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| '''Einstein's constant''' or '''Einstein's gravitational constant''', denoted κ ([[kappa]]), is the coupling [[Physical constant|constant]] appearing in the [[Einstein field equations|Einstein field equation]] which can be written:
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| <center><math>G^{\alpha \gamma} = \kappa \, T^{\alpha \gamma}~</math></center>
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| where ''G<sup>αγ</sup>'' is the [[Einstein tensor]] and ''T<sup>αγ</sup>'' is the [[stress-energy tensor]].
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| This equation relates to the [[curvature]] of [[space]] and [[Time in physics|time]], telling that [[Stress-energy tensor|stress-energy]] is what causes the disturbance of [[spacetime]], thus [[gravitation]]. [[Albert Einstein|Einstein]] used [[Newton's law of universal gravitation]] in his field equations, and the constant of κ is found to have a value of:<ref>
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| {{cite book |title= Introduction to General Relativity |author= Ronald Adler, Maurice Bazin, Menahem Schiffer |publisher= McGraw-Hill |location= New York |year= 1975 |edition= 2nd edition |isbn= 0-07-000423-4}}<br />
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| (see Chapter 10 "The Gravitational Field Equations or Nonempty Space", section 10.5 " Classical Limit of the Gravitational Equations" p. 345)</ref>
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| <center><math>\kappa \, = \, - { 8 \, \pi \, G \over c^2 }~</math></center>
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| N.B.: Writing Einstein's constant depends on how the stress-energy tensor is defined, so the Einstein field equations are always invariant (see details in the section ''"About the two possible writings"'' further).
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| == Calculation ==
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| In the following, the value of Einstein's constant will be calculated. To do so, at the beginning a field equation where the [[cosmological constant]] Λ is equal to zero is taken, with a [[Steady state theory|steady state hypothesis]]. Then we use the Newtonian approximation with hypothesis of a weak field and low velocities with respect to the speed of light.
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| The Newton law will arise and its corollary [[Poisson's equation]].
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| In this approximation, Poisson's equation appears as the approached form of the field equation (or the field equation appears as a generalization of Poisson's equation). The identification gives the expression of Einstein's constant related to quantities ''G'' and ''c''.
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| === The Einstein field equations in non-empty space ===
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| We have to obtain a suitable [[tensor]] to describe the geometry of space in the presence of an energy field. Einstein proposed this equation in 1917, written as:
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| <center><math>G^{\alpha\gamma} + \Lambda \mathrm{g}^{\alpha\gamma} = (\mathrm{const}) T^{\alpha\gamma}~</math></center>
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| (const) is what will become Einstein's constant. We will take the cosmological constant Λ equal to zero (one of the requirements of the properties of the gravitational equations is that they reduce to the free-space field equations when the density of energy in space ''T<sup>αγ</sup>'' is zero, therefore that the cosmological constant λ appearing in this equation is zero) so the field equation becomes:
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| <center><math>G^{\alpha\gamma} = \left( R^{\alpha\gamma} - \frac{1}{2} \mathrm{g}^{\alpha\gamma} \, R \right) = \kappa \, T^{\alpha\gamma}~</math></center>
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| where ''R<sup>αγ</sup>'' s the Ricci tensor, ''g<sup>αγ</sup>'' is the metric tensor, ''R'' the scalar curvature and κ is Einstein's constant we will calculate in the next section..
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| This equation can be written in another form, contracting indexes:
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| <center><math>{R^\alpha}_\alpha - \frac{1}{2} \, {\mathrm{g}^\alpha}_\alpha \, R = {\kappa \, T^\alpha}_\alpha~</math></center>
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| Thus:
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| <center><math>R = -{\kappa \, T^\alpha}_\alpha = -\kappa \, T~</math></center>
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| where ''T'' is the scalar ''T<sup>α</sup><sub>α</sub>'' which we shall refer to as the Laue scalar.
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| Using this result we can write the field equation as:
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| {| class="wikitable" style="margin: 1em auto 1em auto"
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| |<math>R^{\alpha\gamma} = \kappa \left(T^{\alpha\gamma} - \frac{1}{2} \, \mathrm{g}^{\alpha\gamma} \, T \right)</math>
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| |}
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| === The classical limit of the gravitational equations ===
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| We will show that the field equations are a generalization of Poisson's classical field equation. The reduction to the classical limit, besides being a validity check on the field equations, gives as a by-product the value of the constant κ.
