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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35 views

The metric and Kronecker's delta

I am reading some lecture notes for GR and it is currently showing how we are going to derive the field equations using a metric for a massive free particle with a metric ...
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2answers
54 views

Is Relativity a specific instance of Riemannian geometry?

If I am a mathematician and do not anything about Special/General Relativity, then should I study Riemannian geometry to learn Relativity? Is Relativity just an instance/example of some particular ...
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1answer
16 views

How to isolate the variable “v” buried deep in a relativity formula?

finding v in v = L[1-(v^2/c^2)]/t closest attempt: [1-(c^2/v^2)]v = L/t I've been working on this since yesterday. I think I need some help.
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45 views

Simply-connected Lorentzian manifold and event horizon

Can a simply connected Lorentzian manifold admit an event horizon? Or does the event horizon makes it non-simply connected?
2
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1answer
23 views

Computing for $\theta$ component of the geodesic equation

The schwarzschild metric is given by: $ds^2=-(1-\frac {2GM}{r})dt^2+(1-\frac{2GM}r)^{-1} dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2$ Here is the well known geodesic equation: $0=\frac ...
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3answers
80 views

Prove: Gravitation operator is invertible.

Let $(M,g)$ be a Riemannian manifold and $\Gamma(S^2M)$ the space of symmetric 2-covariant tensors. Define the gravitation operator as the map \begin{align*} ...
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0answers
19 views

Maxwell’s equations in the Curved Space-time

I've been told that Maxwell’s equations in the curved space-time $(\mathscr{M},g)$ take the form $$\nabla^a F_{ab} =0 \, \,(*), \quad \nabla_a F_{bc} + \nabla_b F_{ca} + \nabla_c F_{ab} = 0 \, ...
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1answer
70 views

Is it possible to build a tensor with the following properties?

I am searching for a tensor in 4-dimensional space-time with two indices that satisfy: \begin{eqnarray} M_{;\mu }^{\mu \nu } &=&0, \\ M^{\mu \nu } + M^{\nu\mu}&=&0, \nonumber \\ ...
2
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1answer
94 views

Prove: $(\delta^\nabla\text{d}^\nabla+\text{d}^\nabla\delta^\nabla)h=\nabla^*\nabla h-\mathring{R}_gh+h\circ\text{Ricc}_g$

Let $(M,g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$ and adjoint $\nabla^*$, and exterior derivative $\text{d}^\nabla$ and adjoint $\delta^\nabla$. For a symmetric 2-covariant ...
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0answers
43 views

Distance function under diffeomorphism of manifolds

community, I am concerned with measuring distances of systems under diffeomorphisms. Concrety, I consider a smooth diffeomorphism $\varphi: M\rightarrow N$ from the smooth differentiable manifold ...
0
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1answer
32 views

What is a thin loop?

I read one definition of a thin loop: $\gamma$ is a thin loop if there exists a homotopy of $\gamma$ to the trivial loop with the image of the homotopy lying entirely within the image of $\gamma$. ...
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1answer
26 views

Differentiating a rank-2 tensor to some power in index notation.

If I have some rank-2 tensor $g_{ab}$ with components dependent on some coordinate system $x^a$, how do I do the following differentiation in index notation (assuming the $\dot x^d$ are independent of ...
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0answers
64 views

Some questions about synthetic differential geometry

I've been trying to read Kock's text on synthetic differential geometry but I am getting a bit confused. For example, what does it mean to "interpret set theory in a topos"? What is a model of a ...
1
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1answer
24 views

Parallel Transporting a vector

I want to parallel transport a vector $V^{\mu}$ with the initial condition $V^{\mu} = (V^{\theta},V^{\phi}) = (1,0)$ along a closed curve parameterized by $ \lambda \in [0,1]$ and determine the ...
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0answers
9 views

Why is sinus amplitudinus the solution to a polynomial differential equation?

So I followed the derivation in https://en.wikipedia.org/wiki/Schwarzschild_geodesics and understand how to derive the equation $$ \left( \frac{du}{d\varphi} \right)^{2} = \frac{1}{b^{2}} - \left( 1 ...
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0answers
12 views

Help with 4-velocity

The 4-velocity $\overline{U}=(U^0,U^1,U^2,U^3)$ corresponds to the 3-velocity, $\underline{v}$, in the sense that $\overline{U}=\gamma(v)(1,\underline{v})$. Express (a) $U^0$ in terms of ...
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1answer
32 views

Show $A_{ab}$ are the components of a tensor.

The question asks: "If $v_a$ are the components of a vector, show that in an arbitrary coordinate system that $A_{ab}$ are components of a rank-2 tensor, where:" $$A_{ab}= ...
1
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1answer
36 views

How to solve system of Differential Equations with 1 independent and 3 dependent variables

How can one solve this set of three differential equations in one independent variable "t" and three dependent variables A, B and F, which are functions of only t? $$ \frac{F(t) B''(t)+B'(t) ...
5
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2answers
60 views

About the term $-\nabla_{[u,v]}w$ in the definition of Riemann curvature tensor

As we know, in the definition of Riemann curvature tensor, we require $$ R(u,v)w=\nabla_u\nabla_v w-\nabla_v\nabla_u w-\nabla_{[u,v]}w $$ Could somebody tell me why we need $-\nabla_{[u,v]}w$ ...
1
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1answer
76 views

Mathematical aspects of General Relativity

I was wondering what mathematical subjects are used in the study of the theory of General Relativity (black holes ...) I assume mostly differential geometry, Riemann Geometry ... but is there also ...
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1answer
81 views

Partial derivative with respect to metric tensor $\frac{\partial}{\partial{g_{kn}}}(g_{pj}g_{ql})$

$$-\frac{1}{4\mu_0}F^{pq}F^{jl} \frac{\partial}{\partial{g_{kn}}}(g_{pj}g_{ql})=+\frac{1}{4\mu_0} F^{pq} F^{lj} 2 \delta^k_p \delta^n_j g_{ql}$$ I need to know how to derive ...
0
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0answers
11 views

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 ...
3
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1answer
72 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 ...
2
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1answer
62 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 ...
2
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1answer
89 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. ...
2
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1answer
68 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 ...
0
votes
1answer
23 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 ...
0
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0answers
40 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 ...
0
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1answer
34 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 ...
0
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0answers
13 views

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 ...
2
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1answer
73 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 ...
2
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2answers
157 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 ...
7
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1answer
112 views

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
69 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 ...
2
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2answers
56 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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0answers
67 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
64 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 ?
0
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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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0answers
34 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?
2
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0answers
76 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 ...
4
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0answers
161 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 ...
1
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1answer
51 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 ...
2
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0answers
28 views

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) : ...
2
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1answer
37 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 ...
4
votes
2answers
190 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
21 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?
4
votes
1answer
86 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
votes
1answer
123 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 ...
3
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0answers
57 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 ...