Killing vector Definition and 52 Threads

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. cianfa72

    I 3-parameter group of symmetries on the plane

    Consider ##\mathbb R^2## as the Euclidean plane. Since it is maximally symmetric it has a 3-parameter group of Killing vector fields (KVFs). Pick orthogonal cartesian coordinates centered at point P. Then the 3 KVFs are given by: $$K_1=\partial_x, K_2=\partial_y, K_3=-y\partial_x + x...
  2. bobrubino

    B How do you project a Killing Vector onto a Schwarzschild field?

    What is the math for projecting a Killing vector onto a Schwarzschild field of spacetime? How would you do it?
  3. Onyx

    B Solve General Geodesics in FLRW Metric w/ Conformal Coordinates

    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...
  4. cianfa72

    I Minkowski 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...
  5. cianfa72

    I Synchronous Reference Frame: Definition and Usage

    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...
  6. ergospherical

    A Massless Particle Action under Conformal Killing Vector Transformation

    For a massless particle let\begin{align*} S[x,e] = \dfrac{1}{2} \int d\lambda e^{-1} \dot{x}^{\mu} \dot{x}^{\nu} g_{\mu \nu}(x) \end{align*}Let ##\xi## be a conformal Killing vector of ##ds^2##, then under a transformation ##x^{\mu} \rightarrow x^{\mu} + \alpha \xi^{\mu}## and ##e \rightarrow e...
  7. 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...
  8. cianfa72

    I Parallel transport vs Lie dragging along a Killing vector field

    Hi, I would like to ask for a clarification about the difference between parallel transport vs Lie dragging in the following scenario. Take a vector field ##V## defined on spacetime manifold and a curve ##C## on it. The manifold is endowed with the metric connection (I'm aware of it does exist...
  9. 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...
  10. George Keeling

    I Understanding Killing Vectors & Schwarzschild Geodesics

    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 $$...
  11. W

    I Getting Used to Killing Vector Fields: Explained

    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?
  12. PeterDonis

    A Extra Killing Vector Field in Kerr Spacetime?

    In a recent thread, the following was posted regarding the "no hair" theorem for black holes: In the arxiv paper linked to, it says the following (p. 2, after Theorem 1.1): "Hawking has shown that in addition to the original, stationary, Killing field, which has to be tangent to the event...
  13. 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...
  14. 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...
  15. J

    A Why is Killing vector field normal to Killing horizon?

    In p.244 of Carroll's "Spacetime and Geometry," the Killing horizon ##\Sigma## of a Killing vector ##\chi## is defined by a null hypersurface on which ##\chi## is null. Then it says this ##\chi## is in fact normal to ## \Sigma## since a null surface cannot have two linearly independent null...
  16. 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...
  17. binbagsss

    General Relativity geodesics, killing vector, conserved quantities

    Homework Statement Homework EquationsThe Attempt at a Solution [/B] Let ##k^u## denote the KVF. We have that along a geodesic ##K=k^uV_u## is constant , where ##V^u ## is the tangent vector to some affinely parameterised geodesic. ##k^u=\delta^u_i## , ##V^u=(\dot{t},\vec{\dot{x}})## so...
  18. binbagsss

    A Killing vector notation confusion time translation

    Okay so when there is time-translation symmetry because the metric components do not have any time- dependence, ##\partial_x^0## is a Killing vector. I'm just confused what this means explicitly, since a derivative doesn't make sense without acting on anything really? But by 'spotting the...
  19. 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...
  20. L

    A Question about Killing vector fields

    I am trying to follow Nakahara's book. From the context, it seems that the author is trying to say if moving a point along a flow always give a isometry, the corresponding vector field X is a Killing vector field. am I right? then the book gives a proof. It only considers a linear approximation...
  21. F

    Topology Sources about Killing vector fields?

