Basic Summation Indices

If [itex]j^\mu = ( j^0 , \vec{j} )[/itex], why does

[itex]\partial_\mu j^\mu = \partial_0 j^0 + \vec{\nabla} \cdot \vec{j}[/itex]

surely when you take a dot product of four vectors you get a subtraction as in
[itex]a^\mu b_\mu = a^0 b_0 - \vec{a} \cdot \vec{b}[/itex]

Maybe I'm forgetting something
 
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Suppose your metric is (+1,-1,-1,-1). Then

[tex]a^\mu b_\mu= a^0b_0+a^ib_i=a^0b^0-\sum_{i=1}^3a^ib^i[/tex]

You flip the sign when you rise space-like indices. But with your continuity equation

[tex]\partial_\mu j^\mu=\partial_0 j^0+\sum_{i=1}^3\partial_i j^i[/tex]

there is no reason to rise or lower the indices.
 
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