Frequency in general relativity.

Ok. It's clear what's the role of frequency in special relativity. Frequency does not change essentially from its origin in classical mechanics. Frequency has units of inverse of time, which means it's something with no dimensions that is measured per unit of time. In case of frequency that 'something with no dimension' is a repetition.

Quite clear, right? Better explanations here, and don't forget to donate. I will when I get paid for what I do, which has not come yet.

Back to the subject, in special relativity time changes and so does frequency. If there's a time dilation according to velocity value, frequency changes as time does. With an inverse proportionality relation, of course. 

In general relativity, all references I've found so far point out frequency is the same. They talk about red-shifting and the same time dilation issues special relativity does.

But, how come?

We have said before time is just another dimension in general relativity (a special one because our perception, but just another dimension). Several questions arise, then:

1. Are there spatial frequencies the same as a temporal one? If so, what's their use?
2. What does it make time so special? Why are we so used to talk about temporal frequency as frequency?

Easy answer for a physicist, but you have to remember I'm not one them.

In any case, what's the point of this? Well, that picture of the response to a step input in a second order system has made me think about frequency and an old friend/foe concept known as resonance. Which I'll go deeper later. Not now.

Spatial frequencies

Yes, there are spatial frequencies. And their origin is not binded to relativity but for daily use physics and engineering. Thinking about it a little it turns out it's quite normal, right? If you have repetitions of something along a distance, you can characterize that something with a spatial frequency. Think in location markers along a road, for instance. This kind of thing is useful in image processing. Summarizing a lot, picture out a chess board.

A chess board from here.

Every dark square is repeated along x and y axis exactly (more or less) the same distance. Setting up an origin in the first dark square, we could say dark squares have a frequency of  the inverse of the sum of clear and dark squares longitudes.

If we go a little forward, we discover there's a characterization called wavenumber. More than that, if you pick up a kind of wave, you can convert into energy a spatial frequency, similarly to energy equivalence to temporal frequency. Funny thing. Just don't forget about it. It will be important at the end.

Is time so special?

The answer is, yes and no. A generalization of wavenumber leads to the wave vector, and a wave vector is, roughly, a way of characterizing waves which takes into account not only temporal frequency but spatial frequencies too!

So no, time is not special.

But wait for a moment... It turns out, there's a relationship between spatial frequencies and the temporal one (in a monochromatic light beam, at least). The 4-frequency modulus is a null vector, which means, temporal frequency squared (and conveniently normalized with speed of light) is the sum of spatial frequencies each one of them squared.

 In a wave with no mass, modulus defined this way is zero, which makes this equivalence true. You can check it here. We are using "easy" coordinates. No metric conversion, no scalar product surprises.

So, sign is changed in temporal frequency so, maybe time is special after all. That remembers me the funny definition of  Riemannian manifold and that (3,1) signature we saw a couple of posts ago.

But we are not done yet.

More than that, in a matter wave, this null modulus 4-vector, is not null at all. Instead of summing zero, relationship between spatial frequencies and the temporal one is our old friend...

mc squared... normalized by Planck constant.

That means, if you pick up a matter wave (a wave with mass, weird concept, but they exist)  no matter (free pun here) where your reference frame is, you can always get the relationship between the temporal and the spatial frequencies if you know its mass, and furthermore, if you check out wavenumber definition and the definition of momentum...

It turns out we have arrived, without intending it, to the complete Einstein equivalence between mass and energy...

Unexpected and sweet; logical too: we have been talking about the same concepts one time and another with different names and in different orders, but always the same in the end.

Concluding, I've learned not only frequency does exist in relativity with almost the same meaning as classic physics, but it does exists in all dimensions and (at least in the case of waves) there's a close relationship between energy-mass and frequency. Two final questions for next posts:

1. How does all of this translates into things that are not waves? Can we consider 4-frequencies for macroscopic-human-sized-things? If so, is it true we can pump energy in by playing with frequencies (temporal and spatial ones)?
2. What did you say about resonance? Cause it sounds like bullsh*t. Or weird science but I bet nonsenses in the end. There's a ton of sh*t I've got to filter before posting any reference about resonances, so you'll have to be very patient.

See you!