*or-, *r(ei)-

This root is a bit contested, so there is no 100% consensus in the etymological field about its existence.

The core idea of it was:

start moving, start flowing, being aggrevated

The notion of motion is pretty visible in run and its German translations rennen and rinnen as well as in reiten and to ride and reisen (to travel).
The idea of ernst (earnest, serious) came from the notion of being determined, ready to get going possibly for battle. Like… someone who is looking like they’re ready for a fight usually does look pretty serious.  

The branch around arg and Ärger probably started out from the idea of shivering, be it with fear or rage.

Another interesting member of the family is “river“, which comes from the Latin branch and which is related to “rival” – your friendly, competitive neighbor on the riverbank.
And to arrive also belong here, as it originally meant “get to shore”.