thermodynamics - Really how does the entropy of the universe increase?

Universe means the system along with its surroundings. I have always got this statement while studying the second law; be it a thermodynamics book (Sears, Salinger) , physical chemistry book (Atkins, Paula) or any site. But really how does this happen?

Take for example, the isothermal expansion of gas; the increase in entropy of the gas(which is our system) is given as $$\Delta S_{\text{sys}} = nR\ln\dfrac{v_f}{v_i}$$ Since the the surroundings remain at constant pressure, the change in enthalpy is same as the heat energy taken by the system. Therefore the entropy of the surroundings is given by $$\Delta S_\text{surr} = -nR\ln\dfrac{v_f}{v_i}$$ Therefore the entropy if the universe is what? $0$; Then how can the entropy of the universe increase? It remains the same! Then why is the statement telling otherwise?

Answer

Then how can the entropy of the universe increase?

Because the universe is like your gas with no surroundings. But see the stress–energy–momentum tensor, and note that it "describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics". See the shear stress term? Space is more like a gin-clear ghostly elastic solid than a gas. And note the energy-pressure diagonal? Space has this innate "pressure". If you've got one, squeeze a stress-ball down in your fist, then let go. Watch it expand.

Oh, and note that there is no actual evidence for "the impossibility of boundaries". I think that's just a failure of imagination myself. In the old days some people couldn't conceive of a world that didn't have an edge. Nowadays some people can't conceive of a world that does.