What is wrong with the following equation?

Correct. It defies Newton’s Third Law, but that’s fine because I am Accelerator! When Jeka showed me To Aru Majutsu no Index, I thought it was great, it had really lovable characters and interesting story, but when I saw Accelerator–that was the point when I simultaneously went: “You can’t do that” and “I wish I had your power; I love this show”. NOTE: Spoilers follow. 

So Accelerator has the power of “Vector change”; and when I told my friends taking physics, they said, “what’s that supposed to mean”. Although the show is a little vague on it, I’m pretty sure Accelerator has basically the ability to change anything humans can attach a vector quantity to whatever he wants (I believe however, only one of the three examples of types of vectors that characters mention in the show are actually vectors: “Electric Charge, Heat, Momentum”. Now this may be a mistranslation, but only Momentum is a vector. Electric charge could mean electric vector field, but I’m pretty sure heat, which is energy, is scalar). In the show, Accelerator deflects bullets, changes the direction of gravity to make him float, changes the flow of MISAKA’s blood vessels, defies Newton’s Third Law by switching the reaction force as in the above equation (he actually explains it) and (this is the part where Vendredi and I went “errrrr”) forces the wind’s direction vector to point inwards causing the air molecules to condense so much such that it changes into plasma!

Vector change...yep

Vector Change...yep

I have to admit that this is perhaps the most creative and unbelievably cheap power ever conceived. I don’t think Accelerator’s powers have been fully explored yet–even at his Level 5 power level, from what I’ve seen, he could do so much more; literally, vectors are everywhere–we’re talking about controlling all of physics. 

  1. Deflecting Bullets. We know at least that Accelerator has a great deal of control over his power, at the very least able to passively deflect many bullets; which in itself requires a lot of force to change the momentum vector of the incoming bullet in a short period of time (note that in the episode where MISAKA fires at Accelerator, the bullet is deflected almost instantly). 
  2. Changing direction of gravity. One can only wonder how the entire world is affected by Accelerator’s slight changes in forces–it may have catastrophic consequences. Anyway, I presume when Accelerator says changing the vector on his feet, he means the force of the earth upon his body–by changing the downward vector forward, he can almost move as fast as he would when falling. Although, it doesn’t make much sense that he was moving at a constant velocity–maybe he was changing the initial gravitational force direction first, then destroying it completely so he moves at terminal velocity. 
  3. Changes the flow of MISAKA’s blood vessels. I wonder why he chose to kill her this way–with the amount of power he had, he could have killed her a lot faster; perhaps by increasing the gravitational force upon her, or causing the flow to go outwards radially (explosive!).
  4. Defies Newton’s Third Law. Its pretty interesting that he can keep track of the vectors he’s controlling, since his body can passively control vectors, how is he sure he won’t suddenly change vectors towards his own body?
  5. Plasma. I’m not quite sure how plasma works, so I’m probably not going to attempt it in order to save me from being mocked by my friends. You could refer to Wikipedia‘s page on Plasma-scaling, and find that plasmas have a lot of different parameters and differ by many orders of magnitude. The point is, Accelerator was able to take very, very, thinly distributed gases and compress them into a plasma. The particles are moving extremely rapidly because of the high pressure, and they are incredibly dense.

Ok, now, given a general idea of the degree and magnitude of his power, I think we can safely say that his powers haven’t been exploited fully.

  • His fight against Toma would have been quickly solved by sending wind at (great, great) speeds to slice him into pieces. 
  • He could have, in a single punch, split Toma’s heart. 
  • Assuming he could create plasma, he could probably cause a nuclear explosion by forcing a small amount of hydrogen very close together…
  • At the very least, he has Misaka Mikoto’s power over electricity–this is enough to perhaps utterly destroy the direction of electric field in Toma’s body. Or even better- he can possibly change the direction of Earth’s magnetic field, magnify it and disturb the electric waves in Toma’s brain; or, with enough magnification of the magnetic field, he could probably lift Toma and utterly destroy him, without actually touching him as he tried to do in the show.
  • Alternatively, he could control the gravitational field of the Earth and launch Toma into space.

To summarize, Accelerator has the most awesome, most over-arching power ever. It doesn’t make much sense to say that Accelerator can get any more powerful–we haven’t even covered the possibility that Accelerator can “micromanage” forces between atoms–that would probably make him…completely unbeatable.