Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.

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Complex tetrad vs. Real metric

I asked this question almost a month ago on mathoverflow (http://mathoverflow.net/q/228138/) but received no response. I thought I could have better luck here: I have a question on the relationship ...
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1answer
86 views

Is it possible to integrate a function over a null hypersurface?

For $f$ a compactly supported function on a spacetime $(\mathcal{M},g)$, one may define its integral over $\mathcal{M}$ by $$f\longmapsto \int_\mathcal{M}\star f,$$ where $\star$ is the Hodge star of ...
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2answers
84 views

How do you know what kind of space(time) you have when solving the Einstein Field Equations?

I'm experimenting with the EFE, and I ''invented'' a metric; a diagonal non-zero metric, and I discovered that the Riemann tensors are equal to zero which implies the Einstein tensor $G_{mn}$ equals ...
3
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0answers
54 views

Physical meaning of the Morse functions? [closed]

What is the physical correspondence of the Morse functions in a physical system? Currently I am studying Mirror symmetry but I can not get a physical intuition out of it.
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1answer
64 views

How does the gravitational field behave inside a star?

The interior gravitational field of a star with constant density is given by ...
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0answers
25 views

Manifold corners and M theory

I am currently trying to understand a paper by Hisham Sati on manifold corners and M theory. The background is that M theory admits manifolds with corners. One of the results in the paper is that the ...
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1answer
62 views

Derivation of the Cartan Field equation

Please help me understand how, in this introduction to spacetime and fields, the Einstein Cartan equation: $$C^k_{\hspace{2mm} [ji]}-\delta_{[i}^{k}C^l_{\hspace{2mm} ...
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51 views

Why Dirac monopole is a topological defect in a $U(1)$ gauge theory? [duplicate]

How does $U(1)$ gauge group at long distances, give rise to magnetic monopoles? Also why is it said that Dirac monopole is a topological defect in a compact $U(1)$ gauge theory?
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4answers
933 views

Coordinates vs. Geometries: How can we know two coordinate systems describe the same geometry?

Specifically, I'm asking this because I'm taking a class on General Relativity, and in Hartle's book Gravity, in Ch. 12, after having spent some time using Schwarzschild coordinates, we are introduced ...
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0answers
50 views

Metric defining an sphere [closed]

I want to find for which cases this metric can define an sphere: $$\frac{1}{P^2}\left(\mathrm d\theta^2+\sin^2 \theta\; \mathrm d\phi^2\right)$$ where $P=\sin^2 \theta+K\cos^2 \theta$, with $K$ the ...
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0answers
65 views

Differential geometry and tensors using Cartan method: advantages over other methods in Physics? [closed]

Let me begin with a simple example. I am trying to calculate the Christoffel symbols, the Ricci and the Curvature tensor for the metric of the surface (parabolic-like): $ds^2=(1+u^2)du^2+u^2d\theta$ ...
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1answer
55 views

Problem on parallel transport [closed]

I've been trying to go through an example for parallel transport but I cannot quite follow the solution. A surface (paraboloid) is given by the parametric equation $r(ρ, φ)$ = $ρ \cos(φ)\hat{i}$ + $ρ ...
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1answer
44 views

Is there a parametrization for the shape of space?

I was thinking about how the space is curved. And how do we know that the shape of space arround a singularity is something like that: So I was trying to make a similar parametrization of this kind ...
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1answer
84 views

Use partial or covariant derivatives when deriving equations of a field theory?

I feel like this question has been asked before but I can't find it. would the Euler Lagrange equation for, say, the standard model Lagrangian be $$\frac{\partial L}{\partial \phi}=\partial_\mu ...
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3answers
119 views

Is the local Lorentz transformation a general coordinate transformation?

