Questions tagged [general-relativity]

Questions related to the mathematical aspects of Einstein's theory of relativity. For the physics and its interpretations, please ask at the physics.SE. You may also consider the tags (differential-geometry) and (pde).

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Why do we need to use the chain rule in the transformation of a partial derivative of a vector component from different coordinate systems?

Let us consider the partial derivative of a vector component $A^i$, this is $\partial_jA^i = \frac{\partial A^i}{\partial X^j}$ We also know that we can write the transformed contravariant vector ...
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Killing vectors in Minkowsky Metric

Firstly, I know this is a physics-related problem, and I have posted here, but the physics forum seems so much more empty then this one, so here it goes: I was in the process to find the Killing ...
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How to prove that a Riemann Manifold is conformally flat?

In Ray D'Inverno's book - "Einstein's Relativity" he states a theorem that says the following - "Any two-dimensional Riemann manifold is conformally flat". How can I prove this? I ...
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Riemann curvature tensor identity proof

Let $R_{abcd}$ be the Riemann curvature tensor defined by a torsion free and metric compatible covariant derivative. Ultimately I am trying to prove that $R_{abcd} = - R_{abdc}$ The prove in my ...
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30 views

Computation of Laplace operator of Einstein tensor

I'm new to Relativity and I'm trying to understand this computation: $$ (\nabla^\mu G)_{\mu\nu}=0 $$ where $G=Ric-\frac{1}{2}\mathcal{R}g $ is the Einstein tensor, and $\nabla^\mu=\nabla_{\partial^\...
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Lie derivative of Lagrangian

We are on a Lorentzian manifold $(M,g)$ and we have a Lagrangian $L(\psi, d\psi, g^{-1})$ so $L:(\mathbb{R},(TM)^*, g^{-1})\rightarrow \mathbb{R}$. We fix $\psi$ and for a vector field $X$ would like ...
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Tensor Calculus swapping out repeated indices question

Consider the following result from General relativity: $$(1) \space\space\Gamma^{j}_{ij} = \frac{1}{2} g^{jk} (\partial_i g_{jk} + \partial_j g_{ki} - \partial_{k}g_{ij} ) $$ My question is whether it ...
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Normal coordinates on a flat manifold

The context: I'm trying to show that any geodesic normal coordinates in Minkowski spacetime form an inertial coordinate system. I believe it should be the case that on any pseudo-Riemannian manifold ...
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In what sense do vector fields generate isometries?

The question is literally this simple. I would like to know, in not extremely complicated terms, in what sense does a Killing vector field generate an isometry. I am somewhat familiar with the concept ...
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How to find vector normal to a one-form?

Question: Suppose I have some one-form (linear functional), $f = \alpha \; \omega^1 + \beta \; \omega^2 + \gamma \omega^3$ for some basis forms $\{\omega^i\}$ and coefficients $\{\alpha, \beta, \gamma\...
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Can you use black hole theorems to solve navier stokes problem?

To answer whether the navier-stokes equations don't "blow up" for some initial input. It seem like the same type of problem as to whether a particle in GR will hit a singularity. Except in ...
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Plane-fronted (gravitational) waves

What is the meaning of plane-fronted gravitational waves? and how it can be a generalization of plane waves? I could not imagine the wave with variable wave-vector, and the importance of this ...
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Christoffel Symbols Different Representarion?

I was given the $\mathbb{R^2}$ metric in polar coordinates, as follows: $$ ds^2=dr^2+r^2d\theta^2. $$ In this context we denote $e_1=\partial_r=(\cos(\theta), \sin(\theta))$, $e_2=\partial_{\theta}=(-...
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Partial differential equation with laplacian squared

I am working in extensions of General Relativity Theory and at the moment of taking the newtonian limit of this extension theory (essentialy, mathematically speaking, this is just linearizing the ...
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Pertubation theory in sagemath

I have seen the documentation on how to truncate polynomials using sage but I am stuck as to how I can actually apply this in my work... I am currently trying to find the...say Ricci tensor for a ...
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Riemannian compact six-dimensional manifolds Ricci-flat

Are there Real compact six-dimensional manifolds Real Ricci-flat? It is known that Calabi-Yau manifolds exist, that is, Complex compact three-dimensional Ricci-flat, but I don't know if Real compact ...
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particular part of non-linear second order ODE

I ma trying to find a solution for the following ODE: $$y''+p y'+q y=r y^{2},$$ where prime depicts the derivative with respect to $x$ and $p$, $q$ and $r$ are constant. The left side of the equation ...
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Topological Manifolds and Physics: a rapid doubt on Hausdorff, Second-Countability and Paracompactness.

