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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Spacelike surface lying in a null hypersurface

Why is it possible to be able to pick a spacelike 2 surface S that lies in a null hypersurface N? We know that all the tangents vectors to N are either spacelike or parrelel to the normal vector. I ...
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1answer
61 views

Special reference for differential geometry

I am not entirely sure how to formulate the question, but here it is. I am looking to start a self study on general relativity, and of course I need a good grasp on semi-riemannian geometry (I am ...
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1answer
37 views

Divergence in terms of Levi-Civita connection

The divergence of a vector field $X$ on a manifold $M$ is defined usually as the function $\text{Div}(.)$ such that $(\text{Div} X) \;\mu =L_X \mu$ for $\mu$ a volume form. I know that there is also ...
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1answer
64 views

What is contracting a tensor actually doing?

I'm learning about tensors, and have a vague idea regarding what contracting a tensor means—but I'm still not sure of exactly what it's doing. Maybe someone here can put it in more intuitive terms. ...
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1answer
55 views

Wormhole - How to model it?

I am trying to model wormhole between two points in 3D space, but do not know/understand how to do so. A concrete example: Think of something like a game where we have a 3D world with a size of ...
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1answer
18 views

(Strong) causality condition

I'm studying the causality in Lorentz Manifold with the book "Semi-Riemann Geometry", B.O'Neill. I have the followig problem: He says that, picked a subset A of a Lorentzian Manifold M, the causality ...
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38 views

Pseudo-scalar product on Manifold

I'm trying to study the Semi-Riemannian Manifold and the relativity (I use the book Semi-Riemannian Manifold- O'Neill). But I don't understand the following thing: In a Semi-Riemannnian Manifold, I ...
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1answer
27 views

Taylor's Theorem in 2 variables

I've never been 100% happy with using taylors theorem, mainly because I see it used in a bunch of different ways and I'm never sure in which situations it is valid. The way I was introduced to it was ...
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Can the change in a parallel transported vector be obtained from the Riemann tensor R( ∙ ,B ⃗,C ⃗,D ⃗ )

If we do not feed the Riemann type (1,3) tensor a 1-form in the first slot, we get a vector that looks like R( ∙ ,B ⃗,C ⃗,D ⃗ )=R_μλη^α e ⃗_α⊗e ̃^μ⊗e ̃^λ⊗e ̃^η ( ∙ ,B^τ e ⃗_(τ ),〖 C〗^ρ e ⃗_ρ,D^σ e ...
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1answer
70 views

Using metric to raise and lower indices

Everything I read on tensors makes it clear that using the metric matrix $g_{ab}$ and its inverse $g^{ab}$ to respectively lower and raise indices of a tensor is very important. As far as I know (and ...
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2answers
129 views

Hodge star operator

Again I have issues with notations. The hodge star operator is defined as : (m is the dimension of the manifold) $$\star: \Omega^{r}(M) \rightarrow \Omega^{m-r}(M)$$ $$\star(dx^{\mu_{1}} \wedge ...
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Ellipticity of Ricci tensor, does it depend on coordinates?

Well, I am afraid this is a silly question because I know the answer must be 'yes, it does'. But I don't see why. I put the problem in context. The ricci tensor can be regarded as a differential ...
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0answers
67 views

Resources for learning Relativity

I´m looking for books to the study of Relativity. I know that this is math stack schange and not physics stack schage, but I believe that some of the users here are interesed in physical-mathematical ...
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2answers
55 views

Establishing compactness of manifolds for the purposes of applying Chern-Gauss-Bonnet

A unit sphere possesses an induced metric, $$ds^2=d\theta^2 + r^2\sin^2\theta d\phi^2$$ By applying the Cartan formalism, for a basis $e^\theta = d\theta$ and $e^\phi=r\sin\theta d\phi$, I found, ...
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66 views

Planetary motion integral

I was reading Planetary Motion (page 117) in Barry Spain's Tensor calculus, and stupidly enough, I didn't understand this. The equations are: $$\frac{d^2\psi}{d\sigma^2} + ...
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1answer
57 views

Differential geometry unit vector

Why is $$e_\mu=\partial_\mu$$always said to be the unit vector ? Doesn't the size of the vector $\partial_\mu$ kindoff depend on the underlying manifold ?
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1answer
30 views

Covariant vectors

As far as I'm aware, covariant vectors are defined by how they transform: But I've also heard that the covariant components of a vector are defined as the dot product of the vector and the various ...
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32 views

Motivation for tensor density

Wiki has provided the basic definitions of the tensor density, but what I really want to know is the motivation and the advantage of this concept. Could anyone give me some ideas?
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71 views

Imaginary number in relativistic speeds

I am layman in field of mathematics but when I was reading about theory of special relativity I have come across speed limit of light and the book said that no one can cross that limit because ...
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150 views

How to balance learning and researching as a new PhD student?

