Gravity: Riemann Tensor

The Riemann-Christoffel Tensor

Second covariant derivatives generally are not independent of order, and their commutated value depends linearly on the original tensor, and the coefficients form a matrix-valued 2-form called the Riemann-Christoffel tensor. Its "official" definition is:

    A[j][/k][/L] - A[j][/L][/k] = A[m] g'[m][i] R[i][j][k][L]

    R[i][j][k][L] = C[j][L][i][!k] + C[j][k][m] g'[m][n] C[i][L][n]
                  - C[j][k][i][!L] - C[j][L][m] g'[m][n] C[i][k][n]

Let's see if we can get this by formally differentiating the Christoffel symbols as if they were a tensor. It will be seen later that the Christoffel symbols as a matrix-valued 1-form can be uniquely segregated into the sum of a tensor and a maximally non-tensor part, and justification will be given for relegating the latter to oblivion.

Lemma 1: Expansion of g[i][k][!j].

    C[i][j][k] = 1/2 (g[i][k][!j] + g[j][k][!i] - g[i][j][!k])
    C[k][j][i] = 1/2 (g[k][i][!j] + g[j][i][!k] - g[k][j][!i])
    Sum:       = g[i][k][!j]

Lemma 2: To calculate g'[m][j][!k], from the metric inverse:

    g[i][m] g'[m][j] = (i==j)
    g[i][m][k!] g'[m][j] + g[i][m] g'[m][j][!k] = 0
    g'[m][j][!k] = - g'[m][n] g[n][p][!k] g'[p][j]  (m->p in the last sum)
        = - g'[m][n] C[n][k][p] g'[p][j] - g'[m][n] C[p][k][n] g'[p][j]

Now let's compute the commutated second covariant derivative of a 1-form:

    A[i][/j] = A[i][!j] - A[m] g'[m][n] C[i][j][n]  (first deriv.)

    A[i][/j][/k] - A[i][/k][/j] =                   (. = cancels)
        A[i][!j][!k]. - A[m][!k] g'[m][n] C[i][j][n]. 
                     - A[m] g'[m][n][!k] C[i][j][n]
                     - A[m] g'[m][n] C[i][j][n][!k]
                     - A[m][!j] g'[m][n] C[i][k][n].
                     + A[m] g'[m][n] C[p][j][n] g'[p][q] C[i][k][q]
                     - A[i][!m] g'[m][n] C[j][k][n].
                     + A[m] g'[m][n] C[i][p][n] g'[p][q] C[j][k][q].
        -A[i][!k][!j].+ A[m][!j] g'[m][n] C[i][k][n]. 
                     + A[m] g'[m][n][!j] C[i][j][n]
                     + A[m] g'[m][n] C[i][k][n][!j]
                     + A[m][!k] g'[m][n] C[i][j][n].
                     - A[m] g'[m][n] C[p][k][n] g'[p][q] C[i][j][q]
                     + A[i][!m] g'[m][n] C[k][j][n].
                     - A[m] g'[m][n] C[i][p][n] g'[p][q] C[k][j][q].
        = A[m] g'[m][n] (- C[i][j][n][!k] + C[i][k][n][!j] 
            + C[p][j][n] g'[p][q] C[i][k][q] - C[p][k][n] g'[p][q] C[i][j][q]
            + g[n][p][k!] g'[p][q] C[i][j][q] - g[n][p][j!] g'[p][q] C[i][j][q])

Using the lemma the g[n][p][k!] term combines with -C[p][k][n] giving C[n][k][p] and similarly for j!.

        = A[m] g'[m][n] (- C[i][j][n][!k] + C[i][k][n][!j] 
            + C[n][k][p] g'[p][q] C[i][j][q] - C[n][j][p] g'[p][q] C[i][k][q])
        = - A[m] g'[m][n] R[n][i][k][j]

(meeting up with the standard definition of R, the Riemann-Christoffel tensor.) Now collect terms, remembering that g[m][n][/k] = 0:

    g'[m][n] R[n][i][k][j] = 
        + g'[m][n][!k] C[i][j][n]
        + g'[m][n] C[i][j][n][!k]
        - g'[m][n] C[p][j][n] g'[p][q] C[i][k][q]
        - g'[m][n][!j] C[i][j][n]
        - g'[m][n] C[i][k][n][!j]
        + g'[m][n] C[p][k][n] g'[p][q] C[i][j][q]
    = (g'[m][n] C[i][j][n])[/k] - (g'[m][n] C[i][k][n])[/j]

Get rid of the contraction with g'[m][n] leaving just the Riemann-Christoffel tensor:

    R[L][i][k][j] = C[i][j][L][/k] - C[i][k][L][/j]
    R[i][j][k][L] = C[j][L][i][/k] - C[j][k][i][/L]  (L->i, i->j, k->k, j->L)

(It being understood that C does not transform as a tensor.) It's a curious fact that in 2 or 3 dimensions the covariant corrections all cancel out, but it's not likely so convenient in higher dimensions.

Now let's substitute the definition of C[...] in terms of the distance function:

    R[i][j][k][L] = 0.5(r²[2/j][1/i][2/L][/k] - r²[2/j][1/i][2/k][/L])

(Expand [/k] as [1/k] + [2/k] since x2 == x1:)

    R[i][j][k][L] = 0.5(r²[2/j][1/i][2/L][1/k] + r²[2/j][1/i][2/L][2/k] 
                      - r²[2/j][1/i][2/k][1/L] - r²[2/j][1/i][2/k][2/L])
                  = 0.5(r²[2/j][1/i][2/L][1/k] - r²[2/j][1/i][2/k][1/L])
                  = g(x2,x1)[j][i][2/L][1/k] - g(x2,x1)[j][i][2/k][1/L]

(with the restriction that x2 == x1.)

Some important coordinate-related symmetries can be seen from the expression involving 0.5 r²:

    R[i][j][k][L] = -R[j][i][k][L] = -R[i][j][L][k] = R[k][L][i][j]

since r²[2/i] = -r²[1/i] if x2 == x1.

In terms of the exterior derivative,

    R[i][j][k][L] = C[j][L][i][/k] - C[j][k][i][/L]	(L->i, i->j, k->k, j->L)
    R[i][j] = dC[j]..[i]

where C[j]..[i] means the matrix-valued 1-form C[j][m][i] dx[m]. After differentiation and subtraction it is seen that the value is a matrix-valued 2-form.

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