Understanding One Forms in General Relativity: A Beginner's Guide

[resurgance2001]

Hi

I am a new(ish) student of general relativity. I am currently reading 'Relativitiy DeMystified'

However this guys explanation of one forms is completely mystifying to me.

He says that basis vectors

$$e_a = ∂_a = {\frac{∂a}{∂X^a}}$$


And then says that this type of basis is called a coordinate basis, and that is allows us to 'think of a vector as an operator, one that maps a function into a new function

Then

$$Vf = (V^a e_a) = V^a ∂_a f$$

The vector V can be represented by covariant components V_a and this vector is called a 'one form'

I just did not get that. Can anyone explain in really simple terms what a one form is?

Thanks

Peter

 

[Nugatory]

The vector V can be represented by covariant components V_a and this vector is called a 'one form'

I just did not get that. Can anyone explain in really simple terms what a one form is?

Stab an onion with an icepick, and the icepick will pierce some number of layers of the onion. The one-form tells you how many layers of the onion will be pierced by a stab of unit length in a particular direction. Higher numerical values of the components of the one-form correspond to more closely spaced and thinner layers of the onion.

The traditional gradient of a function can be treated as a one-form. It tells you how the value of the function changes when you move an infinitesimal distance in a given direction. The surfaces of constant value of the function are analogous to the layers of the onion.