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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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Clarification regarding the transformation law of the Christoffel symbols

I'm learning about general relativity from Sean M. Carroll's textbook. I recently encountered the transformation law for the Christoffel symbols, and I'm confused, as it seems like I'm seeing two ...
Aidan Beecher's user avatar
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Local cartesian coordinates on Riemannian manifold

I'm wondering is possible for every given metric $g=g_{ij}dx^i \otimes dx^j$ on $M$ and for every given $p\in M$ to find such chart $(U, \varphi)$ around $p\in U \subset M$ that the metric $g|_U$ in ...
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Double Trace of the tensor product of the metric tensor with vector fields.

So I am currently preparing for an exam on General Relativity and while reading the notes I stumbled upon this: $$ tr[tr[g \otimes X \otimes Y]]= g(X,Y) $$ Where $$ g=g_{ij} dx^{i}\otimes dx^{j} $$ is ...
Geotrael's user avatar
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Conditions on positivity of Riemann squared products in Euclidean space?

I wonder if it is known whether there are simple conditions on $u,v$ such that: $$R_{abcd}R^{abcd}+ u R_{ab}R^{ab} + vR^2 = A_{abcd}A^{abcd}$$ and thus proving that the combination is always positive ...
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A question about the definition of the normal variation of a tensor field

My question comes from the proof of Prop. 7.32 in Dan Lee's book Geometric relativity. The setting is we have a hypersurface $\Sigma$ with unit normal $\nu$ in an initial data set $(M, g, k)$. We now ...
user354113's user avatar
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General relativity problem from Wald's book

I'm self-studying General Relativity using Carroll's and Wald's book, and I'm confused about this problem from Wald (Chapter 3, Problem 2). Suppose that $M$ is a manifold with metric $g_{ab}$ and ...
Alan Chung's user avatar
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Show that the conformally complete Schwarzschild spacetime is asymptotically flat at null infinity

I am trying to show that the conformal factor used to conformally complete the Schwarzschild spacetime renders it asymptotically flat at null infinity (according to the mathematical definition given ...
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Sign error in the computations of $R_{\theta\phi\theta\phi}$ for the Schwarzschild metric

I am computing the components of the Riemann tensor for the Schwarzschild metric using the following formula $R_{\alpha\beta\mu\nu}$=$(\partial_\alpha\Gamma^l_{\beta\mu}-\partial_\beta\Gamma^l_{\...
darkside's user avatar
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Are there different types of metric?

I don't know if this is the right place to ask this question. I'm reading Bernard Schutz's First Course In General Relativity (2nd edition) and in page 61 under the title 'Picture of a one-form', ...
Aniruddha Bhattacharya's user avatar
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What condition of the Hawking singularity theorem fails for de Sitter and Minkowski spacetimes?

I am trying to figure out what condition of the Hawking singularity theorem fails for de Sitter and Minkowski spacetimes. Some has to fails otherwise we would had geodesic incompleteness which its ...
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Contradictions in "A diagonalizable energy-momentum tensor $T$ satisfies the SEC iff $\rho + p_1+p_2+p_3 \ge 0$ and $\rho + p_i ≥ 0 (i = 1, 2, 3)$

I am trying to understand the prove of this proposition : Let $T$ be a diagonalizable energy-momentum tensor, that is, (T_{µν}) = diag$(\rho, p_1, p_2, p_3)$ on some orthonormal frame $\{E_0, E_1, E_2,...
some_math_guy's user avatar
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1 answer
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Derivation of a spin connection in general relativity

On my journey to understand the mathematical structure behind general relativity, I came across the concept of a spin connection. which (I understand) is a connection defined in a spinor bundle. The ...
Tomás's user avatar
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Is time-orientability a condition on the metric, smooth or topological structure of a manifold?

