If you are still interested, feel free to explore the following links for more information on this topic. Some sites are more interesting than others.

http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/earthole.html - Great introduction to problem, assumes uniform sphere

http://www.wired.com/wiredscience/2012/11/how-long-would-it-take-to-fall-through-the-earth/ - Also great source - Explores question without air resistance

http://www.grc.nasa.gov/WWW/k-12/airplane/falling.html - Describes air resistance/drag

http://en.wikipedia.org/wiki/Vertical_pressure_variation - Describes the general model for how the pressure/density of air changes as a function of altitude/height

http://en.wikipedia.org/wiki/Structure_of_the_Earth - Explains the reason for why Earth is not uniformly dense

http://upload.wikimedia.org/wikipedia/commons/3/36/Antipodes_LAEA_inverted.png - Interesting picture showing how few places a tunnel through Earth could be located and be connected on the crust with two land masses

http://answers.yahoo.com/question/index?qid=20101018141220AAfbQqa - Used for approximation of Earth density

http://www.wolframalpha.com/input/?i=nitrogen+phase+diagram - 3 phase diagram of Nitrogen, shows that it, and most likely the other air will solidify if pressurized enough

http://en.wikipedia.org/wiki/Sound_barrier - While not discussed in the video, the first reduced atmosphere was chosen to avoid the sound barrier, which is discussed in this source

http://en.wikipedia.org/wiki/Tungsten - Describes the material properties of Tungsten, a relatively normal building choice that might be good to use for this tunnel, although it would still need major modifications in order to function