The PS4 won't play Gran Turismo 4. Never Ever.
Your best bet would be to do the same thing that i did: just get yourself a 60GB PS3 (or any other backwards compatible PS3) and spare yourself all the fiddling with adapters to get the game running on a modern HD TV. Just plug the HDMI cable in and you are done! Even though you'll read a lot of contrary stuff, i'd say that GT4 plays and looks good on a PS3 (besides some flickering in the upper left corner, which can be ignored).
I know these early PS3s are somehow infamous because of their reliability, but i use mine strictly for PS2 games and therefore it won't use all of its power... so it never gets really hot and loud either. Just have that in mind, when you are planning to buy one and come across another horror story
One more thing: loading times are practically non existent on the PS3 (compared to the original PS2)