As a new PhD student, how to balance learning and researching? I am in Australia and here we don't have any course in PhD period. I know I need to learn something about my programme, but sometimes ...
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1answer
41 views

How to proof Frobenius Theorem in general?

The general Frobenius Theorem stating that Let $u_1,\dots,u_k$ be $k$ smooth linearly independent vector field on $M$. Let $$ W=\operatorname{Span}(u_1,\cdots,u_k) $$ Then $[u_i,u_j]\in W$ for ...
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Expression of $2\int_{0}^{\frac{1}{r_{0}}}\frac{du}{\sqrt{\frac{r_{0}-r_{s}}{r_{0}^{3}}-u^{2}\left(1-u r_{s}\right)}}$ in terms of elliptic integrals

In Gravitation by Misner et al. 1973 the authors state that (calculus related to the Schwarzschild metric page 678) : ...
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1answer
34 views

Metric Derivative Proof

How do I go about proving $\partial_{\mu} g^{\nu \rho}=-g^{\nu \sigma}g^{\rho \lambda}\partial_{\mu} g_{\sigma \lambda}$? I've tried using the covariant derivative and the Christoffel symbols but ...
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2answers
179 views

Definite integral $\int_{R_0}^{R}\frac{dr}{r^2\sqrt{\frac{R_0-R_S}{R_0^3}-\frac{1}{r^2}\left(1-\frac{R_{s}}{r}\right)}}$

In general relativity, null geodesics (in the unbounded case) can be written under the following form : ...
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1answer
20 views

Plus and Cross polarizations

The plus and cross polarizations of a gravitational wave are at 45 degree to each other. However, I find no explanation of this angle. Can somebody help?
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1answer
77 views

Schwarzschild metric tensor normal vectors

The Euclidean Schwarzschild metric describing a manifold (a black hole, though this is not relevant to the question) is given by, $$\mathrm{d}s^2 = \left( 1-\frac{2GM}{r}\right)\mathrm{d}\tau^2 + ...
3
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1answer
102 views

Tensor Laplacian

For a general tensor $T_{\mu_1 \dots \mu_n}$ on a (pseudo-)Riemannian manifold, is it true that $\Delta (T_{\mu_1 \dots \mu_n})= (\Delta T)_{\mu_1 \dots \mu_n}$? In general, it is not true that ...
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54 views

Tensor differential equations

I am reading Ringstrom's book The Cauchy problem in General Relativity, But I don't really understand Chapter 12 associating to tensor equations. I want to read some other material about this. Could ...
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1answer
78 views

doing research/project on spacetime curvature

So I recently undertook the daunting task of presenting a project on general relativity for a differential geometry course. Does anyone have any suggestions for topic or topics to narrow it down to? ...
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1answer
67 views

Computing Normals from Metric Tensor

I asked a similar question on the Physics Stack Exchange, but unfortunately I have had no reply. It may be more suited for the Math section, as it focuses on the mathematical interpretation of GR. ...
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1answer
36 views

Property of stress-tensor in flat spaces

Let $T_{ab}$ be a stress-tensor in a flat space satisfying conservation equations. Define $$ P^i=\int T^{oi}d^3x, \;\; D^i=\int T^{00}x^id^3x $$ Can anyone show me how to prove $$ \frac{dD^i}{dt}=P^i ...
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Isometries of a general metric

For a general (pseudo-)Riemannian manifold, i.e. in which the interval $ds$ can be written $ds^2 = g_{ab}\,dx^a \,dx^b$, is there a general prescription for finding the group of isometries- by ...
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1answer
41 views

Twin paradox spacetime intervals

I've very recently started to try to understand special relativity. I want to get a decent understanding of the twin paradox. I'll post what I've done so far and highlight what's gone wrong for me. ...
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1answer
109 views

Maths undergrad dissertation - minimal surfaces

I'm a third year maths undergrad writing a dissertation on minimal surfaces, and their application in space. Would anyone be willing to read through it (so far) and give me any feedback? positive or ...
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3answers
81 views

