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| {{General relativity}}
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| The '''Einstein–Hilbert action''' (also referred to as [[Relativity priority dispute|Hilbert action]]) in [[general relativity]] is the [[action (physics)|action]] that yields the [[Einstein field equations]] through the [[principle of least action]]. With the [[Sign_convention#Relativity|(− + + +) metric signature]], the gravitational part of the action is given as<ref>Richard P. Feynman, Feynman Lectures on Gravitation, Addison-Wesley, 1995, ISBN 0-201-62734-5, p. 136, eq. (10.1.2)</ref>
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| :<math>S = {1 \over 2\kappa} \int R \sqrt{-g} \, \mathrm{d}^4x \;,</math>
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| where <math>g=\det(g_{\mu\nu})</math> is the determinant of the [[metric tensor]], <math>R</math> is the [[Ricci scalar]], and <math>\kappa = 8\pi Gc^{-4}</math>, where <math>G</math> is the Newton's [[gravitational constant]] and <math>c</math> is the [[speed of light]] in vacuum. The integral is taken over the whole [[spacetime]] if it converges. If it does not converge, <math>S</math> is no longer well-defined, but a modified definition where one integrates over arbitrarily large, relatively compact domains, still yields the Einstein equation as the [[Euler–Lagrange equation]] of the Einstein–Hilbert action.
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| The action was first proposed by [[David Hilbert]] in 1915.
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| == Discussion ==
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| The derivation of equations from an action has several advantages. First of all, it allows for easy unification of general relativity with other classical fields theories (such as [[Maxwell theory]]), which are also formulated in terms of an action. In the process the derivation from an action identifies a natural candidate for the source term coupling the metric to matter fields. Moreover, the action allows for the easy identification of conserved quantities through [[Noether's theorem]] by studying symmetries of the action.
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| In general relativity, the action is usually assumed to be a [[functional (mathematics)|functional]] of the metric (and matter fields), and the [[connection (mathematics)|connection]] is given by the [[Levi-Civita connection]]. The [[Palatini action|Palatini formulation]] of general relativity assumes the metric and connection to be independent, and varies with respect to both independently, which makes it possible to include fermionic matter fields with non-integral spin.
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| The Einstein equations in the presence of matter are given by adding the matter action to the Hilbert–Einstein action.
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| ==Derivation of Einstein's field equations==
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| Suppose that the full action of the theory is given by the Einstein–Hilbert term plus a term <math>\mathcal{L}_\mathrm{M}</math> describing any matter fields appearing in the theory.
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| :<math>S = \int \left[ {1 \over 2\kappa} \, R + \mathcal{L}_\mathrm{M} \right] \sqrt{-g} \, \mathrm{d}^4 x </math>
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| The [[action principle]] then tells us that the variation of this action with respect to the inverse metric is zero, yielding
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| :<math>
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| \begin{align}
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| 0 & = \delta S \\
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| & = \int | |
| \left[
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| {1 \over 2\kappa} \frac{\delta (\sqrt{-g}R)}{\delta g^{\mu\nu}} +
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| \frac{\delta (\sqrt{-g} \mathcal{L}_\mathrm{M})}{\delta g^{\mu\nu}}
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| \right] \delta g^{\mu\nu}\mathrm{d}^4x \\
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| & = \int
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| \left[
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| {1 \over 2\kappa} \left( \frac{\delta R}{\delta g^{\mu\nu}} +
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| \frac{R}{\sqrt{-g}} \frac{\delta \sqrt{-g}}{\delta g^{\mu\nu} }
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| \right) +
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| \frac{1}{\sqrt{-g}} \frac{\delta (\sqrt{-g} \mathcal{L}_\mathrm{M})}{\delta g^{\mu\nu}}
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| \right] \delta g^{\mu\nu} \sqrt{-g}\, \mathrm{d}^4x.
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| \end{align}
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| </math>
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| Since this equation should hold for any variation <math>\delta g^{\mu\nu}</math>, it implies that
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| :<math> \frac{\delta R}{\delta g^{\mu\nu}} + \frac{R}{\sqrt{-g}} \frac{\delta \sqrt{-g}}{\delta g^{\mu\nu}} | |
| = - 2 \kappa \frac{1}{\sqrt{-g}}\frac{\delta (\sqrt{-g} \mathcal{L}_\mathrm{M})}{\delta g^{\mu\nu}},</math>
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| is the [[equation of motion]] for the metric field. The right hand side of this equation is (by definition) proportional to the [[stress–energy tensor]],
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| :<math> T_{\mu\nu}:= \frac{-2}{\sqrt{-g}}\frac{\delta (\sqrt{-g} \mathcal{L}_\mathrm{M})}{\delta g^{\mu\nu}}
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| = -2 \frac{\delta \mathcal{L}_\mathrm{M}}{\delta g^{\mu\nu}} + g_{\mu\nu} \mathcal{L}_\mathrm{M}.</math>
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| To calculate the left hand side of the equation we need the variations of the Ricci scalar R and the determinant of the metric. These can be obtained by standard text book calculations such as the one given below, which is strongly based on the one given in {{Harvnb|Carroll|2004}}.
