A Jackson: justification of the Poynting vector by GR

[coquelicot]

The Poynting vector is a definition, that is supposed to represent the energy flow at each point. Unfortunately, the only observable effect caused by the Poynting vector is through the energy variation in a volume subject to an energy flux through its surface, that is, the Poynting theorem.

As a curl could be added to the Poynting vector without changing the Poynting theorem, it can not be decided by EM only that this should be the actual flow of energy at each point. Feynman, commenting other proposed forms of energy flow, just said that "no one has ever found something bad with the Poynting vector". Jackson seemed to be aware of this issue, because he said, and apparently demonstrated in his book, that "the Poynting vector is the only expression of the energy flow compatible with GR".

I cannot follow Jackson in his argument. Could someone try to simplify/explain Jackson argument? Is there a better argument for the form of the Poynting vector?

Jackson 3d edition, section 12.10 (claim about the uniqueness of the energy flow representation in sec 6.7 relative to the Poynting vector, where the author sends us to sec 12.10).

 

[JimWhoKnew]

I'm currently away from my copy of Jackson's, so I can't look it up. However, note that in classical physics, QM and QFT we can add a complete time derivative to the Lagrangian without affecting the physics (equations of motion). Consequently, we can derive by Noether theorem many expressions for the energy-momentum tensor. They are all equivalent, since the mentioned theories don't care about the absolute values of the components - only the differences affect measurable quantities. This is not the case in GR, where the EMT is the "source of curvature" (RHS of Einstein's field equations), and its absolute values do matter(!). Only one specific form can be compatible.

 

[coquelicot]

I'm currently away from my copy of Jackson's, so I can't look it up. However, note that in classical physics, QM and QFT we can add a complete time derivative to the Lagrangian without affecting the physics (equations of motion). Consequently, we can derive by Noether theorem many expressions for the energy-momentum tensor. They are all equivalent, since the mentioned theories don't care about the absolute values of the components - only the differences affect measurable quantities. This is not the case in GR, where the EMT is the "source of curvature" (RHS of Einstein's field equations), and its absolute values do matter(!). Only one specific form can be compatible.

Thank you for your answer. I would appreciate something more detailed though, as I am not that good at GR.

 

[JimWhoKnew]

Thank you for your answer. I would appreciate something more detailed though, as I am not that good at GR.

Einstein's field equations, which govern GR, use the Stress-Energy Tensor. The SET is required to be symmetric and gauge-invariant. Moreover, at regions where there is only EM radiation, SET has to be traceless. Alternatively, Jackson postulates eq. 12.109 and derives the symmetry requirement from it. We can write the total SET as$$T^{\mu\nu}=\Theta^{\mu\nu}+T'^{\mu\nu}\quad, \tag{1}$$where $~\Theta~$ is the EM part and $~T'~$ is the rest (including interactions of charges with the EM field). We want each term to satisfy the above requirements independently, because in some regions of space the other term may vanish. The $~\Theta~$ that satisfies the requirements is unique (not proved in Jackson, look for Belinfante-Rosenfeld in the references) and is given by eq. 12.114. You can now infer from this equation, combined with (1), that the conservation property$$\partial_\nu T^{0\nu}=\partial_\nu\Theta^{0\nu}+\partial_\nu T'^{0\nu}=0 \tag{2}$$is actually Poynting theorem (eq. 12.118+12.119).
You can now still define Poynting vector $~\mathbf{S}~$ by adding a curl, but you already have (2), so you get nothing new out of it. Moreover, it adds the burden of clarification in any further discussion.

 

[coquelicot]

Einstein's field equations, which govern GR, use the Stress-Energy Tensor. The SET is required to be symmetric and gauge-invariant. Moreover, at regions where there is only EM radiation, SET has to be traceless. Alternatively, Jackson postulates eq. 12.109 and derives the symmetry requirement from it. We can write the total SET as$$T^{\mu\nu}=\Theta^{\mu\nu}+T'^{\mu\nu}\quad, \tag{1}$$where $~\Theta~$ is the EM part and $~T'~$ is the rest (including interactions of charges with the EM field). We want each term to satisfy the above requirements independently, because in some regions of space the other term may vanish. The $~\Theta~$ that satisfies the requirements is unique (not proved in Jackson, look for Belinfante-Rosenfeld in the references) and is given by eq. 12.114. You can now infer from this equation, combined with (1), that the conservation property$$\partial_\nu T^{0\nu}=\partial_\nu\Theta^{0\nu}+\partial_\nu T'^{0\nu}=0 \tag{2}$$is actually Poynting theorem (eq. 12.118+12.119).
You can now still define Poynting vector $~\mathbf{S}~$ by adding a curl, but you already have (2), so you get nothing new out of it. Moreover, it adds the burden of clarification in any further discussion.

