4-velocity, the mass shell and potential energy

  • Thread starter pellman
  • Start date
666
2

Main Question or Discussion Point

4-velocity, the "mass shell" and potential energy

For a free relativistic particle we have the condition

[tex]E^2-\mathbf{p}^2c^2=m^2c^4[/tex].

putting [tex]p^0 = c^{-1}E[/tex] we may write this as

[tex]p^\mu p_\mu = m^2c^2[/tex]

And since [tex]p^\mu = m\frac{dx^\mu}{d\tau} = mu^\mu[/tex] we may express the same condition as

[tex]u^\mu u_\mu = c^2[/tex]

At least, this is true for a free particle. Is this condition affected by the presence of a potential?

In the presence of a potential [tex]A^\mu[/tex] the momentum-energy relation becomes

[tex](p^\mu-A^\mu)(p_\mu-A_\mu) = m^2c^2[/tex]

doesn't it? But then in that case, we are talking about the canonical momentum and we no longer have [tex]p^\mu = mu^\mu[/tex]. But is [tex]p^\mu = mu^\mu+A^\mu[/tex]? So that we still have [tex]u^\mu u_\mu = c^2[/tex]? Can't find this in my texts and just wanted someone to confirm I understand it correctly.
 

Answers and Replies

979
1
Always, [tex]u^\mu u_\mu = c^2[/tex]. Mechanical momentum is m times that. Canonical momentum is determined from what the Lagrangian of the system is.
 
2,945
0
For a free relativistic particle we have the condition

[tex]E^2-\mathbf{p}^2c^2=m^2c^4[/tex].

putting [tex]p^0 = c^{-1}E[/tex] we may write this as

[tex]p^\mu p_\mu = m^2c^2[/tex]

And since [tex]p^\mu = m\frac{dx^\mu}{d\tau} = mu^\mu[/tex] we may express the same condition as

[tex]u^\mu u_\mu = c^2[/tex]

At least, this is true for a free particle. Is this condition affected by the presence of a potential?

In the presence of a potential [tex]A^\mu[/tex] the momentum-energy relation becomes

[tex](p^\mu-A^\mu)(p_\mu-A_\mu) = m^2c^2[/tex]

doesn't it? But then in that case, we are talking about the canonical momentum and we no longer have [tex]p^\mu = mu^\mu[/tex]. But is [tex]p^\mu = mu^\mu+A^\mu[/tex]? So that we still have [tex]u^\mu u_\mu = c^2[/tex]? Can't find this in my texts and just wanted someone to confirm I understand it correctly.
If you look in Jackson's text Classical Electrodynamics you will find something very close.

Pete
 
666
2
Thanks, guys.
 

Related Threads for: 4-velocity, the mass shell and potential energy

  • Last Post
Replies
11
Views
2K
Replies
15
Views
3K
Replies
4
Views
2K
Replies
202
Views
22K
  • Last Post
Replies
9
Views
2K
Replies
14
Views
1K
Replies
7
Views
2K
Top