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In null hypersurfaces, as null curves lie on the hypersurface, it is said that it is advantageous to take parameter $\lambda$ of the null curve to be one of the coordinates.

While, I am getting a feel that taking $\lambda$ as one of my coordinates is fine, because of general equivalence. As parameterization is arbitrary, we can choose any parameter for my curve. Hence, I think any parameter family would serve as a good coordinate.

But I am still not fully satisfied with it. Is there any condition on our choice of the parameter so as to classify it as a coordinate?

PS: I have been doing exactly the inverse of the above till now in GR. For example, if I have set up my coordinates, then I am at freedom to choose any parameter for my curve including the coordinate basis as parameters of the curve. What I am asking seems to be the exact opposite of this.

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The coordinates are just arbitrary labels. As long as they are continuous functions from a patch of your null hypersurface to a patch of $R^{3}$, they can be whatever you want.

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