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

[jaguar7]

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

 

[tiny-tim]

hi jaguar7!

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

cosh  sinh
sinh  cosh

just as a rotation between two space dimensions is

cos   sin
-sin  cos

… a rotation does not affect dimensions equally!

 

[jaguar7]

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

 

[tiny-tim]

hi jaguar7!

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)

 

[jaguar7]

Thanks. =)

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

 

[pervect]

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.

 

[jtbell]

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)

http://en.wikipedia.org/wiki/Four-velocity

 

[andrien]

I think it is a fourth component because of the simple relation
E²-P²=m₀²,c=1 I have put.