Short answer: there's no way but AJAX + a server-side script (in your case, jQuery + php)
Being a client-side script, javascript gets run on the client's computer, so if the file whose m-time you want to check is on the server, then you are correct to use AJAX and a server-side script. No other way will work.
If the file whose m-time you want to check is on the client's computer, then you're out of luck. Javascript is intentionally designed to be prevented from accessing the client's files. (It can only access cookies, which are on the client's computer, however, because the browser (not any javascript) loads those into its work environment.)