Skip to main content
Physics LibreTexts

15: Relativistic Forces and Waves

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    As we’ve previously discussed in Chapter 14, you can analyze all types of collisions in special relativity without ever making a reference to the forces they exert on each other. In fact, we haven’t talked about force at all so far, and there’s a good reason for that: forces, already frequently less practical than energies in classical mechanics, become veritable nightmares in special relativity. Nonetheless, there are some questions you can only answer with reference to forces - for example, what velocity a particle will get if you exert a certain force on it for a given period of time.

    • 15.1: The Force Four-Vector
      In classical mechanics, Newton’s second law relates momenta and forces, through the time derivative of the momentum. In relativity, we’ll therefore simply define the force four-vector as the derivative of the energy-momentum four-vector with respect to the proper time.
    • 15.2: The Four-Acceleration
      We can of course also define a four-vector version of the acceleration, by taking the derivative of the four-velocity with respect to the proper time. As with the forces, we’ll see that we’re in for some nasty surprises.
    • 15.3: Relativistic Waves
      We’ve seen that in special relativity, space and time are intimately coupled. There is a classical phenomenon for which this is also the case: the waves we discussed in Chapter 9.
    • 15.E: Relativistic Forces and Waves (Exercises)

    Thumbnail: Two-dimensional representation of gravitational waves generated by two neutron stars orbiting each other. (Public Domain; NASA).

    This page titled 15: Relativistic Forces and Waves is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Timon Idema (TU Delft Open) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.