Who wants it?
> a very large capital investment and so would be worth while only on routes between very large population centres.
I give you the last supersonic transport system between large population centres on this planet, viz. Concorde. It didn't sustain itself, and give rise to successor systems, because not enough people (need | want) & can afford to travel that far that quickly.
Also the physics is a bit dicey. If you're accelerating a train at 0.5g, as others have suggested you need to do to reach 4000 kph before you reach halfway to your destination, then the track experiences the reaction, and can't be built on piddly little concrete pylons. Similarly when changing direction: fast aircraft use a "half-rate" turn of 1.5 deg per second. For a 4000 kph train changing direction from N to NE, say, the turn will take 30s, and to limit the sideways acceleration to 0.5g, the radius of the turn is near enough 28.6 km (17.7 miles). Around the curve , that's about 28 miles of track that has to be braced to support a force equivalent to 122% of the weight of the train. See 'piddly pylons', supra.
I don't say that it can't be done. I just don't expect the engineering investment to be justified by the economics of the business case.
 I've simplified. In reality, you have to turn gradually into the curve, and gradually out again, along an 'Euler spiral'