Invisibility Physics: Acceleration without radiation, part I

A couple of years ago, a number of physicists made international news (some descriptions here and here) by proposing that “cloaking devices” were theoretically possible to construct. Two papers appeared consecutively in Science Magazine in May 2006, one by U. Leonhardt of the University of St Andrews, Scotland (Science 23 June 2006: Vol. 312. no. 5781, pp. 1777 – 1780), and the other by J.B. Pendry of Imperial College, London and D. Schurig and D.R. Smith of Duke University (Science 23 June 2006: Vol. 312. no. 5781, pp. 1780 – 1782). Both papers describe how, with the proper materials, one could create devices which ‘guide’ light around a central core region without distortion, effectively making the cloak, and whatever sits in the core, invisible. This idea is illustrated by the figure below, from the Pendry paper, which shows how light rays could be guided around the core:

These papers have generated so much interest that it is fair to say that they have created their own subfield of optical science, what one might call ‘invisibility physics’, and numerous research groups are busy concocting their own invisibility schemes or attempting to construct a Leonhardt/Pendry-style device.

It is interesting to note, however, that the study of objects which are in some sense ‘invisible’ is not really new, and in fact there is a century-long history of scientists studying objects which may be considered, one way or another, undetectable.

I happen to know a lot about the history of such objects, so I thought I’d start yet another long-running series of posts, this one on invisibility physics. We start today with a discussion of what may be the first paper of this type, written by none other than the remarkable physicist Paul Ehrenfest.

Any student of physics is aware of the basic observation that accelerating electric charges radiate electromagnetic radiation: in other words, if the speed or direction of motion of a charged particle is changed, that particle emits energy in the form of electromagnetic waves. For example, radio waves are produced by inducing fluctuating electric currents in a large antenna. When operational, ring-shaped particle accelerators such as the LHC (known as a synchrotron) push charged particles around large circular paths, and these particles emit what is now known as synchrotron radiation. A more mundane example is the low-frequency electromagnetic fields produced by the alternating current used to power all electrical appliances in one’s household.

Students of physics are taught, in essence, that “accelerating charges = radiation”. However, in 1910 Paul Ehrenfest published a short paper (“Phys. Z. 11 (1910), 708-709) entitled, “Ungleichförmige Elektrizitätsbewegungen ohne Magnet- und Strahlungsfeld”. Roughly translated*, the title is, “Irregular electrical movements without magnetic and radiation fields”. In this beautiful and extremely short paper he demonstrates, quite conclusively, that Maxwell’s equations allow for the existence of accelerating charge distributions which emit no radiation.

Ehrenfest’s motivation for writing this paper, though not explicitly stated, seems to be rooted in the prehistory of quantum mechanics. By 1910, physicists still did not have a good understanding of even basic atomic structure, and at least a pair of models had been put forth:

It is to be noted that the “correct” model, the Bohr model of the atom, would not be proposed until 1913. Even Rutherford’s discovery of the nucleus would not be unveiled until 1911. The “plum pudding” model of J.J. Thomson suggested that the atom consisted of electrons moving in a “soup” of positive charge, while the “Saturnian” model of H. Nagaoka suggested that the atom consisted of electrons moving in one or more “rings” around a central nucleus.

Both models suggested localized motion, and hence acceleration, of electrons. As we have noted, Maxwell’s equations demonstrate that accelerating charges should radiate, and this in turn would have suggested to physicists of the time that both Thomson’s atom and Nagaoka’s atom must be constantly losing energy and hence unstable.

