A Dissident’s View: General Relativity in the 21st Century
Hello fellow dissidents, allow me to introduce myself. I am a relatively new enthusiast of General Relativity. Now, by enthusiast I mean in the sense of someone who loves a good riddle, enjoys learning the secret, then going back over WHY you were fooled. Do I think I have the whole riddle figured out? Not by any means, but I do hope to join with my fellow researchers to pique their interest and listening to their understanding of it.
If the start of this trick is General Relativity:
http://www.nbc.com/saturday-night-live/video/guest-performance—penn-and-teller/n9398
in what ways could the reveal be Lorentz’s Aether through the lense of a Cosmological Constant?
Anybody care to ponder what the most basic definition of a tensor is? How can a Constant of Integration fool you in equations? (Without cheating by looking through the proceeds of the first CNPS conference)
As I know it (and that is not much), a tensor is pretty general in that it is any mathematical object used to describe some property of a physical object. That to me is REAL general. But knowing you Jeff, you have a different answer. Could it have to do with AREA???
🙂
Of course a linguist would pick out the important points in an idea, heh. This might be a bit long, but could be important to the dissident who ponders gravity.
You might have heard of a one-form before? Eh, probably not (I hadn’t several years ago). If you were to look at the definition on Wikipedia, the second equation is something like a_x=f_1(x)dx_1+ yada yada yada
The important point is that most people who have taken integral calculus should recognize the f(x)dx. Slap an integral sign on the front of that baby and you have yourself area. In other words, chunks of area can be summed but one-forms can be integrated. When I say integrated, I mean that you are summing up little slices of area, so a one-form is a tiny little (infinitesimal) slice of area.
Now that is nice but so what? Welp, what if you wanted to find out how the area was changing at a point in that integral? Easy peasy, you just find the infinitesimal change of your little slices of area. In other words finding the infinitesimal change of your one forms is the same as finding the infinitesimal change of area. So now you can find out how much your area is changing. We can just call f(x)=y so we end up with dydx as an expression of our infinitesimal change of area. Normalize that puppy and we now have (dydx)/(dxdx) which simplifies down to dy/dx, a measure of a change in area.
That dy/dx gives rise to several names (and more): A first derivative, a directional derivative, a tensor of rank 1. It is important for a dissident scientist to have a broad view of the equations they work with because, using partial derivatives I can’t write here, if you write a classic field equation using directional derivatives then y=φ giving in three “dimensions” ∇φ.
So if you write F=mg and equate this to ∇φ, then you are stating that force is the same thing as a measure of the change in area. Now, if instead of y=f(x) we let x=f(y), things get real fun interpreting ydx, dydx and dy/dx…
Note that I didn’t bother to use partial derivative symbol but they are implied 🙂