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.

 

[JimWhoKnew]

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.

Note that within the framework of SR you already get some restrictions on $\mathbf{V}~$. for example, in empty Minkowski spacetime the SET should vanish. This suggests that the corrections to the SET should be functions of $~j^\mu~$ and $~F^{\mu\nu}~$. Next, we want the modified SET to be Lorentz covariant. If the modification is a simple 3D curl in one frame of reference, it is expected to "spread" to other components by LTs.

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?).

For a start, you may try to explore how the electrovacuum spherical symmetric Reissner-Nordström solution is affected. Can the modified SET be diagonalized by a coordinate transformation to get $~T'^\mu{}_\nu~$ ^* which is a function of $Q$ and $r'$ only? If no, can the corresponding spacetime be both stationary and spherical symmetric? If yes, is it the same (up to an additional coordinate transformation) as the old RN SET? If different, you may integrate to derive the modified solution for the metric. Does it possess an Event Horizon? Divergence of curvature scalars as $r'$ approaches 0?
If the modified solution is stationary, the proper time in which an observer in a circular geodesic orbit completes one revolution along a path of a given spatial proper length, is an example for a physically measurable effect that can be compared between the old RN and modified solutions.

^* Assuming we can find the modified $~T'^\mu{}_\nu~$ in SR and take its general-covariant form.

 

[PeterDonis]

If yes, is it the same (up to an additional coordinate transformation) as the old RN SET?

There is an analogue of Birkhoff's Theorem that shows that the only spherically symmetric electrovacuum solution (i.e., the only stress-energy present is a source-free electromagnetic field) is Reissner-Nordstrom. So if you are suggesting that there might be a different such solution, that's not possible.

 

[PeterDonis]

Note that within the framework of SR you already get some restrictions on $\mathbf{V}~$. for example, in empty Minkowski spacetime the SET should vanish.

That's not just a "restriction" on $\mathbf{V}~$. It means $\mathbf{V}~$ has to vanish. The SET vanishing means no stress-energy can be present anywhere, including electromagnetic fields.

 

[PeterDonis]

Based on the Wald excerpts I feel like I might be on the right track here

I'm not sure you are. The point of the passage from Wald that was quoted is not that there is a difference in the measurable effects of EM fields in curved vs. flat spacetime.

The point of the Wald passage is that EM fields are stress-energy, so if they are present, spacetime must be curved, not flat. In other words, EM fields must have gravitational effects. But if the only stress-energy present is EM fields (for example, EM radiation in vacuum), the gravitational effects predicted from the equations are very, very tiny, much too small for us to measure directly with current or foreseeable future technology. So instead we have to look for indirect evidence, some of which Wald describes.

 

[SiennaTheGr8]

I'm not sure you are. The point of the passage from Wald that was quoted is not that there is a difference in the measurable effects of EM fields in curved vs. flat spacetime.

The point of the Wald passage is that EM fields are stress-energy, so if they are present, spacetime must be curved, not flat. In other words, EM fields must have gravitational effects. But if the only stress-energy present is EM fields (for example, EM radiation in vacuum), the gravitational effects predicted from the equations are very, very tiny, much too small for us to measure directly with current or foreseeable future technology. So instead we have to look for indirect evidence, some of which Wald describes.

Yes, I suppose all I've shown (if my math was right) is that the addition of a "4-curl" term to the electromagnetic stress-energy tensor can "make a difference" in general relativity in a way that it doesn't in special relativity. Interesting maybe, but I think you're right: not what Wald was getting at.

 

[JimWhoKnew]

I'm not sure you are. The point of the passage from Wald that was quoted is not that there is a difference in the measurable effects of EM fields in curved vs. flat spacetime.

The point of the Wald passage is that EM fields are stress-energy, so if they are present, spacetime must be curved, not flat. In other words, EM fields must have gravitational effects. But if the only stress-energy present is EM fields (for example, EM radiation in vacuum), the gravitational effects predicted from the equations are very, very tiny, much too small for us to measure directly with current or foreseeable future technology. So instead we have to look for indirect evidence, some of which Wald describes.

At the bottom line, what I wrote in #2 is correct. Wald goes in the opposite direction to the common practice in textbooks: He postulates that the SET has the given ("minimal") form, and derives Lorentz force from it. In his discussion of the observations due to CMB he says that it is in accord not only with the EM effect on curvature (as expected by GR), but also with the postulated form of SET (within limitations of measurment precision).

