Reflection group - Revision history

en>Rjwilmsi: /* References */Added 1 dois to journal cites using AWB (10081)

2014-04-29T10:54:51Z

References: Added 1 dois to journal cites using AWB (10081)

← Older revision		Revision as of 12:54, 29 April 2014
Line 1:		Line 1:
	~~In [[general relativity]], a '''scalar field solution'''~~ is ~~an [[Exact solutions~~ in ~~general relativity\|exact solution]] of the [[Einstein field equation]] in which the gravitational field is due entirely to the field energy and momentum of~~ a ~~[[scalar field]]. Such a field may or may not be ''massless'', and~~ it ~~may be taken to have ''minimal curvature coupling'', or some other choice, such as ''conformal coupling''~~.		They call me Emilia. What I adore performing is taking part in baseball but I haven't produced a dime with it. South Dakota is her birth location but she requirements to move because of her family. Supervising is my profession.<br><br>Here is my blog :: [http://youulike.com/profile/116412 youulike.com]

	~~==Mathematical definition==~~

	~~In general relativity, the geometric setting for physical phenomena~~ is a [[Lorentzian manifold]], which is physically interpreted as a curved spacetime, and which is mathematically specified by defining a [[metric tensor]] <math>g_{ab}</math> (or by defining a [[frame fields in general relativity\|frame field]]). The [[Riemann tensor\|curvature tensor]] <math>R_{abcd}</math>
	~~of this manifold and associated quantities such as the [[Einstein tensor]] <math>G^{ab}</math>, are well-defined even in the absence of any physical theory,~~ but ~~in general relativity they acquire a physical interpretation as geometric manifestations~~ of ~~the [[gravitational field]]~~.

	~~In addition, we must specify a scalar field by giving a function <math>\psi</math>. This function~~ is ~~required to satisfy two following conditions:~~
	~~# The function must satisfy the (curved spacetime) ''source-free'' [[wave equation]] <math>g^{ab} \psi_{;ab} = 0</math>,~~
	~~# The Einstein tensor must match the [[energy-momentum density\|stress-energy tensor]] for the scalar field, which in the simplest case, a ''minimally coupled massless scalar field'', can be written~~
	~~<math>G^{ab}= 8 \pi \left( \psi^{;a} \psi^{;b} - \frac{1}{2}~~
	~~\psi_{;m} \psi^{;m} g^{ab} \right) </math>~~.

	Both conditions follow from varying the [[Lagrangian#Lagrangians and Lagrangian densities in field theory\|Lagrangian density]] for the scalar field, which in the case of a minimally coupled massless scalar field is
	:<~~math~~> ~~L = -g^{mn} \, \psi_{;m} \, \psi_{;n}~~ <~~/math~~>
	Here,
	~~:<math>\frac{\delta L}{\delta \psi} = 0</math>~~
	~~gives the wave equation, while~~
	~~:<math>\frac{\delta L}{\delta g^{ab}} = 0</math>~~
	~~gives the Einstein equation (in the case where the field energy of the scalar field~~ is ~~the only source of the gravitational field).~~

	~~==Physical interpretation==~~

	Scalar fields are often interpreted as classical approximations, in the sense of [[effective field theory]], to some quantum field. In general relativity, the speculative [[quintessence (physics)\|quintessence]] field can appear as a scalar field. For example, a flux of neutral [[pion]]s can in principle be modeled as a minimally coupled massless scalar field.

	~~==Einstein tensor==~~

	The components of a tensor computed with respect to a [[frame fields in general relativity\|frame field]] rather than the coordinate basis are often called ''physical components'', because these are the components which can (in principle) be measured by an observer.

	~~In the special case of a ''minimally coupled massless scalar field'', an ''adapted frame''~~
	:~~<math>\vec{e}_0, \; \vec{e}_1, \; \vec{e}_2, \; \vec{e}_3</math>~~
	~~(the first is a [[timelike]] unit [[vector field]], the last three are [[spacelike]] unit vector fields)~~
	~~can always be found in which the Einstein tensor takes the simple form~~
	:
	~~<math>G^{\hat{a}\hat{b}} = 8 \pi \sigma \, \left[ \begin{matrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{matrix} \right] </math>~~
	~~where <math>\sigma</math> is the ''energy density'' of the scalar field.~~

