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| 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''.
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| ==Mathematical definition==
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| 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>
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| 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]].
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| In addition, we must specify a scalar field by giving a function <math>\psi</math>. This function is required to satisfy two following conditions:
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| # The function must satisfy the (curved spacetime) ''source-free'' [[wave equation]] <math>g^{ab} \psi_{;ab} = 0</math>,
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| # 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
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| <math>G^{ab}= 8 \pi \left( \psi^{;a} \psi^{;b} - \frac{1}{2}
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| \psi_{;m} \psi^{;m} g^{ab} \right) </math>.
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| 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
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| :<math> L = -g^{mn} \, \psi_{;m} \, \psi_{;n} </math>
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| Here, | |
| :<math>\frac{\delta L}{\delta \psi} = 0</math>
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| gives the wave equation, while
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| :<math>\frac{\delta L}{\delta g^{ab}} = 0</math>
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| gives the Einstein equation (in the case where the field energy of the scalar field is the only source of the gravitational field).
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| ==Physical interpretation==
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| 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.
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| ==Einstein tensor==
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| 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.
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| In the special case of a ''minimally coupled massless scalar field'', an ''adapted frame''
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| :<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)
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| can always be found in which the Einstein tensor takes the simple form
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| <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>
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| where <math>\sigma</math> is the ''energy density'' of the scalar field.
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| ==Eigenvalues==
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| The [[characteristic polynomial]] of the Einstein tensor in a minimally coupled massless scalar field solution must have the form
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| :<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:
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| :<math> a_2 = 0, \; \; a_1^3 + 4 a_3 = 0, \; \; a_1^4 + 16 a_4 = 0 </math>
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| Using [[Newton's identities]], we can rewrite these in terms of the traces of the powers. We find that
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| :<math> t_2 = t_1^2, \; t_3 = t_1^3/4, \; t_4 = t_1^4/4 </math>
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| We can rewrite this in terms of index gymanastics as the manifestly invariant criteria:
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| :<math> {G^a}_a = -R</math>
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| :<math> {G^a}_b \, {G^b}_a = R^2 </math>
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| :<math> {G^a}_b \, {G^b}_c \, {G^c}_a = R^3/4 </math>
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| :<math> {G^a}_b \, {G^b}_c \, {G^c}_d \, {G^d}_a = R^4/4 </math>
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| ==Examples==
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| Notable individual scalar field solutions include
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| :* the [[Janis–Newman–Winicour scalar field solution]], which is the unique ''static'' and ''spherically symmetric'' massless minimally coupled scalar field solution.
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| ==See also==
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| *[[Exact solutions in general relativity]]
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| *[[Lorentz group]]
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| ==References==
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| *{{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}}
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| *{{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.
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| [[Category:Exact solutions in general relativity]]
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