My question is quite rapid. I know that the right thing to do is dig into books, but I don't have much time to spend on a proper mathematical reading. So, considering the standard Differential ...
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Penrose and the meaning of Light-Cone [closed]

In historical foundation of Newman-Penrose (spin-coefficient) formalism, the underlying motivation for the choice of a null basis was Penrose's strong belief that the essential element of a space-time ...
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Reality check on Schwarzschild computations

TL;DR: I have something Ricci-flat which is not turning out to be scalar-flat and this is absurd. Consider $P_I\times_r \Bbb S^2$, where $P_I =\{(t,r) \in \Bbb R^2 \mid r>2m\}$, with $m>0$, and ...
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Some conditions required of a covariant derivative operator

The following content about Differential Geometry comes from the General Relativity book written by Robert M. Wald. Compared with ordinary DG books, its material may not be in the mainstream, but I ...
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Laplace Equation on a Perturbed Manifold

I am reading this article where the author derives an exact solution for the Laplace equation on the $t=0$ hypersurface in Schwarzchild spacetime (ie. for the spatial Schwarzchild metric): the ...
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Uniqueness of solution to the Klein-Gordon equation in Lorentzian Anti-de Sitter space

I am confused with one aspect regarding uniqueness of solutions to the Klein-Gordon equation with specific boundary conditions in the Lorentzian manifold ${\rm AdS}_{d+1}$ and its Riemannian ...
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Integration over pseudo-Riemannian manifolds

Given a pseudo-Riemannian manifold $M$, let say for example Minkowski spacetime in $s>1$ dimensions, often we talk about integration on it. Namely the connection/metric definition gives us a ...
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Better coordinates for maximally extended Schwarzschild spacetime

There are many possible coordinate systems for describing the Schwarzschild spacetime. Eddington–Finkelstein (EF) coordinates have an especially simple form: $$ ds^2 = -\left(1 - \frac{r_s}{r}\right) ...
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Domain issues in transformation of the coordinate representation of a function

Start with a Manifold $M$ and define a function $f:M\rightarrow\mathbb{R}$. As usual, pick two charts $(U,x)$ and $(V,y)$ with $p \in U\cap V$ and $x:M \supset U \rightarrow x(U) \subset\mathbb{R}^n$. ...
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Well-definedness of ADM mass, energy, linear and angular momentum

In asymptotically flat initial data sets, one can define the ADM mass, energy, linear and angular momenta via spatial limits of integrals over 2-spheres. My questions are: Is asymptotical flatness ...
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Book Request - General Relativity (for mathematicians)

Please can someone recommend some books on 'higher-level' (couldn't think of a better way to phrase...) books on GR? I've read over half of Wald (General Relativity) and I'm about to finish Carroll (...
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Closed Curve Through an Event in Spacetime

I'm studying the book Techniques of Differential Topology in Relativity by Roger Penrose and I'm stuck in an exercise he left to the reader. we say that the spacetime $M$ is strongly causal in $p$ if ...
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Existence of a 3D Riemannian manifold with neck pinch singularity and Mobius loop

I could construct this 2D Riemannian balloon manifold. It is a flat plane at infinity, transitioning to a region of negative curvature, followed by the pinch point, and then a balloon with positive ...
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How do you express a steady quantity moving with time-dependent velocity as an equivalent time-dependent quantity?

This question is inherrently physical, but it focuses in on a mathematical concept, so I thought it best to ask it here while still detailing the physics behind the question... Say you have a magnet, ...
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Symmetric tensor satisfying the dominant energy condition has $T^{00} \geq |T^{\alpha\beta}|$.

In Hawking & Ellis's Large-Scale Structure of Spacetime, I've come across the following claim: If $T_{ab}$ is a symmetric tensor on a Lorentzian manifold $(M, g)$ satisfying the dominant energy ...
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How do you move between an observer's frame orbiting a Kerr B-H and the B-H's frame? In context of photon emission from accretion disk.

