I Why vector lengths may not be preserved?

[Nusc]

Given some metric, what is an example where the length of a vector is not preserved?

 

[Nusc]

I presume it's because a metric could have a scalar and that would not preserve the length.

 

[fresh_42]

Given some metric, what is an example where the length of a vector is not preserved?

Not preserved under what?

 

[Orodruin]

Your question is rather ill defined. Preserved when exactly?

It is also not clear what you would mean by a ”metric having a scalar”. A metric is a type (0,2) tensor, not a scalar.

 

[Nusc]

Not preserved under what?

length

 

[Nusc]

Your question is rather ill defined. Preserved when exactly?

It is also not clear what you would mean by a ”metric having a scalar”. A metric is a type (0,2) tensor, not a scalar.

if you define a metric g' = k g where k is some scalar function. you can expresss the lengths of a' = g' a_i a_j = k g a_i a_j

 

[fresh_42]

You need a metric in order to speak of length, not the other way around. If you have two different metrics, they might be equivalent $g'=k\cdot g$, or not. The question about length is directly coupled to the existence of a metric - one metric. As an example you could consider the distance between two real numbers as $|x-y|$ or look at the discrete metric $d(x,y)=1$ as soon as $x\neq y$. These are two different, non-equivalent metrics on one space, $\mathbb{R}$ in this case.

So as long as you only have one metric on a vectorspace, you cannot have two lengths.

 

[Nusc]

I see. thank you