I recently asked a question on Physics Stack Exchange about orientability and time-orientability of a manifold in the language of fiber bundles. This new question is related to, but independent, of ...
Níckolas Alves's user avatar
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Counterexamples to show that "$p < q ⇒ t^{±}(p) < t^{±}(q)$", and "$t^{±}$ are continuous" are not true

About this proposition For a general spacetime $(M, g)$ the volume functions $t^{±}$ (a) $p < q ⇒ t^{±}(p) \le t^{±}(q)$, (b) $t^{±}$ are upper/lower semicontinuous. What counterexamples could I ...
some_math_guy's user avatar
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How do I show that the region {r > 2m} of the Schwarzschild spacetime is globally hyperbolic?

How do I show that the region {r > 2m} of the Schwarzschild spacetime is globally hyperbolic? I am just starting with this so I don't really know how to lay out this arguments. I consulted Beem's ...
darkside's user avatar
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Solving Einstein vacuum equations related to a specific metric

Let the following metric $g = \begin{pmatrix} e^{\frac{2}{3} (\lambda+\mu)} & 0 & 0 & 0\\ 0 & e^{\lambda} & 0 & 0 \\ 0 & 0 & r^2 e^{\mu} & 0 \\ 0 & 0 & 0 &...
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Is causality preserved between two points on an immersed hyperboloid in Minkowski $R^3$?

Consider $\mathbb{R}^{1+2}$ with coordinates $\{t, x, y\}$ and Minkowski metric $g = diag(-1,1,1)$. Suppose we have a hyperboloid $x^2 + y^2 - t^2 = 1$ inside the previous Minkowski spacetime. My ...
Damiano Scevola's user avatar
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If the metric does not depend on a coordinate $i$, then along a geodesic, $g_{i\mu}\overset{.}{x}^\mu$ is conserved

In the title, $\overset{.}{x}^\mu$ is meant to mean the derivative of the geodesic with respect to the curve parameter, that is, the tangent to the geodesic. This is another of my general relativity ...
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2 votes
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Vector field along a geodesic is Killing

This is an exercise from my General Relativity course, so Einstein summation is implied. On a Riemannian manifold with metric $g$, show that if for a vector field $X^{\mu}$ and a geodesic with tangent ...
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The explaination of Einstein-Hilbert action in mathematics-wise

I've recently been studying about the General relativity and Einstein field equation. When I reading of the derivation of the field equation, I encounterd a method called Einstein-Hilbert action. This ...
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Calculation of Cartan structure constants

I'm trying to figure out the step from equation (19) to equation (20) in this document when $\mathcal{F} = 0$ y $\mathcal{A} = 0$. In this case, equation (19) reduces to $$ - e^{\alpha \phi} \hat{\...
David Lazaro's user avatar
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Equality characterization of the reverse Cauchy-Schwarz inequality in a Lorentzian manifold

Let $(M, g)$ be a Lorentzian manifold (signature -++...) with a time orientation and suppose that $v, w \in T_pM$ are causal vectors that are in the same light cone(ie, both future-directed or past-...
some_math_guy's user avatar
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Riemannian geometry and Symmetries, Hyperbolic Spaces

I am currently trying to connect Hyperbolic geometry through several models with Riemannian geometry. At first I have transformed the metric tensor from sphere in $R^3$ and the pseudo-sphere in $R^{2,...
S_d_pap's user avatar
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Position Vector in Relativity

In special relativity, the underlying manifold is the vector space $\mathbb R^4$ equipped with the Minkowski metric. The position of a (point) particle located at $p\in\mathbb R^4$ is the point $p$ ...
Thato Tsabone's user avatar
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How is an autoparallel equation related to Newton's second axiom?

I am currently working through The International Winter School on Gravity and Light lecture series and have had some trouble understanding the following claim made in lecture $9$ at the 22:30 minute ...
Taylor Rendon's user avatar
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Whathappens to the Ricci curvature, the scalar curvature, and Einstein tensor if I change the signs in the def of the Riemann curvature tensor?

