At the opening of my class yesterday on `Radiative Processes in Astrophysics,’ the Harvard Astronomy graduate students mentioned that they are frantically preparing for a general exam in which they will be asked questions on a wide range of topics in astrophysics.
I sympathized with them and promised to give them a tip that will boost their self-esteem. In case the faculty examiners will ask a question that they have difficulty answering, I promised to arm them with a question that will challenge the faculty in return.
To establish the foundation for this magic bullet, I dedicated the subsequent couple of hours to the derivation of the equations for the electromagnetic radiation emitted by a charged particle which is accelerated. This so-called Liénard–Wiechert solution for the electromagnetic scalar and vector potentials was derived independently by Alfred-Marie Liénard in 1898 and by Emil Wiechert in 1900. The velocity of the charged particle is inevitably changed by the radiation it emits for the same reason that a rocket is pushed by the gas coming out of its exhaust. This raises an interesting puzzle if the acceleration of the charged particle is triggered by gravity.
As reported in Viviani’s biography, Galileo Galilei realized after dropping objects from the Leaning Tower of Pisa, that all objects free-fall in the same way under the influence of gravity. This realization was confirmed in 1971 by the Apollo 15 astronaut David Scott who dropped a hammer and a feather which reached the surface of the Moon simultaneously. The MICROSCOPE satellite verified the result recently to one part in a quadrillion. Isaac Newton formulated this so-called Equivalence Principle in 1687 (Principia, Book III, Proposition VI, Theorem VI):
“It has been, now for a long time, observed by others, that all sorts of heavy bodies (allowance being made for the inequality of retardation which they suffer from a small power of resistance in the air) descend to earth from equal heights in equal times; and that equality of times we may distinguish to a great accuracy, by the help of pendulums. I tried the thing in gold, silver, lead, glass, sand, common salt, wood, water, and wheat. I provided two wooden boxes, round and equal: I filled the one with wood, and suspended an equal weight of gold (as exactly as I could) in the centre of oscillation of the other. The boxes hanging by equal threads of 11 feet made a couple of pendulums perfectly equal in weight and figure, and equally receiving the resistance of the air. And, placing the one by the other, I observed them to play together forward and backward, for a long time, with equal vibrations.”
In 1915, Albert Einstein used the Equivalence Principle as the foundation for the General Theory of Relativity, in which gravity is formulated as the curvature of spacetime — the same for all particles.
This leads to the silver bullet question that I armed the students with. Let us consider an electrically charged particle and a neutral particle, both accelerated in free-fall under gravity. They would not follow the same trajectory because the charged particle will emit electromagnetic radiation as a result of its gravitational acceleration whereas the neutral particle does not couple to electromagnetism. As a result, the two particles will follow different trajectories under the influence of gravity. Does this violate the Equivalence Principle and puts into question Einstein’s theory of General Relativity?
The students were baffled. They recognized the potential of using this question to embarrass their examiners. But first they needed to find the correct answer.
After a few minutes of silence, a brilliant student in the front row figured out the answer. She said: “the electromagnetic fields carry energy and must be included in the discussion.” I congratulated her for the insight and explained the context to the rest of the class. A static charge has an electric field around it. This electric field carries an extended distribution of energy around the particle. When the charged particle is accelerated, the electric field is distorted in its vicinity and the distortion signal propagates out at the speed of light, taking the form of electromagnetic radiation. In describing the effect of gravity on the entire system, one needs to include both the particle and the extended electromagnetic fields around it because they also respond to gravity. The radiation moves away from the accelerated particle at the speed of light and is part of the system as a whole. The center of mass of the entire system, including the charged particle and the electromagnetic fields around it, follows expectations in terms of its motion in the curved spacetime according to Einstein’s theory of gravity. However, restricting attention to part of the system, such as the charged particle alone, is akin to considering the motion of a rocket without paying attention to the gas emitted through its exhaust. The total momentum of the rocket and its ejected gas is conserved, even though the rocket gets accelerated when viewed separately. In the case under consideration, the neutral particle is much simpler than the charged particle because it is not surrounded by electromagnetic fields and so its mass-energy is localized.
The lesson from this silver bullet question is simple. Being neutral or charged makes a difference in the context of gravity, as much as it does — metaphorically speaking — in politics.
Avi Loeb is the head of the Galileo Project, founding director of Harvard University’s — Black Hole Initiative, director of the Institute for Theory and Computation at the Harvard-Smithsonian Center for Astrophysics, and the former chair of the astronomy department at Harvard University (2011–2020). He is a former member of the President’s Council of Advisors on Science and Technology and a former chair of the Board on Physics and Astronomy of the National Academies. He is the bestselling author of “Extraterrestrial: The First Sign of Intelligent Life Beyond Earth” and a co-author of the textbook “Life in the Cosmos”, both published in 2021. The paperback edition of his new book, titled “Interstellar”, was published in August 2024.
