Q:

How to transport gigawatt-level power across a 100km distance?

I'm investigating the idea of transporting gigawatts of electrical power to the first stage booster of an electrical rocket.
The first stage booster, like Falcon 9's recoverable first stage, stages at 70km altitude, 60km downrange, for a diagonal distance of 92km.
I'm using 100km for a nice, round figure.
If we use a 1000 Isp electro-thermal rocket to launch a 10 ton payload with 25 ton upper stage and 20 ton lower stage (ignoring engine and fuel tank masses), and assume a TWR of 1.2 at liftoff, we would need 3.17GW of power at liftoff.
How can this level of power be transported over a wire 100km long? Low mass is critical.

A:

By wire is going to be extremely difficult. Let us first have a look at the required cross section of the wire:
$$A = \frac{\rho lP}{V^2}$$
We can use the resistivity value of graphene for a very optimistic scenario. Even with a voltage over 20kV, the wire is still going to outmass the rocket by far. Not to mention the wire has to tolerate exposure to rocket exhaust.
edit: The above calculation is possibly incorrect, as pointed out by Uwe.
A simpler argument is that for the wire not to outmass the spacecraft, it can be no more than $0.5kg/m$. Given the size of high voltage wires carrying an order of magnitude less power, that is simply not possible.
/edit
Beaming up the power, as others have mentioned, is not likely to be any more successful, as current laser technology simply is not sufficient. Continuous lasers are at the moment only a few kW, roughly five orders of magnitude short.
Having the energy source onboard, however, is not as impossible as it sounds. A Nuclear thermal rocket may be something you want to look at. Reactor tests at your magnitude of power were conducted during Project Rover.

