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There's the general space expansion due to dark energy and it acts in areas that aren't gravitationally bound. There's gravitational distortion of space around massive bodies, which also adds space. Lorentz Contraction would be an effect that "reduces" amount of space along the path of motion of the relativistic observer, but it contracts the entire universe for that observer, not locally.

Meanwhile, there's the concept of Alcubierre drive, where a spacecraft would traverse distances by contracting space in front of it and expanding space behind it, resulting in effective faster-than-light travel.

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What operation / physical effect / influence could make the space to contract in front of it? Expansion is easy, place a large mass, done. But how does one go about locally contracting space?

SF.
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You need to create negative mass, that is energy density less then the vacuum has in front of the drive, that will contract space.

That would not work with matter that we know, we would need matter that has the ability to create that kind of vacuum.