Online resources for special relativity

I wasn't sure where to post this, but I'm on a mathematics course that has basically brushed over special relativity. I'm also doing an out of department module called philosophy of physics and as you ...
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Using Invariance of Lorentz interval and constant speed of light to prove the Lorentz transformations

By the invariance of the Lorentz interval and the fact that the speed of light is the same in both frames we have \begin{align*} -c^2 dt^2 + dx^2 = -c^2 dt'^2 + dx'^2 \end{align*} By considering the ...
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2answers
64 views

Compute the hyperbolic angle subtended to the origin by the unit hyperbola through (ct, x) = (0, 1)

I'm trying to find the angle subtended by the unit hyperbola through the point $(ct,x)=(1,0)$. I think that I should be integrating something, but I'm not sure how to set it up. I've been trying to ...
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0answers
28 views

geodesic sphere in hyperbolic space

I have a question if it possible. Let us consider a geodesic sphere of center a and geodesic radius R in the hyperbolic space H. I want to know how I can define the vector joint the center a and a ...
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1answer
95 views

How must I understand concepts equations of physics?

I teach myself mathematics, but those days I wanted to learn about General relativity (not to pursue in it but only to have some background), perhaps because I am very curious to learn why exactly We ...
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Is there any convenient identity for $\mathrm{Tr}[\mathbf M^{-1}(\partial_\mu \mathbf M)\mathbf M^{-1}(\partial_\nu \mathbf M)]$?

For a matrix $\mathbf M$ it is well known that $\mathrm{Tr}(\mathbf M^{-1} \partial_\mu \mathbf M) = \partial_\mu(\ln(\det(\mathbf M)))$, which is rather convenient because it enables one to write ...
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2answers
84 views

Geodesics Through a Singularity

A singularity on a manifold with metric is defined to be a point at which some geodesic cannot be continued through. For example in Schwarzchild spacetime, $r=0$ defines such a point. Is it the case ...
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Multivariable calculus along with tensors …etc to start studying General Relativity

I bought Spivak Calculus on Manifolds last time and I was really really disappointed... I opened the first chapters and I understood nothing of what he was saying. But i need to understand ...
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2answers
171 views

Most important aspects of differential geometry for general relativity

I'm an undergraduate getting ready to take a graduate course in general relativity next quarter. I purchased Wald's General Relativity (who incidentally will be teaching the class) in order to get a ...
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1answer
155 views

Riemann tensor symmetries

The Riemann tensor has its component expression: ...
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FLRW metrics (isotropic and homogeneous space)

Consider a spacetime with metric $$ ds^2 = -dt^2 + a^2(t)d\Omega_k^2, \quad k=0,\pm1$$ where $a(t)$ is any regular function and $d\Omega_k^2$ is the 3-dimensional metric of the 3-sphere $S^3$, if ...
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Covariant derivative in abstract index notation

Spose $f,h$ functions, where $\nabla _af = \epsilon _{ab}\nabla ^bh$. Then $\nabla ^af=g^{ac}\epsilon _{cb}\nabla ^bh$. My question is then does $\nabla _a\nabla ^af=\nabla ^c\epsilon _{cb}\nabla ^bh$ ...
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Global conformal equivalence of two regions of Minkowski spacetime

I am wondering whether the region $H:=\{(t,x) : x^2-t^2<1\}$ of $(1+1)$-dimensional Minkowski spacetime, equipped with the restriction $g_H$ of the standard Minkowski metric $g=-\mathrm{d}t \otimes ...
2
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1answer
76 views

About the symmetry of Riemann Tensor

It is a problem in my homework. First I was asked to show $$ \nabla_a\nabla_bA_c-\nabla_b\nabla_aA_c=R_{a,b,c}^{\;\;\;\;\;d}A_d $$ where $A$ is a (0,1)-tensor and $R_{a,b,c}^{\;\;\;\;\;d}$ is the ...
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1answer
68 views

Examples of manifolds foliated by $S^2$

I have come across the Frobenius theorem in my study of GR, which for the special case of $S^2$ roughly means, that every point of a manifold with spherical symmetry can be foliated by spheres. I know ...
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Differentiation along a curve on a manifold (Re: Schutz's intro to GR)

I am trying to show (Schutz chpt. 6 prob 13) that if two vector fields $\vec{A}$ and $\vec{B}$ are parallel transported along a curve $\gamma:\mathbb{R}\to M$ with real parameter $\lambda$ ($M$ a ...