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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If $\mathcal{M}$ is a manifold, what is meant by "the standard pairing on $\mathcal{M}^2$"?

I've been taking a look at the paper arXiv:2202.07580 [math-ph] , which discusses the so-called Wetterich equation in Lorentzian manifolds. I believe the Physics details behind the Wetterich equation ...
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Can the equivalence of the covariant and contravariant "connection coefficients" be proved simply by using the flatness of the tangent plane?

Do the derivations below prove the equivalence of the "connection coefficients" (5.78) and (5.78) for Riemannian and pseudo-Riemannian manifolds? Are they correct, and justified by my stated ...
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Showing metric is coordinate independent implies Killing vector field.

I am studying general relativity out of Sean Carroll's textbook, and I had trouble proving the following assertion (which is more about differential geometry than physics per se). Suppose we have a ...
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The killing vectors of flat space in polar coordinates

The metric of flat three-dimensional space is written in usual Cartesian coordinates ds^2 = dx^2 + dy^2 + dz^2 There are 3 killing vectors (1,0,0) , (0,1,0) and (...
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The Rindler metric - show that a quantity is constant along geodesics using coordinate independence.

I have some questions regarding the author's solution to parts $(\mathrm{ii})$ and $(\mathrm{iii})$ to the following question on the Rindler metric: The Rindler metric is defined by $$ds^2=\alpha^2x^...
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Is there a general linear group version of general relativity (not diffeomorphism)?

General relativity is strongly based on the concept of diffeomorphism. However diffeomorphisms are not linear. I wish to recover as much of general relativity as possible, but while remain linear. I ...
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Einstein field equation from covariant derivative

I am trying to derive the Einstein field equations from this approach: Let $\psi(t) = e^{-tM}$, where $M$ is a $4\times4$ matrix. Then $$ \frac{d}{dt}\psi(t) = -M\psi(t) $$ Let us now suppose that I ...
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Why is $R_{\rho \mu} = \eta^{\sigma \nu} R_{\rho \sigma \mu \nu}$?

Given the tetrad basis $\{(e_{\mu})\}$, i.e. smooth vector fields for which: $$ (e_{\mu})^a (e_{\nu})_a = \eta_{\mu \nu} $$ we wish to find the components of the Ricci tensor in this basis, in terms ...
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Does every globally hyperbolic spacetime admit a maximal slicing?

A globally hyperbolic spacetime is (roughly speaking) one with the topology of $\Sigma \times \mathbb{R}$ and a physically reasonable causal structure (no closed causal curves, that sort of thing) and ...
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Using $g^{ab}\nabla_a\nabla_b\phi$ show that $\nabla^2 \phi=\left(\partial_r^2+\frac1r\partial_r+\frac{1}{r^2}\partial_\theta^2\right)\phi$

I am having a hard time understanding the proof to the following question: In a two dimensional space with the usual metric $\mathrm{d}s^2 = \mathrm{d}x^2 +\mathrm{d}y^2$ the Laplacian can be written ...
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Why are the red parts of this expression present $\int_a^b \sqrt{g_{cd}\frac{dx^c}{\color{red}{d\lambda}}\frac{dx^d}{\color{red}{d\lambda}}}d\lambda$?

I am now trying to teach myself about geodesics, and a passage of my notes reads: In this chapter geodesics have been introduced as generalizations of straight lines through $$\frac{Du^a}{ds}=0$$ ...
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World line and light cones in two dimensions

The following is an example of hartle : Consider the two dimensional metric ds^2 = -X^2 dT^2 + dX^2 And the world line X(T)=A cosh(T) where A is a constant with the dimensions of length. The light ...
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Two ways of defining connection coefficients

I am relatively new to general relativity, and I am puzzled about the relationship between two ways of introducing connection coefficients on a differentiable manifold. One way (e.g. in S. Carroll's &...
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Variation of Pontryagin density $*R^{abcd}R_{abcd}$ with respect to inverse metric $g^{ab}$

I am computing the variation of $*R^{abcd}R_{abcd}$ with respect to inverse metric $g^{ab}$, where $R^{abcd}$ is Riemann tensor and $*R^{abcd}$ is its dual. I hope to express the result in terms of ...
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Prove that $\Gamma_{abc}=\frac{1}{2}\left(\partial_bg_{ac} + \partial_cg_{ab}-\partial_ag_{bc}\right)$

I am tasked with the following problem Use the equation $$\nabla_ag_{bc}=\partial_ag_{bc}-\Gamma_{cba}-\Gamma_{bca}=0\tag{1}$$ where $$\Gamma_{abc}=g_{ad}\Gamma^d_{bc}\tag{A}$$ and the (no torsion) ...
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How to choose basis vectors for a given metric to calculate curl of a vector in General Relativity?

