Killing vector Definition and 17 Discussions

In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point on an object the same distance in the direction of the Killing vector will not distort distances on the object.

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  1. Onyx

    B Solving for General Geodesics in FLRW Metric

    Once having converted the FLRW metric from comoving coordinates ##ds^2=-dt^2+a^2(t)(dr^2+r^2d\phi^2)## to "conformal" coordinates ##ds^2=a^2(n)(-dn^2+dr^2+r^2d\phi^2)##, is there a way to facilitate solving for general geodesics that would otherwise be difficult, such as cases with motion in...
  2. cianfa72

    I Spacetime KVF symmetries

    Hi, reading Carrol chapter 5 (More Geometry), he claims that a maximal symmetric space such as Minkowski spacetime has got ##4(4+1)/2 = 10## indipendent Killing Vector Fields (KVFs). Indeed we can just count the isometries of such spacetime in terms of translations (4) and rotations (6). By...
  3. cianfa72

    I Synchronous reference frame

    Hi, reading the Landau book 'The Classical theory of Field - vol 2' a doubt arised to me about the definition of synchronous reference system (a.k.a. synchronous coordinate chart). Consider a generic spacetime endowed with a metric ##g_{ab}## and take the (unique) covariant derivative operator...
  4. cianfa72

    I Is acceleration absolute or relative - follow up

    Hello, Some doubt arose me reading this thread https://www.physicsforums.com/threads/is-acceleration-absolute-or-relative-revisited.999420/post-6454462 currently closed. Sorry, I have not be able to quote directly from it :frown: Your claim is not , however, asserting that the spacetime...
  5. Aemmel

    Finding killing vector fields of specific spacetime

    I have been at this exercise for the past two days now, and I finally decided to get some help. I am learning General Relativity using Carrolls Spacetime and Geometry on my own, so I can't really ask a tutor or something. I think I have a solution, but I am really unsure about it and I found 6...
  6. George Keeling

    I Using Killing vectors

    I'm on to section 5.4 of Carroll's book on Schwarzschild geodesics and he says stuff in it which, I think, enlightens me on the use of Killing vectors. I had to go back to section 3.8 on Symmetries and Killing vectors. I now understand the following: A Killing vector satisfies $$...
  7. W

    I Getting Used to Killing Vector Fields

    I'm struggling to get the hang of killing vectors. I ran across a statement that said energy in special relativity with respect to a time translation Killing field ##\xi^{a}## is: $$E = -P_a\xi^{a}$$ What exactly does that mean? Can someone clarify to me?
  8. T

    I Invariance of timelike Killing vector of Schwarzschild sol.

    I use the ##(-,+,+,+)## signature. In the Schwarzschild solution $$ds^2=-\left(1-\frac{2m}{r}\right)dt^2+\left(1-\frac{2m}{r}\right)^{-1}dr^2+r^2d\Omega^2$$ with coordinates $$(t,r,\theta,\phi)$$ the timelike Killing vector $$K^a=\delta^a_0=\partial_0=(1,0,0,0)$$ has a norm squared of...
  9. D

    I Killing vectors in isotropic space-times

    I've been reading up on Killing vectors, and have got on to the topics of homogeneous, isotropic and maximally symmetric space-times. I've read that for an isotropic spacetime, one can construct a set of Killing vector fields ##K^{(i)}##, such that, at some point ##p\in M## (where ##M## is the...
  10. F

    I Timelike Killing vectors & defining a vacuum state

    I've read that if a given spacetime possesses a timelike Killing vector, then it is possible to define a unique vacuum state by constructing positive and negative frequency modes with respect to this timelike Killing vector. My question is, what does this mean? Explicitly, how does one use a...
  11. T

    Change in Energy and Angular Momentum Upon Decreasing Orbit

    Homework Statement A stationary, axisymmetric, spacetime has two Killing vector fields [ξt, ξφ] corresponding to translation along t or φ directions. A particle of unit mass moving in this spacetime has a four-velocity u = γ[ξt + Ωξφ]. (i) Explain why we can interpret this as a particle moving...
  12. Augbrah

    Show that killing vector field satisfies....

    I'm trying to do past exam papers in GR but there are some things I don't yet feel comfortable with, so even though I can do some parts of the question I would be very happy if you could check my solution. Thank you! 1. Homework Statement Spacetime is stationary := there exists a coord chart...
  13. D

    Covariant derivative of Killing vector and Riemann Tensor

    I need to prove that $$D_\mu D_\nu \xi^\alpha = - R^\alpha_{\mu\nu\beta} \xi^\beta$$ where D is covariant derivative and R is Riemann tensor. ##\xi## is a Killing vector. I have proved that $$D_\mu D_\nu \xi_\alpha = R_{\alpha\nu\mu\beta} \xi^\beta$$ I can't figure out a way to get the required...
  14. N

    Conservation laws with Killing fields

    Hi, In general relativity we have no general conservation of energy and momentum. But if there exists a Killing-field we can show that this leads to a symmetry in spacetime and so to a conserved quantity. Thats what the mathematic tells us. But I don't understand what's the meaning of an...
  15. loops496

    Levi-Civita Connection and Derivative of The Ricci Scalar

    I have a question about the directional derivative of the Ricci scalar along a Killing Vector Field. What conditions are necessary on the connection such that K^\alpha \nabla_\alpha R=0. Is the Levi-Civita connection necessary? I'm not sure about it but I believe since the Lie derivative is...
  16. loops496

    Killing Vector field question

    Homework Statement Suppose v^\mu is a Killing Vector field, the prove that: v^\mu \nabla_\alpha R=0 Homework Equations 1) \nabla_\mu \nabla_\nu v^\beta = R{^\beta_{\mu \nu \alpha}} v^\alpha 2) The second Bianchi Identity. 3) If v^\mu is Killing the it satisfies then Killing equation, viz...
  17. S

    Some questions about isometry

    Isometry is the symmetry s.t. g^\prime_{\mu\nu}(x)=g_{\mu\nu}(x) under the transformation x^\mu\to x^{\prime\mu}(x). This means under infinitesimal transformation x^\mu\to x^\mu+\epsilon \xi^\mu(x) where \epsilon is any infinitesimal constant, the vector field \xi^\mu(x) satisfies Killing...
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