All the mathematics discussed in this article were known before Einstein's general theory of relativity.
For an introduction based on the specific physical example of particles orbiting a large mass in circular orbits, see Newtonian motivations for general relativity for a nonrelativistic treatment and Theoretical motivation for general relativity for a fully relativistic treatment.
Vectors and Tensors
Vectorsmathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric 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.
Tensorsscalar, that is, a simple set of numbers without magnitude, 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 tensor extends this concepts to additional dimensions. A two dimensional tensor would be called a second order tensor. This can be viewed as a set of related vectors, moving in multiple directions on a plane.
ApplicationsVectors 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.
Common applications of tensors in physics include:
- 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
DimensionsIn relativity, four-dimensional vectors, or four-vectors are required. These four dimensions are length, height, width and time. In this context, a point would be an event, as it has both a location and a time. Similar to vectors, tensors require four dimensions. One example is the Riemann curvature tensor.
Coordinate transformationIn physics, as well as mathematics, a vector is often identified with a tuple, or list of numbers, which depend on some auxiliary coordinate system or reference frame. When the coordinates are transformed, for example by rotation or stretching, then the components of the vector also transform. The vector itself has not changed, but the reference frame has, so the components of the vector (or measurements taken with respect to the reference frame) must 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. In general, contravariant vectors are "regular vectors" with units of distance (such as a displacement) or distance times some other unit (such as velocity or acceleration); covariant vectors, on the other hand, have units of one-over-distance such as gradient. If you change units (a special case of a change of coordinates) from meters to millimeters, a scale factor of 1/1000, a displacement of 1 m becomes 1000 mm–a contravariant change in numerical value. In contrast, a gradient of 1 K/m becomes 0.001 K/mm–a covariant change in value.
The importance of coordinate transformation is because relativity states that there is no one correct reference point in the universe. 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, take the signing of the Declaration of Independence. To an modern observer on Mt Rainier looking East, Revere 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 or down, and in the future. The event itself has not changed, the location of the observer has.
Curvilinear coordinates and curved spacetimeCurvilinear 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, gravity has curvature effects on the four dimensions of the universe. A common analogy is placing a heavy object on a stretched out rubber sheet which causes the rubber to bend downward. This creates a curved coordinate system around the object, much like an object in the universe creates a curved coordinate system. The mathematics here are much more complex than on Earth, as it results in four dimensions of curved coordinates instead of two as there are on Earth.
The interval in a high dimensional spaceImagine our four-dimensional, curved spacetime is embedded in a larger N dimensional flat space. Any true physical vector lies entirely in the curved physical space. In other words, the vector is tangent to the curved physical spacetime. It has no component normal to the four-dimensional, curved spacetime.
In the N dimensional flat space with coordinates the interval between neighboring points is
The relation between neighboring contravariant vectors: Christoffel symbolsThe difference in for two neighboring points in the surface differing by is
Now shift the vector to the point keeping it parallel to itself. In other words, we hold the components of the vector constant during the shift. The vector no longer lies in the surface because of curvature of the surface.
The shifted vector can be split into two parts, one tangent to the surface and one normal to surface, as
Christoffel symbol of the second kindThe Christoffel symbol of the second kind is defined as
This allows us to write
The constancy of the length of the parallel displaced vectorFrom Dirac:
The constancy of the length of the vector follows from geometrical arguments. When we split up the vector into tangential and normal parts ... the normal part is infinitesimal and is orthogonal to the tangential part. It follows that, to the first order, the length of the whole vector equals that of its tangential part.
The covariant derivativeThe partial derivative of a vector with respect to a spacetime coordinate is composed of two parts, the normal partial derivative minus the change in the vector due to parallel transport
The covariant derivative of a product is
GeodesicsSuppose we have a point zμ that moves along a track in physical spacetime. Suppose the track is parameterized with the quantity τ. The "velocity" vector that points in the direction of motion in spacetime is
DefinitionThe curvature K of a surface is simply the angle through which a vector is turned as we take it around an infinitesimal closed path. For a two dimensional Euclidean surface we have
This expression can be reduced to the commutation relation
Symmetries of the curvature tensorThe curvature tensor is antisymmetric in the last two indices
Bianchi identityThe following differential relation, known as the Bianchi identity is true.
Ricci tensor and scalar curvatureThe Ricci tensor is defined as the contraction
- Differentiable manifold
- Christoffel symbol
- Riemannian geometry
- Differential geometry and topology
- List of differential geometry topics
- General Relativity
- Gauge gravitation theory
- ^ Ivanov 2001
- ^ Heinbockel 2001
- ^ Latin: vectus, perfect participle of vehere, "to carry"/ veho = "I carry". For historical development of the word vector, see "vector n.". Oxford English Dictionary. Oxford University Press. 2nd ed. 1989. and Jeff Miller. "Earliest Known Uses of Some of the Words of Mathematics". http://jeff560.tripod.com/v.html. Retrieved 2007-05-25. .
- P. A. M. Dirac (1996). General Theory of Relativity. Princeton University Press. ISBN 0-691-01146-X.
- Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0-7167-0344-0.
- Landau, L. D. and Lifshitz, E. M. (1975). Classical Theory of Fields (Fourth Revised English Edition). Oxford: Pergamon. ISBN 0-08-018176-7.
- R. P. Feynman, F. B. Moringo, and W. G. Wagner (1995). Feynman Lectures on Gravitation. Addison-Wesley. ISBN 0-201-62734-5.
- Einstein, A. (1961). Relativity: The Special and General Theory. New York: Crown. ISBN 0-517-02961-8.