That's not just a "restriction" on $~\mathbf{V}~$. It means $~\mathbf{V}~$ has to vanish. The SET vanishing means no stress-energy can be present anywhere, including electromagnetic fields.

By "empty" I meant no charge (nor matter) and no fields. But $~\mathbf{V}~$ may still be some function of these, that vanishes when they are absent. A little nitpicking: $~\mathbf{V}~$ can be any real constant even in empty SR space (the curl vanishes). Moreover, if you add to the SET a term $~\Delta T^\mu{}_\nu=c\cdot \delta^\mu{}_\nu~~$ (gauging vacuum zero point energy), which is symmetric and Lorentz + gauge invariant, it is a catastrophe in GR, but has no observable implications within SR EM.

There is an analogue of Birkhoff's Theorem that shows that the only spherically symmetric electrovacuum solution (i.e., the only stress-energy present is a source-free electromagnetic field) is Reissner-Nordstrom. So if you are suggesting that there might be a different such solution, that's not possible.

Wald postulates the "minimal" form for the SET. He writes "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". Meaning that an empirical corroboration (or disprove) of that postulate can only come from GR.
Suppose for example that the SET postulated by Wald is not exactly the correct form after all. Let's consider the hypothetical case where in the diagonalized form of the electrovacuum SET we need to introduce everywhere the corrective replacement$$\frac{Q^2}{r^4}~~\rightarrow~~ \frac{Q^2}{r^4}+(n-1)\frac{Q^n}{r^{n+2}} \quad , \quad n>2 \quad.$$It has the solution$$ds^2=-\left(1-\frac{2m}r+\frac{Q^2}{r^2}+\frac{Q^n}{r^n}\right)dt^2+\left(1-\frac{2m}r+\frac{Q^2}{r^2}+\frac{Q^n}{r^n}\right)^{-1}dr^2+r^2d\theta^2+r^2\sin^2\theta ~d\phi^2 \quad.$$Any problem with Birkhoff's Theorem or its analogues?

 

[PeterDonis]

At the bottom line, what I wrote in #2 is correct.

Yes, and what Wald wrote is consistent with what you wrote in #2. Wald agrees that adding a term to the SET has gravitational effects.

an empirical corroboration (or disprove) of that postulate can only come from GR.

Well, of course, since we're talking about observing the gravitational effects of EM fields, which means we're using GR. Isn't that what this thread is about?

 

[PeterDonis]

Wald postulates the "minimal" form for the SET.

I'm not sure it's just a matter of "postulating". The Reissner-Nordstrom metric, according to the theorem I referred to, is the unique solution of the coupled vacuum Einstein Field Equation (i.e., no SET put in "by hand") and source-free Maxwell Equations. So whatever SET comes out of that is derived, not postulated.

 

[JimWhoKnew]

I'm not sure it's just a matter of "postulating". The Reissner-Nordstrom metric, according to the theorem I referred to, is the unique solution of the coupled vacuum Einstein Field Equation (i.e., no SET put in "by hand") and source-free Maxwell Equations. So whatever SET comes out of that is derived, not postulated.

Since MTW and Wald leave the RN as an exercise, I looked at the books by Chandrasekhar, Poisson and d'Inverno. They all insert "by hand" the "minimal" form of SET (Wald's postulate) as if it was obvious. They certainly don't derive it from the Einstein-Maxwell equations.

Are you sure it can be done without? Can you demonstrate it yourself, or provide references?

 

[PeterDonis]

Are you sure it can be done without? Can you demonstrate it yourself, or provide references?

I'll see if I can find a reference to the theorem I mentioned.

 

[PeterDonis]

MTW and Wald leave the RN as an exercise

Wald has it as part (c) of Problem 3 in Chapter 6. But note that the SET for this part is not "postulated". It is derived by first completing parts (a) and (b) of the same problem. As far as I can tell, the solutions of parts (a) and (b) do not leave any room for a different form of the SET such as you hypothesized in post #17. (At the very least, I would want to see how you derive what you hypothesized in post #17 from the 4-potential given in part (b) of the Wald problem.)

 

[PeterDonis]

MTW and Wald leave the RN as an exercise

For reference, in MTW it's exercise 32.1.

 

[JimWhoKnew]

Wald has it as part (c) of Problem 3 in Chapter 6. But note that the SET for this part is not "postulated". It is derived by first completing parts (a) and (b) of the same problem. As far as I can tell, the solutions of parts (a) and (b) do not leave any room for a different form of the SET such as you hypothesized in post #17. (At the very least, I would want to see how you derive what you hypothesized in post #17 from the 4-potential given in part (b) of the Wald problem.)

