Why vector lengths may not be preserved?

  • I
  • Thread starter Nusc
  • Start date
  • Tags
    tensors
  • #1
753
2
Given some metric, what is an example where the length of a vector is not preserved?
 

Answers and Replies

  • #2
753
2
I presume it's because a metric could have a scalar and that would not preserve the length.
 
  • #3
13,457
10,516
Given some metric, what is an example where the length of a vector is not preserved?
Not preserved under what?
 
  • #4
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
16,829
6,643
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.
 
  • #6
753
2
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
 
  • #7
13,457
10,516
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.
 
  • #8
753
2
I see. thank you
 

Related Threads on Why vector lengths may not be preserved?

Replies
9
Views
2K
Replies
1
Views
1K
Replies
5
Views
2K
  • Last Post
Replies
0
Views
2K
  • Last Post
Replies
9
Views
3K
Replies
11
Views
1K
  • Last Post
Replies
5
Views
5K
Replies
1
Views
2K
Replies
18
Views
15K
Top