Why "time part" represents energy in Four-momentum?

[PLAGUE]

TL;DR

Four-momentum has four parts. But why it is energy that is represented by time part?

I was going through Spacetime Physics by Taylor and Wheeler and came to a point where they said, and I quote,

In what follows we find that momenergy is indeed a four-dimensional arrow in spacetime, the momenergy 4-vector (Box 7-1). Its three "space parts" represent the momentum of the object in the three chosen space directions. Its "time part" represents energy. The unity of momentum and energy springs from the unity of space and time.

This part feels too abrupt for me and I am looking for a more elaborated explanation.

Here is a link to that chapter.

 

[Ibix]

It's a definition of energy in the context of relativity, effectively.

If you like, you can observe that the four momentum for a massive object is its rest mass times its four velocity. The zeroth component is therefore $\gamma mc^2$. You can Taylor expand that and show that it reduces to $mc^2+\frac 12mv^2$ when $v\ll c$, linking it back to the Newtonian concept of kinetic energy plus a constant term that we don't care about in Newtonian physics.

 

[Dale]

From Noether’s theorem we know that there is a conserved quantity for every differential symmetry in the laws of physics. Momentum is the conserved quantity associated with space translation symmetry. Energy is the conserved quantity associated with time translation symmetry.

So when you combine space and time into spacetime you have a spacetime translation symmetry. That leads to a conserved four vector. So it should come as no surprise that the spacelike part of this four vector is momentum and the timelike part is energy. It really couldn’t be any other way given both Noether’s theorem and the fact that special relativity must reduce to Newtonian physics in the non-relativistic limit.

 

[robphy]

TL;DR Summary: Four-momentum has four parts. But why it is energy that is represented by time part?

I was going through Spacetime Physics by Taylor and Wheeler and came to a point where they said, and I quote,


This part feels too abrupt for me and I am looking for a more elaborated explanation.

Here is a link to that chapter.

Presumably, you seek an answer using the storyline that you are following,
as opposed to another viewpoint that develops relativity in a [possibly completely-] different way.

What did you think about the preceding subsection 7.1, particularly the Question and Answer?

 

[PeroK]

So it should come as no surprise that the spacelike part of this four vector is momentum and the timelike part is energy.

I was surprised. I'd never heard of Noether's theorem when I was learning SR.

 

[Dale]

I was surprised. I'd never heard of Noether's theorem when I was learning SR.

Thanks for the reminder, maybe that was phrased poorly. I meant that given Noether’s theorem and the classical conservation relationships of space with momentum and time with energy, once you have that then it is not surprising that momentum gets automatically connected to energy

 

[Sagittarius A-Star]

Four-momentum has four parts. But why it is energy that is represented by time part?

The energy appears as factor in all 4 components of the 4-momentum. You can see this in the following notation:
$$\mathbf P= {E\over c^2} {d \over dt} \begin{pmatrix}
ct \\
x \\
y \\
z
\end{pmatrix}$$