(Line-of-sight co-ordinate / Curtate cycloid / sharp cusp)
In this section, Feynman discusses the
concept of line-of-sight co-ordinate,
curtate cycloid, and sharp cusp that help to explain the emission of
synchrotron radiation. This section could also be titled as “Finding the retarded lateral motion”
instead of “Finding the apparent motion” because the lateral motion (or distance) matters more than the horizontal
distance.
1.
Line-of-sight coordinate:
“In
this way we get a new motion, in which the line-of-sight coordinate is ct,
as shown at the right… The point is that the horizontal (i.e., line-of-sight)
distance now is no longer the old z, but is z+cτ, and
therefore is ct (Feynman et
al., 1963, p. 34–3).”
It may not be clear why Feynman defines horizontal (line-of-sight)
distance in terms of ct = z+cτ. However, the main purpose
is to deduce the retarded field of a relativistic electron using the second
derivative of its retarded (lateral) position (x¢ or y¢). Pedagogically, it is
useful to draw a graph of lateral distance with respect to time (or line-of-sight
distance) to illustrate the variation of d2er′/dt2. That is, we can use the large apparent transverse acceleration at a certain
point to explain the synchrotron radiation. It may be worth mentioning that the
retarded field does not depend on z (except the time delay) since it is practically constant. In essence, the rule is “we take the actual motion, translate it backwards
at speed c, and that gives us a curve whose curvature measures the
electric field (Feynman et al, 1963, section 34-5).”
We
take the actual motion of the charge (shown at left) and imagine that as it is going
around it is being swept away from the point P at the speed c (there
are no contractions from relativity or anything like that; this is just a
mathematical addition of the cτ) (Feynman et al., 1963, p. 34–2).
Note that the analysis is simplified such that it
does not involve Lorentz contraction, but there is a geometrical effect that is
similar to Doppler effect. During a circular motion, an electron is moving away
from the observer over one half of a period, but it is moving towards the
observer over the other half of the period. As a result, we expect a red shift
or blue shift in the radiation depending on whether the electron is moving away
or towards the observer. The most intense radiation occurs when many electrons are
moving closely behind the photons that are emitted at the earlier (retarded)
times. One may also idealize two photons emitted by an electron at two different
positions and times to reach the observer at the same time (Kim, 1989).
“So
the final answer is: in order to find the electric field for a moving charge,
take the motion of the charge and translate it back at the speed c to
“open it out”; then the curve, so drawn, is a curve of
the x′ and y′ positions of the function of t (Feynman et al., 1963, p. 34–3).”
Feynman explains that there is an “open it out” effect in the curve of lateral distance with respect to time. However, the horizontal distance (= ct) is not the distance between the charge and observer because it is continuously increasing when an electron returns to its original position. Furthermore, there is a time squeezing effect (Kim, 1989) due to the electron “makes a U-turn to catch up” with the photons that were emitted at earlier times. In other words, the “time of photon emission” appears to be compressed when the electron is moving toward the observer, and it results as a sharp pulse of radiation. Perhaps it is good to include a diagram (see below) that shows the distortion of the curve of the x′ from a sinusoidal curve. Interestingly, Feynman suggests, “imagine that this whole ‘rigid’ curve moves forward at the speed c through the plane of sight.”
Note: In section 34-3, Feynman uses the phrase piling up effect to describe Fig. 34-3.
![]() |
Source: Patterson, 2011 |
2. Curtate
cycloid:
“Therefore, when we
translate the motion back at the speed of light, that corresponds to having a
wheel with a charge on it rolling backward (without slipping) at the speed c;
thus we find a curve which is very close to a cycloid—it is called a hypocycloid
(Feynman et
al., 1963, p. 34–3).”
