# Metric tensor at the earth surface

by tm007
Tags: earth, metric, surface, tensor
 P: 2 I want to find the ricci tensor and ricci scalar for the space-time 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.
 C. Spirit Sci Advisor Thanks P: 5,584 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.
 Sci Advisor P: 5,436 The Kretschmann scalar is non-zero M2/r6
 Sci Advisor PF Gold P: 5,059 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.
 P: 2 The mass of the Earth should curve the space surounding the Earth. Why is then the ricci tensor equal to zero.
 Mentor P: 6,242 Outside the Earth, there is a vacuum, so Ricci is zero. Inside the Earth, Ricci is non-zero. A somewhat crude model takes the Earth as a constant density, non-rotating sphere. Then, Schwarzschild's interior solution can be used. When $G=c=1$, $$ds^{2}=\left( \frac{3}{2}\sqrt{1-\frac{2M}{R}}-\frac{1}{2}\sqrt{1-\frac{2Mr^{2}}{R^{3}}}\right) ^{2}dt^{2}-\left( 1-\frac{2Mr^{2}}{R^{3}}\right) ^{-1}dr^{2}-r^{2}\left( d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}\right),$$ where $R$ is the $r$ 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.
Physics
PF Gold
P: 6,126
 Quote by tm007 The mass of the Earth should curve the space surounding the Earth.
It does, but the curvature produced in the vacuum region around the Earth is Weyl curvature, not Ricci curvature.
P: 5,436
 Quote by tm007 The mass of the Earth should curve the space surounding the Earth. Why is then the ricci tensor equal to zero.
Spacetime is curved, that's why several curvature invariants and Weyl curvature are non-zero, whereas Ricci-curvature is zero
C. Spirit