Now that we have a better understanding of Maxwell's equations, we can study this classic problem again.
A charge in a constant acceleration in a Minkowski space
Edward M. Purcell's approach fits this case. In the "real" physical world there do not exist eternally accelerated charges. The acceleration has started some time in the past and will end at some time in the future.
The diagram at top right of the link, and also within the text of the link, has the growing circle which contains the end result of the acceleration. Then there is a "transition area", and outside is the field before any acceleration started.
For a long smooth acceleration, the transition area is very wide.
How does Purcell accomplish to connect the lines of force inside the circle and outside the circle?
He assumes that there is one "arc" (half a wavelength) of an electromagnetic wave traveling outward at the speed of light. That half-wave pulls the electric lines of force to the right direction, so that they can connect to the lines outside the circle. The magnetic field lines are like in the diagram at the top of the Wikipedia link.
The Purcell approach looks at the system in the frame of a static Minkowski observer.
When the charge is being accelerated, the static observer interprets the electromagnetic field as having a single arc of a forming electromagnetic half-wave. The arc grows in length at the speed of light as long as the acceleration continues. The Poynting vector E × B which the static observer sees, shows energy flowing outward from the charge at the speed of light. We can say that the static observer sees "radiation" flowing out.
When the acceleration ends, the half-wave starts moving outward at the speed of light. Then it is clearly an "electromagnetic wave" moving outward, and everyone would agree that the charge has radiated.
When the acceleration ends, the half-wave starts moving outward at the speed of light. Then it is clearly an "electromagnetic wave" moving outward, and everyone would agree that the charge has radiated.
Any inertial observer in the Minkowski space can fill the entire space with static spatial coordinates. By static in this case, we mean that the distances between the intersections of the grid do not change in time. Such a coordinate grid is good for analyzing energy flow.
However, an observer who co-accelerates along with the charge, cannot define such a global static grid which would move along him, because for him, part of the Minkowski space is behind a horizon. He can, however, define local static coordinates. In his coordinates, the electric field of the charge looks static. The field looks static also for an inertial observer who just at this moment t moves along with the charge. The inertial observer does not see any magnetic field close to him. He interprets that the charge is not radiating close to him. The Poynting vector E × B is zero.
Thus, the accelerated charge does radiate from the point of the view of a global inertial Minkowski observer, but in a local partial coordinate system of a co-accelerating observer (or an inertial observer briefly moving along) it does not radiate.
David G. Boulware (1980) writes about the significance of a horizon in the accelerating frame.
Radiation reaction
Does the radiation of an accelerated charge produce a force which slows it down?
The Abraham-Lorentz force is proportional to the time derivative of the acceleration. In a constant acceleration, there would be no Abraham-Lorentz force.
We have previously noted that the Abraham-Lorentz formula tries to derive the radiation reaction from incomplete information: it only contains the time derivative of the acceleration of the charge, and does not contain full information of the complete electromagnetic field around the charge.
Let us look at our analogy of a flexible solid object being accelerated. We start pushing the object from the center, and the central part starts pulling static outer parts of the object up to the pace. When the central part pulls on a small outer part, the central part loses some of its kinetic energy. Some of that energy goes to the kinetic energy of the small part. But momentum conservation dictates that some energy is left over. That energy goes to the vibrations of the object, if we assume that the object is perfectly elastic.
The force which slows down the central part is due to the momentum flow to the small part. In that sense, there is no "radiation reaction". The radiation gains its energy from the leftover kinetic energy, when the small part slows down the central part.
In the case of a flexible object, some push energy is lost to vibrations of the object. The "radiation reaction" in this case might also be defined as the energy difference relative to pushing a rigid object of the same mass. In that sense, there is a radiation reaction.
In an inelastic collision, some kinetic energy is lost to deformations. We might define that energy as the radiation reaction.
Suppose that we push a rigid object with a 1 newton force a distance of 1 meter. We give 1 joule of kinetic energy to the object.
Suppose that we push a same mass flexible object in the same way. The push happens faster because the object is "soft": we give less momentum to the object. Only part of the 1 joule I gave ends up being kinetic energy, the rest is spent on vibrations.
The flexibility of the object initially lowers the inertia which we feel when we start the push. If we push for 1 second with a 1 newton force, we need to push a longer distance than we would with a same mass rigid object. That extra distance is the source of the radiation energy. The inertial mass of an electron varies with time during the push.
