en>DH85868993: +cat

2012-12-23T07:39:27Z

+cat

← Older revision		Revision as of 09:39, 23 December 2012
Line 1:		Line 1:
	The ~~author~~'s ~~name~~ is ~~Andera~~ and ~~she believes it sounds quite~~ [~~http~~:~~//www~~.~~prograd~~.~~uff~~.br/~~novo/facts~~-~~about~~-~~growing~~-~~greater-organic-garden good psychic~~]. ~~My wife~~ and ~~I reside~~ in ~~Kentucky~~. The ~~preferred hobby for him~~ and ~~his kids~~ is to ~~perform lacross~~ and he would ~~never give it up~~. ~~My working day job is an invoicing officer but I~~'~~ve currently applied [http~~://~~www.edmposts~~.~~com~~/~~build~~-a-~~beautiful~~-~~organic~~-~~garden~~-~~using~~-~~these~~-~~ideas~~/ ~~accurate psychic predictions] for an additional one~~.<br><br>~~Feel free to surf to my website psychic love readings~~ - [http://~~gcjcteam~~.org/~~index.php?mid~~=~~etc_video&document_srl~~=~~696611~~&~~sort_index=regdate~~&~~order_type=desc gcjcteam.org~~] -		The '''initial value formulation of general relativity''' is a reformulation of [[Albert Einstein]]'s theory of [[general relativity]] that describes a [[universe]] evolving over [[time]].

			Each solution of the [[Einstein field equations]] encompasses the whole history of a universe – it is not just some snapshot of how things are, but a whole [[spacetime]]: a statement encompassing the state of matter and geometry everywhere and at every moment in that particular universe. By this token, Einstein's theory appears to be different from most other physical theories, which specify [[evolution equation]]s for physical systems; if the system is in a given state at some given moment, the laws of physics allow you to extrapolate its past or future. For Einstein's equations, there appear to be subtle differences compared with other fields: they are self-interacting (that is, [[non-linear]] even in the absence of other fields); they are [[diffeomorphism invariance\|diffeomorphism invariant]], so to obtain a unique solution, a fixed background metric and gauge conditions need to be introduced; finally, the metric determines the spacetime structure, and thus the domain of dependence for any set of initial data, so the region on which a specific solution will be defined is not, a priori, defined.<ref>Cf. {{Harvnb\|Hawking\|Ellis\|1973\|loc=sec. 7.1}}.</ref>

			There is, however, a way to re-formulate Einstein's equations that overcomes these problems. First of all, there are ways of rewriting spacetime as the evolution of "space" in time; an earlier version of this is due to [[Paul Dirac]], while a simpler way is known after its inventors [[Richard Arnowitt]], [[Stanley Deser]] and [[Charles Misner]] as [[ADM formalism]]. In these formulations, also known as "3+1" approaches, spacetime is split into a three-dimensional hypersurface with [[interior metric]] and an embedding into spacetime with [[exterior curvature]]; these two quantities are the dynamical variables in a [[Hamiltonian mechanics\|Hamiltonian formulation]] tracing the hypersurface's evolution over time.<ref>{{Harvnb\|Arnowitt\|Deser\|Misner\|1962}}; for a pedagogical introduction, see {{Harvnb\|Misner\|Thorne\|Wheeler\|1973\|loc=§21.4–§21.7}}.</ref> With such a split, it is possible to state the ''initial value formulation of general relativity''. It involves initial data which cannot be specified arbitrarily but needs to satisfy specific [[Constraint (mathematics)\|constraint]] equations, and which is defined on some suitably smooth three-manifold <math>\Sigma</math>; just as for other differential equations, it is then possible to prove [[existence theorem\|existence]] and [[uniqueness theorem\|uniqueness]] theorems, namely that there exists a unique spacetime which is a solution of Einstein equations, which is [[globally hyperbolic manifold\|globally hyperbolic]], for which <math>\Sigma</math> is a [[Cauchy surface]] (i.e. all past events influence what happens on <math>\Sigma</math>, and all future events are influenced by what happens on it), and has the specified internal metric and extrinsic curvature; all spacetimes that satisfy these conditions are related by [[isometry\|isometries]].<ref>{{Harvnb\|Fourès-Bruhat\|1952}} and {{Harvnb\|Bruhat\|1962}}; for a pedagogical introduction, see {{Harvnb\|Wald\|1984\|loc=ch. 10}}; an online review can be found in {{Harvnb\|Reula\|1998}}.</ref>

			The initial value formulation with its 3+1 split is the basis of [[numerical relativity]]; attempts to simulate the evolution of relativistic spacetimes (notably merging [[black hole]]s or [[gravitational collapse]]) using computers.<ref>See {{Harvnb\|Gourgoulhon\|2007}}.</ref> However, there are significant differences to the simulation of other physical evolution equations which make numerical relativity especially challenging, notably the fact that the dynamical objects that are evolving include space and time itself (so there is no fixed background against which to evaluate, for instance, perturbations representing gravitational waves) and the occurrence of singularities (which, when they are allowed to occur within the simulated portion of spacetime, lead to arbitrarily large numbers that would have to be represented in the computer model).<ref>For a review of the basics of numerical relativity, including the problems alluded to here and further difficulties, see {{Harvnb\|Lehner\|2001}}.</ref>

