Relativity makes sense if everything is wavelike

 When I first encountered relativity theory, I took an unusual approach. The point is that c can always be constant while the wavelength (λ) and frequency (f ) can change accordingly, because c = λ x f. This is a direct consequence of v = s/t (where v is velocity, s distance and t time). We all know that velocity (or speed) is just distance divided by time. And λ represents distance while f represents 1/t. 

So, here’s the thing: it occurred to me that while wavelength and frequency would change according to the observer’s frame of reference (meaning relative velocity to the source), the number of waves over a specific distance would be the same for both, even though it’s impossible to measure the number of waves. And a logical consequence of the change in wavelength and frequency is that the observers would ‘measure’ different distances and different periods of time.

 

One of the first confirmations of relativity theory was to measure the half-lives of cosmic rays travelling through the Earth’s atmosphere to reach a detector at ground level. Measurements showed that more particles arrived than predicted by their half-life when stationary. However, allowing for relativistic effects (as the particles travelled at high fractional lightspeeds), the number of particles detected corresponded to time dilation (half-life longer, so more particles arrived). This means from the perspective of the observers on the ground, if the particles were waves, then the frequency slowed, which equates to time dilation - clocks slowing down. It also means that the wavelength was longer so the distance they travelled was further. 

 

If the particles travelled slower (or faster), then wavelength and frequency would change accordingly, but the number of waves would be the same. Of course, no one takes this approach - why would you calculate the Lorentz transformation on wavelength and frequency and multiply by the number of waves, when you could just do the same calculation on the overall distance and time.

 

Of course, when it comes to signals of communication, they all travel at c, and changes in frequency and wavelength also occur as a consequence of the Doppler effect. This can create confusion in that some people naively believe that relativity can be explained by the Doppler effect. However, the Doppler effect changes according to the direction something or someone is travelling while relativistic effects are independent of direction. If you come across a decent mathematical analysis of the famous ‘twin paradox’, you’ll find it allows for both the Doppler effect and relativistic effects, so don’t get them confused.

 

Back to the cosmic particles: from their inertial perspective, they are stationary and the Earth with its atmosphere is travelling at high fractional lightspeed relative to them. So the frequency of their internal clock would be the same as if they were stationary, which is higher than what the observers on the ground would have deduced. Using the wave analogy, higher frequency means shorter wavelength, so the particles would ‘experience’ the distance to the Earth’s surface as shorter, but again, the number of waves would be the same for all observers.

 

I’m not saying we should think of all objects as behaving like waves - despite the allusion in the title - but Einstein always referred to clocks and rulers. If one thinks of these clocks and rulers in terms of frequencies and wavelengths, then the mathematical analogy of a constant number of waves is an extension of that. It’s really just a mathematical trick, which allows one to visualise what’s happening.