If is the tangent to the curve [ the four- velocity of the particle, see Figure 3.2 ] then:
since [see section 2.2].
In three dimensions one thinks of a gradient as a vector [ normal to surfaces of constant ] but is a one- form and specifies a vector only if there is a metric.
Now how do the components of transform?
But we also have by the chain rule:
which means that
and since we have