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For a scalar field and a world line of some particle , we have
Figure 3.2: World line of particle with four- velocity .
If is the tangent to the curve [ the four- velocity of the particle, see Figure 3.2 ] then:
since [see section 2.2].
This defines a one- form since it maps into real numbers and represents the rate of change of along a curve with tangent .
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
This is a useful result, that the basis one- form is just .