Thanks for the suggestion, but i will still need to know when to send the open/close signal and for that i need the proximity to the gate. I do not know how the lawnmower thinks and I do not want to hack the lawnmower. ;)
The lawn mower may have a predetermined path, or it may moving randomly. In both cases, the mower is likely to get near to the gate without wanting to go through it (e.g. when mowing the grass in the vicinity of the gate). So besides detecting the mower's presence, you still need to have a way to find out when the mower wants to go through that gate, or when it is just passing by.
My first thought was indeed also in the lines of Bluetooth or similar short-range wireless connection, and the mower actively asking the gate to open for it, when the mower is done with one half of the property and then wants to move on to the next.
If the gate is inside a small alleyway or so, and the mower only gets in there to go through the gate, you may consider installing an IR beam in front of the gate. If that beam is broken, the gate can send a short range wireless request to the mower to confirm the mower's presence, and open the gate upon a positive response. If the mower is out of range (this of course should be limited to maybe a few meters, or whatever you think appropriate) the gate doesn't open as it's probably the kid, the dog, or a bird.