A passing knowledge in digital forensics informs us of the fact that all activity on machines leaves evidence, just like evidence in a crime scene, that can be detected. This is why the famous "gutman wipe" involves 35 passes of wiping the data. Intuitively, we'd think only one, but at the lower level it's far more complicated.
As for how this applies to your particular situation, I'd have to know what you mean by trustable. However, within the confines of a court, a timestamp that is made by your operating system actually should suffice, save you do silly things which call it into question.
We can see this with existence of the tool timestomp, which is used to counter forensics. Even though with that tool we can change timestamp entires to believable values, this still only will serve to pass undetected through casual glance. A forensics investigator who takes a closer look at a file will be able to tell that it was changed, and conversely, if it has not been.
Therefore, it's fair to conclude that if you present an unaltered file, it would be possible to prove with relative accuracy that it is indeed unaltered.
Now, to tie that all the way back in to truly proving that your entire footage is real, and that each frame is indeed occurring at that exact time: You could grab one of the open-source security-cam softwares that timestamp the footage as it's rolling. It's a standard security cam feature where each frame actually has a timestamp in the image. Since it's open source it can be verified that the software is indeed functioning correctly, and since you're a good boy and having it write straight to the as it goes, we'd be able to reasonably conclude that the footage stored in that file was unedited, and that the frames listed in the footage were accurate.
HOWEVER, nothing is perfect, and I could go into more detail into ways in which you could up it's reliability further. But, for most applications, just having a good open source software which timestamps as it's recording, and leaving it in it's original unaltered state coming right off the software should make it good enough for most cases.