Abstract
The electromagnetic waves, which are inevitably studied in Special Relativity, do not exhaust all types of waves. Indeed, as it has been remarked already in many cases, Special Relativity is a theory of all physical phenomena including electromagnetism. Therefore within Special Relativity one must be able to deal with e.g. thermal waves, acoustic waves etc. In this section we discuss the generic concept of a wave in Special Relativity. Waves are rather difficult to understand properly, even in Newtonian Physics where one has a direct sensory observation of space, time and motion. Moreover even there when the subject is approached at a slightly higher level demands rather heavy mathematical formalism and one gradually creates the opinion that somewhere Physics is lost and mathematics prevail. The situation is even worse in Special Relativity where one is obliged to work with a geometric concept of motion right from the beginning. Therefore in order to study the waves in Special Relativity it is best to reformulate the Newtonian wave theory in a geometric rather than in the standard formalism. The subject of relativistic waves is vast and in this book we aim to deal only with the basic elements of it, in fact we shall take the subject up to the point that one can apply properly optics and understand the deBroglie waves, which are required for a proper understanding of Schröndinger equation and consequently of Quantum Mechanics.
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Notes
- 1.
Because the geometry of the space is Euclidian the normal |∇S|≠ 0.
- 2.
Note the difference between w and u a!
- 3.
This result is different form the composition of 3-velocities we met before, that is, the phase velocity does not behave as the 3-velocity of a particle. This is due to the fact that the phase velocity is a ‘geometric velocity’ concerning surfaces in spacetime and not particles or other physical systems. For this reason the phase speed a. is not restricted by the speed of light in vacuum and b. does not obey the relativistic rule of composition of 3-velocities.
- 4.
Unfortunately the role of the Doppler effect frequently is not properly understood and people approach it in a different way for electromagnetic and non-electromagnetic waves. See for example R. Bachman ‘Relativistic acoustic Doppler effect‘ (1982) Am. J. Phys. 50, 816 and R. Bachman ‘Relativistic acoustic Doppler effect in the optical limit‘ (1986) Am. J. Phys. 54, 848.
- 5.
When the mirror is moving towards the observer O the velocity u →−u and the relations we derived become \(\nu _{refl,O}= \frac {1+\frac {u}{c}}{1-\frac {u}{c}}\nu \) while again w refl,O = c, that is the phase velocity w remains the same. This holds only for the photons whose kinematic and particle nature are completely equivalent or, equivalently, the phase velocity of the deBroglie wave associated (to be discussed below) with a photon and the ‘particle’ velocity (speed c) are equal. This is not true for other types of waves and certainly not for the deBroglie wave associated with a particle with mass.
- 6.
\(\frac {1}{ \sqrt {1-x^{2}}}=1+\frac {x^{2}}{2}+\frac {1\cdot 3}{2^{2}\cdot 2}x^{4}+\frac { 1\cdot 3\cdot 5}{2^{3}\cdot 3}x^{6}+\cdots \) where |x| < 1.
- 7.
H. Ives and G.R. Stilwell, “An experimental study of the rate of a moving clock” J. Opt. Soc. Am 28 215–226 (1938) and part II. J. Opt. Soc. Am.31, 369–374 (1941).
- 8.
To date, only one inertial experiment appears to have verified the redshift effect for a detector actually aimed at 90∘ to the object. See a. D. Hasselkamp, E. Mondry, and A. Scharmann (1979) b. ‘Direct Observation of the Transversal Doppler-Shift‘, Z. Physik A 289, 151–155. For a more recent account see Walter Köndig (1963), ‘Measurement of the Transverse Doppler Effect in an Accelerated System‘ Phys. Rev. 129, 2371–2375.
- 9.
The interested reader can find more information in K. Gordon (1980), ‘The Doppler effect: A consideration of quasar redshifts‘ Am. J. Phys. 48,514.
- 10.
See R. Bachman Am. J. Phys. (1989), 57, 628.
- 11.
The Cěrenkov radiation is due to the polarization of the atoms of the material due to the passage of the charged particle. It is an electromagnetic shock-wave phenomenon, the direct optical analogue of the supersonic bang, or the bow wave from a swiftly moving vessel on water. It must not be confused with he Bremsstrahlung process which is due to the deflection of a particle (i.e. the acceleration) caused by the strong Coulomb field of the atomic nucleus. For a detailed explanation of the Cě renkov radiation see J. V. Jelley (1963) ‘Cěrenkov radiation: its Origin, Properties and Applications’ The Physics Teacher I, 203–209.
- 12.
Recall that in order two null four-vectors to be parallel it is enough that their spatial parts be parallel in one coordinate frame. Here we do not assume that the frequency four-vectors are null, that is, we do not restrict our considerations to light waves propagating in vacuum.
- 13.
Louis deBroglie (1923) Compte rendus, 1977 pp. 507–510 and (1924) PhD Thesis “Resherches sur la theorie des Quanta“ University of Paris.
- 14.
Recall that a null vector in a RIO Σ is defined in terms of one scalar A 0 and a unit 3-direction \(\hat {\mathbf {e}}\) by the formula \( A^{0}(1,\hat {\mathbf {e})}_{\Sigma }.\)
- 15.
Universal constants always relate physical quantities of different nature.
- 16.
SeeJ. Haslett ‘Phase waves of Louis deBroglie‘ (1972) Amer.J. Phys. 40, 1315–1320. This paper is a translation of the first chapter of Louis deBroglie’s thesis.
- 17.
See C.Davisson, L.H. Germer (1927). “Reflection of electrons by a crystal of nickel”. Nature Vol. 119: 558–560 and C. Davisson (1928) “Are Electrons Waves?,” Franklin Institute Journal 205, 597.
- 18.
See R. Newburgh (1956), Lett. Nuovo Cimento 29, 195–196, P. Dirac (1924) Proc. Cambridge Philos. Soc.22,432.
- 19.
See A. Einstein (1950) Out of my later Years” (Philosophical Library N.Y. and A. Einstein (1979) ‘A Centenary Volume’ edited A.P. French Harvard University, Cambridge, MA, 319).
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Tsamparlis, M. (2019). Waves in Special Relativity. In: Special Relativity. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-27347-7_18
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