Why is energy the time counterpart of momentum

[lolgarithms]

why is energy the "time" counterpart of momentum

my question is. how did they get the time component of the 4-momentum vector to be energy.

i try to derive it to really find out but i can't .

 

[Hepth]

Well, I'm not sure the historical justification, but if you think about it, momentum is the generator of spatial translations (3 momentum), and energy/the hamiltonian is the generator of time translations, if you can call it that. How a wavefunction changes in time depends on the energy of the system very much like how its position depends on its 3-momentum. So the 4 momentum is really just a set of the 4 space-time translation generators.
Look up generators.

 

[Fredrik]

Hepth's answer is accurate, but it requires that you understand either Noether's theorem of classical mechanics, or quantum mechanics and representations of the Poincaré group on the Hilbert space of state vectors. A good place to learn about the latter is chapter 2 of Weinberg's QFT book. Noether's theorem says that there's a conserved quantity for each continuous symmetry of the Lagrangian. The components of momentum are the conserved quantities associated with translations in space, and energy is the conserved quantity associated with translations in time. Unfortunately, I have only seen this done for non-relativistic point particle theory and relativistic field theory. I have never seen it done for relativistic point particle theory.

If you're looking for an answer that requires less mathematical sophistication, I think you'll have to settle for the fact that it seems reasonable to define four-momentum as mu, where m is the (rest) mass and u the four-velocity. Think of it as a tentative definition that turned out to be useful. This reduces your problem to understanding why four-velocity is defined the way it is, and why the definition is useful. (Why is four-momentum conserved in classical particle interactions?). This post (along with this correction) can probably help you with the first part.

 

[tiny-tim]

my question is. how did they get the time component of the 4-momentum vector to be energy.

i try to derive it to really find out but i can't .

Hi lolgarithms!

Simple answer …

if you accept that the other 3 components are the 3-momentum,

then we know that 3-momentum is conserved, so, if it's a genuine 4-vector, the 4th component must be conserved also …

and that's a scalar, and the only velocity-related scalar that's conserved is energy, so it must be (a multiple of) energy!

 

[Dale]

I like a step-by-step from the basic position 4-vector following classical physics.

First, we start with the four-position (ct,x,y,z) which is the relativistic (technical term is "Lorentz covariant") analog of position. We note that the norm of the four-position is an invariant quantity and we call it proper time to distinguish it from the coordinate time.

Now, we want a relativistic analog of velocity which is the time derivative of position. But in relativity we can choose from either the coordinate time or the proper time. Since a relativistic quantity must be Lorentz covariant we must use the invariant proper time in order to get a relativistic velocity analog which we call the four-velocity.

Finally, the rest mass is also a relativistic invariant so if we multiply the Lorentz covariant four-velocity by the invariant rest mass then we get another Lorentz covariant quantity. Now, if we examine the spacelike components we notice that they have the same formula as the relativistic momentum. This isn't too surprising since we took the rest mass times the four-velocity, so we will call this new quantity the four-momentum. If we examine the timelike component then we notice that it has the same formula as relativistic total energy (divided by c), so we say that energy is the timelike component of the four-momentum.