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I have a few questions regarding the metric in GR, if you wish to answer some of them, I would appreciate a detailed answer.

(1) General relativity is a theory in which the metric plays a crucial role and I would like to know why we use it, or alternatively, is there a good reason to assume that spacetime is curved? Is it related to the equivalence principle? In SR, it seems just like a neat "trick".

Please keep in mind that Im a beginner in GR and still don't know differential geometry at a mathematician level.

Qmechanic
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Gauss_fan
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1 Answers1

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In relativity, the metric is used to determine distances in spacetime. This is required as the geometry is no longer simply Euclidean - in SR, for example, the spacetime is flat but hyperbolic. Another simple example is calculating distance on the surface of a sphere with radius $R$ using spherical polar coordinates. Clearly you can't use just Pythagoras' theorem, $ds^2 = R^2dθ^2 + R^2dφ^2$, and instead have to use $ds^2=R^2dθ^2+R^2sin^2θdφ^2$. This is the basic reason for needing a metric.

The equivalence principle showed that free-fall in a gravitational field was equivalent to inertial motion, meaning that locally everything looked like SR. This motivated the idea that gravity is not a force but the result of spacetime curvature, with an inertial observer following geodesics, which can be thought of as generalized "straight lines".

If you want to properly understand GR, I recommended following these notes. They provide the background differential geometry needed for the subject.

dk30
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