How can I handle physics in a multiplayer shooter

Hi everyone!

I've already heard that the physics in Unity do not behave equally in all computers, and that could be a problem in a multiplayer game.

So I kinda decided that I will do a basic physics by myself, so the physics will behave equally in all computers, and to ensure everything is syncronized, the server will send updates from time to time.

The game will be simple, not too much dynamic objects (probably only the players). So I started to ask myself if it will worth the time to make a hole new physics system... Could somebody help me?

I never have developed I multiplayer before, but I'm a little bit concerned about lag. So the server will send the clients updates from time to time and make client side prediction. However, the server will decide wether or not someone kill a player.

And I'm not concerned about hacking now, so that's out of question.

Thank you for any help!

TL;DR: Is writing a new physics worthwhile to don't have issues with lag or client/server discrepancy?