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| <math>|i</math> and <math>|i|j</math> respectively indicate <math>\tfrac{\partial}{\partial x^i}</math> and <math>\tfrac{\partial^2}{\partial x^i \partial x^j}</math>. Thus, <math>|i|i</math> means <math>\tfrac{\partial^2}{(\partial {x^i})^2}</math>
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| We will consider a field of matter with low proper density ρ, moving at low velocity ''v''. The stress-energy tensor can be written:
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| <center><math>T_{{\mu}v}= \rho
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| \begin{pmatrix}
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| 1 & v_x / c & v_y / c & v_z / c\\
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| v_x / c & v^2_x / c^2 & v_x v_y / c^2 & v_x v_z / c^2\\
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| v_y / c & v_y v_x / c^2 & v^2_y / c^2 & v_y v_z / c^2\\
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| v_z / c & v_z v_x / c^2 & v_z v_y / c^2 & v^2_z / c^2
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| \end{pmatrix}</math></center>
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| If the terms of order <math>(\tfrac{v}{c})^2</math> and <math>\rho (\tfrac{v}{c})</math> are neglected, it becomes:
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| <center><math>T_{\mu\nu}=
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| \begin{pmatrix}
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| \rho_0 & 0 & 0 & 0\\
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| 0 & 0 & 0 & 0\\
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| 0 & 0 & 0 & 0\\
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| 0 & 0 & 0 & 0
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| \end{pmatrix}</math></center>
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| We assume the flow to be stationary and therefore expect the metric to be time-independent. We use the coordinates of special relativity ''ct'', ''x'', ''y'', ''z'' that we write as ''x''<sup>0</sup>, ''x''<sup>1</sup>, ''x''<sup>2</sup>, and ''x''<sup>3</sup>. The first coordinate is time, and the three others are the space coordinates.
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| Applying a [[perturbation theory|perturbation method]], we will consider a metric appearing through a two-terms summation. The first is the Lorentz metric, ''η<sub>μν</sub>'' which is that of the [[Minkowski space]], locally flat. Formulating we get:
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| <center><math>\displaystyle \mathrm{d}s^2 = (\mathrm{d}x^0)^2 - (\mathrm{d}x^1)^2 - (\mathrm{d}x^2)^2 - (\mathrm{d}x^3)^2</math></center>
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| The second term corresponds to the small perturbation (due to the presence of a gravitating body) and is also time-independent:
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| <center><math>\displaystyle \varepsilon\gamma_{\mu\nu}</math></center>
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| Thus we write the metric:
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| <center><math>\displaystyle g_{\mu\nu} = \eta_{\mu\nu} + \varepsilon\gamma_{\mu\nu}</math></center>
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| Clarifying the length element:
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| <center><math>\displaystyle \mathrm{d}s^2 = (\mathrm{d}x^0)^2 - (\mathrm{d}x^1)^2 - (\mathrm{d}x^2)^2 - (\mathrm{d}x^3)^2 + \varepsilon\gamma_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}</math></center>
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| If we neglect terms of order <math>\displaystyle \varepsilon\rho_0</math>, the Laue scalar <math>T^{\mu}_{\mu}</math> is:
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| <center><math>T^{\mu}_{\mu} = \operatorname{Tr} \begin{pmatrix} \rho_0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0& 0\\ 0 & 0 & 0 & 0 \end{pmatrix} = \rho_0</math></center>
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| And the right side of the field equations is to first order in all the small quantities <math>\rho_0</math>, <math>\tfrac{v}{c}</math> and <math>\varepsilon\gamma_{\mu\nu}</math> is written:
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| <center><math>\begin{align}C\left(T_{\mu\nu}-\frac{1}{2} g_{\mu\nu}T \right)&\simeq C\left(T_{\mu\nu}-\frac{1}{2} g_{\mu\nu} T \right) \\ & \simeq C \left[ \begin{pmatrix} \rho_0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix} - \frac{1}{2}\begin{pmatrix} \rho_0 & 0 & 0 & 0\\ 0 & -\rho_0 & 0 & 0\\ 0 & 0 & -\rho_0 & 0\\ 0 & 0 & 0 & -\rho_0 \end{pmatrix} \right] \\ & \simeq \frac{C\rho_0}{2} \delta_{\mu\nu} \end{align}</math></center>
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| Neglecting second-order terms in <math>\varepsilon\gamma_{\mu\nu}</math> gives the following approximate form for the contracted Riemann tensor:
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| <center><math>R_{\mu\nu} \cong \frac{1}{2}\left[\ln(-g)\right]_{|\mu|\nu|}-[\mu\nu,\beta]_{|\beta}</math></center>
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| Thus the approximate field equations may be expressed as:
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| <center><math>\frac{1}{2}\left[\ln(-g)\right]_{|\mu|\nu|}-[\mu\nu,\beta]_{|\beta} = \frac{\kappa \, \rho_0}{2} \, \delta_{\mu\nu}</math></center>
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| At first let us consider the case μ = ''ν'' = 0. As the metric is time-independent, the first term of the equation above is zero. What remains is:
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| <center><math>[00,\beta]_{|\beta} = \left(g^{\alpha\beta}\left[00,\alpha\right]\right)_{|\beta} = -\frac{\kappa \, \rho_0}{2}\qquad\qquad (*)</math></center>
