From elementary general relativity

1. Sep 20, 2014

fu11meta1

A map h: = T.(M) ---> T.(M) is defined by h(X) = X + g(U,X)U where U ε T.(M) is a fixed vector with g(U,U) = -1.

i: Give an expression for the components h^i (sub) j (This is "h" with a superscript i and subscript j) of h regarded as a tensor type (1,1)

ii: Prove that h^2 = h. Interpret h geometrically.

So I've been playing around with this but I'm getting no where. I could use some guidance on where to really get started. I'm also VERY new to general relativity, so every step/hint/anything would be great

2. Sep 22, 2014

Matterwave

So, the first step is probably to figure out what the question is really asking. So we have this map $h$ which takes a vector $X$ and gives you the back the vector $h(X)=X+g(U,X)U$. Since this mapping is taking a vector into another vector, then we know that $h$ is a type (1,1) tensor which has components $h^i_{~~j}$. So we know then that in component notation $h(X)\equiv h^i_{~~j}X^j e_{(i)}$ where $e_{(i)}$ are the basis vectors (the parenthesis mean that they are not components of a vector, but each $i$ denotes a different vector). Can you perhaps turn the right hand side of the first equation into component notation to see what it looks like?

3. Sep 22, 2014

Orodruin

Staff Emeritus
Regarding ii, this should be a simple matter of insertion and using the linear property of g. For the interpretation, what happens if you set X = U? What happens when you use X such that g(U,X) = 0? (I suggest not using the coordinate representation for ii as it really is not necessary.)