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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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Signature of an off-diagonal metric/unique diagonalisation

Consider a 2-dimensional space, with a metric of the form $$ds^2 = dx^0 dx^1$$ From what I understand of metrics, this should correspond to a matrix of $$\begin{pmatrix} 0 & \frac{1}{2} \\ \...
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1answer
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Differential geometry textbook to supplement a General Relativity self-study/refresher course

I have taken an undergraduate course in GR via Thomas Moore's A General Relativity Workbook. To prepare for a research experience, I have been advised to take a GR course, so I will be refreshing ...
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1answer
15 views

Symmetrizing terms of Christoffel symbols

In equation 8 in this paper they claim that in the geodesic equations this can be done: $ \ddot{x}^\mu = \Gamma^\mu_{\alpha\beta} \dot{x}^\alpha \dot{x}^\beta = - \frac{1}{2} g^{\mu\nu}\left( \...
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What is the fractal dimension of the critical scaling factor that just avoids forming a black hole?

In one of Scott Aaronson's lectures he mentions that attempting to scale up any three or even two dimensionally laid out hard drive will eventually result in a black hole. That is, if you've got some ...
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1answer
33 views

Telling the neighborhoods of a smooth manifold apart.

Consider a curved subspace of $\mathbb{R^3}$, in particular an ellipsoid. From an intrinsic geometry point of view, can I tell apart a neighborhood around the pole of the ellipsoid from a neighborhood ...
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33 views

Geodesic equations in polar coordinates (Euclidean space)

I tried to derive the geodesics in polar coordinate system (which should be a straight line since the metric is still Euclidean), and arrived the same equations as in this question: How to calculate ...
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0answers
29 views

Book on tetrads formalism and tetradic formulation of General Relativity

Could anyone give me some references for mathematicians (coordinates free notation, formalism of fiber bundles etc.) about tetrads, Palatini-Cartan theory, stuff about formulation of GR with tetrads? ...
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Adapted coordinates in the entire open set where a congruence is defined

Context: Null congruences that define Kundt spacetimes. However my question is a little bit more general, since the metric is not relevant. Consider an $n$-dimensional differentiable manifold $M$, an ...
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Inverse of the second-order metric in GR

How do we calculate the second-order metric tensor? Given a metric tensor which includes a second order perturbation around a background metric $\bar{g}_{ab}$ $g_{ab} = \bar{g}_{ab} + \epsilon h_{ab}...
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1answer
33 views

How metric on tangent space affects topology

Hallo I'm starting to study manifold in order to understand Einstein's relativity but there is something I don't understand. I already asked something very similar today but now I have another ...
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41 views

Manifold and the topology of $\Bbb R^n$

i'm starting to study manifold in order to understand Einstein's relativity but there is something I don't understand. A manifold $M$ is defined as being locally homeomorphic to $\Bbb R^n$. ...
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Hypersurfaces in Schwarzchild Spacetime

The Schwarzschild Spacetime metric is is given by : $\text ds^2=-\left(1-\frac{2M}{r}\right)dt^2+\left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$ I am currently studying ...
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How to perform these two conversions from a symmetric metric

Suppose in Cartesian coordinate system a Minkowski metric for flat spacetime can be written as : $$ ds^2 = a^2 (t)[-(1 + 2ψ(t,x,y,z))dt^2 - 2B_idx^idt + (1 - 2ψ(t,x,y,z))dx^{(i)2}] $$ This is a ...
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1answer
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How to transfer a metric to the orthonormal coordinate?

Suppose in Cartesian coordinate system a Minkowski metric for flat spacetime can be written as : $$ ds^2 = -[1 + 2ψ(t,x)]dt^2 + a^2(t) [1 - 2ψ(t,x)]dx^2 $$ This is a diagonal metric. How can I ...
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Fundamental question for Initial Data in General Relativity

So I was just wondering about the 3+1 formalism in GR. I know that initial data consists of a three-dimensional Riemannian manifold and a symmetric (0,2) tensor. This data is then a time slice in the ...
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Geodesic Co-ordinate Patches, proving $u^1$ curves are geodesic.