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| The [[Christoffel symbols|Christoffel symbol]] of the first kind is defined by:
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| <center><math>\left[00,\alpha\right] = \frac{1}{2}\left(g_{0\alpha|0} + g_{\alpha 0|0} - g_{00|\alpha}\right)</math></center>
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| Since the [[Pseudo-Riemannian manifold#Lorentzian manifold|Lorentz metric]] is constant in space and time, this simplifies to:
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| <center><math>\left[00,\alpha\right] = -\frac{\varepsilon}{2} \gamma_{00|\alpha}</math></center>
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| Moreover <math>\gamma_{\mu\nu}</math> is time-independent, so [00,0] is zero. Neglecting second-order terms in the perturbation term <math>\varepsilon\gamma_{\mu\nu}</math>, we get:
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| <center><math>g^{\beta\alpha} \left[00,\alpha\right] = \frac{\varepsilon}{2} \gamma_{00|\beta}</math></center>
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| which is zero for β = 0 (which then corresponds to the derivative with respect to time). Substituting inside (*) we obtain the following approximate field equation for <math>\gamma_{00}</math>:
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| <center><math>\varepsilon\sum_{\beta = 0}^{3} \gamma_{00|\beta|\beta} = -\kappa \, \rho_0</math></center>
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| or, by virtue of time independence:
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| <center><math>\varepsilon\sum_{\beta = 1}^{3} \gamma_{00|\beta|\beta} = -\kappa \, \rho_0</math></center>
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| This notation is just a writing convention. The equation can be written:
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| <center><math>\sum_{\beta = 0}^{3} \gamma_{00|\beta|\beta} = \sum_{i=1}^3 \frac{\partial {}^2 \gamma_{00}}{\partial {x_{\beta}}^2} = \frac{\partial {}^2 \gamma_{00}}{\partial {x_1}^2} + \frac{\partial {}^2 \gamma_{00}}{\partial {x_2}^2} + \frac{\partial {}^2 \gamma_{00}}{\partial {x_3}^2} = -\kappa \, \rho_0</math></center>
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| which can be identified to Poisson's equation if we write:
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| <center><math>-\frac{\varepsilon\gamma_{00}}{\kappa} = \frac{\varphi}{4\pi G}</math></center>
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| Therefore we have established that the classical theory (Poisson's equation) is the limiting case (weak field, low velocities with respect to the speed of light) of a relativistic theory where the metric is time-independent.
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| To be complete, gravity has to be demonstrated as a metric phenomenon. In the following, without detailing all calculation the simplistic description of the complete calculation is given. Again, at first we start from a perturbed Lorentz metric:
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| <center><math>\displaystyle g_{\mu\nu} = \eta_{\mu\nu} + \varepsilon\gamma_{\mu\nu}</math></center> | |
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| made explicit:
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| <center><math>\displaystyle \mathrm{d}s^2 = (\mathrm{d}x^0)^2 - (\mathrm{d}x^1)^2 - (\mathrm{d}x^2)^2 - (\mathrm{d}x^3)^2 + \varepsilon\gamma_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}</math></center>
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| We suppose the velocity ''v'' is low with respect to the speed of light ''c'', with a small parameter <math>\beta = \tfrac{v}{c}</math>.
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| We have:
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| <center><math>\displaystyle x^0 = ct</math></center>
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| We can write:
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| <center><math>\begin{align}
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| \left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)^2 &= c^2 - v^2 + \varepsilon\gamma_{\mu\nu} \frac{\mathrm{d}x^{\mu}}{\mathrm{d}t} \frac{\mathrm{d}x^{\nu}}{\mathrm{d}t}\\
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| &= c^2 \left(1 - \beta^2 + \varepsilon\gamma_{\mu\nu} \frac{\mathrm{d}x^{\mu}}{\mathrm{d}x^0} \frac{\mathrm{d}x^{\nu}}{\mathrm{d}x^0}\right)
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| \end{align}</math></center>
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| Limiting to the first degree in β and ε we get:
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| <center><math>\left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)^2 \cong c^2(1 + \varepsilon\gamma_{00})</math></center>
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| Then we write, as a classical calculation, the [[differential equation]] system giving the [[geodesic]]s. Christoffel symbols are calculated. The geodesic equation becomes:
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| <center><math>\frac{\mathrm{d}^2 x^{\alpha}}{\mathrm{d}t^2} + [00,\alpha] c^2 = 0 \qquad (**)</math></center>
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| The approximate form of the Christoffel symbol is:
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| <center><math>[00,i] = \frac{1}{2}\varepsilon\gamma_{00|i}</math></center>
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| Introducing this result into the geodesic equation (**) we get:
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| <center><math>\displaystyle \frac{\mathrm{d}^2 x^i}{\mathrm{d}t^2} = -\frac{c^2}{2} \varepsilon \gamma_{00|i}</math></center>
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| This is a vector equation. Since the metric is time-independent, only space variables are concerned. Therefore the second member of the equation is a gradient.