The flexibility of the object initially lowers the inertia which we feel when we start the push. If we push for 1 second with a 1 newton force, we need to push a longer distance than we would with a same mass rigid object. That extra distance is the source of the radiation energy. The inertial mass of an electron varies with time during the push.
Is there some sense in saying that a "radiation reaction produced a force" which slowed down the flexible object? All the momentum which our push gave to the object did go to building up its momentum. In this sense, there is no radiation reaction.
But energy-wise, part of the push energy did go to building up vibrations.
If we push a rigid object on a floor, then friction eats up both some momentum and some energy.
Conclusion: if we look at the momentum of the flexible object, there is no radiation reaction. All the push momentum goes to the momentum of the object. But energy-wise, there is a radiation reaction. Vibrations eat up part of the push energy.
A supported observer sees the electric field static, except far away where the signal from the current setup has not reached yet. He sees the Poynting vector as zero in the local area.
If we think of flexible object supported in a gravitational field, the upward push at the center of the object and the weight of the outer parts have deformed the object. Far away, gravity is weak and the deformation will be small.
The deformation can be considered as a "wave frozen in time". If the forces on the object would disappear, the deformation would start moving mostly outwards at the speed of light, with some reflection of the wave back.
So, does a supported charge radiate? No, its radiation wave is "frozen in time".
What about a freely falling observer? Locally, he sees the configuration like an inertial observer saw an accelerating charge in a Minkowski space. The Poynting vector says that there is energy flow from the charge outward. Where does this energy end up? Freely falling coordinates are not static, and it is not clear if one can meaningfully analyze energy flows in them.
The charge will not fall freely because it has to pull behind the parts of the electric field which are outside the gravitational field. A freely falling observer will see the charge accelerate upwards. He will see an energy flow. Can we call that radiation? In the local frame, it looks like the accelerated charge Minkowski case which we treated above. We can call that radiation.
Also a supported observer sees the falling charge pull on the outer parts of its electric field, that is, he sees that the charge is not falling freely. He sees radiation.
A supported charge in a gravitational field
A supported observer sees the electric field static, except far away where the signal from the current setup has not reached yet. He sees the Poynting vector as zero in the local area.
If we think of flexible object supported in a gravitational field, the upward push at the center of the object and the weight of the outer parts have deformed the object. Far away, gravity is weak and the deformation will be small.
The deformation can be considered as a "wave frozen in time". If the forces on the object would disappear, the deformation would start moving mostly outwards at the speed of light, with some reflection of the wave back.
So, does a supported charge radiate? No, its radiation wave is "frozen in time".
What about a freely falling observer? Locally, he sees the configuration like an inertial observer saw an accelerating charge in a Minkowski space. The Poynting vector says that there is energy flow from the charge outward. Where does this energy end up? Freely falling coordinates are not static, and it is not clear if one can meaningfully analyze energy flows in them.
An "almost" freely falling charge in a gravitational field
The charge will not fall freely because it has to pull behind the parts of the electric field which are outside the gravitational field. A freely falling observer will see the charge accelerate upwards. He will see an energy flow. Can we call that radiation? In the local frame, it looks like the accelerated charge Minkowski case which we treated above. We can call that radiation.
Also a supported observer sees the falling charge pull on the outer parts of its electric field, that is, he sees that the charge is not falling freely. He sees radiation.
Does an accelerating object see Unruh radiation?
We have in this blog long stressed that the existence of Unruh radiation would break the conservation of momentum. But can we harmonize them with our new ideas?
The accelerating object is carrying an electric charge as a sensor to the thermal photons it is supposed to see. The conservation of energy requires that these photons should get their energy from the kinetic energy of the electron or from the force which is pushing the electron.
We saw that the flexibility of the electron's electric field allowed it to move momentum to the trailing parts of its field and free some kinetic energy which it radiates away as electromagnetic radiation. Can Unruh use the same mechanism?
An outside agent is able to extract energy from the accelerating object without extracting momentum. The idea is to make use of the changing speed of the object. The agent first helps in acceleration for 1 second with a 1 newton force, and then resists for 1 second with a 1 newton force. The object moves faster in the resist phase and the agent can pick up more energy than he gave. Could this be the mechanism of Unruh radiation?
