Tensors and Relativity: Chapter 3

Gradients

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Next: The metric as a Up: One- forms Previous: One- forms

For a scalar field tex2html_wrap_inline1742 and a world line of some particle tex2html_wrap_inline1744 , we have

equation1273

  figure324
Figure 3.2: World line of particle with four- velocity tex2html_wrap_inline1746 .

If tex2html_wrap_inline1746 is the tangent to the curve [ the four- velocity of the particle, see Figure 3.2 ] then:

equation1279

so

equation1281

since tex2html_wrap_inline1750 [see section 2.2].

This defines a one- form  since it maps tex2html_wrap_inline1746 into real numbers and represents the rate of change of tex2html_wrap_inline1754 along a curve with tangent tex2html_wrap_inline1746 .

equation1291

In three dimensions one thinks of a gradient as a vector [ normal to surfaces of constant tex2html_wrap_inline1754 ] but tex2html_wrap_inline1760 is a one- form and specifies a vector only if there is a metric.

Now how do the components of tex2html_wrap_inline1762 transform?

equation1294

But we also have by the chain rule:

equation1298

which means that

equation1302

so

equation1306

and since tex2html_wrap_inline1764 we have

equation1310

This is a useful result, that the basis one- form is just tex2html_wrap_inline1766  .