Wikipedia says,
"In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional space-time. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in space-time."
Does this mean that four-momentum is the type of vector resulting from three spatial dimensions being placed, as a whole, into a greater level of momentum? As in three dimensions traveling along a fourth axis (time), correct? And this extra dimensional relationship is entirely relativistic, correct (I apologize if I am not making myself clear, I am very fascinated with relativity, spending hours on Wikipedia trying to understand it, but I need a teacher to walk me through a few things that confuse me)? What I am trying to say in that last statement I will hopefully make clear in the following example:
A train is traveling along it's track so smoothly that there is no way for its passengers to tell that they are in motion. A man in the train tosses a ball up in the air and catches it. From the perspective of the man, the ball has gone straight up and down. The path of that ball relative to the inside of the train can be calculated using classical three dimensional momentum. However, relative to someone living outside the train, the ball has traveled in an arc, not a straight line. The balls paths (up and down) relative to the inside of the train was a path that could be described on one axis. But, that same path relative to the outside of the train requires two axes to describe (up and down and side to side), and an extra vector.
So the dimensional path of the ball went from one (a line) to two (a plane), and held a new vector given to it by the trains momentum, just by taking our perspective out of the train.
Does this accurately depict relative levels of momentum?
If the answer is yes, then here is where my confusion begins. I can take that same basic scenario and apply it to a man on earth who tosses a ball straight into the air. Relative to the man, the ball went up and down. Relative to the sun, the earth is moving so the ball traveled in an arc (well, the earth is also rotating so that gives the path of the ball another vector and its curvature greater complexity). Yet the sun is in motion, so relative to the space the sun travels through the ball moves in a sort of partial helix. Yet the system of motion containing the sun is contained by an even greater system of motion, and so on, adding further levels of relativistic spatial dimensions to the balls momentum. Will not the relative momentum of the ball quickly reach and exceed special relativity's "4-vector" as we continue to place our perspective into the greater levels of gravity that smaller systems of motion are always contained by?
So my questions is:
How is it that while observing the universe realistically, Einsteins equations for space-time use four-momentum, and not rather five or six-momentum, seven, or even greater levels of spatial dimensions?
What am I not seeing?
My first thought was that Einstein was describing the motion of the entire universe through time, and not the motion of bodies through space, but I then discovered that this is not entirely true as Einstein showed that these two descriptions are inseparable, as time is dynamically linked to motion itself.
My only idea is that the train scenario I described is somehow disconnected from what the building of multiple vectors actually is, and in order to understand relativity I would be incredibly grateful to anyone who could complete my understanding of the above "ball in train" depiction.
Best Answer
All four dimensions are present in both examples. All that you mean when you say that space-time is four dimensional is that you need four numbers to describe when and where an event happens.
The path of a particle is a string of such events--the ball is one inch above my hand, the ball is two inches above my hand, the ball is at it's peak, it's two inches above my hand, etc.
What is novel about describing the two reference frames that you do is that in one, the ball only travels vertically and in time, while in the second, it also has a horizontal component to its motion.