I don't see any 4-potential in part (b). The derivation in (b) is of $~F_{ab}~~$, which we all agree on.
Now, in (c) Wald writes:

Write down and solve Einstein's equation, ##~G_{ab}=8\pi T_{ab}~#, with electromagnetic stress-energy tensor corresponding to the solution of part (b).

So the reader is left with the task of incorporating the known and undisputed $~F_{ab}~$ into $~T_{ab}~$ , which is a major point in this thread. That's not a derivation. It is the same as in the other books I've mentioned in #20, only without writing it explicitly.

If you still insist that it is a valid derivation, please post your detailed answer to Wald's exercise step by step, so I can see explicitly how you top Chandrasekhar's and Poisson's treatments.

For reference, in MTW it's exercise 32.1.

Exercise 32.1 uses the result obtained in exercise 31.8, which in turn has the same F to T thing as in Wald.

Please don't send me to all the literature on GR. Only when you have a good debugged reference, which you've carefully verified that it really derives the minimal SET uniquely, post it.
(I truly apologize if I sound impolite. No intention to offend nor disrespect.)

(At the very least, I would want to see how you derive what you hypothesized in post #17 from the 4-potential given in part (b) of the Wald problem.)

I'll try to clarify: The references in #7 suggest that there might be possible non-trivial modifications of the minimal SET which are compatible with SR, and only GR can arbitrate which is the correct form. I didn't read the references within the references.
In #10 @SiennaTheGr8 seemed to try to construct such a modification. So in #12 I was trying to suggest that in case of success, the modification may be tried on a "simple" case, namely RN. From there, I just reviewed possible logical outcomes. Nothing more than that.
Suppose there is a non-minimal form of SET consistent with SR, which can be expressed in a general covariant form. In this case I see three possibilities. The first is that it can't be transformed into the form required for a spherical symmetric and stationary electrovacuum solution. The second is that it can be transformed, but then for the specific problem at hand, its transformed diagonalized form coincides with the minimal SET (meaning that RN is not a good case study - it is insufficient to tell the two apart). The third option is that we may get a "good" modification that can yield a solution whose observable predictions may be compared with RN.
Then in #13 you said that the third possibility is ruled out. In #17 I gave an example that Birkhoff's Theorem allows for a range of solutions, hinting that we will need some more arguments in order to decide that none of them may fit. The specific example given in #17 certainly doesn't qualify. That's why it is hypothetical. No point in digging further.

Clearer?

 

[PeterDonis]

Can you demonstrate it yourself

I'll take a stab at it here. I'll be making use of some equations from two of my Insights articles, on the EFE and Maxwell's Equations in a spherically symmetric spacetime:

https://www.physicsforums.com/insig...-in-a-static-spherically-symmetric-spacetime/

https://www.physicsforums.com/insig...-in-a-static-spherically-symmetric-spacetime/

Note that those articles focus on the static case, but the equations I'll be using from them don't assume that the spacetime is static, only that it's spherically symmetric. As with the standard Birkhoff's Theorem, the presence of an additional Killing vector field will be part of what we derive.

We adopt Schwarzschild coordinates, which are adapted to the spherical symmetry of the spacetime. The line element in these coordinates for a general spherically symmetric spacetime is:

$$
ds^2 = - J(r) dt^2 + \frac{1}{1 - \frac{2m(r)}{r}} dr^2 + r^2 d\Omega^2
$$

where $J(r)$ and $m(r)$ are functions of $r$ that we will try to determine. The most general stress-energy tensor for a general spherically symmetric spacetime has only three nonzero components:

$$
T^t{}_t = - \rho(r)
$$

$$
T^r{}_r = p(r)
$$

$$
T^\theta{}_\theta = T^\varphi{}_\varphi = s(r)
$$

which gives us three more functions of $r$ to determine.

We also can derive the following equations by computing the Einstein tensor of the metric above and applying the EFE, as shown in the first article referenced above:

$$
\frac{dm}{dr} = 4 \pi r^2 \rho
$$

$$
\frac{1}{2J} \frac{dJ}{dr} = \frac{m + 4 \pi r^3 p}{r \left( r - 2m \right)}
$$

$$
\frac{dp}{dr} = - \left( \rho + p \right) \frac{1}{2J} \frac{dJ}{dr} - \frac{2}{r} \left( p - s \right)
$$

Finally, we have from the second article that the electromagnetic 4-potential, given the constraint of spherical symmetry, must be

$$
A_t = \Phi(r)
$$

with all other components zero.

