We know heavy objects bend space-time, but does the curvature only depends on the mass? Or different shapes bend space time differently?
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Spacetime is curved by the presence of mass in a perticular point and not by the body's center of mass... so the shape of the body affects the overall curvature because it affects the distribution of mass in the spacetime... the magnitude of the curvature is porpotional to the mass present in a point and the overall "shape" of the curvature is related to the mass distribution on the body...
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$\begingroup$ What would happen in a scenario where a large 5 point star (for example) is orbited by a small sphere? The pointy ends of the star would distort the space time "shape" from the tips to the center with increasing "magnitude of disturbance". How would this affect the orbit? Perhaps it would create a rounded pentagon type orbit? Classical mechanics (or that part which im familiar with) doesn't account for the shape, merely the center of mass. Is my interpretation of this wrong? $\endgroup$– 22134484Jun 20, 2018 at 10:41
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1$\begingroup$ @22134484: Classical mechanics does account for the shape of an extended object, it's just that these effects aren't usually considered in intro courses. For example, the gravitational field of a star-shaped object differs from that of a point mass. Usually we can get away with this approximation, though, because of the fact that the gravitational field outside a uniform sphere is the same as that of a point mass. Many celestial objects are pretty close to spherical, so we can get away with considering them as point masses. $\endgroup$ Jun 20, 2018 at 18:07
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The Einstein field equations read,
$$R_{\mu\nu}-\frac12 g_{\mu\nu}R = 8\pi G\, T_{\mu\nu}$$
where $T_{\mu\nu}$ is the stress-energy-momentum tensor. This is not a constant matrix, it is a tensor field which can depend on space-time.
As such, how the matter is distributed will influence the curvature of space-time and hence ultimately the resultant metric. As a particular example, $T_{\mu\nu}$ with a delta function like distribution often give rise to brane line geometries.
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The shape of the space around a body depends on its geometry. This Image shows the "shape of the space" in 2 dimensions caused by a spherical symmetric mass. In 3 dimension its a sphere around this mass. So the shape of the space is similar to the geometry of the body. From this you can approximately assume e.g. the shape of the space in the vicinity of a cylinder.
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$\begingroup$ This Image shows the "shape of the space" in 2 dimensions caused by a spherical symmetric mass. Those images are just metaphors. There is no "shape of the space." $\endgroup$– user4552Jun 20, 2018 at 17:57
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$\begingroup$ Yes agreed and therefor I used "". But those pictures are often used to give an impression and as such they are useful in popular science in my opinion. In this case they show that the "shape" doesn't "depend only on the mass". $\endgroup$– timmJun 20, 2018 at 20:19