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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:<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.