	~~==Eigenvalues==~~

	~~The~~ [~~[characteristic polynomial]] of the Einstein tensor in a minimally coupled massless scalar field solution must have the form~~
	:~~<math> \chi(\lambda) = (\lambda + 8 \pi \sigma)^3 \, ( \lambda - 8 \pi \sigma )<~~/~~math>~~
	In other words, we have a simple eigvalue and a triple eigenvalue, each being the negative of the other. Multiply out and using [[Gröbner basis]] methods, we find that the following three invariants must vanish identically:
	~~:<math> a_2 = 0, \; \; a_1^3 + 4 a_3 = 0, \; \; a_1^4 + 16 a_4 = 0 <~~/~~math>~~
	~~Using [[Newton's identities]], we can rewrite these in terms of the traces of the powers~~. ~~We find that~~
	~~:<math> t_2 = t_1^2, \; t_3 = t_1^3~~/~~4, \; t_4 = t_1^4~~/~~4 </math>~~
	~~We can rewrite this in terms of index gymanastics as the manifestly invariant criteria:~~
	~~:<math> {G^a}_a = -R</math>~~
	~~:<math> {G^a}_b \, {G^b}_a = R^2 </math>~~
	~~:<math> {G^a}_b \, {G^b}_c \, {G^c}_a = R^3/4 </math>~~
	~~:<math> {G^a}_b \, {G^b}_c \, {G^c}_d \, {G^d}_a = R^4/4 </math>~~

	~~==Examples==~~

	~~Notable individual scalar field solutions include~~

	~~:* the [[Janis–Newman–Winicour scalar field solution]], which is the unique ''static'' and ''spherically symmetric'' massless minimally coupled scalar field solution.~~

	~~==See also==~~

	*[[Exact solutions in general relativity]]
	*[[Lorentz group]]

	~~==References==~~

	*{{cite book \| author=Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; & Herlt, E. \| title=Exact Solutions of Einstein's Field Equations (2nd edn.) \| location=Cambridge \| publisher=Cambridge University Press \| year=2003 \| isbn=0-521-46136-7}}

	*{{cite book \| author=Hawking, S. W.; and Ellis, G. F. R. \| title = The Large Scale Structure of Space-time \| location= Cambridge \| publisher=Cambridge University Press \| year = 1973 \| isbn=0-521-09906-4}} See ''section 3.3'' for the stress-energy tensor of a minimally coupled scalar field.

	~~[[Category:Exact solutions in general relativity]~~]

en>Monkbot: /* References */Fix CS1 deprecated date parameter errors

2014-01-29T04:58:56Z

References: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 06:58, 29 January 2014
Line 1:		Line 1:
	~~The author~~ is ~~called Irwin Wunder but~~ it'~~s not~~ the ~~most masucline title out there~~. ~~Doing ceramics is what my family~~ and ~~I enjoy~~. ~~Hiring~~ is ~~her day occupation now but she~~'s ~~usually needed her own company~~. ~~For many years he~~'~~s been living~~ in ~~North Dakota and his family members loves it~~.<br><br>~~Here~~ is ~~my blog~~: [~~http~~://~~www~~.~~hotporn123.com~~/~~blog~~/~~154316 home std test kit~~]		In [[general relativity]], a '''scalar field solution''' is an [[Exact solutions in general relativity\|exact solution]] of the [[Einstein field equation]] in which the gravitational field is due entirely to the field energy and momentum of a [[scalar field]]. Such a field may or may not be ''massless'', and it may be taken to have ''minimal curvature coupling'', or some other choice, such as ''conformal coupling''.

			==Mathematical definition==

			In general relativity, the geometric setting for physical phenomena is a [[Lorentzian manifold]], which is physically interpreted as a curved spacetime, and which is mathematically specified by defining a [[metric tensor]] <math>g_{ab}</math> (or by defining a [[frame fields in general relativity\|frame field]]). The [[Riemann tensor\|curvature tensor]] <math>R_{abcd}</math>
			of this manifold and associated quantities such as the [[Einstein tensor]] <math>G^{ab}</math>, are well-defined even in the absence of any physical theory, but in general relativity they acquire a physical interpretation as geometric manifestations of the [[gravitational field]].

			In addition, we must specify a scalar field by giving a function <math>\psi</math>. This function is required to satisfy two following conditions:
			# The function must satisfy the (curved spacetime) ''source-free'' [[wave equation]] <math>g^{ab} \psi_{;ab} = 0</math>,
			# The Einstein tensor must match the [[energy-momentum density\|stress-energy tensor]] for the scalar field, which in the simplest case, a ''minimally coupled massless scalar field'', can be written
			<math>G^{ab}= 8 \pi \left( \psi^{;a} \psi^{;b} - \frac{1}{2}
			\psi_{;m} \psi^{;m} g^{ab} \right) </math>.

