I Does Physics Allow for a Time Component in Vector Quantities?

[Delta2]

Are there any theories in physics that allow for a time component of the various vector quantities besides the x,y,z components? For example the velocity of a particle to have a time component $v_t$ besides the x,y,z components $v_x,v_y,v_z$

 

[Dale]

Yes, this is common in relativity. They are called four-vectors. For example, energy and momentum together form a four-vector where energy is the time component.

https://en.m.wikipedia.org/wiki/Four-vector

 

[Delta2]

Yes, this is common in relativity. They are called four-vectors. For example, energy and momentum together form a four-vector where energy is the time component.

https://en.m.wikipedia.org/wiki/Four-vector

I knew about this but this is not quite what I was thinking.. For example we put the energy together with the momentum as a 4-vector for reasons that suit our computations and equations to be expressed in a compact and elegant form. I mean i just view Energy as just the 4th component of the energy-momentum 4-vector. OR is there some deep conceptual reason that you called the energy the time component of the energy momentum vector?

I expected the time component to be defined in some sort of very special way...

 

[Dale]

I mean i just view Energy as just the 4th component of the energy-momentum 4-vector.

I don’t see the difference with this and what you said previously

I expected the time component to be defined in some sort of very special way...

The time component is just the one with the opposite sign in the signature.

 

[PeroK]

I knew about this but this is not quite what I was thinking.. For example we put the energy together with the momentum as a 4-vector for reasons that suit our computations and equations to be expressed in a compact and elegant form. I mean i just view Energy as just the 4th component of the energy-momentum 4-vector. OR is there some deep conceptual reason that you called the energy the time component of the energy momentum vector?

You can't transform 3-momentum between frames. To put it crudely, the "energy" in one frame is made up of some of the energy and some of the momentum, as measured in another frame. That's pretty deep.

 

[Delta2]

The time component is just the one with the opposite sign in the signature.

Huh? Excuse me I don't understand. I guess deep special humour hehe?

You can't transform 3-momentum between frames. To put it crudely, the "energy" in one frame is made up of some of the energy and some of the momentum, as measured in another frame. That's pretty deep.

Ok fine, that's an interesting property, the ability to transform 4-momentum between different frames. I was expecting that you would tell me that conservation of the 3-momentum relates to the translational spatial symmetry, while conservation of energy relates to time symmetry, that's another deep reason I can think of.

 

[Dale]

I was expecting that you would tell me that conservation of the 3-momentum relates to the translational spatial symmetry, while conservation of energy relates to time symmetry, that's another deep reason I can think of

Yes, that is true too.

Huh? Excuse me I don't understand. I guess deep special humour hehe?

Sorry, I mistakenly assumed since you knew about four-vectors you also knew about signatures. I am not sure now what you know and what you don’t, so please forgive me if I under or over explain.

In relativity everything stems from the metric. In an inertial frame (and in units where c=1) the metric can be written $ds^2=-dt^2+dx^2+dy^2+dz^2$. As you can see, there are three terms with a + sign and one term with a - sign, so this metric has a (-+++) signature. The only thing that distinguishes time from space is that there is only one time component and the signature is opposite.