I noticed some books define the Riemann curvature tensor with opposite signs on all $3$ terms. I wonder what happens to the Ricci curvature, the scalar curvature, and the Einstein tensor when using ...
darkside's user avatar
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3 votes
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Prove that the Riemann curvature tensor is a tensor

How would I prove that the Riemann curvature tensor $R: \scr X(M)^3 → \scr X(M)$, $R(X, Y )Z := \nabla_X\nabla_Y Z − \nabla_Y \nabla_XZ − \nabla_{[X,Y ]}Z$, is indeed a tensor ? I thought I could use ...
some_math_guy's user avatar
2 votes
2 answers
72 views

Is $\partial_k g_{ij}=\partial_j g_{ik} $?

I Found the following identity: $\partial_k g_{ij}=\partial_j g_{ik} $ ? or at least that is what it looks that it is being used. g is a riemannian metric by the way. Why is it true? .I know the ...
darkside's user avatar
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1 answer
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Determining Conformal Map from Conformal Factor

I am working on a project and need to apply a certain (Lorentzian) conformal transformation, $\psi:\mathbb{R}^2\mapsto \mathbb{R}^2$, to a figure which I have generated numerically in Mathematica. I ...
Daniel Grimmer's user avatar
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Prove that maximally symmetric submanifold of maximally symmetric manifold is umbilical

I come form General Relativity background and have a question about differential geometry. Assume a pseudo-Riemannian manifold $M$ equipped with a metric tensor field $g$ inducing inner product on the ...
P.L.'s user avatar
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1 vote
1 answer
81 views

Covariant Derivative of a Covariant Derivative

I am taking a course in General Relativity at university and am struggling with getting to grips with tensors which are new to me. I am struggling with the idea of taking the covariant derivative of a ...
Thomas's user avatar
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1 vote
1 answer
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Request for sources concerning tensor densities and their covariant derivative

I am studying an article (Geodesically equivalent metrics by V. Matveev) that uses covariant derivatives of tensor densities, but I am failing to find some literature that deals with how this is ...
Ioannes's user avatar
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Proof of covariant derivative becoming partial in a torsion free connection

for my homework I have to prove that inside a Lie derivative of a tensor along a vector field, if the connection is torsion free, I can swap the partial derivatives with covariant ones. My approach ...
Filippo Pavarino's user avatar
1 vote
1 answer
78 views

Can a Riemannian submanifold effectively act as a local tangent space?

I have been studying rudimentary gauge gravity, using the soldering equation to show how local P-translations can be related to global Lorentz and coordinate transformations. My question relates to ...
Gabriel Turner's user avatar
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1 answer
32 views

Number of positive and negative eigenvalues

The Jacobian of the inversion of the Lobachevsky space is given by $$J_{\mu\nu} = \delta_{\mu\nu} - 2 \frac{z_\mu z_\nu}{z^2}, $$ where $z_\mu$ are real numbers, $z_0 > 0$ and $z^2 = \delta^{\alpha\...
Geigercounter's user avatar
2 votes
1 answer
75 views

Show that continuity-preserving maps are homeomorphisms

Let $X$ and $Y$ be topological spaces. Call $f:X \rightarrow Y$ continuity-preserving if, for all continuous functions $\gamma: \mathbb{R} \rightarrow X$, $f \circ \gamma$ is also continuous. I know ...
paad89's user avatar
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1 vote
0 answers
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Writing momentum 4vector as an integral over the EM stress-energy tensor

I have been watching a series of lectures on general relativity by Neil Turok and I have run into a problem. In one of the lectures, the professor writes the momentum 4-vector as a contraction of the ...
Jesse Van Der Kooi's user avatar
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What is exactly is a continuous curve in a Lorentzian manifold?

Often, a continuous function $f: \mathbb{R} \rightarrow M$ from the real numbers $\mathbb{R}$ to a metric space $M$ is called continuous at $r \in \mathbb{R}$ if and only if, for any $\epsilon>0$, ...
paad89's user avatar
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Why does $\frac{dx^\mu}{d\lambda}(\frac{\partial}{\partial{x^\mu}}\frac{dx^\nu}{d\lambda})=\frac{d^2x^\nu}{d\lambda^2}$?