			Both conditions follow from varying the [[Lagrangian#Lagrangians and Lagrangian densities in field theory\|Lagrangian density]] for the scalar field, which in the case of a minimally coupled massless scalar field is
			:<math> L = -g^{mn} \, \psi_{;m} \, \psi_{;n} </math>
			Here,
			:<math>\frac{\delta L}{\delta \psi} = 0</math>
			gives the wave equation, while
			:<math>\frac{\delta L}{\delta g^{ab}} = 0</math>
			gives the Einstein equation (in the case where the field energy of the scalar field is the only source of the gravitational field).

			==Physical interpretation==

			Scalar fields are often interpreted as classical approximations, in the sense of [[effective field theory]], to some quantum field. In general relativity, the speculative [[quintessence (physics)\|quintessence]] field can appear as a scalar field. For example, a flux of neutral [[pion]]s can in principle be modeled as a minimally coupled massless scalar field.

			==Einstein tensor==

			The components of a tensor computed with respect to a [[frame fields in general relativity\|frame field]] rather than the coordinate basis are often called ''physical components'', because these are the components which can (in principle) be measured by an observer.

			In the special case of a ''minimally coupled massless scalar field'', an ''adapted frame''
			:<math>\vec{e}_0, \; \vec{e}_1, \; \vec{e}_2, \; \vec{e}_3</math>
			(the first is a [[timelike]] unit [[vector field]], the last three are [[spacelike]] unit vector fields)
			can always be found in which the Einstein tensor takes the simple form
			:
			<math>G^{\hat{a}\hat{b}} = 8 \pi \sigma \, \left[ \begin{matrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{matrix} \right] </math>
			where <math>\sigma</math> is the ''energy density'' of the scalar field.

			==Eigenvalues==

			The [[characteristic polynomial]] of the Einstein tensor in a minimally coupled massless scalar field solution must have the form
			:<math> \chi(\lambda) = (\lambda + 8 \pi \sigma)^3 \, ( \lambda - 8 \pi \sigma )</math>
			In other words, we have a simple eigvalue and a triple eigenvalue, each being the negative of the other. Multiply out and using [[Gröbner basis]] methods, we find that the following three invariants must vanish identically:
			:<math> a_2 = 0, \; \; a_1^3 + 4 a_3 = 0, \; \; a_1^4 + 16 a_4 = 0 </math>
			Using [[Newton's identities]], we can rewrite these in terms of the traces of the powers. We find that
			:<math> t_2 = t_1^2, \; t_3 = t_1^3/4, \; t_4 = t_1^4/4 </math>
			We can rewrite this in terms of index gymanastics as the manifestly invariant criteria:
			:<math> {G^a}_a = -R</math>
			:<math> {G^a}_b \, {G^b}_a = R^2 </math>
			:<math> {G^a}_b \, {G^b}_c \, {G^c}_a = R^3/4 </math>
			:<math> {G^a}_b \, {G^b}_c \, {G^c}_d \, {G^d}_a = R^4/4 </math>

			==Examples==

			Notable individual scalar field solutions include

			:* the [[Janis–Newman–Winicour scalar field solution]], which is the unique ''static'' and ''spherically symmetric'' massless minimally coupled scalar field solution.

			==See also==

			*[[Exact solutions in general relativity]]
			*[[Lorentz group]]

			==References==

			*{{cite book \| author=Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; & Herlt, E. \| title=Exact Solutions of Einstein's Field Equations (2nd edn.) \| location=Cambridge \| publisher=Cambridge University Press \| year=2003 \| isbn=0-521-46136-7}}

			*{{cite book \| author=Hawking, S. W.; and Ellis, G. F. R. \| title = The Large Scale Structure of Space-time \| location= Cambridge \| publisher=Cambridge University Press \| year = 1973 \| isbn=0-521-09906-4}} See ''section 3.3'' for the stress-energy tensor of a minimally coupled scalar field.

			[[Category:Exact solutions in general relativity]]

92.239.174.70: Corrected external link for new structure of the EoM

2012-02-23T13:18:26Z

Corrected external link for new structure of the EoM

New page

The author is called Irwin Wunder but it's not the most masucline title out there. Doing ceramics is what my family and I enjoy. Hiring is her day occupation now but she's usually needed her own company. For many years he's been living in North Dakota and his family members loves it.<br><br>Here is my blog: [http://www.hotporn123.com/blog/154316 home std test kit]