Killing vectors of minkowski space

In summary, the Killing vectors of Minkowski space are vector fields that preserve the metric structure of the space. They represent the symmetries of the space and are used to generate isometries, or transformations that preserve distances and angles. These Killing vectors can be found by solving the Killing equation, and they play a crucial role in understanding the geometry and physics of Minkowski space.
  • #1
WannabeNewton
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How does one know from the general form of the killing vectors in minkowski space:
X[tex]^{a}[/tex] = [tex]\omega[/tex]_a_b(x[tex]^{a}[/tex]) + t[tex]^{a}[/tex]

that there are 3 rotational isometries, 3 boosts, 3 spatial translations, and 1 time translation from that general form? It has me very confused >.<
 
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  • #2
I can't read your notation so I could be wrong. The four translations are obviously from ta and if [itex]\omega[/itex] is traceless then it has 3 symmetric degrees of freedom ( boosts) and 3 anti-symmetric df (rotation) .
 
  • #3
Mentz114 said:
I can't read your notation so I could be wrong. The four translations are obviously from ta and if [itex]\omega[/itex] is traceless then it has 3 symmetric degrees of freedom ( boosts) and 3 anti-symmetric df (rotation) .

Omega is a rank two anti symmetric tensor sorry I couldn't get latex to work properly. I understand the part with the translation but how can you show that there are 6 components of omega with 3 being boosts and the other 3 being rotations? Also a lot of my books represent the translations in terms of [tex]\partial[/tex] / [tex]\partial[/tex]x[tex]^{a}[/tex] - are they simply representing translations as displacements or is there a way in which you can extract the differentials from t[tex]^{a}[/tex]?
 
  • #5
WannabeNewton said:
Omega is a rank two anti symmetric tensor ...
In that case I'm baffled. I'll have to read George Jones' thread again because I didn't see how spatial rotations and boosts are accommodated.
 
  • #6
George Jones said:
The thread

https://www.physicsforums.com/showthread.php?t=454237

might help.

Thanks I think I get it now. If minkowski space killing vectors are this confusing to me I can't imagine schwarzschild or robertson metrics. Sucks for me considering how important these isometries seem to be.
 
  • #7
Any vector in Minkowski space is a Killing vector, because any directional derivative of the metric with respect to displacement is zero. You can think of it as degenerate eigen vectors. Any complete set spanning the space works as well as any other.

As a consequence, the energy and momentum in Minkowski space are absolutely conserved.
 

1. What are killing vectors of Minkowski space?

Killing vectors are vector fields on a manifold (in this case, Minkowski space) that preserve the metric tensor. This means that when these vectors act on a point in the manifold, they do not change the distance between any two points. In other words, they represent symmetries of the manifold.

2. How do killing vectors relate to special relativity?

Killing vectors are important in special relativity because they represent the symmetries of Minkowski space, which is the mathematical framework for special relativity. These symmetries correspond to the laws of physics that are invariant under different coordinate systems, and are essential for understanding the principles of special relativity.

3. How many killing vectors does Minkowski space have?

Minkowski space has ten killing vectors, corresponding to the ten independent components of the metric tensor. These vectors can be represented in terms of the four-dimensional Lorentz group, which is the group of transformations that preserve the Minkowski metric.

4. What is the significance of killing vectors in general relativity?

In general relativity, killing vectors play a crucial role in determining the symmetries of a given spacetime. They are used to define the concept of isometry, which refers to the invariance of a spacetime under a specific transformation. Killing vectors are also used to solve the Einstein field equations, which describe the curvature of spacetime in the presence of matter and energy.

5. Can killing vectors be used to solve certain mathematical problems?

Yes, killing vectors can be used to solve problems related to symmetry. For example, they can be used to find solutions to differential equations that possess certain symmetries. In the study of differential geometry, killing vectors are also used to classify manifolds and determine their properties.

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