- #1

redstone

- 26

- 0

Where D equals the dimension of the metric -

Step 0: [tex]{{A}^{ab}}=\frac{1}{D}{{g}^{ab}}{{g}_{cd}}{{A}^{cd}}[/tex]

Step 1: [tex]{{g}_{ab}}{{A}^{ab}}={{g}_{ab}}\frac{1}{D}{{g}^{ab}}{{g}_{cd}}{{A}^{cd}} [/tex]

Step 2: [tex]{{g}_{ab}}{{A}^{ab}}=\frac{1}{D}{{g}_{ab}}{{g}^{ab}}{{g}_{cd}}{{A}^{cd}} [/tex]

Step 3: [tex]{{g}_{ab}}{{A}^{ab}}=\frac{1}{D}g_{a}^{a}{{g}_{cd}}{{A}^{cd}} [/tex]

Step 4: [tex]{{g}_{ab}}{{A}^{ab}}=\frac{1}{D}D{{g}_{cd}}{{A}^{cd}} [/tex]

Step 5: [tex]{{g}_{ab}}{{A}^{ab}}={{g}_{cd}}{{A}^{cd}} [/tex]

Step 6: [tex]{{g}_{ab}}{{A}^{ab}}={{g}_{ab}}{{A}^{ab}} [/tex]

So I know that the equation in step 0 is true for any tensor A, is that correct?