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}}</ref> or '''spatial vector''',<ref>{{harvnb|Heinbockel|2001}}</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]]. If the coordinates are transformed, such as by rotation or stretching the coordinate system, the components of the vector also transform. The vector itself doesn'tdoes not change, but the reference frame does. This means that the components of the vector have to 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.

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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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{{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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{{Tensors}}

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[[Category:General relativity| ]]