Suppose $\textbf{x}$ is a co-ordinate patch such that $g_{11} = 1$ and $g_{12} = 0$ at all instances, prove that the $u^1$ curves are geodesics. I can see that $u^1$ curves are unit speed, but I'm ...
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2answers
145 views

Proof that one can replace coordinate derivatives in coordinate formula for Lie derivative with covariant derivatives

How would we show that for a tensor of any rank we can replace the partial derivatives by co-variant (Levi-Civita) derivatives, I was reading this is a GR text where it was left to the reader as an ...
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1answer
50 views

Calculate initial conditions to integrate null geodesic

Suppose, this is the line element of a spherically symmetric FLRW metric, $$ ds^2 = -[1 + 2ψ(t,x)]dt^2 + a^2(t) [1 - 2ψ(t,x)]dx^2 $$ and the geodesic equation is, $$ \frac{d^2x^α}{dλ^2} = - Γ_{βγ}^α \...
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1answer
39 views

Tangent vector to a curve

I am trying to relate things simply. If a curve is on a flat 2D space represented by the parameter $\lambda$. In polar coordinate system $(r,\theta)$ at any lambda the tangent vector components are $$...
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1answer
38 views

Finding a relation between functions according to known constraints

I am solving a problem on geodesics with ideas from General Relativity and got stuck with one step. The simplified version is the following: With notations $$\dot{x}\equiv \frac{dx}{dt}, \quad \...
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1answer
124 views

Einstein field equation derivation

Someone could me explain how Einstein goes from this: \begin{align*} \frac{\partial g^{\sigma\beta} Γ^\alpha_{µ\beta}}{\partial x^\alpha} &=−\Omega\left[(t^\sigma _\mu+T^\sigma_\mu)−\frac12\delta^...
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2answers
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Action of a 1-form on the push-forward and pull-back of a vector

I am studying differential geometry I am trying to proof the expression below. Given that for a map $\phi$ : $M$ $\to$ $M$ the pull-back $\phi$*$\omega$ $\in$ $T^\ast_p M$ of a 1-form $\omega $ $ \...
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Lie derivative and Weyl Tensor

Let's consider a conformal symmetry such as : $\mathcal{L}_X \ g_{\alpha \beta}=2 \psi(x)g_{\alpha \beta}.$ How to prove that ${\mathcal{L}}_X \ W_{\alpha \beta \gamma \delta}=0$, where $W$ is the ...
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2answers
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Metric of a cross-section in General Relativity

Consider a finite closed region $V=(x,y,z)$ as a simply-connected subset of a 3-dimensional flat Euclidean space ${\Bbb R}^3$ with the metric $\text{d}s^2=\text{d}x^2+\text{d}y^2+\text{d}z^2$. A ...
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2answers
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Einstein Summation convention

Let A, B be two vectors. Then A=$a^ie_i$ B=$b^ie_i$ Then why is it in order to taking the inner product of the two I must change the summation of one vector: $a^ie_i$ $\cdot$ $b^je_j$? The summation ...
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1answer
70 views

Schoen & Yau's proof of the positive-mass theorem: Why is the surface S homeomorphic to $\mathbb{R}^2$?

I'm currently reading through Schoen & Yau's 1979 proof of the positive-mass theorem and am trying to understand the following statement on p. 55 of the publication (page 11 of the proof / PDF): ...
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How to relate two definitions of space of 1-forms?

I am trying to understand how to connect following quote from S. Carroll's "Spacetime and Geometry" with another definition of space of 1-forms from this book. Roughly speaking, the space of one-...
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35 views

Can 2 manifolds neighbor each other

I understand the definition of a set being a neighborhood of a point with a disk around it. But can 2 manifolds have elements in common where if m and n are manifolds of the same dimensions, there ...
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2answers
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Does $d x^\mu=g^{\mu\nu} \partial_\nu$, or $\partial_\nu=g_{\nu \mu}dx^\mu$?

As the title suggests: does $d x^\mu=g^{\mu\nu} \partial_\nu$, or $\partial_\nu=g_{\nu \mu}dx^\mu$? This is merely my own guess, from the material that I'm reading; so I'm not sure... The partial ...
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How to flatten a curved space

I know that it is not possible to flatten a generally curved surface without distortion. The distances between points marked on the curved surface will change when the surface is flattened, the ...
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1answer
63 views

Conditions for a given manifold to admit a given metric

It is well-known that every smooth manifold admits a Riemannian metric and it is commonplace to study when certain manifolds admit metrics of some specific type (eg Kahler, Ricci-Flat...), but I am ...
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2answers
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Is the orthogonal distribution of a Killing vector field always involutive?

Let $M, g$ be a semi-Riemannian manifold and $K$ a Killing vector field on $M$. Let $D = \{X \in \Gamma(TM) | g(X, K) = 0\}$ be the orthogonal distribution to $K$. Does $D$ have to be involutive, i. e....
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1answer
51 views

Derivative of metric along curve

Let $(M,g)$ be a semi-Riemannian manifold with Levi-Civita connection $D$. Let $\alpha : [a,b] \rightarrow M$ be a smooth curve on $M$, and let $\frac{D}{dt}$ be the induced covariant derivative on $\...
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0answers
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Decay properties of the Dirac equation in Witten's positive energy proof

In his 1981 paper E. Witten provides a proof of the positive energy theorem by considering the "Dirac" equation $\not D \epsilon \equiv h^{ab}\gamma_a \nabla_b \epsilon=0$ on a spacelike hypersurface ...
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Degrees of freedom of Riemann curvature tensor

