Maybe this can help someone else. I had the same question as you but for a normal user. Let's say I want to start firefox using the user account foo. I'm logged in as bar:
[bar@localhost ~]$ sudo -u foo -H firefox
Sadly that command failed with the same error as in the question (i.e. no protocol specified & cannot open display)
My solution was to simply add the user foo to the list of authorised access to the X server.
And that was it, I was then able to launch Firefox (and other X application) using
sudo and the user foo.
Background: On X Window, there is a client/server architecture. When you launch an application you request the X server authorisation to display it. By default once you open a session (you graphically login), you (your user) are obviously allowed to commmunicate with the server and display applications. Other users do not have this permission unless you specify it.
xhost is a tool to manipulate the list of permissions. The
si indicates that the rule is server side and it authorise the local user
foo to display applications. X Window is very powerful in this regard and you can display remote applications locally by playing with the
DISPLAY environment variable and
xhost (but not limited to them). In older times, when people typed
xhost + and implicitely allowed everyone to use their X session, it was possible to display application on their screen for pranks ;-) not so much nowadays as people are less and less using X Window client/server architecture (at least for what I observe in the past 10 yers).
I hope this helps others.
PS: I did this in order to launch Firefox in a kind of "jail" (to avoid a vulnerability like for pdf.js in the future). But I quickly found out that calling Firefox via sudo won't allow it to access audio nor the video hardware. But there is one guy which explain clearly how to activate video hardware acceleration and audio when calling Firefox via sudo. YMMV with these instructions, e.g. I still have a permission denied with audio but video is fine (tested on Fedora 22 with SELinux ON).