Length of the line element does not change in transformations[subject to the conditions of continuity,differentiability,one-to-one correspondence etc]. "ds^2" is invariant with respect to transformations[subject to the conditions].(adsbygoogle = window.adsbygoogle || []).push({});

Now let us have a look at the following transformation--the projection transformation..

You are standing under a hemispherical roof. The floor is a flat surface. A curve is drawn on the ceiling and we take its projection on the floor[x-y plane].We get[by this transformation] another curve on the floor which is of unequal length.

Integral ds is different for the two curves.Obviously ds is also different.

Lets have a closer look:

Line element for the hemispherical ceiling:

[tex]{ds}^{2}{=}{R}^{2}{[}{d}{\theta}^{2}{+}{sin}^{2}{\theta}{d}{\phi}^{2}{]}[/tex]

Transformation[for the projection]:

[tex]{x}{=}{R}{Sin}{\theta}{Cos}{\phi}[/tex]

[tex]{y}{=}{R}{Sin}{\theta}{Sin}{\phi}[/tex]

For an infinitesimal projection we get from geometrical conditions:

[tex]{ds'}^{2}{=}{dx}^{2}{+}{dy}^{2}{=}{R}^{2}{[}{cos}^{2}{\theta}{d}{\theta}^{2}{+}{Sin}^{2}{\theta}{d}{\phi}^{2}{]}[/tex]

ds and ds' are not identical.

So in General Relativity we must consider only such transformations that leave ds unchanged

For an arbitrary well behaved transformation the the length of the line element may change

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# Transformation of the Line-Element

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