Indeed, we can imagine the following process with virtual photons: the accelerated electron sends a virtual photon which "borrows" some momentum and energy from the vacuum. The virtual photon helps in accelerating the electron. Later, when the electron is moving faster, it absorbs the virtual photon. Some kinetic energy is left over which the electron radiates away.
The above process does not give out any radiation in the accelerated frame. Otherwise, a statically supported electron on Earth would radiate. Is this a contradiction?
We saw in the previous section that in the inertial frame there is an energy flow from the electron outwards but in the accelerated frame there is no energy flow. If we believe the energy flow consists of "photons" of some kind, then the inertial observer sees the electron radiate but the accelerated observer not. The photons cannot be ordinary full electromagnetic waves because then they would be visible to the accelerated observer.
This reminds us of hypothetical Hawking radiation which is not visible to the freely falling observer.
Actually, the virtual photon cannot extract energy in the way that we described. The electron is in a bound state if it is supported by an electric field. The virtual photon may excite the electron, but the virtual photon probably gives very little - if any - momentum to the electron. When the photon is reabsorbed to the electron, there is no leftover kinetic energy. It looks like the inertial observer does not see any radiation in "ordinary photons". He does see a Poynting vector energy flow in the field, though.
Unruh claims that the accelerating observer can see the electron jump to a higher energy state. That does not happen in our example. It is not Unruh radiation.
If there exists Unruh radiation, then an equivalence principle is broken as a supported electron on Earth does not see it. Could it be that some global phenomenon of the field in the Minkowski space would produce Unruh radiation? That global thing would not be present in the gravitational field of Earth.
In the Minkowski case, the electron keeps pulling the outside parts of its field up to the pace. There must be some kind of a self-force which slows down the electron. But also in the gravitational case, the electron must keep pulling the mass-energy of the field up. Also in that case there has to be some self-force.
If Unruh radiation is real, there has to be a derivation of it also in an inertial Minkowski frame. Since quantum mechanics is formulated in inertial frames, the analysis in an inertial frame is much more trustworthy than the controversial field quantization in an accelerating frame.
An outside agent is able to extract energy from the accelerating object without extracting momentum. The idea is to make use of the changing speed of the object. The agent first helps in acceleration for 1 second with a 1 newton force, and then resists for 1 second with a 1 newton force. The object moves faster in the resist phase and the agent can pick up more energy than he gave. Could this be the mechanism of Unruh radiation?
Indeed, we can imagine the following process with virtual photons: the accelerated electron sends a virtual photon which "borrows" some momentum and energy from the vacuum. The virtual photon helps in accelerating the electron. Later, when the electron is moving faster, it absorbs the virtual photon. Some kinetic energy is left over which the electron radiates away.
The above process does not give out any radiation in the accelerated frame. Otherwise, a statically supported electron on Earth would radiate. Is this a contradiction?
We saw in the previous section that in the inertial frame there is an energy flow from the electron outwards but in the accelerated frame there is no energy flow. If we believe the energy flow consists of "photons" of some kind, then the inertial observer sees the electron radiate but the accelerated observer not. The photons cannot be ordinary full electromagnetic waves because then they would be visible to the accelerated observer.
This reminds us of hypothetical Hawking radiation which is not visible to the freely falling observer.
Actually, the virtual photon cannot extract energy in the way that we described. The electron is in a bound state if it is supported by an electric field. The virtual photon may excite the electron, but the virtual photon probably gives very little - if any - momentum to the electron. When the photon is reabsorbed to the electron, there is no leftover kinetic energy. It looks like the inertial observer does not see any radiation in "ordinary photons". He does see a Poynting vector energy flow in the field, though.
Unruh claims that the accelerating observer can see the electron jump to a higher energy state. That does not happen in our example. It is not Unruh radiation.
If there exists Unruh radiation, then an equivalence principle is broken as a supported electron on Earth does not see it. Could it be that some global phenomenon of the field in the Minkowski space would produce Unruh radiation? That global thing would not be present in the gravitational field of Earth.
In the Minkowski case, the electron keeps pulling the outside parts of its field up to the pace. There must be some kind of a self-force which slows down the electron. But also in the gravitational case, the electron must keep pulling the mass-energy of the field up. Also in that case there has to be some self-force.
If Unruh radiation is real, there has to be a derivation of it also in an inertial Minkowski frame. Since quantum mechanics is formulated in inertial frames, the analysis in an inertial frame is much more trustworthy than the controversial field quantization in an accelerating frame.
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