GR: Metric, Inverse Metric, Affine Connection Caluculation Help

[bmb2009]

Homework Statement

Consider the Schwarschield Metric in four dimensional spacetime (M is a constant):

ds² = -(1-(2M/r))dt² + dr²/(1-(2M/r)) + r²(dθ² + sin²(θ)dø²)

a.) Write down the non zero components of the metric tensor, and find the inverse metric tensor.

b.) find all the components of the connection. (you can use arguments of symmetry to set to zero some of these components)

Homework Equations

The Attempt at a Solution

a.) Excluding some work I proved that g_ij = 0 (if I does not equal j)... so the only nonzero components of the metric tensor are g₁₁, g₂₂, g₃₃, g<sub>44... This reduces the metric tensor to

ds² = g11dr² + g22dθ² + g33dø² + g44dt²... by simply equalities I said

g11 = 1/(1-2M/r)

g22 = r²

g33 = r²sin²θ

g44 = 2M/r - 1


I then put these in a diagnol 4x4 matrix because all the other entries are 0. Now for the inverse is it the inverse of each of the diagonal components? The only way I know how to find the inverse of a tensor was in linear algebra when [A|I] was row reduced to [I|A^-1] and that was the inverse matrix... is that the same for this as well?

b.) We did not covver the connection (which I am assuming is the affine connection?) at all in class so I did my best to read up on the topic online but can't really grasp the starting point. Any guidance here? Can somebody point me in the direction to generate the components of the connection from the given spacetime? or maybe a general definition?

Thanks!</sub>

 

[nrqed]

Homework Statement

Consider the Schwarschield Metric in four dimensional spacetime (M is a constant):

ds² = -(1-(2M/r))dt² + dr²/(1-(2M/r)) + r²(dθ² + sin²(θ)dø²)

a.) Write down the non zero components of the metric tensor, and find the inverse metric tensor.

b.) find all the components of the connection. (you can use arguments of symmetry to set to zero some of these components)

Homework Equations

The Attempt at a Solution

a.) Excluding some work I proved that g_ij = 0 (if I does not equal j)... so the only nonzero components of the metric tensor are g₁₁, g₂₂, g₃₃, g<sub>44... This reduces the metric tensor to

ds² = g11dr² + g22dθ² + g33dø² + g44dt²... by simply equalities I said

g11 = 1/(1-2M/r)

g22 = r²

g33 = r²sin²θ

g44 = 2M/r - 1


I then put these in a diagnol 4x4 matrix because all the other entries are 0. Now for the inverse is it the inverse of each of the diagonal components? The only way I know how to find the inverse of a tensor was in linear algebra when [A|I] was row reduced to [I|A^-1] and that was the inverse matrix... is that the same for this as well?

b.) We did not covver the connection (which I am assuming is the affine connection?) at all in class so I did my best to read up on the topic online but can't really grasp the starting point. Any guidance here? Can somebody point me in the direction to generate the components of the connection from the given spacetime? or maybe a general definition?

Thanks!</sub>

<sub>In the special case of a diagonal matrix, its inverse is also diagonal with all the entries being the inverse of the elements of the initial matrix. In other words, it is trivial in that case, g^{\mu \mu} = (g_{\mu \mu})^{-1} (mp summation implied here, this is a set of four equations.)

For the affine connection, you can look up the expression for the connection in any GR book. Each component is given by a combination of derivatives of elements of the metric tensor.</sub>