That isn't a tesla coil. It appears to an ignition coil hooked up to a spark gap (hence "you can hear it in an auditorium") and directly linked to the batteries from there. It's classified as a DC transformer.
If you're going to use a transformer based circuit then AC is needed to work. The principle, in general, is that a coil of wire will create a small EMP when the current changes direction. DC circuit will not do this so energy cannot be transformed via wire coil. Solid state systems can use DC because they don't use transformers. But as far a potato gun ignition is concerned, it doesn't really matter. But remember, DC current is much more deadly than AC current, that's why they use it for the electical chair.
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