Introduction to the mathematics of general relativity: Difference between revisions

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{{Short description|none}}

{{Introductory article|Mathematics of general relativity}}

{{General relativity sidebar}}

The '''mathematics of general relativity''' is complexcomplicated. In [[Isaac Newton|Newton]]'s theories of motion, an object's length and the rate at which time passes remain constant while the object [[Acceleration|accelerates]], meaning that many problems in [[Classical mechanics|Newtonian mechanics]] may be solved by [[algebra]] alone. In [[Theory of relativity|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 [[Vector space|vectors]], [[tensor]]s, [[pseudotensor]]s and [[curvilinear coordinates]].

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

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[[Image:Vector by Zureks.svg|right|thumb|Illustration of a typical vector]]

In [[mathematics]], [[physics]], and [[engineering]], a '''[[Euclidean vector']]'' (sometimes called a '''geometric' vector''<ref>{{harvnb|Ivanov|2001}}{{Citation not found}}</ref> or '''spatial vector''',<ref>{{harvnb|Heinbockel|2001}}{{Citation not found}}</ref> or – as here – simply a vector) is a geometric object that has both a [[Magnitude (mathematics)|magnitude]] (or [[Norm (mathematics)#Euclidean norm|length]]) and direction. A vector is what is needed to "carry" the point {{math|''A''}} to the point {{math|''B''}}; the Latin word ''vector'' means "one who carries".<ref>From Latin ''vectus'', [[perfect participle]] of ''vehere'', "to carry". For historical development of the word ''vector'', see {{OED|vector ''n.''}} and {{cite web|author = Jeff Miller| url = http://jeff560.tripod.com/v.html | title = Earliest Known Uses of Some of the Words of Mathematics | access-date = 2007-05-25}}</ref> The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from {{math|''A''}} to {{math|''B''}}. Many [[algebraic operation]]s on [[real number]]s such as [[addition]], [[subtraction]], [[multiplication]], and [[negation]] have close analogues for vectors, operations which obey the familiar algebraic laws of [[Commutative property|commutativity]], [[Associative property|associativity]], and [[Distributive property|distributivity]].

===Tensors===

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In general relativity, four-dimensional vectors, or [[four-vector]]s, 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]].

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=== Coordinate transformation ===

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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 [[frame of reference|reference frame]]. WhenIf the coordinates are transformed, forsuch exampleas by rotation or stretching of the coordinate system, then the components of the vector also transform. The vector itself hasdoes not changedchange, but the reference frame has,does. soThis means that the components of the vector (or measurements taken with respecthave to the reference frame) must change to compensate.

The vector is called [[Covariance and contravariance of vectors|''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&nbsp;m becomes 1000&nbsp;mm.

* Covariant vectors, on the other hand, have units of one-over-distance (such 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&nbsp;[[Kelvin|K]]/ per m becomes 0.001&nbsp;K/Kelvin per mm.

In [[Einstein notation]], contravariant vectors and components of tensors are shown with superscripts, e.g. {{math|''xⁱ''}}, and covariant vectors and components of tensors with subscripts, e.g. {{math|''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 [[United States Declaration of Independence|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 [[Orthogonality|orthogonal]] to each other; that is, they meet at angles other than [[right angle]]s. When using coordinate transformations as described above, the new coordinate system will often appear to have oblique axes compared to the old system.

==Nontensors==

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:<math>s^2 = \Delta r^2 - c^2\Delta t^2 \,</math>{{spaces|5}}(spacetime&nbsp;interval),

where {{math|''c''}} is the speed of light, and {{math|Δ''r''}} and {{math|Δ''t''}} denote differences of the space and time coordinates, respectively, between the events. The choice of signs for {{math|''s''²}} above follows the [[sign convention#Relativity|space-like convention (−+++)]]. A notation like {{math|Δ''r''²}} means {{math|(Δ''r'')²}}. The reason {{math|''s''²}} is called the interval and not {{math|''s''}} is called the interval is that {{math|''s''²}} can be positive, zero or negative.

Spacetime intervals may be classified into three distinct types, based on whether the temporal separation ({{math|''c''²Δ''t''²}}) or the spatial separation ({{math|Δ''r''²}}) of the two events is greater: time-like, light-like or space-like.

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===Christoffel symbols===

{{main|Christoffel symbolsymbols}}

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.

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{{main|Riemann curvature tensor}}

The [[Riemann curvature tensor]] {{math|''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:

# The [[Riemann curvature tensor]]: {{math|''R^ρ_σμν''}}, which gives the most information on the curvature of a space and is derived from derivatives of the [[metric tensor]]. In flat space this tensor is zero.

# The [[Ricci tensor]]: {{math|''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.

# The [[scalar curvature]]: {{math|''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.

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==References==

* {{cite book | author=P. A. M. Dirac | title=General Theory of Relativity | publisher=[[Princeton University Press]]| year=1996 | isbn=0-691-01146-X}}

*{{Citation

| title = Introduction to Tensor Calculus and Continuum Mechanics

| first = J. H.

| last = Heinbockel

| publisher = Trafford Publishing

| year = 2001

| isbn = 1-55369-133-4

| url = http://www.math.odu.edu/~jhh/counter2.html

}}.

{{sfn whitelist |CITEREFIvanov2001}}

* {{springer|id=V/v096340|title=Vector|first=A.B.|last=Ivanov}}.

* {{cite book |author1=Misner, Charles |author2=Thorne, Kip S. |author3=Wheeler, John Archibald | title=Gravitation | ___location=San Francisco | publisher=W. H. Freeman | year=1973 | isbn=0-7167-0344-0}}

* {{cite book |author1=Landau, L. D. |author2=Lifshitz, E. M.| title=Classical Theory of Fields | ___location=Oxford | publisher=Pergamon | year=1975 | isbn=0-08-018176-7|edition=Fourth Revised English}}

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{{Tensors}}

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[[Category:General relativity| ]]