Finding Killing Vectors for $$ds^2 = dr^2 + r^2d\theta^2$$

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In summary, the conversation discusses the finding of three Killing vector fields for the Euclidean metric in polar coordinates. The three vector fields are found to satisfy the Killing equations and are thus considered Killing vector fields. However, it is noted that these are vector fields, not vectors, as the Killing equation describes assignments of vectors to each point in space. It is also mentioned that the three vector fields are independent and cannot be expressed as multiples of each other. When evaluated at a specific point, they form three linearly dependent vectors.
  • #1
davidge
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I have tried to find the three Killing vectors for the metric $$ds^2 = dr^2 + r^2d \theta^2$$ that is, the Euclidean metric of ##\mathbb{R}^2## written in polar coordinates. I found these to be

$$\bigg(\text{first}\bigg) \ \ \xi_r = \text{Cos} \theta \\
\xi_\theta = -\text{rSin} \theta \\

\bigg(\text{second}\bigg) \ \ \xi_r = \text{Sin} \theta \\
{\xi_\theta = \text{rCos} \theta} \\

\bigg(\text{third}\bigg) \ \ \xi_r = 0 \\
\xi_\theta = \text{r²}$$ As I have found solutions only for 3d on web, I would like to know whether these are correct or not.
 
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  • #2
Why don't you check whether or not they satisfy the Killing equations?
 
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  • #3
Orodruin said:
Why don't you check whether or not they satisfy the Killing equations?
I did
And they do satisfy the Killing equation.
 
  • #4
davidge said:
I did
And they do satisfy the Killing equation.
And thus they are Killing vector fields ...
 
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  • #5
Orodruin said:
And thus they are Killing vector fields ...
:biggrin:
 
  • #6
What bothers me is that in 2d we should have only two independent vectors. So I should be able to get one of those three above by a linear combination of the other two, but when I do that, I get non constant coefficients multiplying them.
 
  • #7
These are vector fields, not vectors.
 
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  • #8
Orodruin said:
These are vector fields, not vectors.
So they are'nt vectors? Can you say a bit more on this please
 
  • #9
There is no such thing as a "Killing vector". The Killing equation is a differential equation and as such describes vector fields, ie, assignments of one vector to each point in the space.
 
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  • #10
Orodruin said:
assignments of one vector to each point in the space
For instance, what could be such one vector?
 
  • #11
You wrote down several vector fields (in coordinate basis) in the firs post.
 
  • #12
Orodruin said:
You wrote down several vector fields (in coordinate basis) in the firs post.
But you say they aren't vectors. I asked for an example of assigment of a vector by a vector field
 
  • #13
Your field is an assignment of a (dual) vector to every point in space!

For example, for ##\theta = 0## (and arbitrary r) your first field takes the value ##\xi = dr##, where ##dr## is the coordinate basis dual vector.
 
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  • #14
Orodruin said:
Your field is an assignment of a (dual) vector to every point in space!

For example, for ##\theta = 0## (and arbitrary r) your first field takes the value ##\xi = dr##, where ##dr## is the coordinate basis dual vector.
I got it. Thanks.
 
  • #15
Is it correct to say that they are three independent vector fields? In the sense that one cannot be expressed as a multiple of another one.

Also, if we evaluate any of them at a particular point ##(r, \theta)## do they form three linearly dependent vectors?
 

1. What is a Killing vector?

A Killing vector is a type of vector field that satisfies a specific equation, known as the Killing equation. This equation relates the metric tensor of a given space to the Lie derivative of the vector field. A Killing vector represents a symmetry or conservation law of a physical system.

2. How do you find Killing vectors for a given metric?

To find Killing vectors for a given metric, you need to solve the Killing equation, which is a set of differential equations. This can be a complex and time-consuming process, and there is no general method for solving it. However, there are some techniques and symmetry arguments that can be used to simplify the process.

3. What is the significance of Killing vectors?

Killing vectors have significant applications in physics and geometry. In physics, they represent symmetries of a system, which can help in solving problems and understanding the behavior of physical systems. In geometry, they are used to define and classify different types of manifolds and to study their properties.

4. How many Killing vectors can a given metric have?

The number of Killing vectors for a given metric depends on the dimension and the symmetry of the space. In general, a metric can have up to n(n+1)/2 Killing vectors, where n is the dimension of the space. For example, a 3-dimensional space can have up to 6 Killing vectors.

5. How are Killing vectors related to isometries?

Killing vectors and isometries are closely related concepts. Isometries are transformations that preserve the metric of a space, while Killing vectors represent symmetries of a space. In fact, every isometry of a space is associated with a Killing vector, and vice versa. This relationship is known as the Killing theorem.

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