Not necessarily, said Ehrenfest. He begins his paper by describing two extended distributions of charge which can accelerate but clearly produce no radiation. The first is an infinite uniform planar sheet of charge, which oscillates along the direction perpendicular to the surface:

Let us assume that the plane is parallel to the x-y plane and oscillating in the z-direction. Which direction does the electric field point? Because the system is infinite and uniform along the x and y directions, there is nothing that could `allow’ the field to point in these directions. In fact, the electric field E must point in the z-direction. Similarly, the only direction in which electromagnetic radiation emitted by the plane can point is the z-direction. However, electromagnetic waves are transverse: like water waves or waves on a string, the electric field vibrates perpendicular to the direction of radiation. Here we have a system where the electric field is parallel to the only possible direction of motion, and so this system cannot radiate electromagnetic radiation. This argument is illustrated below:

Not incidentally, the magnetic field H must also point in the z-direction, but this would imply the existence of magnetic monopoles, which are strictly forbidden in Maxwell’s equations. Therefore this system also produces no magnetic field!

This example flies in the face of “conventional wisdom” that accelerating charges necessarily give off electromagnetic waves. The discrepancy is readily traced to the fact that all textbook examples of accelerating, radiating particles are point particles. Ehrenfest’s example is an extended distribution of accelerating charge, which nobody had previously studied.

It’s a little too extended, though, as the result holds true only for an infinite plane. Since no such infinite charge distribution exists, it is reasonable to object that this example is a pathological, unrealistic one. In Ehrenfest’s own words, “Der Umstand, daß hier die elektrischen Ladungen bis ins Unendliche reichen, kann das Resultat bedenklich erscheinen lassen,” roughly translated as, “The circumstance that the electrical charges are enough here in the infinite, can let the result appear precarious.” He then considers a more realistic example: a spherical surface, with charge uniformly distributed across it, which pulsates:

Now we have a system with rotational symmetry, i.e. it looks the same from any direction. By symmetry, the only direction that an electric field, magnetic field, and radiation field can point is radially outward from the center of the sphere. But again, in a radiation field the electric and magnetic fields must be transverse to the direction of motion, so this system also will not produce any radiation, or have a magnetic field!

So with two easy-to-understand physical examples, Ehrenfest had demonstrated that extended accelerating distributions of electric charges exist which (a) produce no radiation, and (b) produce no magnetic field.

He concludes his paper by going even further, however, and giving a prescription for mathematically deriving radiationless accelerating charge distributions of any shape. The prescription is as follows: (WARNING! Some math content follows!)

Define a volume A in which your accelerating charges will reside, and an exterior region B in which there is to be no radiation.

In region B choose a scalar electrostatic potential \phi(x,y,z) such that it satisfies Laplace’s equation, i.e.

\nabla^2\phi(x,y,z)=0.

In region A let the electric potential vary in time, i.e. \phi(x,y,z,t), and choose it such that at the boundary between A and B it goes continuously to the value in B. Then the electric field E, the charge density ρ, and the convective current density v are given by the usual formulas:

{\bf E} = -\nabla \phi, 4\pi \rho =-\nabla^2 \phi, v = \frac{\frac{\partial}{\partial t}\nabla \phi}{\nabla^2 \phi}.

These choices, together with H = 0, automatically satisfy Maxwell’s equations and represent an electromagnetic source, in an arbitrary volume, which only produces a static electric field outside the source region. (END math content!)

Such radiationless charge distributions are now referred to (oxymoronically) as nonradiating sources. They represent a form of ‘invisibility’ in that they are electromagnetic systems that should emit light, according to conventional wisdom, but don’t. In later posts, we will also see that there is a strong connection between the physics of nonradiating sources and other types of invisible objects.

Ehrenfest’s prescription made plausible the possibility of a stable atom consisting of orbiting or accelerating electrons. Alas for his paper, by 1913 Bohr had proposed his model of the hydrogen atom which could explain the atom’s unusual emission properties. As time progressed, scientists came to realize that ‘loopholes’ in classical physics were inadequate to explain the observed behaviors on the atomic level, and that a new theory was needed to account for all the new and strange observations: quantum mechanics. Ehrenfest’s prescription seems to have been mostly forgotten in the quantum hubbub which ensued.

However, other authors extended and/or rediscovered Ehrenfest’s original ideas, and would attempt to apply them to all sorts of problems. We’ll discuss some of these later discoveries, and their applications, in later posts!

