Differential Forms in GR: Higher Order Derivatives

[kent davidge]

The differential form of a function is
\partial{f(x^1,...,x^n)}=\frac{\partial{f(x^1,...,x^n)}}{\partial{x^1}}dx^1+...+\frac{\partial{f(x^1,...,x^n)}}{\partial{x^n}}dx^nIs there (especially in General Relativity) differential of higher orders, like \partial^2{f(x^1,...,x^n)}? If so, how is it computed?

 

[jedishrfu]

This paper discusses Differential forms, Tensors and uses in General Relativity so I would say yes higher order forms are used and are useful in General Relativity.

http://www.aei.mpg.de/~gielen/diffgeo.pdf

 

[Orodruin]

This paper discusses Differential forms, Tensors and uses in General Relativity so I would say yes higher order forms are used and are useful in General Relativity.

http://www.aei.mpg.de/~gielen/diffgeo.pdf

But at the same time no, there are no forms that are the second derivative of a function (as the OP suggests) as the exterior derivative applied twice gives zero (the any-dimensional equivalent of curl(grad(f))=0).

Of course there are other forms of higher order.

 

[stevendaryl]

But at the same time no, there are no forms that are the second derivative of a function (as the OP suggests) as the exterior derivative applied twice gives zero (the any-dimensional equivalent of curl(grad(f))=0).

Of course there are other forms of higher order.

Yes, d^2 always produces zero. However, you can get something sort of conceptually similar to d^2:

1. Operate on F with d to produce dF.
2. Take the Hodge dual, *dF.
3. Operate on THAT with d, to produce d * dF

This isn't necessarily zero, and is sort of like a higher-order derivative. In 3-D, if F is a scalar function, then * d * d F = \nabla^2 F.

 

[kent davidge]

But at the same time no, there are no forms that are the second derivative of a function (as the OP suggests) as the exterior derivative applied twice gives zero (the any-dimensional equivalent of curl(grad(f))=0).

Of course there are other forms of higher order.

Yes, d^2 always produces zero. However, you can get something sort of conceptually similar to d^2:

1. Operate on F with d to produce dF.
2. Take the Hodge dual, *dF.
3. Operate on THAT with d, to produce d * dF

This isn't necessarily zero, and is sort of like a higher-order derivative. In 3-D, if F is a scalar function, then * d * d F = \nabla^2 F.

In General Relativity one frequently deal with differential forms, say W. What actually is it? I know it has to be a completely antisymmetric (0,p) tensor. But what are its components W_μ1...μp? Would it be some array of derivatives of a function?

 

[haushofer]

The components don't have to be derivatives. They are just antisymmetric, which makes threm taylor-fit for integration over manifolds.