A Shinkansen line (arbitrarily chosen because it was the first thing that popped up in my search) can do 13 trains per hour with 1,323 passengers per train, which is equivalent to one 747 every 1.4 minutes.
That would replace one runway, but it still doesn't compare to the number of planes that the sky can handle. Sky which is cheap/free in comparison to building rails. There is definitely a point where trains are more efficient, particularly between a small number of large cities, but moving a large proportion of US air traffic onto trains, imho, is a different problem given the number of routes (1000s) and airports (100s).
I'd have to see the math, but there is also a question as to carbon emissions per-passenger/distance. I believe that on long flights (more time at high altitude) widebody aircraft might be more fuel efficient than high-speed trains stuck on tracks.
How many airports are landing anything like a 747 every 1.4 minutes? Especially given that the Shinkansen runs in both directions, so your airport is going to have to handle exclusively 747s, with high loadings and a headway of around 40 seconds. That's slightly more planes and massively more passengers than London Heathrow.
In terms of carbon emissions you'd need the flights to be long, and the trains to be pretty inefficient and empty (think current Amtrak trains), and even then there is evidence that the high altitude appears to make the emissions significantly more harmful than the equivalent ground-level emissions
It's hard to compare price because it is very rare for two means to compete directly. It's often apples and oranges. A train that moved as fast as plane isn't going to be very fuel efficient. They can be electric (insert source debate) but planes can also run on biofuel (another debate). And when comparing plane-train, a plane burns much of its fuel taking off and climbing to altitude. Once there (10,000m) it is remarkably fuel efficient in terms of passenger/mile at a given speed. And there are times when rail simply won't ever be an option (over large water). The head of RyanAir gave a great presentation a couple years ago about highspeed rail. His point was that, while great if you live in london and vacation in Paris, most people don't live downtown, that planes can land in places much closer to actual destinations --> reducing total-journey emissions.
I'd love a high-speed rail from Vancouver BC to Calgary Alberta, or Prince George (both places I go every year). But building such a structure is a monumental task (10s of billions) and a fast train would take a vast amount of energy covering that distance. Imho trains just cannot compete in such markets.
I take the train everyday. For me, travelling by train for < 4h trips to a city sounds like a good deal.
It is generaly a smoother experience than taking a low-cost flight. But, I realize it really depends on how 'car-first' and large your country is. Definately not an universal solution.
