Electromagnetic version of the positive energy theorem?

[schieghoven]

Hello,

I've been quite avidly reading about one of the spectacular recent joint achievements of physics and pure math. The positive energy theorem [1,2,3] concerns the large-distance asymptotic behaviour of the gravitational field due to a localised distribution of mass-energy. I think I paraphrase the theorem correctly to say that, provided the source is physical (energy density T_00 is everywhere positive in every local Lorentz frame), then the energy of the system, as inferred from the large-distance gravitational field, is also positive. A neat and powerful result.

I was wondering if a similar result holds for electromagnetic theory. If we take Maxwell's equations and again impose that the source is physical (the current J_\mu is future-timelike), does this place some analogous constraint on the large-distance behaviour of the associated electromagnetic field? This linear system should presumably be much easier than the hard-core nonlinearity of Einstein GR, but nothing obvious jumped out at me.

Thanks,

Dave

* [1] Schoen, R. and Yau, S-T., Commun. Math. Phys, 65 (1979) 45
* [2] Witten, E., Commun. Math. Phys. 80 (1981) 381
* [3] Kazdan, J., Seminaire N. Bourbaki, 24 (1982) 315, Exp 593

[3], a review article, is publicly available at http://www.numdam.org/numdam-bin/browse?j=SB

 

[Stingray]

Assuming that the current density is nonzero and future-directed timelike implies that the system's charge (measured either normally or via the asymptotic fields) is non-negative. This is pretty easy to see from the first method:

$$q = \int_\Sigma J^a \mathrm{d}S_a > 0.$$

Using Stoke's theorem together with Maxwell's equations shows that this also appears in the field at infinity:

$$q = \frac{1}{8\pi} \int_{\partial \Sigma} F^{ab} \mathrm{d}S_{ab} .$$

Unlike in the gravitational case, this kind of restriction on the source is not consistent with observations. Spacelike currents are actually very common. Think of an ordinary wire. There's no net charge, yet there is a 3-current. The 4-current is therefore spacelike. This is possible because there are both positive and negative charges.

 

[Entropia]

.


Hello Dave,

Thank you for sharing your thoughts on the positive energy theorem and its application in both physics and mathematics. The electromagnetic version of this theorem is an interesting topic to consider. While there is no direct equivalent to the positive energy theorem in electromagnetic theory, there are some related results that have been studied.

One such result is the positivity of energy density in electromagnetism, which states that the energy density of the electromagnetic field is always positive in every local inertial frame. This is similar to the condition in the positive energy theorem that the energy density of the source must be positive in every local Lorentz frame.

Another related result is the positivity of energy flux, which states that the energy flux of the electromagnetic field, defined as the Poynting vector, is always positive and points in the direction of energy flow. This is similar to the condition in the positive energy theorem that the energy must be positive in the gravitational field.

However, unlike the positive energy theorem in general relativity, these results in electromagnetism are not as powerful and do not lead to as many far-reaching consequences. This is because electromagnetism is a linear theory, while general relativity is a highly nonlinear theory. The nonlinear nature of general relativity allows for more complex and interesting solutions, and the positive energy theorem plays a crucial role in understanding the global properties of these solutions.

In summary, while there is no direct electromagnetic version of the positive energy theorem, there are related results that have been studied and provide insight into the nature of energy in electromagnetism. I hope this helps answer your question. Thank you for your interest in this topic.