The space-time interval in General Relativity is $ds^2=g_{\mu\nu}dx^\mu dx^\nu$, where $g_{\mu\nu}$ is the metric tensor. If the space-time is stationary and axisymmetric, the corresponding geometry ...
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How to transform a vector written in the language of differential geometry to the usual vector notation?

A vector $\vec{\omega}$ in General Relativity is expressed in the language of differential geometry as $$\omega=\omega_r\frac{\partial}{\partial r}+\omega_\theta\frac{\partial}{\partial\theta}$$. ...
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Lie Derivative weird form

I am reading a physics paper, in which the Lie derivative is presented in this strange (to me) way: $$\mathcal{L}_Y=Y^A\partial_A+\frac{i}{2}D_AY_BS^{AB}$$ where $A,B=\{z,\bar{z}\}$ with $\{z,\bar{z}\}...
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Straight policy on a curved surface in presence of a force field

I have already posted this question yesterday in Physics Stack Exchange: https://physics.stackexchange.com/q/711270/334688 However, they've suggested me to repost it here where it may be more relevant ...
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1 answer
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Definition of static spherically symmetric spacetime as fiber bundle

I am working on a physical paper about solutions of Einstein field equations in case of static spacetimes with perfect fluid spheres and wanted to invent a new definition of spherical symmetry there. ...
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Does the Differential Topology/Geometry frameworks being able to model solutions to diff. eqs. that are Non-Smooth?

I don't have much knowledge about Differential Topology neither Differential Geometry, but working on this another question about solutions to differential equations, and someone recommend me to ...
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How to interpret "Ricci-tensor represents volume gain"?

The Ricci-tensor is told to be the volume gain in comparison to Eucledean (Lorentzian) space (wikipedia-page on Ricci-Tensor). Question part 1: Is that the volume change of the manifold or of an ...
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2 answers
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Non-linearity of Einstein's field equations

How to show that, for two Schwarzschild- metrics, the Ricci tensors of two metric tensors do not sum up linearly: $R_{\mu\nu} (g_1+g_2) \neq R_{\mu\nu} (g_1)+R_{\mu\nu} (g_2)$ while the Ricci tensor ...
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1 answer
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Conjugate points and expansion of the geodesic congruence

I am working in a Lorentzian manifold $(M, g)$ (but I think the problem would be quite similar in a Riemannian manifold) and I am considering a timelike geodesic whose tangent vector field is denoted ...
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Extendibility of a curve in a compact subset (Lemma 14.2 in O'Neill's book)

Lemma $14.2(5)$ of O'Neill's book Semi-Riemannian geometry with applications to relativity states the following: If $\mathcal{C}$ is a convex open set in $M$, then, a causal curve $\alpha$ contained ...
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Is the covariant derivative a tensor?

While studying general relativity, the covariant derivative is constructed (in no rigorous manner) in order to make the derivative of a tensor transform like a tensor. Symbolically, $$ \nabla'_{\mu} V'...
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Checking Metric Tensor relations

I need to check some simple tensor relations for continuing my calculations. Please write me if you think anyone is incorrect. P.S.: I know this is very simple question, but i need to be assured about ...
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Killing Vector Fields of Schwarzschild

It is known that the vector fields $\partial_t$, $\partial_\phi$, $\sin \phi \partial_\theta+\cot \theta \cos\phi \partial_\phi$, and $\cos\phi \partial_\theta-\cot \theta\sin \phi\partial_\phi$ are ...
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3 votes
1 answer
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Inconsistency in the definition of the connection coefficients

I am new to general relativity and I am currently facing an apparent inconsistency in the definition of the connection coefficients. Some references I've been consulting (e.g. the lecture notes by S. ...
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About the Christoffel Symbols

I have a quick but important question about the concept of the Christoffel Symbols, and it is if you can apply another tensor rather than the metric tensor to find the Christoffel Symbols. I am trying ...
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2 votes
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Applying the divergence theorem to the graph of a function

This question relates to a proof of some theorem, and to avoid reckless discussion, I will incorporate some of the statements in the theorem that might seem irrelevant at first glance. Let $f:\mathbb{...
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Curvature Tensor of Robertson Walker spacetime

I am studying Generalized Robertson Walker spacetime from the book "Differential Geometry of Warped Product Manifolds and Submanifolds, Bang Yen Chen". In Chapter 4 the following lemma is ...
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2 votes
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A strange definition of the ADM mass

