In 4-momentum, why is E the 4th component?

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Discussion Overview

The discussion revolves around the concept of 4-momentum in the context of special relativity, specifically addressing why energy is considered the fourth component of momentum. Participants explore mathematical relationships and the implications of using different terms like velocity and 4-velocity.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that if c is a conversion factor between space and time, then there seems to be an inconsistency when trying to derive energy as the fourth component of momentum, leading to a request for clarification on the mathematical proof.
  • Another participant argues that a change in velocity represents a rotation between dimensions, implying that dimensions are affected differently, which may relate to the understanding of 4-momentum.
  • A participant expresses uncertainty about the use of the term "velocity" in the mathematics, questioning whether they should use "4-velocity" instead.
  • There is a suggestion that to show energy as the fourth component of momentum, one might need to demonstrate that the four-velocity is a 4-vector and consider the product of mass and four-velocity.
  • One participant mentions a simple relation involving energy, momentum, and invariant mass, suggesting a mathematical basis for the discussion.

Areas of Agreement / Disagreement

Participants express various viewpoints on the mathematical relationships and terminology used in the context of 4-momentum. There is no consensus on the best approach to demonstrate that energy is the fourth component of momentum, and multiple competing views remain.

Contextual Notes

Participants reference mathematical transformations and relationships without fully resolving the steps or assumptions involved. The discussion includes uncertainty regarding the terminology and the mathematical framework necessary for understanding 4-momentum.

jaguar7
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Assuming that c is a "conversion factor" to convert between space and time,

Then, in 4-vector, we have x_1 through x_3, and t, where, x/c = t

x/c = t, (where t = time, c= lightspeed, x = spatial dimension)

If we do what we did to space to get time, to momentum,

p/c = m*v/c = m (x/t) / c = m(x/c)/t = mt/t = m

we actually end up with mass... not energy...

So, there is an inconsistency, and I must have made a mistake somewhere. Can you please describe the method and mathematical proof and context for calling energy the component of momentum that is in the time dimension?

Thank you very much. -- j
 
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hi jaguar7! :smile:

you're assuming that a change of velocity should affect different dimensions equally

but a change of velocity is a rotation between two dimensions (t and x, say), and its matrix is
Code: 
cosh  sinh
sinh  cosh

just as a rotation between two space dimensions is
Code: 
cos   sin
-sin  cos

… a rotation does not affect dimensions equally! :wink:
 
Hi, tiny-tim. Thank you for your response.

I'm afraid I'm not very good with matrices. I suppose I'll have to review that. I've been looking for my old books. They've been mysteriously difficult to find after I moved...

I understand that a change in velocity is a rotation between dimensions.

Does that mean I can't use the term velocity in the mathematics? Or that I must use a "4-velocity"? I'm not sure how I would go about doing that... though I would very much like to learn how, somehow... :p

Thank you, again, tim. -- j
 
Last edited:
hi jaguar7! :smile:
jaguar7 said:
Does that mean I can't use the term velocity in the mathematics? Or that I must use a "4-velocity"? I'm not sure how I would go about doing that... though I would very much like to learn how, somehow... :p

i don't think I've ever seen the term "4-velocity"

(4-momentum and 4-force, yes)
 
Thanks. =)

Still, though, how would one go about showing mathematically that energy is the 4th component of momentum, I wonder...?
 
jaguar7 said:
Thanks. =)

Still, though, how would one go about showing mathematically that energy is the 4th component of momentum, I wonder...?

You'd just want to show that (E,P) transformed as a 4-vector using the lorentz transform for an isolated point particle.

To do this, you might first prove that the four-velocity is a 4-vector, then consider the prodect mass * 4-velocity, where mass is the invariant mass.
 
jaguar7 said:
Or that I must use a "4-velocity"? I'm not sure how I would go about doing that... though I would very much like to learn how, somehow... :p

tiny-tim said:
i don't think I've ever seen the term "4-velocity"

(4-momentum and 4-force, yes)

http://en.wikipedia.org/wiki/Four-velocity
 
I think it is a fourth component because of the simple relation
E2-P2=m02,c=1 I have put.
 

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