    I'm interested in Killing vector fields and want to ask whether anybody can name me a good textbook or online-source about them, preferably with a general treatment with local coordinates as examples and not at the center of consideration.
  22. 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...
  23. 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...
  24. D

    I Proving Killing Vector of Static Spacetime - David

    Hello, I am reading through some GR lecture notes and have come across the following: "A spacetime is static if there exists a coordinate chart where: ∂0gμν = 0 g0i = 0 This spacetime admits a Timelike Killing vector X that satisfies: X[α∇βXγ] = 0 " How do I go about proving that this...
  25. A

    Independent Killing vector field

    Hello everyone How is it possible that a n-dimensional spacetime admits m> n INDEPENDENT Killing vectors where m=n(n+1)/2 if the space is maximally symmetric?
  26. 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...
  27. loops496

    Directional Derivative of Ricci Scalar: Lev-Civita Connection?

    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...
  28. loops496

    Can a Killing Vector Field Prove v^\mu \nabla_\alpha R=0?

    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...
  29. Andre' Quanta

    Killing vector tangents to geodesics

    Suppose to have a killing vector that its norm is null, so at the same time is also a null geodesic. Does the metric have special propierty? What can i say about the Killing vector and its proprierties?
  30. F

    How do you find the killing vectors for Minkowski space?

    Homework Statement How do you find the killing vectors for Minkowski space(or from any metric as well)? Homework EquationsThe Attempt at a Solution I'm new to GR and I'm going through Carroll's book. I've been alright so far but for some reason I just don't understand what's going on here...
  31. S

    Exploring Isometry: Questions on Symmetry and Transformation in Manifolds

    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...
  32. C

    Killing vector field => global isomorphisms?

    Suppose we have a vector field ##V## defined everywhere on a manifold ##M##. Consider now point ##p \in M##. As a consequence of the existence and uniqueness theorem of differential equations. this implies that ##V## gives rise to a unique local flow $$\theta:(-\epsilon,\epsilon) \times U \to...
  33. T

    Prove Killing vector is an affine collineation

    Homework Statement Given a manifold M with metric gab and associated derivative operator ∇a, let ξ a be a Killing vector on M. Prove that ξ a is an affine collineation for ∇a. Homework Equations (a) For a vector ξ a to be an affine collineation for a derivative operator ∇a, it must...
  34. C

    Symmetry (killing vector) preserving diffeomorphisms

    Suppose that on a Riemannian manifold (M,g) there is a killing vector such that ##\mathcal{L}_{\xi} g = 0.## How would one then characterize the group of diffeomorphisms ##f: M \to M## such that $$\mathcal{L}_{f^* \xi} (f^*g) = 0?$$ How would one describe them? Do they have a name...
  35. I

    Space time with no killing vector

    Is it possible to have a space-time with no killing vector? Alternatively, can I define the metric only with the killing vector of the space time?
  36. M

    Timelike Killing vector field and stationary spacetime

    I am trying to understand why in the definition of a stationary spacetime the Killing vector field has to be timelike. It is required that the metric is time independent, i.e. the time translations x^0 \to x^0 + \epsilon leave the metric unchanged. So the Killing vector is...
  37. P

    Orbits of a Killing vector field

    I was wondering what the orbits of a Killing vector field are. Do you have any good sources or reading material for this?
  38. T

    Calculating the killing vector fields for axial symmetry

    hi, i need to calculate the killing vector fields for axial symmetry for a project so i can study the galaxy rotation curves. i am assuming the galaxy to be a flat disk, in addition to being axially symmetric. so i figured that the killing vector fields with respect to which the metric...
  39. T

    General Relativity - Riemann Tensor and Killing Vector Identity

    Homework Statement I am trying to show that for a vector field Va which satisfies V_{a;b}+V_{b;a} that V_{a;b;c}=V_eR^e_{cba} using just the below identities. Homework Equations V_{a;b;c}-V_{a;c;b}=V_eR^e_{abc}(0) R^e_{abc}+R^e_{bca}+R^e_{cab}=0 (*) V_{a;b}+V_{b;a}=0 (**) The Attempt at a...
  40. P

    Killing Vector Solutions for General Relativity Metric | Self-Study Tips

    Hi. Currently I am self-studying a book on general relativity (Introducing Einstein's Relativity by Ray D'Inverno), I am stuck trying to find a Killing Vector solution to the following problem. ds^2 = (x^2)dx^2 + x(dy)^2 You can easily obtain the metric from the above. Now the question...
  41. Pengwuino

    How Do Killing Vectors Represent Spacetime Symmetries in Stephani's Relativity?

