In the catalog settings dialog window check "Automatically write changes into XMP". Even if these are DNG files without that checked I have that same prompt. Once that is checked I don't get that arrow.
Ctrl+S (save) writes metadata for all the currently highlighted images (except for virtual copies). I expect that's Command+S on Mac.
This is a completely optional Save process, of course.
[Where Lightroom is providing your front-end, Bridge can and should be left untouched IMO. Because actively involving both Bridge and Lightroom on the same images, is a recipe for unnecessarily messy AFAICT. A large part of LR's purpose and point, is to relieve the user from micro-managing what occurs on disk - and not, to complicate that by creating extra tasks.]
So I agree with the other poster that turning on "Automatically write to XMP" covers this aspect perfectly fine; achieving an immediate and silent "reassurance backup" for your work, as you go along.
(edit: this Auto option only writes the metadata for new further edits, carried out after you have turned the feature on. Prior unwritten edits, would still need to be manually written with Ctrl+S.)
(edit 2: the way the Library module works, it may be necessary to highlight the images in Grid - multiple image - mode, rather than highlighting them in the Filmstrip while using Loupe - single image - view.)
I work almost exclusively in RAW (DNG), but I had just shot a rare set in JPG. For some reason LR is telling me that the metadata has been changed by another application (I haven't used any other program!!). Following the above procedure briefly removes the arrows/notifications, but all return after a few seconds (working in Library/Grid module). Should it work with JPGs?
There's an intermittent bug in this area. The normal remedy is first to select all the affected files (your Jpegs in this case), do Ctrl/Cmd + S to write metadata to the files, when that's finished then from the menu bar do Metadata>Read Metadata from Files.
Thank you. I'll try that.