During the plagues in Egypt, wherever an Egyptian would go, the plague would be. If a Jew and an Egyptian simultaneously drank from the same river or even jug of water, the Egyptian would be drinking blood and the Jew water.
The Maharal, Rabbi Yehuda Loew of Prague, does an amazing job (in the introduction to his sefer Gevurot HaShem) explaining the metaphysics of such a miracle in a way simple enough for anyone to understand. There is a dimension called 'nivdal.' (separate and holy) This dimension can envelope and override the physical world. When this happens, those who are a party to this dimension experience something completely other than those who exist only in the physical world. In such a way, all miracles involve the nivdal superceding the mundane.
There's a duality there that even surpasses both Quantum Mechanics and Relativity in it's complexity. Two observers can not only observe two different outcomes, but they can be affected differently even while occupying the same space. It's like a Quantum wave that collapses into two separate solutions simultaneously, depending on two observers.
Anyways, back to my point. In the same way that the Plagues were super-natural, so too is Shabbath super-natural. Wherever a Jew may go, he brings Shabbath with him. On the day of Shabbath, in the same room, a non-jew can be ruled by the laws of nature, and a Jew can be forebidden to relate to and interact with those laws. The Jew is beyond nature, and the non-jew trapped within it.
What's the common link in both cases, the Jew of course. God gave us the Torah and with it the ability to separate. The ability to envelope the physical world within the nivdal - the separate.
Every Jew has a mitzwah to separate himself, to draw down expression of the nivdal into what is otherwise the normal world. [This is a different way to express an idea that is very similar to creating a dwelling for HaShem]
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