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| ===Variation of the Riemann tensor, the Ricci tensor, and the Ricci scalar===
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| To calculate the variation of the [[Ricci scalar]] we calculate first the variation of the [[Riemann curvature tensor]], and then the variation of the Ricci tensor. So, the Riemann curvature tensor is defined as,
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| :<math> {R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma}
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| - \partial_\nu\Gamma^\rho_{\mu\sigma}
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| + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma}
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| - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}.</math>
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| Since the Riemann curvature depends only on the [[Levi-Civita connection]] <math> \Gamma^\lambda_{\mu\nu}</math>, the variation of the Riemann tensor can be calculated as,
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| :<math>\delta{R^\rho}_{\sigma\mu\nu} = \partial_\mu\delta\Gamma^\rho_{\nu\sigma} - \partial_\nu\delta\Gamma^\rho_{\mu\sigma} + \delta\Gamma^\rho_{\mu\lambda} \Gamma^\lambda_{\nu\sigma} + \Gamma^\rho_{\mu\lambda} \delta\Gamma^\lambda_{\nu\sigma} | |
| - \delta\Gamma^\rho_{\nu\lambda} \Gamma^\lambda_{\mu\sigma} - \Gamma^\rho_{\nu\lambda} \delta\Gamma^\lambda_{\mu\sigma}.</math>
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| Now, since <math>\delta\Gamma^\rho_{\nu\mu}</math> is the difference of two connections, it is a tensor and we can thus calculate its [[covariant derivative]],
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| :<math>\nabla_\lambda (\delta \Gamma^\rho_{\nu\mu} ) = \partial_\lambda (\delta \Gamma^\rho_{\nu\mu} ) + \Gamma^\rho_{\sigma\lambda} \delta\Gamma^\sigma_{\nu\mu} - \Gamma^\sigma_{\nu\lambda} \delta \Gamma^\rho_{\sigma\mu} - \Gamma^\sigma_{\mu\lambda} \delta \Gamma^\rho_{\nu\sigma}. </math>
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| We can now observe that the expression for the variation of Riemann curvature tensor above is equal to the difference of two such terms,
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| :<math>\delta R^\rho{}_{\sigma\mu\nu} = \nabla_\mu (\delta \Gamma^\rho_{\nu\sigma}) - \nabla_\nu (\delta \Gamma^\rho_{\mu\sigma}).</math>
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| We may now obtain the variation of the [[Ricci curvature tensor]] simply by contracting two indices of the variation of the Riemann tensor,
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| :<math> \delta R_{\mu\nu} \equiv \delta R^\rho{}_{\mu\rho\nu} = \nabla_\rho (\delta \Gamma^\rho_{\nu\mu}) - \nabla_\nu (\delta \Gamma^\rho_{\rho\mu}).</math>
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| The [[Ricci scalar]] is defined as
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| :<math> R = g^{\mu\nu} R_{\mu\nu}.\!</math> | |
| Therefore, its variation with respect to the inverse metric <math>g^{\mu\nu}</math> is given by
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| :<math> | |
| \begin{align}
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| \delta R &= R_{\mu\nu} \delta g^{\mu\nu} + g^{\mu\nu} \delta R_{\mu\nu}\\
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| &= R_{\mu\nu} \delta g^{\mu\nu} + \nabla_\sigma \left( g^{\mu\nu} \delta\Gamma^\sigma_{\nu\mu} - g^{\mu\sigma}\delta\Gamma^\rho_{\rho\mu} \right).
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| \end{align}
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| </math>
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| In the second line we used the previously obtained result for the variation of the Ricci curvature and the metric compatibility of the covariant derivative, <math>\nabla_\sigma g^{\mu\nu} = 0 </math>.
