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Scalar field solution
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<h2 id="definition">Definition</h2> <p>In general relativity, the geometric setting for physical phenomena is a <a href="/facts/Lorentzian_manifold/VFXsVjqb">Lorentzian manifold</a>, which is physically interpreted as a curved spacetime, and which is mathematically specified by defining a <a href="/facts/Metric_tensor/tbqek1gJ">metric tensor</a> g a b {\displaystyle g_{ab}} (or by defining a <a href="/facts/Frame_fields_in_general_relativity/Lw90nUa6">frame field</a>). The <a href="/facts/Riemann_tensor/KyeHCWEh">curvature tensor</a> R a b c d {\displaystyle R^{a}{}_{bcd}} of this manifold and associated quantities such as the <a href="/facts/Einstein_tensor/ejyGdaFz">Einstein tensor</a> G a b {\displaystyle G_{ab}} , 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 <a href="/facts/Gravitational_field/K1uTmJFy">gravitational field</a>. </p><p>In addition, we must specify a scalar field by giving a function ψ {\displaystyle \psi } . This function is required to satisfy two following conditions: </p> <ol><li>The function must satisfy the (curved spacetime) <i>source-free</i> <a href="/facts/Wave_equation/J2ksD2Az">wave equation</a> g a b ψ ; a b = 0 {\displaystyle g^{ab}\psi _{;ab}=0} ,</li> <li>The Einstein tensor must match the <a href="/facts/Energy-momentum_density/wWKrzjfI">stress-energy tensor</a> for the scalar field, which in the simplest case, a <i>minimally coupled massless scalar field</i>, can be written</li></ol> <p> G a b = κ ( ψ ; a ψ ; b − 1 2 ψ ; m ψ ; m g a b ) {\displaystyle G_{ab}=\kappa \left(\psi _{;a}\psi _{;b}-{\frac {1}{2}}\psi _{;m}\psi ^{;m}g_{ab}\right)} . </p><p>Both conditions follow from varying the <a href="/facts/Lagrangian_density/p9BCUz57">Lagrangian density</a> for the scalar field, which in the case of a minimally coupled massless scalar field is </p> L = − g m n ψ ; m ψ ; n {\displaystyle L=-g^{mn}\,\psi _{;m}\,\psi _{;n}} <p>Here, </p> δ L δ ψ = 0 {\displaystyle {\frac {\delta L}{\delta \psi }}=0} <p>gives the wave equation, while </p> δ L δ g a b = 0 {\displaystyle {\frac {\delta L}{\delta g^{ab}}}=0} <p>gives the Einstein equation (in the case where the field energy of the scalar field is the only source of the gravitational field). </p> <h2 id="physical-interpretation">Physical interpretation</h2> <p>Scalar fields are often interpreted as classical approximations, in the sense of <a href="/facts/Effective_field_theory/oQ2nWoko">effective field theory</a>, to some quantum field. In general relativity, the speculative <a href="/facts/Quintessence_(physics)/8qk8IMut">quintessence</a> field can appear as a scalar field. For example, a flux of neutral <a href="/facts/Pion/x5HamzG3">pions</a> can in principle be modeled as a minimally coupled massless scalar field. </p> <h2 id="einstein-tensor">Einstein tensor</h2> <p>The components of a tensor computed with respect to a <a href="/facts/Frame_fields_in_general_relativity/Lw90nUa6">frame field</a> rather than the coordinate basis are often called <i>physical components</i>, because these are the components which can (in principle) be measured by an observer. </p><p>In the special case of a <i>minimally coupled massless scalar field</i>, an <i>adapted frame</i> </p> e → 0 , e → 1 , e → 2 , e → 3 {\displaystyle {\vec {e}}_{0},\;{\vec {e}}_{1},\;{\vec {e}}_{2},\;{\vec {e}}_{3}} <p>(the first is a <a href="/facts/Timelike/uRIN1x4b">timelike</a> unit <a href="/facts/Vector_field/ccFNuPPp">vector field</a>, the last three are <a href="/facts/Spacelike/uRIN1x4b">spacelike</a> unit vector fields) can always be found in which the Einstein tensor takes the simple form </p> <p> G a ^ b ^ = 8 π σ [ − 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] {\displaystyle 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]} where σ {\displaystyle \sigma } is the <i>energy density</i> of the scalar field. </p> <h2 id="eigenvalues">Eigenvalues</h2> <p>The <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a> of the Einstein tensor in a minimally coupled massless scalar field solution must have the form </p> χ ( λ ) = ( λ + 8 π σ ) 3 ( λ − 8 π σ ) {\displaystyle \chi (\lambda )=(\lambda +8\pi \sigma )^{3}\,(\lambda -8\pi \sigma )} <p>In other words, we have a simple eigenvalue and a triple eigenvalue, each being the negative of the other. Multiply out and using <a href="/facts/Gr%C3%B6bner_basis/efMrjM7w">Gröbner basis</a> methods, we find that the following three invariants must vanish identically: </p> a 2 = 0 , a 1 3 + 4 a 3 = 0 , a 1 4 + 16 a 4 = 0 {\displaystyle a_{2}=0,\;\;a_{1}^{3}+4a_{3}=0,\;\;a_{1}^{4}+16a_{4}=0} <p>Using <a href="/facts/Newton%27s_identities/5J9flXxA">Newton's identities</a>, we can rewrite these in terms of the traces of the powers. We find that </p> t 2 = t 1 2 , t 3 = t 1 3 / 4 , t 4 = t 1 4 / 4 {\displaystyle t_{2}=t_{1}^{2},\;t_{3}=t_{1}^{3}/4,\;t_{4}=t_{1}^{4}/4} <p>We can rewrite this in terms of index gymnastics as the manifestly invariant criteria: </p> G a a = − R {\displaystyle {G^{a}}_{a}=-R} G a b G b a = R 2 {\displaystyle {G^{a}}_{b}\,{G^{b}}_{a}=R^{2}} G a b G b c G c a = R 3 / 4 {\displaystyle {G^{a}}_{b}\,{G^{b}}_{c}\,{G^{c}}_{a}=R^{3}/4} G a b G b c G c d G d a = R 4 / 4 {\displaystyle {G^{a}}_{b}\,{G^{b}}_{c}\,{G^{c}}_{d}\,{G^{d}}_{a}=R^{4}/4} <h2 id="examples">Examples</h2> <p>Notable individual scalar field solutions include </p> <ul><li>the Janis–Newman–Winicour scalar field solution, which is the unique <i>static</i> and <i>spherically symmetric</i> massless minimally coupled scalar field solution.</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Exact_solutions_in_general_relativity/MFb3MFaC">Exact solutions in general relativity</a></li> <li><a href="/facts/Lorentz_group/t1APc2Ye">Lorentz group</a></li></ul> <ul><li>Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C. & Herlt, E. (2003). <i>Exact Solutions of Einstein's Field Equations (2nd edn.)</i>. Cambridge: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-46136-7.</li> <li>Hawking, S. W. & Ellis, G. F. R. (1973). <i>The Large Scale Structure of Space-time</i>. Cambridge: Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-09906-4. See <i>section 3.3</i> for the stress-energy tensor of a minimally coupled scalar field.</li></ul>