Why does $\frac{dx^\mu}{d\lambda}(\frac{\partial}{\partial{x^\mu}}\frac{dx^\nu}{d\lambda})=\frac{d^2x^\nu}{d\lambda^2}$? I have got as far as using the product rule: ($\frac{dx^\mu}{d\lambda}\frac{\...
Peter Petrov's user avatar
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0 answers
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Transformation rule for components of tensor fields

I would like to prove that the multilinearity condition for a tensor field is equivalent to the transformation rule for its components in a coordinate basis. One way is straightforward, but I haven't ...
gusifang's user avatar
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30 views

Evaluating $\frac{d}{d\tau}(\eta_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu})=0$

Evaluating: $\frac{d}{d\tau}(\eta_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu})=0$ $=\eta_{\mu\nu}\ddot{x}^\mu\dot{x}^\nu+\eta_{\mu\nu}\dot{x}^\mu\ddot{x}^\nu$ I understand that we can relabel $\mu$ and $\nu$ ...
Peter Petrov's user avatar
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0 answers
67 views

Equivalent definition of Hawking quasi-local mass

Recently, I came across a strange definition of Hawking quasi-local mass, which states that given a surface $S$ in the spacetime, the Hawking mass of $S$ is defined as $$m(S)=\sqrt{\frac{\mathrm{Area}(...
Boar's user avatar
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0 answers
31 views

Writing the explicit form of antisymmetrising the metric and Ricci tensor

Whilst going through the solutions to a GR worksheet, I struggled to understand a lie in the solutions. The line is: $g_{\sigma[{\mu}}\nabla_{|\rho|}R_{\nu]}^\rho=\frac{1}{2}(g_{\sigma\mu}\nabla_\rho ...
Chris G's user avatar
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Formulating $Spin^c(3,1)$ Connection and Curvature on a $GL+(4,R)/Spin^c(3,1)$ Structured Manifold

I am exploring a geometric framework where the usual metric tensor role (as in $GL^+(4,\mathbb{R})/\text{SO}(3,1) $) is replaced by a structure defined by the quotient $ GL^+(4,\mathbb{R})/\text{Spin}^...
Anon21's user avatar
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2 votes
0 answers
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Confused by the path I am asked to follow in order to solve the killing equation on S2.

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 ...
Alexandre Zagara's user avatar
3 votes
1 answer
120 views

Understanding the Geometry Spawned from Quotient Spaces $GL^+(4,R)/SO(3,1)$, $GL^+(4,R)/Spin(3,1)$, and $GL^+(4,R)/Spin^c(3,1)$

I'm working on a theoretical framework where I explore different quotient spaces formed with GL$^+$(4,R) and various groups. Specifically, I'm interested in the types of geometry that arise from the ...
Anon21's user avatar
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43 views

Is there an easy proof of $d \star S = s \cdot \epsilon$ for two forms S

In https://arxiv.org/pdf/1906.08616.pdf eq. 3.51 the following identity is proposed to hold for two-form $S$ on a manifold with metric $g_{\mu \nu}$. $d \star S = s \cdot \epsilon$, where $d$ is the ...
DerHutmacher's user avatar
2 votes
1 answer
170 views

Naive questions by a beginner in general relativity.

I want to ask some very naive questions in general relativity. I have the background of PDE and few Riemannian geometry. After Schwazchild gave a solution, people study its singularity and predict the ...
Elio Li's user avatar
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How to differentiate a light-cone integral in relativity?

Consider the following integral $$ \int_{V_X(v)}I(Y)\,dY\tag1 $$ In light cone coordinates $x_\pm=x\mp vt$, where $t$ is time and $v\in\mathbb{R}^+$, $V_X(v)$ is the past light cone, given, by $$ V_X(...
sam wolfe's user avatar
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0 votes
1 answer
178 views

Geodesics with a normal distribution

I'm working with the Desmos 3D calculator and I want to find the geodesic across some manifold currently I have a generalized equation for a generalized 3D distribution curve: $$f(x, y) = ab^{-{\left(\...
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