			==See also==

			*[[ADM formalism]]
			*[[Block universe]]

			==Notes==
			{{reflist\|2}}

			==References==

			*<cite
			id="CITEREFArnowittDeserMisner1962">Arnowitt, Richard<nowiki>; </nowiki>Stanley Deser & Charles W. Misner (1962), "The dynamics of general relativity", in Witten, L., ''Gravitation: An Introduction to Current Research'', Wiley, pp. 227–265</cite>
			*<cite
			id="CITEREFBruhat1962">Bruhat, Yvonne (1962), "The Cauchy Problem", in Witten, Louis, ''Gravitation: An Introduction to Current Research'', Wiley, pp. 130</cite>
			*{{Citation
			\| first=Yvonne
			\| last= Fourès-Bruhat
			\| title= Théoréme d'existence pour certains systémes d'équations aux derivées partielles non linéaires
			\| journal= Acta Mathematica
			\| volume=88
			\| year=1952
			\| issue=1
			\| pages=141–225
			\| doi=10.1007/BF02392131
			}}
			*{{Citation
			\| first=Eric
			\| last=Gourgoulhon
			\| title=3+1 Formalism and Bases of Numerical Relativity
			\| arxiv=gr-qc/0703035
			\| year=2007
			\|bibcode = 2007gr.qc.....3035G }}
			*{{Citation
			\| last1=Hawking
			\| first1=Stephen W.
			\| last2=Ellis
			\| first2=George F. R.
			\| title=The large scale structure of space-time
			\| publisher=Cambridge University Press
			\| isbn=0-521-09906-4
			\| year=1973
			}}
			*{{Citation
			\| last=Lehner
			\| first=Luis
			\| title=Numerical Relativity: A review
			\| journal =Class. Quant. Grav.
			\| volume=18
			\| issue=17
			\| year=2001
			\| pages=R25–R86
			\| arxiv=gr-qc/0106072
			\| doi=10.1088/0264-9381/18/17/202
			\|bibcode = 2001CQGra..18R..25L }}
			* <cite
			id="CITEREFMisnerThorneWheeler1973">Misner, Charles W.<nowiki>; </nowiki>Kip. S. Thorne & John A. Wheeler (1973), ''Gravitation'', W. H. Freeman, ISBN 0-7167-0344-0</cite>
			*{{Citation
			\|first= Oscar A.
			\| last=Reula
			\| title=Hyperbolic Methods for Einstein's Equations
			\| journal=Living Rev. Relativity
			\| volume=1
			\| year=1998
			\| url=http://www.livingreviews.org/lrr-1998-3
			\| accessdate=2007-08-29
			}}
			* <cite
			id="CITEREFWald1984">[[Robert M. Wald\|Wald, Robert M.]] (1984), ''[[General Relativity (book)\|''General Relativity'']]'', Chicago<nowiki>:</nowiki> University of Chicago Press, ISBN 0-226-87033-2</cite>

			[[Category:General relativity]]

en>Nickersonl: ←Created page with '{| class=wikitable width=90% align=center |- ! Determining Acute LD₅₀ |- | For purposes of classifying and assigning packing groups to mixtures possessin...'

2007-09-14T22:05:56Z

←Created page with '{| class=wikitable width=90% align=center |- ! Determining Acute LD<sub>50</sub> |- | For purposes of classifying and assigning packing groups to mixtures possessin...'

New page

The author's name is Andera and she believes it sounds quite [http://www.prograd.uff.br/novo/facts-about-growing-greater-organic-garden good psychic]. My wife and I reside in Kentucky. The preferred hobby for him and his kids is to perform lacross and he would never give it up. My working day job is an invoicing officer but I've currently applied [http://www.edmposts.com/build-a-beautiful-organic-garden-using-these-ideas/ accurate psychic predictions] for an additional one.<br><br>Feel free to surf to my website psychic love readings - [http://gcjcteam.org/index.php?mid=etc_video&document_srl=696611&sort_index=regdate&order_type=desc gcjcteam.org] -

Template:HAZMAT Class 6 Acute LD50 - Revision history

en>DH85868993: +cat

en>Nickersonl: ←Created page with '{| class=wikitable width=90% align=center |- ! Determining Acute LD50 |- | For purposes of classifying and assigning packing groups to mixtures possessin...'

en>Nickersonl: ←Created page with '{| class=wikitable width=90% align=center |- ! Determining Acute LD₅₀ |- | For purposes of classifying and assigning packing groups to mixtures possessin...'