Energy-momentum for a point particle and 4-vectors


by 1effect
Tags: 4vectors, energymomentum, particle, point
yuiop
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#37
Mar8-08, 01:57 PM
P: 3,966
Quote Quote by 1effect View Post
You seem to be missing the point. While you can surely align [tex]u[/tex] with the momentum of one particle, all the other momenta will not be necessarily aligned with [tex]u[/tex].
The point is that you split the momentum of each particle into x,y, and z components and then transform just the x component of each particle.

In practice you would sum all the x components and then transform that quantity. The final transformed momentum of the system is obtained from [tex] ||p '|| = \sqrt{ (\Sigma p_x ' )^2 + (\Sigma p_y ' )^2 + (\Sigma p_z ' )^2} [/tex]
1effect
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#38
Mar8-08, 02:10 PM
P: 321
Quote Quote by kev View Post
The point is that you split the momentum of each particle into x,y, and z components and then transform just the x component of each particle.
yes, this would work
yuiop
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#39
Mar8-08, 02:10 PM
P: 3,966
Quote Quote by 1effect View Post
This is also incorrect. You can easily try it for 2 particles , take [tex]\vec{v_1}=-\vec{v_2}[/tex] and look at [tex]E_1E_2-\vec{p_1} \vec{p_2}[/tex]
You will quickly see that the expression is not invariant.

When [tex]\vec{v_1}=-\vec{v_2}[/tex] the term [tex]\vec{p_1} \vec{p_2}[/tex] is zero which leaves just [tex]E^2[/tex]. It is this quantity, the energy squared of the sytem in the rest frame, that is invariant of the [tex]E_1E_2-\vec{p_1} \vec{p_2}[/tex] expression.

[tex]E_1E_2-\vec{p_1} \vec{p_2}[/tex] gives the "rest energy" of the system and not the rest mass. The only time this expression gives the rest mass is when all the particles are stationary in the rest frame.
1effect
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#40
Mar8-08, 03:16 PM
P: 321
Quote Quote by kev View Post
When [tex]\vec{v_1}=-\vec{v_2}[/tex] the term [tex]\vec{p_1} \vec{p_2}[/tex] is zero

I don't think this one is right :[tex]\vec{p_1} \vec{p_2}=-\gamma^2 m_1m_2v_1^2[/tex]
yuiop
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#41
Mar8-08, 03:22 PM
P: 3,966
Quote Quote by 1effect View Post
I don't think this one is right :[tex]\vec{p_1} \vec{p_2}=-\gamma^2 m_1m_2v_1^2[/tex]
Oops. Your right. I was talking about [tex](\Sigma E)^2-c^2||\Sigma \textbf{p}||^2[/tex]
where [tex]||\Sigma \textbf{p}||^2[/tex] goes to zero, because the momenta are summed before they are squared. [tex] [p_1 + (-p_1)]^2 = 0[/tex]
1effect
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#42
Mar8-08, 04:40 PM
P: 321
Quote Quote by kev View Post
Oops. Your right. I was talking about [tex](\Sigma E)^2-c^2||\Sigma \textbf{p}||^2[/tex]
where [tex]||\Sigma \textbf{p}||^2[/tex] goes to zero, because the momenta are summed before they are squared. [tex] [p_1 + (-p_1)]^2 = 0[/tex]

I don't think this is right either, we were taliking about a system of two particles , so [tex]\vec{p_1}+\vec{p_2}=\gamma (m_1-m_2) \vec{v_1}[/tex].
yuiop
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#43
Mar8-08, 07:39 PM
P: 3,966
Quote Quote by 1effect View Post
I don't think this is right either, we were taliking about a system of two particles , so [tex]\vec{p_1}+\vec{p_2}=\gamma (m_1-m_2) \vec{v_1}[/tex].
If [itex] m_1 = m_2 [/itex] then [tex]\vec{p_1}+\vec{p_2}=\gamma (m_1-m_2) \vec{v_1} =0 [/tex]

