All Questions
Tagged with vector-fieldsmetric-tensor
102 questions
1vote
0answers
69views
Double Null Foliation Schwarzschild Metric
Thinking two null rays coming from a timelike geodesic of , say , a star of mass $m$, one future pointing $l_a = \nabla_a u$ and one past pointing $l'_a = \nabla_a u'$ $ [l_al^a = l'_al'^a = 0] $ and ...
0votes
0answers
49views
Calculate normal vector to future light cone
Consider a future light cone in Minkowski spacetime $(-,+,+,+)$ defined by $u(t,x,y,z)=t - \sqrt{x^2+y^2+z^2}$ and $t>0$. Derivative of $u$ is $du=dt-\frac{x}{r}dx-\frac{y}{r}dy-\frac{z}{r}dz$ with ...
- 594
5votes
1answer
211views
Finding geodesic through Killing fields
I am currently reading Wald's General Relativity, but got a bit confused. Given a manifold with a metric, $(M, g_{ab})$, we may find the set of geodesic curves by solving the equation $$ T^a \nabla_a ...
- 330
0votes
1answer
102views
Problem in deriving Killing equation
I am studying derivation of Killing equation by Wald (also reading some other literature) but having some problem in understanding the math. Let $\chi ^a$ is killing vector on the horizon $$\chi _{[a} ...
- 594
1vote
1answer
154views
Finding Killing vectors for hyperbolic space [closed]
I want to find the Killing vectors for the hyperbolic space, which is described by the metric \begin{equation} ds^2 = \frac{dx^2 + dy^2}{y^2}. \end{equation} I have found the Killing equations, which ...
0votes
0answers
38views
Four-divergence of a vector [duplicate]
The covariant derivatives of a four-vector is $$ \nabla_{\nu}U_{\mu} = \partial_{\nu}U_{\mu} - \Gamma^{\lambda}_{\mu\nu}U_{\lambda} $$ It has the following identity: $$ \nabla_{\mu}U^{\mu} = \frac{\...
1vote
0answers
78views
In the frame field construction in GR, how do you get the vector field dual to a co-frame?
I am trying to understand the frame-field construction in General Relativity. We basically have four point-wise orthonormal vector fields, one of them being timelike and the other three being ...
0votes
1answer
80views
Understanding the derivation of Killing horizon surface gravity
In the book "A Relativist's Toolkit" by Eric Poisson, he explains surface gravity in section 5.2.4 The equation 5.40 says $$ (-t^\mu t_\mu)_{;\alpha} = 2 \kappa t_\alpha \tag{5.40}$$ where $...
- 323
5votes
2answers
769views
Resolving an apparent contradiction between Schwarzschild and ingoing Eddington-Finkelstein coordinates
I believe this is basic differential geometry issue. This may be obvious to many, but I was quite confused about it, and it took me quite a while to find the resolution. I'm going to ask and answer ...
- 223
2votes
0answers
728views
Killing vectors on the unit sphere
I am asked to find the Killing vector fields on $S^2$ where the line element is given by $ds^2=d\theta\otimes d\theta +\sin^2\theta d\phi\otimes d\phi$. I know how to solve this problem by considering ...
2votes
0answers
427views
Static spherically symmetric spacetimes
I would like to better understand a hypothesis that Wald uses to derive the general local formula of a static spherically symmetric spacetime. A spacetime is said to be spherically symmetric if its ...
- 31
0votes
1answer
54views
Metric Tensor Grid
Let, we are in a 2d metric where $g_{xx}=1, g_{yy}=x^2$, therefore $|e_x|=1$, $|e_y|=x$. If we try to draw the metric in a grid - it looks something like the image I uploaded. Note that, along the X ...
- 1,296
2votes
0answers
121views
Tetrad formalism: Cartesian-like tetrad?
I'm confused about tetrad formalism. In the article the have the Kerr metric in Boyler-Lindquist coordinates. They then define the tetrad at a point $r,t,\theta,\varphi$ as the one-form basis $$ e^{(t)...
- 290
0votes
0answers
222views
Proving statements about Killing vector fields
I'm proving some identities on Killing vectors and the like and I've stumbled on this one which I can't seem to figure out. Suppose $A^\mu = K^\mu$ is a Killing vector and we have the following field $...
0votes
0answers
82views
Killing field and vector field type
The Killing field condition can be defined as; $$\mathcal{L}_{X}g=0$$ for a metric $g$ and vector field $X$. In this case does it matter the type ($X^{\alpha}$ or $X_{\alpha}$) of the $X$ used in this ...
- 3,170
