# Is power Lorentz invariant

1. Jul 10, 2012

### Heirot

Power, defined as P = dE/dt is Lorentz invariant according to

http://farside.ph.utexas.edu/teaching/em/lectures/node130.html, Eq. 1645

But, considering another equation for the power, P = q E v, where E and v are electric field and velocity vectors, respectively; this is obviously not the case since in the momentarily rest frame of the charge P = 0?

What is wrong with the above reasoning?

Thanks!

2. Jul 10, 2012

### GarageDweller

Don't think so, dE/dt is basically the time derivative of the 0th component of the four momentum integrated over a 3-volume. Components change from frame to frame.
Furthermore, the concept of "lorentz invariance" basically means that the geometric object stays the same in frames, only it's basis' and components transform to fit this new frame.
I could be wrong, anyone bother reading the entire article?

3. Jul 10, 2012

### Heirot

I would agree with you but I can't find an error in Richard Fitzpatrick's reasoning...

4. Jul 10, 2012

### Staff: Mentor

Where are you getting this formula from?

5. Jul 10, 2012

### Heirot

The Lorentz force is given by F = q (E + v x B), and the power is P = F.v. This gives P = q E.v which is nowhere near being Lorentz invariant.

6. Jul 10, 2012

### elfmotat

In general, power is certainly not invariant. It is the 0'th component of four-force, $dp^\mu /d\tau$.

7. Jul 10, 2012

### Heirot

Actually, the power is defined as a derivative wrt the coordinate, and not proper time.

8. Jul 10, 2012

### GarageDweller

Everything needs to be put into covariant form before you can use them in SR, E and B fields are combined into the faraday tensor which is invariant. Classical equations should not be referred to in relavistic calculations.

9. Jul 10, 2012

### PAllen

That website is not arguing that power, in general, is Lorentz invariant (that is absurd, as elfmotat has pointed out). It is arguing that:

Power radiated by a charged dipole (which has, among other features, that no momentum is carried away) is invariant.

The particular features of this problem are used throughout.

Radiated power for a situation where the radiation carries away net momentum would not meet the conditions of the derivation.

10. Jul 10, 2012

### Heirot

I disagree. It is true that covariant Lorentz scalars are (by definition) Lorentz invariant. But that doesn't mean that there aren't other noncovariant Lorentz invariant quantities.

11. Jul 10, 2012

### Heirot

I fail to see where the argument that the power is Lorentz invariant is used only in connection with the radiating charge. As far as I can see the argument is related to the transformation properties of energy and time and not to the radiating charge.

12. Jul 10, 2012

### PAllen

Right here, the logic is only true for radiation carrying away no net momentum:

It follows from Sect. 10.22 that we can write $P^\mu = (d{\bf p}, dE/c)$, where $d{\bf p}$ and $dE$ are the total momentum and energy carried off by the radiation emitted between times $t=0$ and $t=dt$, respectively. As we have already mentioned, $d{\bf p} = 0$ in the instantaneous rest frame $S$. Transforming to an arbitrary inertial frame $S'$, in which the instantaneous velocity of the charge is $u$, we obtain
\begin{displaymath} dE^{'} = \gamma(u) \left(dE + u dp^1\right) = \gamma dE. \end{displaymath} (1644)

Note, this refers back to the earlier observation:

This is known as Larmor's formula. Note that zero net momentum is carried off by the fields (1628) and (1629).

In the general case, there is no frame where radiation carries no momentum (you can't talk about the 'rest frame' of a laser beam).

13. Jul 10, 2012

### Heirot

Right you are! The mystery is solved!

Thank you, PAllen :)