Now we make use of two further constraints that are imposed by spherical symmetry plus the fact that the only stress-energy present is a source-free EM field (i.e., the charge-current 4-vector $j$ vanishes). These are that the SET must be traceless, and that we must have $J(r) = 1 - 2m(r) / r$ (i.e., the $g_{tt}$ metric coefficient must be minus the reciprocal of the $g_{rr}$ component). (I have not been able to find a proof of the latter fact, but we know it's true for the R-N metric, and it seems reasonable given that there are no sources anywhere, so there's no way for the "redshift factor" to vary from its electrovacuum value.)

Now we start applying our constraints:

The trace of the SET being zero means $- \rho + p + 2 s = 0$, or $\rho = p + 2 s$.

Knowing that $J = 1 - 2m(r) / r$ lets us compute

$$
\frac{1}{2J} \frac{dJ}{dr} = \frac{m - r \frac{dm}{dr}}{r \left( r - 2m \right)}
$$

Plugging in the equation we have above for $dm / dr$ gives

$$
\frac{1}{2J} \frac{dJ}{dr} = \frac{m - 4 \pi r^3 \rho}{r \left( r - 2m \right)}
$$

We now have two equations for the same quantity (the LHS of the equation above), and setting the RHS of the two gives $p = - \rho$. From the tracelessness equation above that gives us $s = \rho$.

Now we plug all of this into the equation for $dp / dr$, to get:

$$
\frac{dp}{dr} = - 4 \frac{p}{r}
$$

This has the obvious solution

$$
p = - \frac{Q^2}{r^4}
$$

where the constant in the numerator has been chosen to match what we already know of the form of the R-N solution, and the minus sign is because the sign of $p$ must be negative (because the sign of $\rho$, the energy density, must be positive, and $p = - \rho$).

Note that it is not possible to add another term, as in the hypothesis in post #17. Note also that I made no assumption whatever regarding the form of the SET in terms of the EM 4-potential or EM field tensor. But we can show, of course, that given the 4-potential above, the electric field $E = - d \Phi / dr$, and from the source-free Maxwell Equations, as shown in the second article referenced above, we have

$$
\frac{dE}{dr} = - 2 \frac{E}{r}
$$

which of course has the solution $E = Q / r^2$. So the energy density we found above is just $E^2$, as expected.

 

[PeterDonis]

I don't see any 4-potential in part (b).

Sorry, I misspoke. It's a specific form for the EM field tensor.

 

[PeterDonis]

The references in #7 suggest that there might be possible non-trivial modifications of the minimal SET which are compatible with SR

They would also have to be compatible with the source-free Maxwell Equations.

only GR can arbitrate which is the correct form.

More precisely, since modifications to the SET mean modifications to the gravitational effects of EM fields, measuring those effects can tell us which of different proposed forms of the SET is correct. I'm not disputing that at all. So far, all the evidence we have is that what you call the "minimal" SET is correct.

But theoretically speaking, modifications to the SET can't just be put in arbitrarily by hand. There has to be some theoretical justification for them. What you call the "minimal" SET is in fact the canonical stress-energy tensor derived from the equations for an EM field. It's not just pulled out of a hat. It's derived from certain theoretical principles. That's why all the references you talk about consider it "obvious" that that's the right form of the SET for an EM field. They're not just making an arbitrary choice.

 

[PeterDonis]

Note that those articles focus on the static case, but the equations I'll be using from them don't assume that the spacetime is static, only that it's spherically symmetric.

For some more background on why that's true, see my Insights article giving a short proof of the standard Birkhoff's Theorem:

https://www.physicsforums.com/insights/short-proof-birkhoffs-theorem/

Note that in that derivation, I have let $g_{tt}$ and $g_{rr}$ be functions of $t$ as well as $r$. But as the EFE computation shows, the time dependence only shows up in the off diagonal components of the Einstein tensor--and in a spherically symmetric spacetime, in Schwarzschild coordinates, those must vanish even if the SET as a whole does not vanish. The diagonal components of the Einstein tensor, which don't vanish in the electrovacuum case, still only involve the $r$ dependence.

 

[JimWhoKnew]

... which of course has the solution $E = Q / r^2$. So the energy density we found above is just $E^2$, as expected.

Thanks for posting.