*by Babelfish and, earlier, by multilingual colleague!
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9 Responses to Invisibility Physics: Acceleration without radiation, part I

  1. Personal Demon says:

    Carlos Stroud’s research group at the Institute of Optics was studying how to make an electron behave “classically” by using a specific superposition of wavefunction states. It seems to me that atoms in such a state should emit EM radiation, but that seemed to still be an unresolved area of research. What do you think?

  2. “It seems to me that atoms in such a state should emit EM radiation, but that seemed to still be an unresolved area of research. What do you think?”

    That’s a good question; in essence you’re asking whether the correspondence principle holds for the atom in a high quantum state, as far as atomic radiation is concerned. Another way to phrase it: is the quantum/classical transition of a system’s behavior completely achievable for a single atom, or do classical radiation laws only appear for ensembles of atoms?

    It’s a bit tricky to decide; I suspect that Ehrenfest’s ‘radiationless motions’ were discarded because, once Bohr’s theory came about, it became obvious that atoms don’t radiate according to a classical interpretation of Maxwell’s equations. We know that an atom releases its energy “all at once” as a photon when it jumps from one discrete orbit to another, and this is seemingly hard to reconcile with the continuous classical radiation theory.

    Or is it? For an electron in a high-n quantum state (not a stationary one, though), the energy levels are very narrowly spaced. Presumably for high enough n, the electron can shed photons and “dribble down” to lower energy states in a manner that looks very much like classical radiation. For large n, the particle will be accelerating very slowly, as well, so it doesn’t have to radiate very fast or at a high frequency. I’m guessing that such low-energy emissions would be very hard to detect, though.

    In other words, I’m guessing that the correspondence principle holds for the radiation of high-n quantum atoms, but is tricky to measure.

  3. Seeker says:

    Hi,
    What would happen if you were to rotate the pulsating charged sphere?
    Presumably you would get a magnetic field, but would this sucker radiate?
    And what would be the effects of varying rates of pulsation Vs. angular velocity on any EM radiation?
    Cheers

  4. Hi Seeker,

    I suspect that for this example, adding rotation would actually make it radiate, because the rotation would break the symmetry of the system that makes it radiationless in the first place. However, this was only the first solution for radiationless motion found. Later researchers found more exotic and counterintuitive examples, including examples which include rotation. I’ll hopefully be addressing those as I blog on, so stay tuned! :)

  5. Amnon says:

    hi,

    Thanks for the clear and interesting paper.
    I will appreciate an answer to the question below:

    An individual piece of charge on the infinite uniform planar sheet “feels” no field when the sheet is at rest. Is it still a zero field on the sheet when it is accelerating?

    It seems that if there is no radiation the field does not change. If the field does not change, the information that the entire sheet was accelerated is an immediate knowledge for every point on the sheet. Accordingly the default is: if part of the sheet is accelerating – the entire sheet is accelerating. Field will change (with radiation at retarded time) only if not the entire sheet was accelerated. Is this a correct description of the phenomenon?

    Thanks,
    Amnon

  6. yoron says:

    As always Mr Skull, it’s a pleasure reading you. I do hope to find your other parts too. Eh, that may have came out wrong. What I meant was those parts binding together your discursion. So, what is charge :)

  7. gary says:

    How could you create this senario?…IOW, with area B saisfying laplacian = 0, and time varying E potential inside??

    PS. what is convection current density?

    Thanks,

    • From a physical point of view, nobody knows: it is very difficult to construct a three-dimensional source distribution that has specified properties. From a mathematical point of view, one simply chooses any function phi(r) to represent the potential inside the area B, chosen such that the function goes smoothly to zero on the boundary of the source; such a function will automatically produce a zero potential outside. The current density/charge density follows directly from Maxwell’s equations.

  8. Lee Hively says:

    Very interesting.
    Is there an English translation of Ehrnfest’s 1910 paper?
    Can you point me to any experiments that test the prediction of no radiation from a charged, oscillating sphere? How does one derive v=(time derivative of grad phi)/(del-squared phi)?

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