If ##\sigma## is an affine paramter, then the only freedom of choice we have to specify another affine parameter is ##a\sigma+b##, a,b constants. [1](adsbygoogle = window.adsbygoogle || []).push({});

For the tangent vector, ##\xi^{a}=dx^{a}/du##, along some curve parameterized by ##u##

My book says that ' if ##\xi^{a}\xi_{a}\neq 0##, then by suitable affine parameterizationwe can arrange such that ##\xi^{a}\xi_{a}=\pm1##,

Question:

What does it mean by some suitable affine parameterization? so say if ##\sigma## is a affine parameter and we do not have ##\xi^{a}\xi_{a}=\pm1##, is it saying that we can use [1] and carefully choose ##a## and ##b## such that this is the case?

I've often seen proper time used such that ##\xi^{a}\xi_{a}=\pm1## is the case.

Why is this?

Or Is this part of the definition of proper time, are there any other 'known' parameters for which ##\xi^{a}\xi_{a}=\pm1## or is the affine parameter for which this holds unique?

Thanks in advance.

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# Product of tangent vectors \pm1 affine parameter

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