    In Stephani's "Relativity", section 33.3, equation (33.9), he has the Killing equations for cartesian coordinates as \xi_{a,b}+\xi_{b,a}=0 From there he says upon differentiation, you can get the following three equations \xi_{a,bc}+\xi_{b,ac}=0 \xi_{b,ca}+\xi_{c,ba}=0 \xi_{c,ab}+\xi_{a,cb}=0...
  42. D

    Divergence of product of killing vector and energy momentum tensor vanishes. Why?

    Hi, in my book, it says: ----------------------- Beacause of T^{\mu\nu}{}{}_{;\nu} = 0 and the symmetry of T^{\mu\nu}, it holds that \left(T^{\mu\nu}\xi_\mu\right)_{;\nu} = 0 ----------------------- (here, T^{\mu\nu} ist the energy momentum tensor and \xi_\mu a killing vector. The semicolon...
  43. N

    CD: How can I compute conformal killing vector fields on a 2D manifold?

    Hey guys, I'm working on Polchinski's string book, and I have a problem. Around page 152 he uses an identity I'm not sure how to prove. Essentially he wants to compute conformal killing vector fields. So we have the eq for a CKV: P_{ab}=\Delta_a \xi_b+ \Delta_b\xi_a- g_{ab}\Delta_c\xi^c=0...
  44. T

    How can Killing Vector Equations help find Christoffel symbols?

    I'm working my way through D'Inverno's Understanding GR and have reached Chapter 7 with no real problems. I'm stuck though on p101 (and the corresponding problem 7.7). D'inverno sets out two forms of the geodesic equations, one of which is the Lagrangian with the Killing Vectors and then has a...
  45. O

    How we compute killing vector for two-sphere

    The metric on S^2 is given by, \displaystyle ds^2=d\theta^2 + sin^2\theta d\phi^2 Here's the answer \displaystyle \xi ^{\mu}_{(1)}\partial _{\mu} = \partial_{\phi} \displaystyle \xi^{\mu}_{(2)}\partial_{\mu} = \ -(cos\phi \partial_{\theta} - cot\theta sin\phi \partial_{\phi})...
  46. N

    Killing Vector on S^2: Solving the Killing Equation

    Hi, I'm trying to understand isometries, for example between S^2 (two sphere) and SO(3). For this I need to show that the killing vectors for S^2 ds^2={d\theta}^2+sin^2 {\theta} {d\phi}^2. are: R=\frac{d}{d\phi}} S=cos {\phi} \frac{d}{d\theta}}-cot{\theta} sin {\phi} \frac{d}{d\phi}}...
  47. L

    Ricci tensor along a Killing vector

    In Carrol's text, he shows that the covariant derivative of the Ricci scalar is zero along a Killing vector. He then goes on to say something about how this intuitively justifies our notion of geometry not changing along a Killing vector. This same informal reasoning would seem to imply that...
  48. P

    Killing Vector and Ricci curvature scalar

    Homework Statement I'm currently self-studying Carroll's GR book and get stuck by proving the following identity: K^\lambda \nabla _\lambda R = 0 where K is Killing vector and R is the Ricci ScalarHomework Equations Mr.Carroll said that it is suffice to show this by knowing: \nabla _\mu...
  49. J

    Solving 2D Riemannian Metric Killing Vector Eqns.

    noob here * indicates multiply (or 'operate on'), d_c is partial derivative w.r.t. c tensor indices have always troubled me, my problem this time is I am trying to prove a vector E = (-y*d_x +x*d_y) is a killing vector after having computed the connection coefficients for 2-d riemannian...
  50. kakarukeys

    Killing Vector Fields: Generating Local Transformations?

    Can a killing vector field generate a diffeomorphism that only shifts points inside a small part of the manifold and preserves points outside of it? Rotation preserves the metric of a sphere but shifts every points on the sphere, I'd to find out if there is a killing vector field that...
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