Equal masses and opposite velocities is the simplest case and that was what I assumed we were talking about. Also remember that by definition the total momentum in the centre of mass rest frame is zero so if [itex] u_1 = -u_2 [/itex] then [itex]m_1[/itex] must equal [itex]m_2[/itex].
1effect
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#44
Mar9-08, 11:27 AM
P: 321
Quote Quote by kev View Post
If [itex] m_1 = m_2 [/itex] then [tex]\vec{p_1}+\vec{p_2}=\gamma (m_1-m_2) \vec{v_1} =0 [/tex]

.
There was no specification of any "equal masses" anywhere. This is a system of arbitrary particles.
1effect
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#45
Mar9-08, 11:28 AM
P: 321
Quote Quote by jtbell View Post
The quantity of interest here is

[tex](m_0 c^2)^2 = E^2 - (pc)^2 = E^2 - (p_x c)^2 - (p_y c)^2 - (p_z c)^2[/tex]

A Lorentz boost along the x-direction leaves [itex]p_y[/itex] and [itex]p_z[/itex] invariant. Therefore it suffices in this case to show that [itex]E^2 - (p_x c)^2[/itex] is invariant.
Yes, thank you, this would work.
yuiop
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#46
Mar9-08, 11:41 AM
P: 3,966
Quote Quote by 1effect View Post
...
This is also incorrect. You can easily try it for 2 particles , take [tex]\vec{v_1}=-\vec{v_2}[/tex] and look at [tex]E_1E_2-\vec{p_1} \vec{p_2}[/tex]
You will quickly see that the expression is not invariant.
Quote Quote by kev View Post
...
Also remember that by definition the total momentum in the centre of mass rest frame is zero so if [itex] u_1 = -u_2 [/itex] then [itex]m_1[/itex] must equal [itex]m_2[/itex].
Quote Quote by 1effect View Post
There was no specification of any "equal masses" anywhere. This is a system of arbitrary particles.
You did specify [tex]\vec{v_1}=-\vec{v_2}[/tex] so as I said in my last post that specifies equal masses by definition because the total momentum in the rest frame of the particle system is zero by definition.
yuiop
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#47
Mar9-08, 01:01 PM
P: 3,966
Quote Quote by 1effect View Post
There was no specification of any "equal masses" anywhere. This is a system of arbitrary particles.
If you must do it the hard way then for two particles the invariant rest energy of the system [tex](\Sigma E)^2-||\Sigma \textbf{p}||^2[/tex] can be expressed as:


[tex] m_1+m_2+{2m_1m_2(1-u_1u_2) \over \sqrt{1-u_1^2} \sqrt{1-u_2^2}[/tex]

which reduces to

[tex] {4m_1^2 \over (1-u_1^2)[/tex]

when [itex]m_2=m_1[/itex] and [itex]v_2 = -v_1[/itex]
1effect
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#48
Mar9-08, 04:43 PM
P: 321
Quote Quote by kev View Post
You did specify [tex]\vec{v_1}=-\vec{v_2}[/tex] so as I said in my last post that specifies equal masses by definition because the total momentum in the rest frame of the particle system is zero by definition.

Why is so difficult for you to accept when you make a mistake?
yuiop
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#49
Mar9-08, 05:17 PM
P: 3,966
Quote Quote by kev View Post
You did specify [tex]\vec{v_1}=-\vec{v_2}[/tex] so as I said in my last post that specifies equal masses by definition because the total momentum in the rest frame of the particle system is zero by definition.
Quote Quote by 1effect View Post
Why is so difficult for you to accept when you make a mistake?
You have not pointed out where my "mistake" is.

P.S. No one else has said I have made a mistake ;)
1effect
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#50
Mar9-08, 05:20 PM
P: 321
Quote Quote by kev View Post
Dr Greg is right that the y and z components of momentum do not change under transformation if the frame is moving in the x direction and it is correct mathematically and convenient to align one axis of the momentum components with the motion. Doing it any other way is just making life hard for yourself.

