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# 15: Relativistic Forces and Waves

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. Image used with permission (Public Domain; NASA).