Time in Physics 4-Vectors: Is Time Included in the Position Four-Vector?

[jaketodd]

A quote from an old thread reads "Energy is the time component of the momentum 4-vector"

That quote came from a Science Advisor.

Does this mean that time can be either a) substituted for energy in the momentum 4-vector, or b) seen as equivalent to energy in the momentum 4-vector? Hmm, maybe my a) and b) are the same. Regardless, the question remains. If the answer is "No," then is there a physics 4-vector that includes time? I say "physics" as to discern them from purely mathematical 4-vectors.

Also, when working with physics 4-vectors of the same sort, such as comparing physics 4-vectors of the same sort, do all the constituent vectors combine of a given physics 4-vector? If so, what are the operators of how they combine?

Thanks,

Jake

 

[jfy4]

Well, neither, and I think I see where the confusion comes from. "the time component" is, what I believe, a reference to the coordinate 4-vector x_{\alpha}=\langle ct,x,y,z \rangle where the first component is a time coordinate. Then identifying the first component of each 4-vector as the "time component" simply references the scalar part of the 4-vector.

There are however some amazing connections between time and energy, outside of, perhaps, simple phrasing. One of which is that the energy of a system is identified as the source for temporal-translation.

I hope this helps.

 

[elfmotat]

Neither.

The four-momentum is \vec{p} =(\gamma mc,\gamma m\boldsymbol{v})=(\frac{E}{c},\boldsymbol{p}). It comes from taking the derivative of the position four-vector with respect to proper time and multiplying by the invariant mass. The temporal component is the energy divided by c. The spatial component is the relativistic three-momentum. That doesn't mean time can be substituted for energy (it doesn't make sense, even dimensionally) or vice versa.

 

[Meir Achuz]

The colloquial use of the phrase 'time component' just means the 0 component of a 4-vector in a 0,1,2,3 metric. The only connection with real time is that ct is the 'time component' of the vector x^\mu.

 

[jaketodd]

So there is no physics 4-vector that includes time?

Thanks,

Jake

 

[Meir Achuz]

I wrote that the space-time 4-vector x^\mu=(ct;x,y,z) includes time as its 'timelike component'.

 

[elfmotat]

So there is no physics 4-vector that includes time?

Thanks,

Jake

The position four-vector does. The components are (ct,x,y,z).