Noob time-space thought problem

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MarvinMartian
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this one bothers me - pls point out the flaw in my reasoning:

Time-space may be called a 'fabric', and may be depicted as a 2D 'sheet'. An object with mass deforms the sheet.

If i were to grab one edge of this 'inelastic sheet' and pull it 1 cm, would the movement would be instantaneous across the sheet .. i.e... anyone point of the sheet moves just 1 cm, but the distant edge, many light-years away, also *immediately* moves towards me 1 cm. too?

Or, would the tension I create warp the 'elastic sheet', and so cause my immediately surrounding fabric to move ~1 cm, but stretching the fabric so that the distant part only responded to the tug over some much longer time period, analogous to pulling a rubber sheet of the same size.

Would the transfer of this pull across the sheet be faster/slower/the same speed of light?
 
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Don't take the rubber sheet analogy too literally.

The very quick answer to your question is that curvature of space-time is what gravity is, so gravitational waves, and thus changes in the curvature, are presumed to travel at c.