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| The last term, <math>\nabla_\sigma ( g^{\mu\nu} \delta\Gamma^\sigma_{\nu\mu} - g^{\mu\sigma}\delta\Gamma^\rho_{\rho\mu} ) </math>,
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| multiplied by <math>\sqrt{-g}</math> becomes a [[total derivative]], since
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| :<math>
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| \sqrt{-g}A^a_{;a} = (\sqrt{-g}A^a)_{,a} \;\mathrm{or}\;
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| \sqrt{-g}\nabla_\mu A^\mu = \partial_\mu\left(\sqrt{-g}A^\mu\right)
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| </math>
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| and thus by [[Stokes' theorem]] only yields a boundary term when integrated. Hence when the variation of the metric <math>\delta g^{\mu\nu}</math> vanishes at infinity, this term does not contribute to the variation of the action. And we thus obtain,
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| :<math>\frac{\delta R}{\delta g^{\mu\nu}} = R_{\mu\nu}.</math>
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| ===Variation of the determinant===
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| [[Jacobi's formula]], the rule for differentiating a [[determinant#Derivative|determinant]], gives: | |
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| :<math>\,\! \delta g = \delta \det(g_{\mu\nu}) = g \, g^{\mu\nu} \delta g_{\mu\nu}</math> | |
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| or one could transform to a coordinate system where <math>g_{\mu\nu}\!</math> is diagonal and then apply the product rule to differentiate the product of factors on the main diagonal.
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| Using this we get
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| :<math>\begin{align}
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| \delta \sqrt{-g}
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| &= -\frac{1}{2\sqrt{-g}}\delta g
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| &= \frac{1}{2} \sqrt{-g} (g^{\mu\nu} \delta g_{\mu\nu})
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| &= -\frac{1}{2} \sqrt{-g} (g_{\mu\nu} \delta g^{\mu\nu}) \,. \end{align}</math>
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| In the last equality we used the fact that
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| :<math>g_{\mu\nu}\delta g^{\mu\nu}=-g^{\mu\nu}\delta g_{\mu\nu}</math>
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| which follows from the rule for differentiating the inverse of a matrix
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| :<math>\delta g^{\mu\nu} = - g^{\mu\alpha} ( \delta g_{\alpha\beta} ) g^{\beta\nu} \,.</math>
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| Thus we conclude that
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| :<math>\frac{1}{\sqrt{-g}}\frac{\delta \sqrt{-g}}{\delta g^{\mu\nu} } = -\frac{1}{2} g_{\mu\nu} .</math>
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| ===Equation of motion===
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| Now that we have all the necessary variations at our disposal, we can insert them into the equation of motion for the metric field to obtain,
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| :<math>R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = \frac{8 \pi G}{c^4} T_{\mu\nu},</math>
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| which is [[Einstein's field equation]] and
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| :<math>\kappa = \frac{8 \pi G}{c^4}</math> | |
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| has been chosen such that the non-relativistic limit yields [[Newton's law of universal gravitation|the usual form of Newton's gravity law]], where ''G'' is the [[gravitational constant]] (see [[Einstein_field_equations#The_correspondence_principle|here]] for details).
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| == Cosmological constant ==
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| Sometimes, a [[cosmological constant]] Λ is included in the [[Lagrangian]] so that the new action
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| :<math>S = \int \left[ {1 \over 2\kappa} \left( R - 2 \Lambda \right) + \mathcal{L}_\mathrm{M} \right] \sqrt{-g} \, \mathrm{d}^4 x </math>
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| yields the field equations:
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| :<math>R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu} \,.</math>
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| ==See also==
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| *[[Belinfante–Rosenfeld stress–energy tensor|Belinfante–Rosenfeld tensor]]
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| *[[Brans–Dicke theory]] (in which the constant ''k'' is replaced by a scalar field).
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| *[[Einstein–Cartan theory]]
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| *[[f(R) gravity]] (in which the Ricci scalar is replaced by a function of the Ricci cuvature)
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| *[[Gibbons–Hawking–York boundary term]]
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| *[[Palatini action]]
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| *[[Teleparallelism]]
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| *[[Variational methods in general relativity]]
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| *[[Einstein–Maxwell–Dirac equations]]
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| ==References==
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| ;Notes
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| {{reflist}}
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| ;Bibliography
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| *{{Citation| last = Carroll | first = Sean M. | year = 2004 | title = Spacetime and Geometry | publisher = Addison Wesley | isbn = 0-8053-8732-3 | url = http://spacetimeandgeometry.net}}
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| *Hilbert, D. (1915) [http://einstein-annalen.mpiwg-berlin.mpg.de/related_texts/relativity_rev/hilbert'' Die Grundlagen der Physik'' (German original for free)] [http://www.springerlink.com/content/t2681418480nq841 (English translation for $25)], Konigl. Gesell. d. Wiss. Göttingen, Nachr. Math.-Phys. Kl. 395-407
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| *{{springer|id=C/c026670|last=Sokolov |first=D.D. |title=Cosmological constant}}
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| {{DEFAULTSORT:Einstein-Hilbert action}}
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| [[Category:Variational formalism of general relativity]]
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| [[Category:General relativity]]
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| [[Category:Albert Einstein]]
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