Can I paint over a gloss door with a water based satin without any primer undercoat

yes you can, you will need to key the existing paint with 80/100 grade sandpaper then apply 2 x coats of w/b satinwood, rubbing down in between coats.

best result use sanding machine it make the wood more smooth rather than hand

No electrical sanding or sanding with 80 grit A light sand for key with 180 or 150 grit yellow. Going at it too agressively can cause the gloss finish to break up especially electrically sanded. clean thoroughly. Then an adhesion primer. Not an undercoat as such (no build as its already finished) Painting over a previosly well glossed finished work with prep as above often results in a very good finish

It's not a black and white answer. Some satin products may have better adhesion than others and as long as the existing gloss is keyed the satin should stay put. But it wont look as good as it would if you use an undercoat. You can always speed up a job but the result wont be as good. Things to consider are, if the old gloss was oil based, it may have gone yellow. If you don't seal it properly with a good primer/undercoat, that yellow can still come through in time. Secondly paint adhesion. A high adhesion primer/undercoat is less likely to chip off when it gets knocked. Satin straight on the gloss may not adhere as well and the slightest knock might damage it.

Always use a good primer or undercoat even when using a water based satin. You will get a better and more longer lasting finish