Eloise was Daniel Faraday's lab rat at Oxford University's Queen's College Physics Department in 1996. Daniel explained to Desmond that the rat's consciousness was sent forward in time one hour in the future, allowing the rat to run an unseen maze perfectly. The experiment was a success, but later in the day, Eloise died. The cause was unknown, but Faraday hypothesized she had suffered a brain aneurysm. According to his theory, not having any constant, her consciousness couldn't differentiate between the future and the present.
An Oxford janitorial custodian had the job of incinerating Faraday's dead subject rats so that no one would discover the purposes and results of his experiments. One would assume that order was at the direction of Widmore, who funded Daniel's Oxford experiments.
Widmore's only purpose was finding a way back to the island. So funding Daniel's experiments had to be tied to this purpose. Was it an actual means of finding the island, or was it payment to Daniel's mother for her help in finding the island for him?
There was some connection in Daniel's mind in conscious time travel and the island. When he first arrived on the island, he observed that the light was different. He also found that time would physically skip (after the FDW was turned by Ben). It was during one of those time skips where he met his mother, who was a leader of the Others. During that encounter, Eloise killed him. There is a debate whether at that moment Eloise knew she killed her (unborn or) time traveling son. But in later events, it would seem that Eloise was doing everything in her power NOT to awaken Daniel to his island's past when he was in the sideways (dead) world.
One of the problems with the LOST story was that it had no consistent mythology on time travel. The idea of mental time travel, based upon premonition theories, was one explanation to Daniel's research. But then, the characters physically time skipped to different years on the island. And further inconsistency, only some of the characters skipped while others did not. Selective involuntary mental and physical time travel is a confusing bunch of story line rubble.
We do not know the details of Daniel's lab rat maze experiment. One could think Daniel used highly charged electric magnetic energy to induce some mental time travel in his lab rats. If so, how can he get "inside" the rat's mind to teach it a maze if the rat itself was unconscious? It would make more sense that both the rat and Daniel physically time traveled an hour in the future, Daniel taught the rat the maze, then returned to normal time to start the experiment. At one point, Daniel said that he was "out" for 75 minutes "after" the Eloise experiment.
There are several theories on why Daniel's research was important.
First, if Daniel had harnessed the power of the island (i.e. mental time travel to the future to either see or alter present events), then Widmore would not have needed to return to the island. Since Widmore still demanded to invade the island for its power, we must assume that Daniel's machine was a failure (as shown with the vegetable state of his former lab assistant.)
Second, if the secret of the island was mental time travel, then why did only Desmond have known episodes of it? He was the person who unrealistically survived the massive EM explosion of the Hatch. This allegedly triggered his premonitions (though some of them proved false, such as Claire leaving the island on a helicopter). So if Desmond could see the future, but he could not control it, who could?
Third, if Desmond was the only one aware of this secret power of mental time travel, why did physical time travel harm Charlotte but not the others. The reason was she did not have a "constant" in both time periods. But in reality, a person has a constant in all time periods based upon family bonds alone. Unless, the time travel was before she was born then she had no one living in that time period. But if that was the case, the other time travelers should have succumbed to the time travel constant rule. Another explanation was that those time travelers had each other as constants in both time periods. But again, so did Charlotte. (Without a consistent rule, it comes down to the writers looking for red shirts to kill off for dramatic effect without realizing the story line consequences.)
Fourth, the cause of death of the lab rat and the time travelers who bled was that if a person failed to complete the events in where the conscious mind traveled to would cause death. For example, Eloise the lab rat Eloise died because the "future" event (instruction on how to complete the maze) that was to occur did not take place. (However, we were told she did complete the maze. So just the fact one mentally flashes to another time is enough to kill you?) Minkowski on the freighter died because he went to the island. The island somehow split his consciousness. But his mission was just to go see the island, which he fulfilled. Minkowski never indicated afterward in his flashes that he needed to complete some task. He was told he had no constant between his flash periods. But again, he did have family bonds in both time periods so that "ante dote" does not have a factual basis. He was merely randomizing into the state of dementia before dying of a brain aneurysm.
But then again, time travel and its alleged effects were immaterial and irrelevant to the island events conclusions. Desmond, as a weapon against MIB, was a non-factor. It was Jack who reset the island cork in the light cave. It was Jack and Kate who defeated MIB. Time travel or electromagnetic power did not destroy MIB or allow any survivor to escape the island.
Based upon the ending, was the concept of "constants" really the mental bridge between the island world and the sideways after life existence? If so, then that would assume that the person's soul (conscious and subconscious) were somehow split to be somehow "reunited" (i.e. awakened) in the sideways world. Time travel is then not time travel but spiritual realm travel.
But even that is inconsistent in the sideways world. Charlotte is in the sideways world. She hooks up with Sawyer through Miles (and Miles was the constant between the island patrol from the freighter with Daniel and Charlotte.) But Charlotte has no contact with Daniel in the sideways world. It is like Eloise is keeping Daniel's island experiences away from his conscious memory . . . that is why she berated Desmond to leave her and Daniel alone. She was willing to sacrifice Penny to move on in the after life, but not Daniel. It would seem that Eloise's guilt for killing Daniel in the island time skip flashed through to her after life fantasy life. Is the ticking time bomb for Daniel's awakening just the sight of Charlotte? If so, why did it take so long for Charlie to reconnect with Claire? Or Kate not remember Sawyer from her island trysts? Or Jack not recognizing the Locke's O6 death drove him nearly insane?
If LOST was a maze of clues, it had no logical exit point. The feature of the "constant" as being a critical requirement to understanding the physics and dynamics of the island does not hold up to any scrutiny. It was neither the cause or the effect of the events in the sideways world.
