# What is Killing vector: Definition and 51 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. ### 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?
2. ### 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...
3. ### 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...
4. ### 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...
5. ### 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...
6. ### 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...
7. ### 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...
8. ### 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...

32. ### 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...
33. ### 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...
34. ### 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?
35. ### 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...
36. ### 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?
37. ### 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...
38. ### 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...
39. ### 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...
40. ### Killing vector in Stephani

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...
41. ### 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...
42. ### 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...
43. ### 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...
44. ### 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})...
45. ### 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}}...
46. ### 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...
47. ### 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...
48. ### 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...
49. ### 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...
50. ### Killing vector in kruskal coordinates

Let (U,V,\theta, \phi) be Kruskal coordinates on the Kruskal manifold, where -UV=\left(\frac{r}{2m}-1\right)e^{r/2m},\hspace{1cm} t=2m\ln\left(\frac{-V}{U}\right) and \theta and \phi are the usual polar angles. The metric is ds^2=\frac{-32m^3}{r}e^{\frac{-r}{2m}}dUdV+r^2d\Omega^2. The vector...