It is predicted in string theory that our world has some extra dimensions. I'm wondering if we want to prove this experimentally, what should we do
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One proposed test is measuring the strength of gravity.
Gravity has the property that if there are $n$ dimensions of space, the strength of gravity when distance increases diminishes by $$\frac{1}{r^{n-1}}$$ In our current understanding of the universe, there are 3 spatial dimensions, therefore the strength of gravity reduces in accordance with the well-known factor $$\frac{1}{r^2}$$ But, if in case in large scale measurements we measure that factor to actually be, say for example, $$\frac{1}{r^3}$$then we can deduce that there are $3 + 1 = 4$ spatial dimensions. Note that in this experiment we have to reasonably large distances for measurements.
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$\begingroup$ The extra dimensions of string theory are compactified. Why would they affect gravity only at very large scales? M theory allows something more like it, but again, why the effect only on the very large? $\endgroup$ Commented Jul 29, 2020 at 19:41