Styx

The River Styx (also meaning "hate" and "detestation") is a river located in the Underworld. It circles the Underworld nine times, eventually converging on a great marsh at the center of the Underworld with the rivers Phlegethon, Acheron, and Cocytus(though there is a fifth river: Lethe). According to Percy the river is very polluted, being filled with pretty much anything you could imagine: broken toys, ripped up diplomas etc. The river is filled with broken dreams, lost when the person died. The River Styx is also where Stygian iron is cooled to make it indestructible, as well as where Stygian Ice is formed. As well, Styx was a goddess - similar to the rivers Phlegethon, Acheron, and Lethe being associated with deities - and a daughter of Oceanus and Tethys. She was the wife of Pallas, and bore him Zeus' four winged enforcers: Kratos, Nike, Zelus, and Bia(and sometimes Eos). Styx supported Zeus in the Titanomachy, being the first to aid Zeus in the First Titan War against Kronos and his Titan Army. For this, her name was honored by being used in binding oaths.

Swearing on the River Styx
The River is also part of a powerful promise that can be made between people and breaking an oath on the Styx is extremely unwise for anyone. Gods who break oaths on the Styx apparently "get off lightly" on the account of being immortal such as when Zeus broke the oath between himself and his brothers of having no more demigod children after the diasterous end of World War II. It is unknown what would happen if a lesser entity were to break an oath on the Styx but it is considered unpleasant.

The consequences for breaking an oath made upon it are supposed to be extreme. The penalty for breaking an oath sworn by the Styx is to drink from it, which causes the gods to loose their voices for nine years, and also results in being expelled from the council of gods for nine years afterwards. Despite this, both Zeus and Poseidon avoided this. Since Zeus is the god in charge of enforcing oaths there was no one to force him to go into exile. This was probable brought about by his great arrogance and general dislike for following his own rules. Poseidon's status and power no doubt protected him as well making exiling him too difficult. Ironcially, Hades is the only one of the Big Three that can be trusted to keep his word when there are no direct consequences.

Achilles' Curse
Anyone who bathes in the River Styx will be gifted with the strength, power, speed, and near invincibility of Achilles. The only spot where one who has bathed in the River Styx is vulnerable is in their 'Achilles Heel', a spot of the bather's choosing which will keep them grounded to the real world and their mortal life after they bathe in the river. If the person doesn't focus on that point while bathing in the river, their soul will be incinerated, washed away in the current. Achilles' ghost appears to anyone who wishes to receive the curse, warning them not to. The ghost claims it will affect them highly but shows few signs in Percy.

People known to have bathed in the River Styx
There have only been three people to have bathed in the river so far and lived:
 * Achilles - His mother dipped him into the River by his heel earlier on when he was only a baby boy years before the Trojan War. Therefore, his heel is vulnerable.
 * Luke Castellan - Luke is known to having done so he could be invincible so his body could hold Kronos' spirit. His Achilles spot is under his left armpit. He possibly envisioned himself and Thalia first meeting and starting a romantic relationship and giving his knife to Annabeth Chase.
 * Percy Jackson - Nico di Angelo first suggested to Percy should bathe in the River Styx so that he could defeat the Titan Lord Kronos in the Battle of Manhattan. Percy was first uncertain about this idea but eventually decided to do it. His Achilles's Heel is the small of his back, opposite of the navel. He pictured it like the time when he fell out of the boat and was being pulled out of the water by Annabeth.