Well, you have two paths: implicit or explicit.
An implicit approach would be to rely on implied affordances and user's past experiences. Thus, you could rely on the user noticing the scroll bar denotes additional content, or simply make your screen height's value
90vh. Needless to say these approaches aren't the best UX.
On the other side, an explicit approach would require an explicit (doh!) element, such as text or graphic elements, like the arrow you mention, or a + sign or whatever your creativity brings. And creativity is the keyword here, because your question comes from the fact that somehow you don't like an element, and obviously nobody but you can tell why.
But be very careful: a good UX designer must know when to back off from his/her own taste to apply the best possible solution. So if you want to go crazy... Do it! By all means, it's extremely boring to see everybody using the same rules and techniques, just make sure to test, and if everything fails.... Just back off and do like everybody does