While R+L = J is pretty much confirmed, it still remains a theory as far as books are concerned in absence of any explicit confirmation from the author regarding Lyanna and Rhaegar's relation (Or lack thereof).
Here's what we know:
- Rhaegar outraged the whole realm by crowning the maiden daughter of Lord Rickard Stark, betrothed to Lord Robert Baratheon instead of his legal wife Princess Elia of Dorne. Polygamy is a no-no in faith of the Seven which the Royal family and most of the population adheres to. Only Aegon I's polygamy was tolerated by the faith, and when Maegor tried to follow his father he was exiled and declared a heretic, which eventually sparked the Faith's war against crown. And bear in mind, Aegon and Maegor had dragons.
- Rhaegar openly "abducted" Lyanna while she was travelling with her family. King's guards apparently helped him.
- Rhaegar goes off-grid and disappears following the kidnapping for reasons best known to him or his associates.
- Aerys had no idea what had transpired. Even if someone told him his son had abducted the Stark girl, He'd not mind his son taking up a mistress or two, the King was no stranger to carnal pleasures. Hardly a moral man, Aerys.
- Brandon rides to Red Keep and threatens to kill Prince Rhaegar.
- For crime of threatening the Blood of the Dragon, Aerys arrests Brandon and his friends and calls for their fathers to present themselves and explain the actions of their sons.
- Rickard goes to KL where Aerys burns him and kills Brandon and his friends (Except his squire). The seeds of the war are sown.
- Still not satisfied, the mad King commands Lord Jon Arryn to kill his wards Eddard Stark and Robert Baratheon.
- Jon Arryn calls his banners and fighting begins in Gulltown.
- Robert and Eddard make it to their domains.
- Robert defeats his royalist vassals and marches his men to Riverlands.
- Tyrells defeat Robert at Ashford.
- Robert retreats into the Riverlands.
- Lord Jon Conington, Hand of the King, surrounds the town Robert was hiding in and Battle of the Bells happens.
- Robert is rescued by sudden arrival of Hoster Tully and Eddard Stark's men. Royal army is shattered and scattered to four winds. Rhaegar and Lyanna are still nowhere to be found.
- Aerys exiles Lord Jon Conington, commands Prince Doran Martell to send forth his armies, dispatches Sers Barristan Selmy and Jonothor Darry to regroup the shattered remains of Ex-Hand's army, sends Lord Commander Gerold Hightower to find Prince Rhaegar.
- Rhaegar finally returns after Ser Gerold finds him, rides to Ruby Ford and dies.
Relationship between Rhaegar and Aerys
There is no indication that Rhaegar and Aerys had any conversation between the abduction and his return. Furthermore, Aerys did not trust Rhaegar and suspected that Rhaegar was trying to usurp his throne with aid of Lord Tywin Lannister. Rhaegar on the other hand was aware that his father was unhinged and he most likely was in league with Lord Tywin Lannister to become Prince Regent for his father, just as Aerys suspected. It is unlikely that he confided in Aerys.
"Rhaegar fought nobly, Rhaegar fought honourably....and Rhaegar was thick as a brick"
As to why couldn't he send a raven? Did he send a raven? No one knows. I'd not make too much of Rhaegar's wisdom though. Pure speculation, Rhaegar was an idiot when it came to Lyanna.
"All men are knights and all men are fools when it comes to women".
His actions show that he didn't even expect it to blow up that much. He might have expected that if he disappears until things cool down a bit, he might have a chance of explaining himself in front of all the parties concerned.
He possibly didn't even know that his little stunt had caused a civil war until it was too late. Rhaegar had always been dutiful after all, it's unlike him to knowingly shirk off his duty to his family and the realm. Pure Speculation, I don't think Rhaegar told anyone. He was secretive by nature and his only real friend was Ser Arthur Dayne.
Why not go public?
He knew he had acted foolishly but he also knew the cost of going public; Condemnation and excommunication from the Sept (doubly so if he tried to marry her), Wrath of Houses Martell, Stark and Baratheon. Martells would be dishonoured by him taking Lyanna as a concubine just as Starks would be. Baratheons would be angered since Lyanna was promised to Robert. And if Baratheons and Starks are angry, that means Tullys and Arryns are also angry. Rhaegar could not set aside Elia to appease the Starks or take Lyanna as a concubine/second wife/mistress because:
- It would cause Martells to rise up in Rebellion.
- Faith doesn't allow a divorce or annulment if the marriage is consummated. High Septon would have been livid.
- It would sow seeds of another Dance of the Dragons, this time between Rhaegar's children from Lyanna and his children from Elia.
- Wrath of houses Stark, Baratheon, Tully and Arryn.