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The classical theory of spacetime geometry that we call gravity consists of the Einstein equation, which relates the curvature of spacetime to the distribution of matter and energy in spacetime. $ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$ Mathematically, how do the Einstein's differential equation of the curvature of spacetime come out of the Einstein's field equations, $$G_{\mu\nu}=8{\pi}T_{\mu\nu}$$. ?

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I'm almost a layman in General Relativity, but I don't understand the question. The Einstein's field equations are a set of equations for the metric tensor, and the metric tensor is kind of a machine that gives you the scalar product of two 4-vectors. As I see it, the Einstein's field equations define in some sense the metric tensor as a solution of themselves, and the metric tensor can be used to compute scalar products. It's not that one implies the other or something like that. However nice question, I'll be waiting for the experts ;) –  Jorge Nov 28 '12 at 13:42
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What do you mean with "Einstein's differential equation of the curvature of spacetime"? The Einstein field equations are equations for the metric and the metric gives you all the local information including the information of the curvature ( Riemann curvature tensor, ricci tensor,weyl tensor ...) –  ungerade Nov 28 '12 at 13:43
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So maybe the answer would be: the Einstein's field equations are the equations for the curvature so they don't have to come out –  ungerade Nov 28 '12 at 14:05
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just came to mind: the geodesic equation(search up on wiki) might be a more appropriate differential equation to ask for such a relationship –  namehere Nov 28 '12 at 16:09
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@Neo Please consider what everyone is saying outside of the answer you accepted. The expression you wrote for $ds^2$ is a definition, nothing more. It no more comes out of $G = 8\pi T$ than $\pi$ comes out of that equation. –  Chris White Dec 4 '12 at 6:13
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The classical theory of spacetime geometry that we call gravity consists of the Einstein equation, which relates the curvature of spacetime to the distribution of matter and energy in spacetime. $$ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$$

This equation is not specific to GR. It is a result from Riemann Geometry. It does not relate the curvature to the StressEnergyMomentum tensor.

Mathematically, how do the Einstein's differential equation of the curvature of spacetime come out of the Einstein's field equations, $$G_{\mu\nu}=\frac{8{\pi}G}{c_0^4} T_{\mu\nu}$$. ?

The EFE is what relates the geometry of spacetime and the StressEnergyMomentum tensor. . . It can be derived from the EH action.

"Einstein's differential equation" is the EFE, it is a differential, polynomial equation in the metric tensooor.

The expression describing the geometry of the spacetime is the metric tensor, of course. Outside the reigon of energy ( or mass, same thing, almost. ), / the value of $T_{\mu\nu}$ is 0. The Ricci Curvature $R_{\mu\nu}$ is thus, by the EFE, also 0. The EFEs then reduce tof:

$$R_{\mu\nu}=0$$

Which is the vacuum field equation. Wait a minute, let me clarify something. The Ricci Curvature is 0, but the Riemann Curvature is not, so there is still some curvature (obviously, or else there would be no gravity outside the earth (but note, this is due to the Weyl tensor, which is only non-zero in 4 or greater dimensions, so in a 3-dimensional spacetime, the Ricci Curvature would be all there is to the Riemann Curvature. ) ).

But then, you may ask, why is there a difference between, say, the Schwarzschild metric and the Kerr metric? Shouldn't all metrics outside any object be the same? And why is there a difference between the gravitational field of the Earth, and that of an apple? The vacuum EFE only tell you that the Ricci Curvature is 0, and absolutely nothing about the Weyl tensor $C_{abcd}$, right? .

It is not the EFE which tells you about the Weyl tensor. You have to use the known principles about the object. m n . Schwarz's pedia tells you more. .

Anyway, to answer your question, leave the EFE untouched and you'll get your desired equation, which is the EFE.

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From Wiki:

$G_{\alpha \beta}= (\delta^\gamma_\alpha \delta^\zeta_\beta - \frac{1}{2} g_{\alpha\beta}g^{\gamma\zeta})(\Gamma^\epsilon_{\gamma\zeta,\epsilon} - \Gamma^\epsilon_{\gamma\epsilon,\zeta} + \Gamma^\epsilon_{\epsilon\sigma} \Gamma^\sigma_{\gamma\zeta} - \Gamma^\epsilon_{\zeta\sigma} \Gamma^\sigma_{\epsilon\gamma})$

Where

$\Gamma^\alpha_{\beta\gamma} = \frac{1}{2} g^{\alpha\epsilon}(g_{\beta\epsilon,\gamma} + g_{\gamma\epsilon,\beta} - g_{\beta\gamma,\epsilon})$

So, you see, the Einstein tensor, G, is a function of the metric g and the 1st and 2nd derivatives of the metric.

The simple looking equation, relating the Einstein tensor to the stress-energy tensor, that you quote in your question is, in fact, a system of differential equations written in compact form.

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$ds^{2} = g_{ab}dx^{a}dx^{b}$ isn't the Einstein equation. It's just the equation for what arc length is. It's the definition of the metric tensor, pretty much. You're implicitly using it if you've ever done calculus in three dimensions.

The Einstein equation and the Einstein field equations are the same thing and they are expressible as:

$$R_{ab} - \frac{1}{2}Rg_{ab} = 8\pi T_{ab}$$

Full stop.

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This got downvoted? Seriously? –  Jerry Schirmer Feb 5 '13 at 17:02
    
Before I comment, I should mention that I didn't downvote your answer. However, I would like to point out that you have not said what a and b are (or anything else) so your full stop is premature. –  Magpie Mar 1 '13 at 5:03
    
@Magpie: $a$ and $b$ are obviously indices of the tensors. The downvoter clearly can't use that as an excuse. –  Dimensio1n0 Jul 17 '13 at 19:59
    
@Dimension10 I don't think it's deserving of a downvote either but a and b could just as easily be subscript and g R and T have not been declared so they could also be anything at all, really. It is better to write things out explicitly so that equations have meaning and the answer is more complete. –  Magpie Jul 18 '13 at 1:19
    
@Magpie: But most people recognise the equation as the EFE immediately, and know what each term is, metric, ricci curvature, SEM tensor and so on. . –  Dimensio1n0 Jul 18 '13 at 4:58
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