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## Summary:

- How are the spacetime interval and the energy-momentum four vector related? How is velocity in the space-direction reconciled with the fact that the time-direction component of the energy-momentum four vector contains a kinetic energy term if the object is not at rest?

## Main Question or Discussion Point

In a spacetime diagram the spatialized time direction is the vertical y-axis and the pure space direction is the horizontal x-axis, ct and x, respectively.

The faster you go and therefore the more kinetic energy you have, you'll have a greater component of your spacetime vector in the xx-direction. More of your energy and forward "motion" through spacetime is devoted to traveling through space than through time. A consequence is time dilation with which we are familiar.

My question arises because I am confused about how this relates to the energy-momentum 4-vector where the time component of this is mc^2 + 1/2mv^2. The rest mass energy plus the kinetic energy. If the kinetic energy term is rather large, you have a large time component in the energy-momentum 4-vector, but if your kinetic energy is large shouldn't you be traveling "less" through time and "more" through space? There is some subtle disconnect here for me and I would appreciate it if someone could help me to think about this properly.

If the kinetic energy is zero we are left with simply mc^2, the energy that mass has on its own at rest. This tells me that it is the energy that the mass has as it moves through spacetime solely in the time direction. So if you add kinetic energy, the time component becomes larger, and more energy is devoted to travel in the time direction in spacetime. How do we reconcile that with larger energies and velocities means you travel more in the space-direction in the spacetime diagram where we consider the spacetime interval?

Furthermore, would this inability to reconcile this have to do with the hyperbolic geometry of Minkowski spacetime and how it changes the Euclidean Pythagorean relationship to a Non-Euclidean geometry?

The faster you go and therefore the more kinetic energy you have, you'll have a greater component of your spacetime vector in the xx-direction. More of your energy and forward "motion" through spacetime is devoted to traveling through space than through time. A consequence is time dilation with which we are familiar.

My question arises because I am confused about how this relates to the energy-momentum 4-vector where the time component of this is mc^2 + 1/2mv^2. The rest mass energy plus the kinetic energy. If the kinetic energy term is rather large, you have a large time component in the energy-momentum 4-vector, but if your kinetic energy is large shouldn't you be traveling "less" through time and "more" through space? There is some subtle disconnect here for me and I would appreciate it if someone could help me to think about this properly.

If the kinetic energy is zero we are left with simply mc^2, the energy that mass has on its own at rest. This tells me that it is the energy that the mass has as it moves through spacetime solely in the time direction. So if you add kinetic energy, the time component becomes larger, and more energy is devoted to travel in the time direction in spacetime. How do we reconcile that with larger energies and velocities means you travel more in the space-direction in the spacetime diagram where we consider the spacetime interval?

Furthermore, would this inability to reconcile this have to do with the hyperbolic geometry of Minkowski spacetime and how it changes the Euclidean Pythagorean relationship to a Non-Euclidean geometry?