Why is there ambiguity of the field energy?

Question

I was reading Field Energy & Field Momentum of Feynman's Lectures on Physics Vol.II. There he deduced energy density $u$ & Poynting's vector as $$u = \frac{\epsilon_0}{2} \mathbf E\cdot\mathbf E + \frac{c^2\epsilon_0}{2} \mathbf B\cdot \mathbf B\\ \mathbf S= \epsilon_0c^2\mathbf E \times \mathbf B\;.$$

But then he told that there can be infinite number of possible values of $u$ and $\bf S$. Here is the concerned excerpt:

Before we take up some applications of the Poynting formulas we would like to say that we have not really “proved” them. All we did was to find a possible “u” and a possible “S.” How do we know that by juggling the terms around some more we couldn’t find another formula for "$u$" and another formula for "$\bf S$"? The new $\bf S$ and the new $u$ would be different, but they would still satisfy Eq. (27.6). It’s possible. It can be done, but the forms that have been found always involve various derivatives of the field (and always with second-order terms like a second derivative or the square of a first derivative). There are, in fact, an infinite number of different possibilities for $u$ and $\bf S$, and so far no one has thought of an experimental way to tell which one is right! People have guessed that the simplest one is probably the correct one, but we must say that we do not know for certain what is the actual location in space of the electromagnetic field energy. So we too will take the easy way out and say that the field energy is given by Eq. (27.14). Then the flow vector $\bf S$ must be given by Eq. (27.15). It is interesting that there seems to be no unique way to resolve the indefiniteness in the location of the field energy. It is sometimes claimed that this problem can be resolved by using the theory of gravitation in the following argument. In the theory of gravity, all energy is the source of gravitational attraction. Therefore the energy density of electricity must be located properly if we are to know in which direction the gravity force acts. As yet, however, no one has done such a delicate experiment that the precise location of the gravitational influence of electromagnetic fields could be determined. That electromagnetic fields alone can be the source of gravitational force is an idea it is hard to do without. It has, in fact, been observed that light is deflected as it passes near the sun—we could say that the sun pulls the light down toward it. Do you not want to allow that the light pulls equally on the sun? Anyway, everyone always accepts the simple expressions we have found for the location of electromagnetic energy and its flow. And although sometimes the results obtained from using them seem strange, nobody has ever found anything wrong with them—that is, no disagreement with experiment. So we will follow the rest of the world—besides, we believe that it is probably perfectly right.

I really didn't get what he is saying.

In the previous section, he deduced the formula for $\bf S$ and $u$ & now he is saying they are just possible values; there can be actually infinite number of different values of them!

So, my questions are:

$\bullet$ Why can there be infinite number of different values of $\bf S$ and $u\;?$ Why is there the ambiguity of the field energy?

$\bullet$ What did Feynman mean by indefiniteness in the location of field energy?

$\bullet$ Can anyone please tell me how this problem can be solved by theory of gravitation as said by Feynman?

Answer 1

The main point of Feynman's analysis is to find a pair $u,\;\bf S$ such that $$ \partial_tu+\bf \nabla\cdot\bf S=-\bf E\cdot \bf J $$ which is a necessary condition for $u,\;\bf S$ to be considered as the energy density and the energy flux density vector (i.e., this is the continuity equation for the energy).

But the pair $u,\;\bf S$ derived by Feynman is not unique: we can find new pairs $u',\;\bf S'$ that can also be interpreted to be the energy density and energy flux density vector of the system. To see this, let's define $$ u'=u+f $$ and $$ \bf S'=\bf S+\boldsymbol g $$ with $f$ and $\boldsymbol g$ chosen such that $\partial_t f+\nabla\cdot \boldsymbol g=0$. Then it is easy to see that the new pair satisfies the old continuity equation: $$ \partial_tu'+\bf \nabla\cdot\bf S'=-\mathbf E\cdot \bf J $$

The new $u',\;\bf S'$ and the old $u,\;\bf S$ are equally good to be interpreted as an energy density and energy flux density vector for the system. As there exists an infinite number of functions $f,\boldsymbol g$ that satisfy $\partial_t f+\bf \mathbf\nabla\cdot\boldsymbol g=0$, there exist an infinite number of possible pairs $u,\;\bf S$.