Thank you again for this very insightfull answer.
I now understand that the "proof" of Jackson for the form of the Poynting vector is of the form: "You have an equation (the Poynting theorem) that is deduced from the EST. You can see in this equation an energy density like term, hence its exact form should be adopted for the energy density vector, as this is the simplest."

 

[JimWhoKnew]

Thank you again for this very insightfull answer.
I now understand that the "proof" of Jackson for the form of the Poynting vector is of the form: "You have an equation (the Poynting theorem) that is deduced from the EST. You can see in this equation an energy density like term, hence its exact form should be adopted for the energy density vector, as this is the simplest."

I'm not sure I follow. The stress-energy tensor is also called the energy-momentum tensor. In SR it is the energy (and momentum, and...) density, and the 4 conservation equations are the same as the ones derived by other means (like in chapter 6). The form in eq. 12.114 is derived uniquely, so its adoption is "justified". The property of "being the simplest" (as well as matching the historical result that was derived in chapter 6) is a welcome bonus that supports this adoption.

 

[SiennaTheGr8]

I briefly looked at some GR books for an explanation but didn't find anything. I had better luck with graduate-level E&M books; in addition to Jackson, there's at least Wald (Chapter 1), Garg (Section 25), and Zangwill (Section 15.4.3). Perhaps a satisfactory answer is in the paper that Zangwill refers the reader to: U. Backhaus and K. Schäfer, "On the uniqueness of the vector for energy flow density in electromagnetic fields," American Journal of Physics 54, 279 (1986).

Wald has a bit more to say than the others (unsurprising, perhaps, given his GR background). On p. 8: "formulas for [the Poynting vector] that differ by a curl of a vector field will have different gravitational consequences, so if one has two formulas for [the Poynting vector] that differ by a curl, at most one of them can be valid." And p. 6:

In principle, the validity of [the equations for the Poynting vector, etc.] could be tested by observing the gravitational effects of electromagnetic fields. Electromagnetic fields make nontrivial contributions to the mass-energy of ordinary matter—certainly large enough to produce observable gravitational effects for macroscopic bodies. However, there is no way to observe these effects separately from the gravitational effects of the nonelectromagnetic constitutents of matter. Thus, it would be necessary to observe the gravitational effects of free electromagnetic fields if one wishes to test [those equations]. The gravitational effects of free electromagnetic fields are far too small to be measured in laboratory experiments. However, in the early universe, the thermally distributed electromagnetic radiation that presently constitutes the cosmic microwave background made a dominant contribution to the energy density and pressure in the universe, both of which affect the expansion of the universe. The expansion history of the universe is observed to be in accord with the electromagnetic energy density and pressure of thermal radiation obtained from the above formulas.

 

[coquelicot]

I'm not sure I follow. The stress-energy tensor is also called the energy-momentum tensor. In SR it is the energy (and momentum, and...) density, and the 4 conservation equations are the same as the ones derived by other means (like in chapter 6). The form in eq. 12.114 is derived uniquely, so its adoption is "justified". The property of "being the simplest" (as well as matching the historical result that was derived in chapter 6) is a welcome bonus that supports this adoption.

Yes, but it depends on what you call "justified". The fact that an equation is derived from a well established theory certainly means it is true. But the equation remains true even after one adds a curl to S, as you pointed out. So, formally speaking, one is not allowed to decide arbitrarily that one term of the equation "should" be adopted as the right form of the energy density flux (even if this may be seen as "justified"). What is boiling down is gauge invariance. Sure, by adding a curl to S, you obtain no additional essential result, exactly as choosing different gauges for an EM problem leads to different equations, which describe essentially the same result. Put even more simply, one can choose a different coordinate system for a problem and this will lead essentially to the same physics. But my point is that by setting an arbitrary gauge/coordinate system etc. as an absolute not only would hide the symmetries of the physics, but would make research and computations difficult, at the very least. Fortunately, physicists have not decided that such or such coordinate system, or such or such EM gauge, had to be used; but (perhaps) curiously, that's exactly what they did for the Poynting vector. This was my motivation for asking this question: is there a decisive proof for the form of the Poynting vector? Thanks to your answer, I now understand that Jackson's argument is insufficiently decisive (in my opinion). I will try to examine other arguments.