[tex]p_x ' = {m_o (v+u_x)\over \sqrt{1-u_x^2}\sqrt{1-v^2}} = { p_x\over \sqrt{1-v^2}}+{m_o v\over \sqrt{1-u_x^2}\sqrt{1-v^2}} [/tex]

[tex]p_y ' = {m_o u_y\sqrt{1-v^2}\over \sqrt{1-u_y^2}\sqrt{1-v^2}} = p_y[/tex]


[tex]p_z ' = {m_o u_z\sqrt{1-v^2}\over \sqrt{1-u_z^2}\sqrt{1-v^2}} = p_z[/tex]
I finally found the time to sit down and derive the momentum transformations.

[tex]\vec{p}=\gamma(v)*m_0 \vec{v}[/tex]

[tex]m_0[/tex] is the invariant mass.

In two dimensions:

[tex]p_x= \gamma(v)*m_0 v_x[/tex]

[tex]p_y= \gamma(v)*m_0 v_y[/tex]

In frame S' , moving with speed [tex]u[/tex] along the aligned x axes:

[tex]\vec{p'}=\gamma(v')*m_0 \vec{v'}[/tex]

[tex]p'_x= \gamma(v')*m_0 v'_x[/tex]

[tex]p'_y= \gamma(v')*m_0 v'_y[/tex]


where :

[tex]v'^2=v'_x^2+v'_y^2[/tex]

[tex]v'_x=\frac{v_x+u}{1+v_xu/c^2}[/tex]
[tex]v'_y=\frac{v_y \sqrt(1-u^2/c^2)}{1+v_xu/c^2}[/tex]

[tex]\gamma(v')=\gamma(v) \gamma(u) (1+v_xu/c^2)[/tex]

so:

[tex]\gamma(v')v'_x=\gamma(v) \gamma(u) (v_x+u)[/tex]

[tex]\gamma(v')v'_y=\gamma(v) v_y[/tex]

so, indeed:

[tex]p'_y=p_y[/tex]

[tex]p'_x=m_0 \gamma(v')v'_x=\gamma(v) \gamma(u) (m_0v_x+m_0u)=
\gamma(u)(p_x+\gamma(v)m_0u)=\gamma(u)(p_x+\frac {uE}{c^2})[/tex]
1effect
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#51
Mar9-08, 05:23 PM
P: 321
Quote Quote by kev View Post
You have not pointed out where my "mistake" is.
[tex]m_1[/tex] is different from [tex]m_2[/tex] :-)
yuiop
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#52
Mar9-08, 05:26 PM
P: 3,966
Quote Quote by 1effect View Post
[tex]m_1[/tex] is different from [tex]m_2[/tex] :-)
In that case refer to post #47
yuiop
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#53
Mar9-08, 05:36 PM
P: 3,966
Quote Quote by kev View Post
You did specify [tex]\vec{v_1}=-\vec{v_2}[/tex] so as I said in my last post that specifies equal masses by definition because the total momentum in the rest frame of the particle system is zero by definition.
Quote Quote by 1effect View Post
Why is so difficult for you to accept when you make a mistake?

You do not seem to accept my definition of the rest frame of a system of particles. If we have a cloud of particles all with random masses and velocities, how would you define the rest frame of that system?

If you were watching this cloud of particles and the cloud as a whole was drifting away from you (ignore any expansion) then would you still claim to be in the rest frame of the cloud of particles?

If the total momentum of the cloud is not zero by your measurements, do you accept that the cloud as a whole will drift away from you?
1effect
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#54
Mar9-08, 05:44 PM
P: 321
Quote Quote by kev View Post
You do not seem to accept my definition of the rest frame of a system of particles. If we have a cloud of particles all with random masses and velocities, how would you define the rest frame of that system?
Why would you care about defining a "rest frame"? The problem is asking you to derive the transformation for the momentum and energy, in any arbitrary frame because you need this for computing the norm [tex]E^2-...[/tex]

If you were watching this cloud of particles and the cloud as a whole was drifting away from you (ignore any expansion) then would you still claim to be in the rest frame of the cloud of particles?

If the total momentum of the cloud is not zero by your measurements, do you accept that the cloud as a whole will drift away from you?
All I can tell you is that , your calculations for [tex]p'_x[/tex] seem incorrect.the ones for [tex]p'_y[/tex] are also incorrect but you got the desired result :-)


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