lost photos after time machine back up

Have you been using iCloud Photos with "Optimise Storage" enabled? Then Time Machine will only have backed up a subset of your photos and you need to enable iCloud Photos again to restore the missing photos from iCloud.

Are you trying to open the Time Machine backup of the library on the TM drive? That can't be done. You must restore the library to your boot drive or another volume and open it there. Here's how to restore the library:

Did you have more than one Photos Library? You may want to check, if Photos has opened the correct library after you restored your Mac from the backup. You can see, which librar Photos is using by looking at the Photos > Preferences > General > Library Location.

To switch between Photos Libraries quit Photos, then hold down the options key ⌥ while launching Photos again. Release the options key ⌥ when the library chooser dialog appears and select all libraries in turn.