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| Coding the position-vector by the letter ''X'' and the gradient by the vector ∇ we can write:
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| <center><math>\displaystyle \frac{\mathrm{d}^2 X}{\mathrm{d}t^2} = \frac{c^2}{2} \varepsilon \gamma_{00}</math></center>
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| This is no more than Newton's law of universal gravitation in classical theory, derivating from the gravitational potential φ if we make the identification:
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| <center><math>\varphi = -\frac{c^2}{2} \varepsilon \nabla \gamma_{00}</math></center>
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| Conversely, if we set a gravitational potential φ, the movement of a particle will follow a space-time geodesic if the first term of the metric tensor is like:
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| <center><math>g_{00} = 1 + \frac{2\varphi}{c^2}</math></center>
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| That step is important. Newton's law appears as a particular aspect of the general relativity with the double approximation:
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| * weak gravitational field
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| * low velocity with respect to the speed of light
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| With the calculation above, we have made the following statements:
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| * A metric ''g'', solution of the Einstein field equation (with a cosmological constant Λ equal to zero).
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| * This metric would be a weak perturbation in relation to a Lorentz metric η (relativistic and flat space).
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| * The perturbation term would not depend of time. Since the Lorentz metric does not depend of time neither, that metric ''g'' is time-independent too.
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| * The expansion into a series gives a linearization of the Einstein field equations.
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| * This linearized form is found to identify to Poisson's equation using the fact that a field is a curvature, linking the perturbation term to the metric and to the gravitational potential thanks to the relation:
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| <center><math>\varphi = \frac{c^2}{2} \varepsilon \gamma_{00}</math></center>
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| And this rewards the value of the constant κ, called "Einstein's constant" (which is ''not'' the cosmological constant Λ nor the speed of light ''c''):
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| <center><math>\kappa \, = \, - { 8 \, \pi \, G \over c^2 }~</math></center>
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| We can then write the Einstein field equation:
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| <center><math>G^{\alpha\gamma} + \Lambda g^{\alpha\gamma} = -\frac{8\pi G}{c^2} T^{\alpha\gamma}~</math></center>
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| == About the two possible writings ==
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| We have seen, neglecting the terms of order <math>(\tfrac{v}{c})^2</math> and <math>\rho (\tfrac{v}{c})</math>, that the Laue scalar could be written:
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| <center><math>T^{\mu}_{\mu} = \operatorname{Tr} \begin{pmatrix} \rho_0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0& 0\\ 0 & 0 & 0 & 0 \end{pmatrix} = \rho_0</math></center>
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| which gives the corresponding Einstein's constant:
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| <center><math>\kappa \, = \, - { 8 \, \pi \, G \over c^2 }~</math></center>
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| But another valid choice for writing the form of the stress-energy tensor is:
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| <center><math>T_{{\mu}v}= \rho \begin{pmatrix} c^2 & v_x c & v_y c & v_z c\\ v_x c & v^2_x & v_x v_y & v_x v_z\\ v_y c & v_y v_x & v^2_y & v_y v_z\\ v_z c & v_z v_x & v_z v_y & v^2_z \end{pmatrix}~</math></center>
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| Neglecting the same term orders, the corresponding Laue scalar is:
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| <center><math>T^\mu_\mu=\operatorname{Tr} \begin{pmatrix} \rho_0 c^2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix}=\rho_0 c^2~</math></center>
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| which owns an additional term c<sup>2</sup>, so the corresponding Einstein's constant in the field equations is then:
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| <center><math>\kappa \, = \, - { 8 \, \pi \, G \over c^4 }~</math></center>
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| This is just a question of similar choices, since for each chosen writing the Einstein field equations are the same.
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| == About constants ==
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| The Einstein field equation has zero divergence. The zero divergence of the stress-energy tensor is the geometrical expression of the conservation law. So it appears constants in the Einstein equation cannot vary, otherwise this postulate would be violated.
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| However since Einstein's constant had been evaluated by a calculation based on a time-independent metric, this by no mean requires that ''G'' and ''c'' must be unvarying constants themselves, but that ''the only absolute constant is their ratio'':
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| <center><math>G \over c^2~</math></center>
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| == References ==
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| <references />
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| ==Further reading==
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| *{{cite book |title=Theoretical Foundations of Cosmology: Introduction to the Global Structure of Space-time |author=Michael Heller |url=http://books.google.com/books?id=JJ1aNeTzmLQC&pg=PA63 |page=63 |isbn=981-02-0756-5 |year=1992 |publisher=World Scientific}}
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| [[Category:General relativity]]
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| [[Category:Albert Einstein]]
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