Themathematics of general relativity is complicated. InNewton's theories of motion, an object's length and the rate at which time passes remain constant while the objectaccelerates, meaning that many problems inNewtonian mechanics may be solved byalgebra alone. Inrelativity, however, an object's length and the rate at which time passes both change appreciably as the object's speed approaches thespeed 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 asvectors,tensors,pseudotensors andcurvilinear coordinates.
For an introduction based on the example of particles followingcircular orbits about a large mass, nonrelativistic and relativistic treatments are given in, respectively,Newtonian motivations for general relativity andTheoretical motivation for general relativity.
Inmathematics,physics, andengineering, aEuclidean vector (sometimes called ageometric vector[1] orspatial vector,[2] or – as here – simply a vector) is a geometric object that has both amagnitude (orlength) and direction. A vector is what is needed to "carry" the pointA to the pointB; the Latin wordvector means "one who carries".[3] The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement fromA toB. Manyalgebraic operations onreal numbers such asaddition,subtraction,multiplication, andnegation have close analogues for vectors, operations which obey the familiar algebraic laws ofcommutativity,associativity, anddistributivity.
A tensor extends the concept of a vector to additional directions. Ascalar, 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.
Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has both a magnitude and direction, such asvelocity, the magnitude of which isspeed. For example, the velocity5 meters per second upward could be represented by the vector(0, 5) (in 2 dimensions with the positivey axis as 'up'). Another quantity represented by a vector isforce, since it has a magnitude and direction. Vectors also describe many other physical quantities, such asdisplacement,acceleration,momentum, andangular momentum. Other physical vectors, such as theelectric andmagnetic field, are represented as a system of vectors at each point of a physical space; that is, avector field.
Tensors also have extensive applications in physics:
In general relativity, four-dimensional vectors, orfour-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 theRiemann curvature tensor.
In physics, as well as mathematics, a vector is often identified with atuple, or list of numbers, which depend on a coordinate system orreference 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 calledcovariant orcontravariant depending on how the transformation of the vector's components is related to the transformation of coordinates.
InEinstein 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.xi. 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 theDeclaration of Independence. To a modern observer onMount 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.
Anoblique coordinate system is one in which the axes are not necessarilyorthogonal to each other; that is, they meet at angles other thanright angles. When using coordinate transformations as described above, the new coordinate system will often appear to have oblique axes compared to the old system.
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 theLandau–Lifshitz pseudotensor.
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 infour dimensions of curved coordinates instead of three as used to describe a curved 2D surface.
In aEuclidean 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 theinvariant 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:
wherec is the speed of light, andΔr andΔt denote differences of the space and time coordinates, respectively, between the events. The choice of signs fors2 above follows thespace-like convention (−+++). A notation likeΔr2 means(Δr)2. The reasons2 and nots is called the interval is thats2 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 ofworld lines are calledgeodesics 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.[4][5] The concept of geodesics becomes central ingeneral 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 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 pointP, and (2) a vector field,v, defined in a neighborhood ofP. The output is a vector, also at the pointP. 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.
Given the covariant derivative, one can define theparallel transport of a vectorv at a pointP along a curveγ starting atP. For each pointx ofγ, the parallel transport ofv atx will be a function ofx, and can be written asv(x), wherev(0) =v. The functionv is determined by the requirement that the covariant derivative ofv(x) alongγ is 0. This is similar to the fact that a constant function is one whose derivative is constantly 0.
The equation for the covariant derivative can be written in terms of Christoffel symbols. The Christoffel symbols find frequent use in Einstein's theory ofgeneral relativity, wherespacetime is represented by a curved 4-dimensionalLorentz manifold with aLevi-Civita connection. TheEinstein field equations – which determine the geometry of spacetime in the presence of matter – contain theRicci 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 bysolving the geodesic equations in which the Christoffel symbols explicitly appear.
Ingeneral relativity, ageodesic generalizes the notion of a "straight line" to curvedspacetime. Importantly, theworld 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 thestress–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 thetangent vector of the curve at any point is equal to theparallel transport of thetangent vector of the base point.
TheRiemann curvature tensorRρσμν 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:
The Riemann curvature tensor can be expressed in terms of the covariant derivative.
The Einstein tensorG is a rank-2tensor defined overpseudo-Riemannian manifolds. In index-free notation it is defined as
whereR is theRicci tensor,g is themetric tensor andR is thescalar curvature. It is used in theEinstein field equations.
Thestress–energy tensor (sometimesstress–energy–momentum tensor orenergy–momentum tensor) is atensor quantity inphysics that describes thedensity andflux ofenergy andmomentum inspacetime, generalizing thestress tensor of Newtonian physics. It is an attribute ofmatter,radiation, and non-gravitationalforce fields. The stress–energy tensor is the source of thegravitational field in theEinstein field equations ofgeneral relativity, just as mass density is the source of such a field inNewtonian 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.
TheEinstein field equations (EFE) orEinstein's equations are a set of 10equations inAlbert Einstein'sgeneral theory of relativity which describe thefundamental interaction ofgravitation as a result ofspacetime beingcurved bymatter andenergy.[6] First published by Einstein in 1915[7] as atensor equation, the EFE equate local spacetimecurvature (expressed by theEinstein tensor) with the local energy andmomentum within that spacetime (expressed by thestress–energy tensor).[8]
The Einstein field equations can be written as
whereGμν is theEinstein tensor andTμν is thestress–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).
InEinstein's theory ofgeneral relativity, theSchwarzschild metric (alsoSchwarzschild vacuum orSchwarzschild solution), is a solution to theEinstein field equations which describes thegravitational field outside a spherical mass, on the assumption that theelectric charge of the mass, theangular momentum of the mass, and the universalcosmological constant are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as manystars andplanets, including Earth and the Sun. The solution is named afterKarl Schwarzschild, who first published the solution in 1916, just before his death.
According toBirkhoff's theorem, the Schwarzschild metric is the most generalspherically symmetric,vacuum solution of theEinstein field equations. ASchwarzschild black hole orstatic black hole is ablack hole that has nocharge orangular 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.
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