I know the argument that uses the symmetries $R_{a b c d} = -R_{b a c d} = R_{c d a b}$ $R_{a b c d} + R_{b c a d} + R_{c a b d} = 0$ of the Riemann curvature tensor $R$ of an $n$-dimensional ...
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Integrals over the space of Riemannian metrics on $M$

Let $M$ be a closed smooth $n$-manifold. In the Hartle-Hawking proposal of Euclidean quantum gravity, the authors consider functional integrals of the form $$ I_M=\int_{\mathcal{Met}(M)}e^{-S(g)}\;\...
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1answer
64 views

Black Holes and Diagrams of Different Coordinate Systems

In General Relativity by Woodhouse there are the three following diagrams in Chapter 9 about Black Holes. Despite a (very brief) description of these diagrams in the book itself, I am struggling to ...
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Coordinate transformation second order partial derivative is zero using chain rule?

The question uses Einstein notation: In a coordinate transformation $ x^{\mu} \rightarrow x'^{ \mu '}$ is $$ \frac{\partial ^2 x'^{ \mu '}}{\partial x^{\sigma} \partial x^{\mu}} \frac{\partial x^{\...
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2answers
107 views

Black Holes and the Schwarzschild Solution

From Section 9.1, in General Relativity by Woodhouse: For a normal star, the Schwarzschild radius is well inside the star itself. As it is not in the vacuum region of space-time, the Ricci tensor ...
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1answer
59 views

Energy and the Schwarzschild Metric

In the Schwarzschild metric, the geodesic Lagrangian is $$L=\frac{1}{2} \left[ \left( 1-\frac{2m}{r} \right) \dot{t}^2-\frac{\dot{r}^2}{1-2m/r}-r^2(\dot{\theta}^2+\sin^2\theta\dot{\psi}^2) \right]$$ ...
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1answer
24 views

Stationary Observers Question

An observer in a fixed location relative to our coordinate system has a worldline with constant $r, \theta, \phi$, and thereofre has four velocity $U$ with only the first component non zero. Because $...
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1answer
72 views

Further deriving the weak field limit in general relativity

Consider the motion of a slow moving particle with worldline $x^a = x^a(t)$. We have $$\frac{dx^0}{dt}=1, \frac{dx^1}{dt}=u_1,\frac{dx^2}{dt}=u_2,\frac{dx^3}{dt}=u_3$$ where $(u_1,u_2,u_3)$ ...
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1answer
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The weak field limit metric setup

Assume that $$g_{ab}=m_{ab}+h_{ab}$$ where $m_{ab}=\text{diag}(1,-1,-1,-1)$ is the Minkowski space metric in an inertial coordinate system $x^a$, and $h_{ab$} is small and slowly varying. $1)$ I ...
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1answer
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The Metric Tensor, A Body of Mass m and Minkowski Space

By saying the body has mass m, we mean that the metric approaches that of Minkowski space for large r and that $$g_{00} \sim 1-2m/r$$ This was under a section in General Relativity by Woodhouse ...
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0answers
49 views

Tidal forces in General Relativity

In local inertial coordinates in which the observer is instantaneously at rest, $V = (1,\textbf{0})$ and $Y = (0,\textbf{y})$, where y is the position vector of the second particle. The acceleration ...
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1answer
52 views

Number of independent components in a general four index tensor

A general four-index tensor has $4^4=256$ indepedent components. Why does something like $R_{abcd}$ mean $4^4$ components? Why are these components independent (for a general tensor)? What are ...
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Geodesic Deviation Derivation

From Section 5.8 in Woodhouse's General Relativity: Let $Y^a(\tau)$ be a vector field. Its covariant derivative $DY^b$ along $\omega$ is defined by the following equivalent expressions, $$DY^b ...
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1answer
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I wanted to know of book suggestions that can help me overcome my fear of indices

I want to go deeper into General Relativity and Tensor Analysis. However, manipulating the indices always seems to overwhelm me. I wanted to know if there is a good book that covers up that and also ...
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2answers
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Deriving Riemann tensor with four lower indices

In terms of the connection coefficients $$R_{abc}^d = \partial_a \Gamma_{bc}^d-\partial_b \Gamma_{ac}^d - \Gamma_{be}^d\Gamma_{ac}^e+\Gamma_{ae}^d\Gamma_{bc}^e$$ Pick an event $A$ and choose ...
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1answer
71 views

General Relativity and the Wave Equation

Suppose that u is a function on space-time. We can define a vector field with components $\triangledown^a u = g^{ab}\partial_b u$. Why have they chosen $g^{ab}$ and not $g_{ab}$ here? The wave ...