Matrix Index Inversion: Clarification Needed

[gentsagree]

is it true that $\frac{1}{g_{ab}}=g^{ba}$? I am a bit confused by the index notation. I especially wonder about the inversion of the indices. Could somebody clarify this please?

 

[homeomorphic]

No, that's not true. That would be the matrix with reciprocal entries, which is obviously not the inverse.

It would take me a while to explain the index notation and lowering and raising indices (and some Latex work), which I am not feeling up to right now.

 

[Fredrik]

$g^{ab}$ is the number on row a, column b of the inverse of the matrix that has $g_{ab}$ on row a, column b.

It's not true in general that if A is an invertible matrix, then $(A^{-1})_{ij}=1/A_{ji}$. Even when A is diagonal, it's only true for the numbers on the diagonal.

 

[HallsofIvy]

IF $g^{ij}$ is intended as the fundamental metric tensor, $ds^2= g^{ij}dx_idx_j$, then it is true that $g_{ij}= (g^{ij})^{-1}$ but, again, that is NOT the same as $\frac{1}{g_{ij}}$.

 

[blue_raver22]

Yes, it is true that \frac{1}{g_{ab}}=g^{ba}. This is known as the inverse property of the metric tensor. In index notation, the inverse of a tensor is denoted by raising the indices using the inverse metric tensor. This notation can be confusing at first, but it is a useful shorthand for representing mathematical operations in tensor calculus.

To better understand this, let's first clarify the notation. The metric tensor, denoted by g_{ab}, is a matrix that encodes the properties of a given space, such as curvature and distance. The indices a and b represent the rows and columns of this matrix, and they can take on values from 0 to 3 (or n, for an n-dimensional space).

Now, when we take the inverse of the metric tensor, we are essentially flipping the matrix over. This means that the rows and columns are swapped, and the indices are raised to indicate this change. So, g^{ba} means that the b index is now the row and the a index is the column, while g_{ab} represents the original matrix with a as the row and b as the column.

In summary, the notation \frac{1}{g_{ab}}=g^{ba} is a shorthand way of expressing the inverse property of the metric tensor. It may seem confusing at first, but with practice, it becomes a useful tool in tensor calculus. I hope this clarifies your confusion.