Given an asymptotically flat manifold $(M^{(n)},g)$ with ends $M_1,\ldots,M_\ell$, we commonly define the ADM mass of each $M_k$ by the following coordinate expression: $$m_{ADM}(M_k,g)=\lim_{\rho\to\...
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Partition of unity for calculating the volume of a paracompact manifold

Looking at this question/answers: Volume of a paracompact manifold it is used a partition of unity $\{\eta_j : j \in J\}$ corresponding to the coutable, locally finite open cover $\{ U_j : j \in J\}$, ...
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2 votes
1 answer
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Definition of asymptotic flatness

This question is about an idea in general relativity, but its underlying realm is basically geometric. Let me begin with a snapshot of Geometric Relativity by Dan A. Lee: I'd like to ask two ...
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1 vote
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Why does parameterizing a curve with its length yield $|g(T,T)|=1$?

Let $l(t)=\int^t_{t_0}|T|(t')dt'$, where $T$ is a tangent vector to some curve $C(t)$. Why does setting this function as a parameterization of the curve $C$, hence letting $l(t)=\psi C(t)$, imply $|T|^...
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7 votes
3 answers
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Conceptual understanding of Christoffel symbols

The basis vectors on a manifold are defined as partial derivative operators of any function that can be locally mapped around a point to $\mathbb R^n.$ The Christoffel symbols come about when a vector ...
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Why is the Riemann curvature tensor a rank 4 tensor?

The Riemann curvature tensor is defined as: $ R(X,Y) = [\nabla_X, \nabla_Y] $ when there is no curvature (no loss of generality in the question). If we expand this to coordinate notation, we get the ...
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2 votes
1 answer
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Alternative expression for Riemann curvature tensor

There is the usual expression for the Riemann tensor $$R_{abcd}=\partial_c\Gamma_{adb}-\partial_d\Gamma_{acb}+\Gamma_{ace}{\Gamma^e}_{db}-\Gamma_{ade}{\Gamma^e}_{cb}.$$ However, in the last page of ...
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General way to calculate Gaussian Curvature in 4D

I am kind of confused by the vast number of formulas for computing the Gaussian Curvature. Having a metric tensor / an expression for the line element in 4D (e.g. $t,x,y,z$ or in spherical coordinates ...
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Geometric interpretation of Ricci tensor acting on two vectors?

I've been looking into the geometric interpretation of the Ricci tensor, and the standard idea is that, for example in 3D, $Ricci(u,u)$ corresponds to the rate of change of volumes in the direction u ...
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Riemann Curvature of Schwarzschild with curvature 2-forms

I have been studying Chapter 14 of Misner, Thorne, Wheeler's Gravitation, in particular the method of computing curvature using exterior differential forms. Is there a reference that computes the ...
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Gram-Schmidt orthogonalization for tetrads

I have asked this question in Physics SE but I suppose that MSE is a better home for this question. If we assume that the coordinates basis of a manifold is linearly independent and hence take the ...
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1 vote
1 answer
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Geometrical meaning of the commutator of vectors on a manifold

On a manifold, vectors do not describe finite displacements, unlike in euclidean geometry, but they do describe infinitesimal displacements, so we can take two vectors, $v^a$ and $w^a$ to span an ...
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1 vote
1 answer
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What happens when contracting a lightlike with spacelike/timelike vectors?

I would like to understand what result I get contracting a lightlike four-vector with a spacelike and/or a timelike four-vectors (I'm interested in both cases) in terms of sign. Is the result of these ...
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Studying Geodesic Deviation and Tidal Forces

I have been studying general relativity from a mathematical point of view and I was wondering what would be a good material (lecture notes, books, and so on) to study geodesic deviation and tidal ...
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What is the partial derivative of a differential form

This here Wikipedia article https://en.m.wikipedia.org/wiki/Proper_reference_frame_(flat_spacetime) has a derivative of the square of a differential form $\frac {\partial}{\partial \tau} d\sigma ^2=0$ ...
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Schwarschilds radius and paramaterizing path

Consider the metric $$ds^2=(1-\frac{2m}{r})dt^2+(1-\frac{2m}{r})^{-1}dr^2 + r^2d\theta^2 + r^2\sin^2\theta d\phi^2.$$ Suppose a particle very large starts at the initial radius $R$ and then radially ...
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2 votes
1 answer
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Understanding why a symmetric matrix can always be diagonalised

I'm reading through some general relativity notes. I have reached a part that I don't understand, probably because my linear algebra is not good enough. I don't really understand what the maths is ...
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2 votes
1 answer
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Why does differentiation of a tensor increase its rank?

There is a statement in both Wald and Carroll's GR texts that, in short, state that the derivative of a $(k,l)$-tensor is a $(k,l+1)$-tensor. In both places this as stated as though it should be ...
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