As an example, consider $f=\partial_t \phi$ (with $\phi=\phi(\mathbf r,t)$), and $\boldsymbol g=\bf{\nabla} \phi$. If we choose $\phi$ such that $\partial_t^2\phi+\mathbf \nabla^2\phi=0$ (i.e., a wave), then it is obvious that $\partial_tf+\mathbf \nabla\cdot\boldsymbol g=0.$ As there exist an infinite number of solutions$^1$ of $\partial_t^2\phi+\mathbf \nabla^2\phi=0$, then there exists an infinite number of possible $f,\;\boldsymbol g$. This in turns proves that there exists an infinite number of possible redefinitions of $u,\;\bf S$.

The total energy of the system is, by definition, $$ U=\int\mathrm d^3\mathbf r \ u(\mathbf r) $$ which, after the redefinition $u=u'-f$ changes into $$ U=\int\mathrm d^3\mathbf r \ u'(\mathbf r,t)-\int\mathrm d^3\mathbf r \ f(\mathbf r,t) $$ but we know that if $\partial_t f+\nabla\cdot\boldsymbol g=0$, then$^2$ $$ \int\mathrm d^3\mathbf r \ f(\mathbf r,t)=\text{constant} $$ which means that the new energy only differs in a constant $U'=U+\text{const}$, which is irrelevant, as we can only measure relative energies and never the absolute energy.

This is almost always true. In GR, we can measure the absolute amount of energy, as the Einstein field equations depend on the actual value of $u\;, \bf S$ (these pair is contained in the object $T_{\mu\nu}$ that you'll find in the wikipedia article). This means that if our choice of $\;u,\bf S$ (the one given by Feynman) is not the right one, then we could detect this by disagreement with experiments where the electromagnetic field is the source of gravitation (this is not easy to actually measure, but possible in principle).

EDIT

What does Feynman mean by "indefiniteness in the location of field energy"?

The function $u(\bf r)$ is the energy density, i.e., it's a function that assigns to each $\bf r$ the amount of electromagnetic energy per unit volume at that point in space. If $u(\bf r)$ is very high at a certain point $\bf r_0$, we may say that in that point (location) there is a high amount of electromagnetic energy. But as $u$ is not unique, we may redefine $u'=u+f$ (as explained above), and it may happen that the new $u'$ is not very high at $\bf r_0$, but at another point $\bf r'_0$. So, where is the electromagnetic energy? at $\bf r_0$ or $\bf r_0'$?. There is not a single possible answer: it depends on the function we choose to use as the energy density $u$. Had we chosen another one (and we can if we want to), we would get another answer.

This means that we cannot in general say where the electromagnetic energy is, because this depends on whether we use $u$ or $u'$. The location energy of the electromagnetic field is not well defined.

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$^1$ for example, if we take $\omega$ to be any real number, and $\boldsymbol k$ any vector such that $|\boldsymbol k|=\omega$, then $\mathrm e^{i\ \omega t-\boldsymbol k\cdot \mathbf r}$ is a solution of the wave equation.

$^2$ this is actually kind of easy to prove: \begin{align} \frac{\mathrm d}{\mathrm dt}\int\mathrm d^3\mathbf r \ f(\mathbf r,t)&= \int\mathrm d^3\bf r \ \partial_tf(\mathbf r,t) \\&=-\int\mathrm d^3\mathbf r \ \mathbf \nabla \cdot\boldsymbol g(\mathbf r,t) \\&=-\oint\mathrm d^2\boldsymbol s \cdot\boldsymbol g(\mathbf r,t)=0\;. \end{align} where I used the Gauss Theorem, and then the fact that $\boldsymbol g(\mathbf r\to\infty)=0$.