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Metric tensor at the earth surface 
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#1
Jul3113, 06:13 PM

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I want to find the ricci tensor and ricci scalar for the spacetime curvature at the earth surface. Ignoring the moon and the sun. I have used the scwharzschilds metric, but then the ricci tensor and the scalar where equal to zero.



#2
Jul3113, 06:17 PM

C. Spirit
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The Schwarzschild metric is a vacuum solution so of course the Ricci tensor and Ricci scalar will vanish. The Schwarzschild metric is valid for the exterior of the Earth, ignoring the Earth's rotation and the presence of the other celestial bodies.



#3
Jul3113, 07:09 PM

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The Kretschmann scalar is nonzero M^{2}/r^{6}



#4
Jul3113, 08:47 PM

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PF Gold
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Metric tensor at the earth surface
Another thing you could look at is the Weyl curvature tensor. This specifically describes the type of curvature in vaccuum regions in GR. Unfortunately, it is really hard to compute by hand, even for a metric as simple as the Scwharzschild.



#5
Aug113, 09:35 AM

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The mass of the Earth should curve the space surounding the Earth. Why is then the ricci tensor equal to zero.



#6
Aug113, 09:46 AM

Mentor
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Outside the Earth, there is a vacuum, so Ricci is zero. Inside the Earth, Ricci is nonzero. A somewhat crude model takes the Earth as a constant density, nonrotating sphere. Then, Schwarzschild's interior solution can be used. When [itex]G=c=1[/itex],
[tex] ds^{2}=\left( \frac{3}{2}\sqrt{1\frac{2M}{R}}\frac{1}{2}\sqrt{1\frac{2Mr^{2}}{R^{3}}}\right) ^{2}dt^{2}\left( 1\frac{2Mr^{2}}{R^{3}}\right) ^{1}dr^{2}r^{2}\left( d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}\right), [/tex] where [itex]R[/itex] is the [itex]r[/itex] coordinate at the surface of the Earth. This metric is treated in many relativity texts, e.g., texts by Schutz, by Hobson, Efstathiou, Lasenby, and by Misner, Thorne, Wheeler. 


#7
Aug113, 09:55 AM

Physics
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PF Gold
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#8
Aug113, 02:32 PM

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#9
Aug113, 03:42 PM

C. Spirit
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Both the Ricci and Weyl curvatures can be made sense of physically in GR by looking at geodesic congruences and seeing which of the three kinematical quantities (expansion, shear, and twist) are dominated by which curvature quantity. See here for a start: http://en.wikipedia.org/wiki/Weyl_tensor 


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