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You're misreading this:  the analogy is "like a heavy object would stretch a rubber sheet."  Yes, that is an analogy: nobody's saying that space is actually a rubber sheet.


But spacetime does curve in the presence of matter, and yes - it even twists when matter is spinning.


Think of it this way: spacetime is a field, and the presence of matter distorts that field.  This curvature of the spacetime metric is real and physical and it has been confirmed by a very wide range of experiments.  The simplest and most direct test of this is the deflection of starlight around the Sun during a solar eclipse, first conducted by Sir Arthur Eddington in 1919, and subsequently confirmed by far more precise observations.  If gravity were simply a force acting in a Euclidean spacetime, the deflection of light would only be 1/2 of the observed value, but because gravity is an actual curvature of the background geometry of spacetime, light follows that geometry around a massive body, resulting in a value twice as great as the value predicted by Newtonian physics.


We recently recorded a Physics Frontiers episode called "The Parameterized Post-Newtonian Formalism" which we'll publish in a few weeks, that discusses the wide range of experiments and theoretical alternatives to general relativity.  Those experimental and theoretical arguments can be found in this excellent 2014 review paper:  The Confrontation between General Relativity and Experiment, Sections 3 and 4 (pp. 26-53).


Have a look at it: all non-metric theories of gravitation have been ruled out.  That means that even if the curvature of spacetime produced by matter is some kind of illusion produced by other physics, then that illusion is so complete that it's physically indistinguishable from a metric curvature of spacetime - so we might as well treat it that way, because a more complicated underlying theory couldn't possibly explain our observations any better than general relativity anyway.  But there's also something to be said for simplicity: if it looks like a duck and quacks like a duck, then it makes more sense to treat it as a duck, than to try to explain how a pigeon could look like a duck and quack like a duck and float like a duck.


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