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I have a field : $\phi(x^\mu+\delta x^\mu)$.
Where $x^\mu$ is a quadriposition.
In my course we wrote :
$$ \phi(x^\mu+\delta x^\mu) \approx \phi(x^\mu)+\delta x^\mu \partial_\mu \phi(x^\mu)$$
What I don't understand is that we will have a "minus" sign that will occur for this taylor developpment.
Indeed :
$$ \phi(x^\mu+\delta x^\mu) \approx \phi(x^\mu)+[\delta x^0 \partial^0 -\delta x^i \partial^i ]\phi(x^\mu)$$
Can you help me to understand this ?
The first step $\phi(x^\nu + \delta x^\nu) = \phi(x^\nu) + \delta x^\mu \partial_\mu\phi(x^\nu)$ is just a simple taylor expansion. In the second step the contraction over $\mu$ is written out using the definition of the metric $$\delta x^\mu \partial_\mu\phi(x^\nu) = \eta^{\mu\kappa}\delta x_\kappa \partial_\mu\phi(x^\nu)\\\text{with}\\\eta^{\mu\kappa} = diag(+ ---)$$
You can also use the matrix notation from your other question: Quadrivectors in relativity if you want to avoid using the metric.
ps: You might want to say 4-vectors instead of quadrivectors because it generalizes a bit better to other dimensions...