So we have the 2 equations:$$\frac{dp}{dr} = - 4 \frac{p}{r}$$solved by$$p = - \frac{Q'^2}{r^4} \quad,$$and$$\frac{dE}{dr} = - 2 \frac{E}{r}$$solved by$$E = Q / r^2 \quad.$$What condition forces $~Q'=Q~$ ?

Note that

where the constant in the numerator has been chosen to match what we already know of the form of the R-N solution

is insufficient as a proof of derivation. All freedom of choice should be eliminated.

Note also that the minimal form of the SET was assumed and used in the second insight article, so the results that follow should be treated with caution.

Yet another thing to note is that Q' affects geodesics of neutral test particles while Q also affects accelerations of charged ones (we expect Q' to be a function of Q, since we want to recover the Schwarzschild solution when Q vanishes).

But theoretically speaking, modifications to the SET can't just be put in arbitrarily by hand. There has to be some theoretical justification for them. What you call the "minimal" SET is in fact the canonical stress-energy tensor derived from the equations for an EM field. It's not just pulled out of a hat. It's derived from certain theoretical principles. That's why all the references you talk about consider it "obvious" that that's the right form of the SET for an EM field. They're not just making an arbitrary choice.

Of course the "minimal" SET form is not "pulled out of a hat". It is the preferred candidate. That's why Wald postulates it. Yet Wald is aware that it deserves a discussion, and that alternative possibilities are not entirely ruled out.

For some more background on why that's true, see my Insights article giving a short proof of the standard Birkhoff's Theorem:

Surprising as it may be, I've heard about Birkhoff's Theorem before

 

[PeterDonis]

What condition forces $Q' = Q$?

The condition that the energy density of an electric field $E$ is $E^2$ (modulo some constant factors that depend on your choice of units).

All freedom of choice should be eliminated.

I think you misunderstood the statement of mine that you quoted. The value of the constant $Q$ is of course a free parameter in the solution. The name I chose to give it was chosen to match the usual naming convention of the R-N solution. I could have chosen any name at all and it still would be the same as the R-N solution; the functional form is all that's necessary for that.

 

[PeterDonis]

Surprising as it may be, I've heard about Birkhoff's Theorem before

Of course. Again, I think you misunderstood me. I was providing the link so that more of the equations that I was using would be seen there. I was not providing it because I thought Birkhoff's Theorem would be new to you.

 

[PeterDonis]

Note also that the minimal form of the SET was assumed and used in the second insight article

But I didn't make use of that part of the article in my derivation in this thread.

 

[PeterDonis]

Wald is aware that it deserves a discussion, and that alternative possibilities are not entirely ruled out.

Not entirely ruled out mathematically.

But, for example, any alternative possibility, if it's going to lead to a different solution from R-N for the spherically symmetric electrovacuum case, would have to imply that the energy density of an electric field $E$ is not $E^2$ (again, modulo constant factors that depend on your choice of units). But what else could it be? $E^2$ is what makes sense on dimensional grounds; any other function of $E$ wouldn't. Nor would it make sense for the energy density to depend on something other than $E$.

 

[JimWhoKnew]

The condition that the energy density of an electric field $E$ is $E^2$ (modulo some constant factors that depend on your choice of units).

If, for example, $~Q'=(1+\epsilon)Q~~$ (in the same choice of units), then $~T^t{}_t~$ is still proportional to $~E^2~$. But Q' is what we have in the metric while Q is used in Maxwell's equations. Insisting that $~\epsilon=0~$ necessarily, is the same as assuming the minimal SET. For $~\epsilon\neq 0~$ the metric will still have the RN form, but charged test particles will behave differently (a physically observable effect, in principle).

Edit:
If you meant that the RN form of the metric is unique, your derivation seems OK (to me, as far as I checked). If you meant to show that the RN solution to the Einstein-Maxwell equations is unique, without assuming the minimal SET (or anything equivalent), you didn't convince me.

 

[PeterDonis]

For $~\epsilon\neq 0~$ the metric will still have the RN form, but charged test particles will behave differently (a physically observable effect, in principle).

This would amount to a different prediction for how much energy density a given amount of electric field produces; there would be an extra factor in the energy density that doesn't arise from the charge producing the field. So what does it arise from?

 

[PeterDonis]

If you meant to show that the RN solution to the Einstein-Maxwell equations is unique

The possible departure from uniqueness is pretty small: all we have, by your argument, is one additional free parameter in the solution, that specifies how much energy density a given amount of electric field produces.