Introduction to the mathematics of general relativity: Difference between revisions

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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'''<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 | accessdateaccess-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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{{main|Einstein field equations}}

The '''Einstein field equations''' ('''EFE''') or '''Einstein's equations''' are a set of 10 [[equation]]s in [[Albert Einstein|Albert Einstein's]] [[general relativity|general theory of relativity]] which describe the [[fundamental interaction]] of [[gravitation]] as a result of [[spacetime]] being [[curvature|curved]] by [[matter]] and [[energy]].<ref name=ein>{{cite journal|last=Einstein |first=Albert |title=The Foundation of the General Theory of Relativity |journal=[[Annalen der Physik]] |volume=354 |issue=7 |pages=769 |year=1916 |url=http://www.alberteinstein.info/gallery/gtext3.html |doi=10.1002/andp.19163540702 |format=[[PDF]] |bibcode=1916AnP...354..769E |url-status=dead |archiveurlarchive-url=https://web.archive.org/web/20060829045130/http://www.alberteinstein.info/gallery/gtext3.html |archivedatearchive-date=2006-08-29 }}</ref> First published by Einstein in 1915<ref name=Ein1915>{{cite journal|last=Einstein| first=Albert| authorlinkauthor-link = Albert Einstein| date=November 25, 1915| title=Die Feldgleichungen der Gravitation| journal=Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin| pages=844–847 | url=http://nausikaa2.mpiwg-berlin.mpg.de/cgi-bin/toc/toc.x.cgi?dir=6E3MAXK4&step=thumb | accessdateaccess-date=2006-09-12}}</ref> as a [[tensor equation]], the EFE equate local spacetime [[curvature]] (expressed by the [[Einstein tensor]]) with the local energy and [[momentum]] within that spacetime (expressed by the [[stress–energy tensor]]).<ref>{{Cite book

| last1=Misner |first1=Charles W. |authorlink1author-link1=Charles W. Misner

| last2=Thorne |first2=Kip S. |authorlink2author-link2=Kip Thorne

| last3=Wheeler |first3=John Archibald |authorlink3author-link3=John Archibald Wheeler

| year=1973

| title=Gravitation

| url=

| publisher=[[W. H. Freeman]] |___location=San Francisco

| isbn=978-0-7167-0344-0