 

[coquelicot]

I briefly looked at some GR books for an explanation but didn't find anything. I had better luck with graduate-level E&M books; in addition to Jackson, there's at least Wald (Chapter 1), Garg (Section 25), and Zangwill (Section 15.4.3). Perhaps a satisfactory answer is in the paper that Zangwill refers the reader to: U. Backhaus and K. Schäfer, "On the uniqueness of the vector for energy flow density in electromagnetic fields," American Journal of Physics 54, 279 (1986).

Wald has a bit more to say than the others (unsurprising, perhaps, given his GR background). On p. 8: "formulas for [the Poynting vector] that differ by a curl of a vector field will have different gravitational consequences, so if one has two formulas for [the Poynting vector] that differ by a curl, at most one of them can be valid." And p. 6:

Thank you so many for these references. I was unaware of them. Regarding the paper of U. Backhaus and K. Schäfer, it can be read online (here), and as far as I understood, their authors think (and justify) that the arguments against the "non uniqueness" of the Poynting vector are not decisive. I will try to have a look at Wald.

 

[SiennaTheGr8]

Thinking about this a little more...

For the laws of physics to be expressed covariantly, we'd have to add not just a curl-term to the Poynting 3-vector, but rather some "curl-like" 4-tensor term to the electromagnetic stress-energy that gives rise to that 3-curl term in the 3+1 split. In flat spacetime this added 4-tensor term should have no physical consequence, probably by virtue of some Minkowski equivalent of the "div-of-curl-vanishes" rule. A 3-curl is $\epsilon^{abc} \nabla_{b} V_c$, so I believe the added 4-tensor term would have to be something like $\epsilon^{\mu \nu \alpha \beta} \nabla_{\alpha} V_{\beta}$.

If that's right, then perhaps the flat vs. curved spacetime difference comes into play when you take the 4-divergence of the stress-energy tensor, relevant for local 4-momentum conservation. With the added "4-curl" contribution, you'd get a term like $\epsilon^{\mu \nu \alpha \beta} \nabla_{\mu} \nabla_{\alpha} V_{\beta}$ (covariant derivative of the Levi-Civita tensor vanishes). That's identically zero in flat spacetime, yeah? (Because covariant derivatives then commute, giving symmetry/antisymmetry annihilation for the contracted indices?) But not if there's curvature, I think.

Based on the Wald excerpts I feel like I might be on the right track here, but I'm really not sure, and if so then I still don't know how to extrapolate from it (i.e., how does this manifest as a physically measurable difference?).

 

[coquelicot]

Thinking about this a little more...

For the laws of physics to be expressed covariantly, we'd have to add not just a curl-term to the Poynting 3-vector, but rather some "curl-like" 4-tensor term to the electromagnetic stress-energy that gives rise to that 3-curl term in the 3+1 split. In flat spacetime this added 4-tensor term should have no physical consequence, probably by virtue of some Minkowski equivalent of the "div-of-curl-vanishes" rule. A 3-curl is $\epsilon^{abc} \nabla_{b} V_c$, so I believe the added 4-tensor term would have to be something like $\epsilon^{\mu \nu \alpha \beta} \nabla_{\alpha} V_{\beta}$.

If that's right, then perhaps the flat vs. curved spacetime difference comes into play when you take the 4-divergence of the stress-energy tensor, relevant for local 4-momentum conservation. With the added "4-curl" contribution, you'd get a term like $\epsilon^{\mu \nu \alpha \beta} \nabla_{\mu} \nabla_{\alpha} V_{\beta}$ (covariant derivative of the Levi-Civita tensor vanishes). That's identically zero in flat spacetime, yeah? (Because covariant derivatives then commute, giving symmetry/antisymmetry annihilation for the contracted indices?) But not if there's curvature, I think.

Based on the Wald excerpts I feel like I might be on the right track here, but I'm really not sure, and if so then I still don't know how to extrapolate from it (i.e., how does this manifest as a physically measurable difference?).

This seems interesting. I sent you a direct message to your inbox, as this kind of discussion usually leads to non main-stream consequences prohibited in this forum.