I'm struggling with the concept of moving to and from reference frames in GR. I'm doing a problem in Spacetime and Geometry by S. Carroll (Chapter 6, Question 6 - the text for the question is very ...
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Averaging of unit vectors in spacetime with Minkowski signature

Averaging of unit vectors in Euclidean space over $S_1$ is a well-known result. If we consider unit vector $\vec{n}=(\cos(\phi),\sin(\phi))$ such as $\vec{n}^2=1$, then the following average: \begin{...
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Confusion about notation over Einstein summation notation

So when evaluating the length of a curve, say $\gamma$, one uses the arc length paramete $s$: $$\int_{\gamma}ds=\int_{\tau_1}^{\tau_2}\sqrt{\sum_{\mu=1}^{3}(\frac{dx^{\mu}}{d\tau})^2} d\tau$$ which is ...
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Coordinate Choice for a Perturbed Metric

I am considering trying to prove something for a perturbation of the spatial part of the Schwarzchild metric. $g_{ij} = g^{SCH}_{ij} + h_{ij}$ The unperturbed spatial metric is conformally related to ...
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Persistence of metric components and null geometry

The following setup is from Rendall's 1990 paper regarding the characteristic initial value problem for Einstein's equations, and also from material on null geometry. Given a 2-dimensional spacelike ...
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Computation of Mean Curvature with respect to a Normal Vector?

I am reading through this article which has a lot of differential geometry calculations. On page 5, I see how they define the mean curvature with respect to $N$ but I'm not sure how they get to the ...
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Relation between point conjugate to spacelike hypersurface and expansion of congruence

Suppose that you have $\Sigma$ a spacelike hypersurface in a spacetime (a $4$-dimensional Lorentzian ($-+++$) manifold). Along a timelike geodesic $\gamma$, $p$ is conjugate to $q$ if there exists a ...
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101 views

Synchronous or Gaussian normal coordinates

Suppose you have a spacetime $(M,g)$, i.e. a $4$-dimensional Lorentzian ($-+++$) manifold. Let $S$ be a spacelike hypersurface. If $p\in M$, in a neighborhood of $p$ you can find a smooth timelike ...
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Hypersurface-orthogonal vector field and helicity

In a (3+1)-dimensional Lorentzian manifold equipped with a metric $g_{ab}$ (context: general relativity), I define a vectof field $k^a$ to be a Helical Killing Vector (HKV) if i. it is a Killing ...
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How do I extract a formula for distance from a metric that employs curvature?

Let's say I have a metric for spacetime that looks like this: $$ ds^2 = -c^2dt^2+a^2(t) \left[\frac{dr^2}{1-kr^2}+r^2d\theta^2+r^2\sin^2\theta d\phi^2\right] $$ As I understand it, this is the ...
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How does topology work when taking charts on a Psuedo-Riemannian manifold?

I'll first explain why I think taking charts is sane when working with Riemannian manifolds, and then show what I believe breaks down in the Pseudo-Riemannian case with a particular choice of a Pseudo ...
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Is it possible to formulate general relativity via quaternions?

Background I was fiddling around and wondered about the below. Consider the following quaternions: $$i^2 = j^2 = k^2 = -1$$ Consider the analog of the line element: $$ ds = dx i + dy j + dz k + \alpha ...
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What are Minkowski space and Lorentzian manifolds, formally speaking?

I am in general confused about what Minkowski space is. I'll write down what I know and what I believe Minkowski space is. I'd appreciate any corrections. A Riemannian manifold is a manifold (so it ...
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Hypersurface in a Perturbation of Schwarzchild Spacetime

I have a fairly simple question about perturbations of Schwarzchild spacetime in general relativity but cannot seem to find the answer anywhere. Start with the standard Schwarzchild metric $g^{SCH}= -\...
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The metric coefficients in cylindrical coordinates are (a) $(1,1,1)$ (b) $(1,0,1)$ (c) $(1,r,1)$ (d) neither?

In Differential and Riemannian Geometry we study metric cylindrical coordinates $(r, \theta, z).$ I am preparing its Multiple choice questions, Q. The metric coefficients in cylindrical coordinates ...
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Symmetry of a Perturbed Metric

If a metric has a particular symmetry, will a perturbation of that metric enjoy the same symmetry? By perturbation of a metric, I mean $g_{\mu \nu}^{perturbed} = g_{\mu \nu} + h_{\mu \nu},$ where $...
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What is the Minkowski Metric?

I recently asked "What is the Metric Tensor?" and a very helpful answer from @R.N.Raia gave me a much better understanding as to what it is. The only problem is that there are a few terms their ...
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The range of the angular coordinates in (asymptotically) AdS$_5$ spacetime

In the papers I’ve seen with GR solutions in (asymptotically) AdS$_5$ spacetimes, when spherical-like coordinates $(t,r,\theta,\phi,\psi)$ are used, the range of the angular coordinates is as follows ...

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