Confusion regarding the terms in the covariant derivative of a Tensor

I am learning General Relativity from Leonard Susskind's Lectures. In Lecture three, he introduces to covariant derivatives, and I understood it's meaning. But when he applies it to a Tensor, I am having some confusion. If we consider the Tensor $$T_{mn}$$ , it's covariant derivative $$D_sT_{mn}$$ , he says is given as $$D_sT_{mn}=\partial_sT_{mn}-\Gamma^t_{ms}T_{tn}-\Gamma^t_{ns}T_{tm}$$ Where $$\partial_sT_{mn}$$ represents the derivative of the Tensor with respect to $$x^s$$, while the Gamma's are the Christoffel Symbols. My question is that why are the contravariant indices of both the Christoffel symbols (The index t in this case) the same.

• It's a dummy index. You can rename them if it confuses you. – ApolloRa Apr 18 at 11:09
• @ApolloRa I know that it is a dummy index, but the equation won't remain the same if the said indices were different. If they were different, say t and u, then the sum (the above represents sums of equations) would contain terms like t=0 with u=1, but if there were only t, it wouldn't have such terms – SK Dash Apr 18 at 11:27

The left hand side has three indices: $$m,n,s$$. These are free indices. In order for this equation to be correct the right hand side must have the same $$m,n,s$$ free indices. Indeed it does. The $$t's$$ are dummy indices. It doesn't matter how you name them. You can change the $$t's$$ (of course both $$t's$$ of each term or else they will not mean summation) to whatever you like. Examples:
$$D_sT_{mn}=\partial_sT_{mn}-\Gamma^u_{ms}T_{un}-\Gamma^t_{ns}T_{mt}$$ $$D_sT_{mn}=\partial_sT_{mn}-\Gamma^u_{ms}T_{un}-\Gamma^u_{ns}T_{mu}$$ $$D_sT_{mn}=\partial_sT_{mn}-\Gamma^a_{ms}T_{an}-\Gamma^b_{ns}T_{mb}$$
• Although, shouldn't the last term have $T_{mt}$ instead of $T_{tm}$, for the covariant derivative to be correct? – Elias Riedel Gårding Apr 18 at 12:02
• @Elias Riedel Gårding Indeed yes! I didnt check the equation in the question! Thanks for pointing it out! For an arbitary tensor $T_{mn} !=T_{nm}$ – ApolloRa Apr 18 at 12:09