## Friday, 2 July 2021

### 2.5 Differential forms

 Vector $U$ (solid) and1-form $\widetilde{U}$ (dashed)
In section 2.5 of Misner, Thorne, Wheeler 'differential forms' or '1-forms' are introduced and there is soon a fairly elaborate recipe for constructing 1-forms $\widetilde{U}$ from a vector $U$. I reproduce it below. I tried out the recipe on some of the sample vectors shown and it seemed to work. A one form $\widetilde{U}$ is shown as a set of planes, whose spacing is defined, and which have a positive sense, or direction, as shown by a dashed arrow (which is not a vector). It would seem that you could equally well show the 1-form by a dashed arrow perpendicular (in the Euclidean sense) to the planes whose direction and length would show the positive sense and the spacing of the planes of $\widetilde{U}$. You can. And I implemented it in a spreadsheet and made the amusing gif on the right. Then I saw the joke.

### The recipe

Figure 2.7. Several vectors, $A,B,C,D,E$, and corresponding 1-forms $\widetilde{A},\widetilde{B},\widetilde{C},\widetilde{D},\widetilde{E}$. The process of drawing $\widetilde{U}$ corresponding to a given vector $U$ is quite simple. 1) Orient the surfaces of $\widetilde{U}$ orthogonal to the vector $U$ . (Why? Because any vector $V$ that is perpendicular to $U$ must pierce no surfaces of $\widetilde{U}$ ($0=U\bullet V=U,V$) and must therefore lie in a surface of $\widetilde{U}$.) 2) Space the surfaces of  $\widetilde{U}$ so the number of surfaces pierced by some arbitrary vector $Y$ (e.g., $Y=U$) is equal to $Y\bullet U$.

Note that in the figure the surfaces of $\widetilde{B}$ are, indeed, orthogonal to $B$; those of $\widetilde{C}$ are, indeed, orthogonal to $C$, etc. If they do not look so, that is because the reader is attributing Euclidean geometry, not Lorentz geometry, to the spacetime diagram. He should recall, for example, that because $C$ is a null vector, it is orthogonal to itself ($C\bullet C=0$), so it must itself lie in a surface of the 1-form  $\widetilde{C}$. Confused readers may review spacetime diagrams in a more elementary text, e.g., Taylor and Wheeler (1966)."

Read it here, including punchline: 2.5 Differential forms.pdf. (6 pages)