v(x) along γ is 0. This is similar to the fact that a constant function is one whose derivative is constantly 0.

Christoffel symbols

Main article: Christoffel symbols

The equation for the covariant derivative can be written in terms of Christoffel symbols. The Christoffel symbols find frequent use in Einstein's theory of general relativity, where spacetime is represented by a curved 4-dimensional Lorentz manifold with a Levi-Civita connection. The Einstein field equations – which determine the geometry of spacetime in the presence of matter – contain the Ricci tensor. Since the Ricci tensor is derived from the Riemann curvature tensor, which can be written in terms of Christoffel symbols, a calculation of the Christoffel symbols is essential. Once the geometry is determined, the paths of particles and light beams are calculated by solving the geodesic equations in which the Christoffel symbols explicitly appear.

Geodesics

Main article: Geodesics in general relativity

In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational force, is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic.

In general relativity, gravity can be regarded as not a force but a consequence of a curved spacetime geometry where the source of curvature is the stress–energy tensor (representing matter, for instance). Thus, for example, the path of a planet orbiting around a star is the projection of a geodesic of the curved 4-dimensional spacetime geometry around the star onto 3-dimensional space.

A curve is a geodesic if the tangent vector of the curve at any point is equal to the parallel transport of the tangent vector of the base point.

Curvature tensor

Main article: Riemann curvature tensor

The Riemann curvature tensor R^ρ_σμν tells us, mathematically, how much curvature there is in any given region of space. In flat space this tensor is zero.

Contracting the tensor produces 2 more mathematical objects:

1. The Ricci tensor: R_σν, comes from the need in Einstein's theory for a curvature tensor with only 2 indices. It is obtained by averaging certain portions of the Riemann curvature tensor.
2. The scalar curvature: R, the simplest measure of curvature, assigns a single scalar value to each point in a space. It is obtained by averaging the Ricci tensor.

The Riemann curvature tensor can be expressed in terms of the covariant derivative.

The Einstein tensor G is a rank-2 tensor defined over pseudo-Riemannian manifolds. In index-free notation it is defined as

${\displaystyle \mathbf{G}=\mathbf{R}-\frac{1}{2}\mathbf{g}R,}$

where R is the Ricci tensor, g is the metric tensor and R is the scalar curvature. It is used in the Einstein field equations.

Stress–energy tensor

Main article: Stress–energy tensor

Contravariant components of the stress–energy tensor

The stress–energy tensor (sometimes stress–energy–momentum tensor or energy–momentum tensor) is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. The stress–energy tensor is the source of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity. Because this tensor has 2 indices (see next section) the Riemann curvature tensor has to be contracted into the Ricci tensor, also with 2 indices.

Einstein equation

Main article: Einstein field equations

The Einstein field equations (EFE) or Einstein's equations are a set of 10 equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy. First published by Einstein in 1915 as a tensor equation, the EFE equate local spacetime curvature (expressed by the Einstein tensor) with the local energy and momentum within that spacetime (expressed by the stress–energy tensor).

The Einstein field equations can be written as

${\displaystyle G_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu },}$

where G_μν is the Einstein tensor and T_μν is the stress–energy tensor.

This implies that the curvature of space (represented by the Einstein tensor) is directly connected to the presence of matter and energy (represented by the stress–energy tensor).

Schwarzschild solution and black holes

Main article: Schwarzschild metric

In Einstein's theory of general relativity, the Schwarzschild metric (also Schwarzschild vacuum or Schwarzschild solution), is a solution to the Einstein field equations which describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, the angular momentum of the mass, and the universal cosmological constant are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun. The solution is named after Karl Schwarzschild, who first published the solution in 1916, just before his death.

According to Birkhoff's theorem, the Schwarzschild metric is the most general spherically symmetric, vacuum solution of the Einstein field equations. A Schwarzschild black hole or static black hole is a black hole that has no charge or angular momentum. A Schwarzschild black hole is described by the Schwarzschild metric, and cannot be distinguished from any other Schwarzschild black hole except by its mass.