There's been a lot of discussion about transmissions in this thread. Some maintain that MS would gain a lot by adding a transmission, and someone said that "with an electric motor, there's many ways to skin the cat". There are purely electrical ways to skin this cat:
The purpose of a transmission is to keep the motor in its sweet spot, allowing it to deliver full power. If you can efficiently deliver full power at any speed, then you don't need a transmission. In an ICE, the transmission has the additional purpose of allowing you to reduce fuel consumption when you don't need power, but that's not needed in an EV anyway.
The ideal motor delivers constant power from 0 to infinite speed. This is obviously impossible, torque will be limited by tire grip, and at some speed the motor will tear itself apart. A more realistic ideal is a motor that delivers as much torque as the tires can handle until max power is reached, and then stays at max power up to the rpm limit. In the constant power region, torque is inversely proportional to speed, so torque must necessarily drop as speed increases.
The battery has a specific voltage, and there is a limit to how much current it can safely deliver.
The motor also has limitations, and it behaves in a certain way. Torque depends only on current - if you pour more current through it, it will pull harder. More amps also mean much more heat (proportional to square of current), so to avoid melting the windings, including those in the difficult-to-access rotor, you have to be able to remove the heat at a sufficient rate.
As the motor spins up, it starts generating its own voltage, opposing the voltage fed to it. This is called "counter electromotive force" or "back emf". This means that the effective voltage across the motor will drop as the rpms increase if the supplied voltage is constant. This causes less current to flow through the motor, or rather, it gets harder and harder to push the current through the motor as its speed increases.
So at low rpm, the system is limited by how many amps can be sent through the motor without either melting it or blowing up the battery. Full battery voltage can't be applied, that would result in too much current, and this means that you can't get full power.
As rpms increase, due to back emf at some point the effective voltage drops below the voltage required to push max amps through the windings even when you apply the full battery voltage. Torque starts to drop. Then you have almost the reverse situation - you have the capacity to deliver more amps, but can't, because the battery voltage is too low.
When Tesla decided to drop the transmission, they made up for it in a very clever way. They realized that they weren't using all the available voltage - significantly less than battery voltage is required to push max amps through the motor at low rpms. So the battery would not deliver max power (i.e. max volts * max amps) in the lower rpm range. They fixed this by adding a voltage converter that traded unneeded voltage for more amps. In other words, the battery would supply something like 600 A at 500 V into the voltage converter, which would output something like 900 A at 333 V into the motor. Those numbers are pure guesswork and just an example.
When back emf from the motor resists the applied voltage so much that max battery amps can no longer be forced through the motor at the battery voltage, one could add another voltage converter to increase voltage to counter the back emf. In principle, nothing prevents you from continuing to push the full battery power through the motor up to the rpm limit - albeit at lower amps and higher voltage. Torque would still drop, but not as much, and the power curve can stay flat, instead of peaking.
For all I know, maybe Tesla already does this, and what we see in the drag strip videos is just the result of lower max power and higher weight of MSP vs M5.
Why add a transmission when the same thing can be achieved with power electronics? I think there are reasons to do that too. If you choose a very long gear ratio and make up for it with more current, you will also increase heat loss, so I think you will lose some range. This could be countered by increasing the system voltage. On the other hand, a transmission also wastes some energy. I'm sure there are many more tradeoffs in motors, power electronics and economics that must be considered. But as the Roadster and Model S have proved, the transmission is already almost as obsolete as the ICE, at least for automotive use, and there might still be room for improvement.