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Invariance of interval in Minkowski space under coordinate transformation was proved by the postulates of special relativity. (https://physics.stackexchange.com/a/453536/213658 .see this answer) Is there any theory or postulates which proves that length is an invariant quantity in Euclidean space?

BioPhysicist
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walber97
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  • What do you mean by "in Euclidean space". Under what transformation are you referring to? – BioPhysicist Feb 18 '19 at 02:53
  • Under Galilean transformation. – walber97 Feb 18 '19 at 02:58
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    Well have you tried to do it on your own? – BioPhysicist Feb 18 '19 at 03:00
  • I did that using Galilean transformation. But I want to know is there any theory which proves the result Like the invariant interval I mentioned in the link. – walber97 Feb 18 '19 at 03:02
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    I'm not sure what you mean by "theory that proves the result". Just write out the length in the transformed coordinates and see that you get the same length as in the original coordinates. – BioPhysicist Feb 18 '19 at 03:04
  • Can you please tell me that, which way we are we deriving, length invariance from Galilean transform or Galilean transform from length invariance. I learned in SR that lorentz transformation is derived from the invariance of interval. If Galilean transform is derived from the invariant length concept ,How this invariance of length should be proved. This is the reason for my confusion. – walber97 Feb 18 '19 at 03:11

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I'm not sure I fully understand you still, but the Galilean transformation is (for transformation in one dimension, I'll leave multiple dimensions to you): $$x'=x-vt$$ $$y'=y$$ $$z'=z$$ $$t'=t$$

So the spatial length between two points $(x_1',y_1',z_1')$ and $(x_2',y_2',z_2')$ is given by $$L=\sqrt{(x_2'-x_1')^2+(y_2'-y_1')^2+(z_2'-z_1')^2}$$ Applying the transformation rules above: $$L=\sqrt{((x_2-vt)-(x_1-vt))^2+(y_2-y_1)^2+(z_2-z_1)^2}$$ $$L=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$$

Which is the length between the unprimed coordinates. Therefore, the length is invariant under Galilean transformation.

BioPhysicist
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  • Thank you for that. You proved the invariance from the rules of Galilean transform. My confusion was if rules of galilean transform came from length invariance , How should we prove the length invariance. I think I am asking some blunder. Any way I assumes that rules Galilean transform proves the invariance. – walber97 Feb 18 '19 at 03:33
  • The invariance of interval can be derived by two ways. First by the lorentz transformation rules, analogous to what you have done here. Second from the postulates of SR as per the link provided in the question. I wonder is there any answer to this question analogous to the latter one. – walber97 Feb 18 '19 at 03:42
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    @walber97 The Galilean transformation is not derived from length invariance. There are plenty of transformations that preserve length that are not the Galilean transformation. – BioPhysicist Feb 18 '19 at 03:58