Feynman is correct in using the term cycloid, but
the curve was revised to be called a curtate cycloid in the latter
edition. In general, a cycloid is defined as a
curve formed by a point on the circumference of a circle or
wheel as it rolls along a straight line without slipping. To be specific, a curtate cycloid is the curve formed by a fixed point
inside the rolling circle without slipping (See below). It is also known as contracted
cycloid because the distance from the fixed point to the center of a
rolling circle is shorter (corresponds to a lower speed) whereby the curve is formed
by the point inside (instead of on) the circle. However, the x-axis
of the graph could be ct instead of t, i.e., it is just a scaling
of time-axis as a useful geometric representation for an electron moving at
almost the speed of light.
![]() |
Source: Weisstein, Eric W. "Curtate Cycloid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CurtateCycloid.html |
A hypocycloid is a curve traced by a fixed point on a smaller circle that rolls within a larger circle instead of a straight line (See below). Although the term hypocycloid is incorrectly used in this chapter, we may still apply it in the study of electromagnetic waves. For example, it has been shown that the path traced by an electric field vector in space as an electromagnetic wave propagates can be described by a hypocycloid. This can be useful in the analysis and modeling of complex electromagnetic systems, such as antennas and waveguides (Singh, Prasad, & Ojha, 2003). In modern physics, the concept of hypocycloid may be applicable in superstring compactifications of Calabi-Yau threefolds (Bartolo & Cogolludo-Agustín, 2017).
![]() |
Source: Weisstein, Eric W. "Hypocycloid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Hypocycloid.html |
3. Sharp cusp:
“If the charge is going very
nearly at the speed of light, the “cusps” are very sharp indeed; if it went at
exactly the speed of light, they would be actual cusps, infinitely sharp.
“Infinitely sharp” is interesting; it means that near a cusp the second derivative
is enormous (Feynman et
al., 1963, p. 34–3).”
The term cusp may be incorrectly used in the context
of normal cycloid, curtate cycloid, or prolate cycloid (see below). Firstly, one
may describe a curve of the x′ with respect to time as a normal cycloid if there is a cusp in the curve.
However, the cusp of the normal cycloid of an electron is an idealized concept
because it is impossible for the electron to move at the speed of light. Thus, some
may consider the term curtate cycloid because the electron moves slower than
the speed of light. Perhaps mathematical physicists prefer using higher
curvature (instead of cusp) to explain synchrotron radiation because a
curtate cycloid does not really have any cusp or vertical tangent.
![]() |
Source: Weisstein, Eric W. "Prolate Cycloid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ProlateCycloid.html |
Review
Questions:
1.
Do you agree with Feynman’s use of the term horizontal distance or line-of-sight
co-ordinate? What is the physical meaning of horizontal distance?
2.
Would you explain synchrotron radiation using the concept of general cycloid, curtate
cycloid, or prolate cycloid?
3. Does
a curtate cycloid has sharp cusps?
The
moral of the lesson: By finding the lateral motion of an electron, it helps to explain how more photons are squeezed into a shorter
period of time such that it results in a very short and higher-intensity pulse
of radiation.
References:
1. Bartolo, E. A.,
& Cogolludo-Agustín, J. I. (2017). On the topology of hypocycloids. arXiv preprint
arXiv:1703.08308.
2. Feynman, R. P., Leighton, R. B., & Sands, M.
(1963). The Feynman Lectures on
Physics, Vol I: Mainly mechanics, radiation, and heat. Reading,
MA: Addison-Wesley.
3. Kim, K. J. (1989).
Characteristics of Synchrotron Radiation, Physics of Particle Accelerators. American
Institute of Physics (AIP), 184. 565-632.
4. Patterson, B. D. (2011). A simplified
approach to synchrotron radiation. American Journal of Physics, 79(10),
1046-1052.
5. Singh, V.,
Prasad, B., & Ojha, S. P. (2003). Theoretically obtained dispersion
characteristics of an annular waveguide with a guiding region cross section
bounded by two hypocycloidal loops. Microwave and Optical Technology
Letters, 37(2), 142-145.