There is a saying in Nakahara's Geometry, Topology and Physics P371 about principal bundles and associated vector bundles: In general relativity, the right action corresponds to the local Lorentz ...
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1answer
55 views

Integrals of Chern class, $c_i$ in YM theories

I am a bit confused with the definition of the 1st (and 2nd by extension) Chern class in YM theories. I understand that in general $c_i \in H^{2i}(M,\mathbb{Z})$ where $M$ is a smooth manifold. Then, ...
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1answer
75 views

GR Tetrads & ZAMO example

I am self-learning GR. Intro: Tetrads are a way of representing general relativity in a coordinate-independent fashion. I am having trouble understanding tetrad notations. Basically, I know that I ...
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1answer
65 views

Difference between local inertial frame and coordinate chart

In the most cases the local inertial frame is definied "physically" but I'm searching for a mathematically meaningful definition of the local inertial frame to solve my problem: Is the local ...
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1answer
58 views

Monodromy matrix and differential equations

What is the significance of monodromy matrix in the context of differential equations? I have seen some papers(1,2,3 etc) in CFT which use the monodromy method to compute conformal blocks at large ...
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0answers
47 views

Spherical metric multiply by a function

I know that if I want to get the metric for a two sphere I consider a Cartesian flat space, I change to spherical coordinates and then I consider that the radio is constant (so the space is not flat ...
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1answer
32 views

Displacement vector in terms of a position vector and velocity [closed]

Is it correct to say that, given $t_0\in\mathbb{R}$, a point on a curve $\gamma$ in $\mathbb{R}^3$ would be given by $$\gamma(t)=\gamma(t_0)+(t-t_0)\dot{\gamma}(t)$$ for all $t>t_0$? I'm guessing ...
2
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1answer
143 views

How can I understand $\mathrm ds^2 = -c\,\mathrm dt^2 + [\mathrm dx-v_s(t)f(r_s)\mathrm dt]^2 +\mathrm dy^2 +\mathrm dz^2 $ in the simplest way?

How can I understand this equation $$\mathrm ds^2 = -c\,\mathrm dt^2 + [\mathrm dx-v_s(t)f(r_s)\mathrm dt]^2 +\mathrm dy^2 +\mathrm dz^2 $$ in the simplest way? I am a 13 year old boy who is totally ...
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1answer
55 views

Homogenuous Maxwell Equations in the Language of Differential Forms

I understand that if I define electric field to be $E=E_i dx^i$, magnetic field to be $B=B_1 dx^2 \wedge dx^3 + B_2 dx^3 \wedge dx^1 + B_3 dx^1 \wedge dx^2 $, and field strength to be $F= dx^0 ...
2
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1answer
114 views

Why only gauge transformations in electromagnetism?

first of all, I need to say that I'm a mathematician, so this question may sound a little stupid. Keeping this is mind, please, try to use coordinate-free notations. Along this question, I will use ...
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2answers
443 views

Stokes theorem in Lorentzian manifolds

I've fallen accross the following curious property (in p.10 of these lectures): in order to be able to apply Stokes theorem in Lorentzian manifolds, we must take normals to the boundary of the volume ...
2
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1answer
93 views

Geometric derivation of quantum mechanics from Lagrangian mechanics

I have used classical Lagrangian mechanics for quite a while, and what I like about it is that everything can be derived from a very small number of geometric principles. There are just three things ...
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3answers
144 views

Why is the Ricci tensor defined as $R^\mu _{\nu \mu \sigma}$?

The Ricci tensor is defined as the contraction of the Riemann tensor in its upper and the second lower index. I was wondering why it is defined this way. What happens if the Ricci tensor is defined ...
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1answer
35 views

Why are the integrability conditions necessary and sufficient for the existence of a canonical transformation's generating function?

Consider a canonical transformation $(p,q) \rightarrow (P,Q)$ under a generating function $F$. The condition for form invariance of Hamiltonian equations of motion looks like : $$\sum_{s}P_s\dot{Q_s} ...
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23 views

Numerical simulation of strong field gravitational vacuum solutions colliding

I am interested in the current state of knowledge of strong field General Relativity learned from numerical investigations of gravitational wave packets colliding with each other or black holes. If ...
0
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1answer
54 views

Schwarzschild manifold

I am given the following metric $$ds^2 = \frac{dr^2}{1-2m/r} + r^2dS,$$ where $dS$ is the standard metric on the unit sphere $S^2$. I am told that this is isometric to $\mathbb{R}^3$ or (taking its ...
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2answers
76 views

Black hole singularity from collapsing light vs dust

Consider two black holes, one formed from a spherical cloud of electromagnetic radiation, and one formed from a non-interacting dust solution. The stress energy tensor is traceless for ...
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1answer
76 views

Meaning of $R=0$, $R_{ab}=0$. $R_{abcd}=0$

First let me state some definition The Einstein tensor is given by \begin{align} G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R \end{align} and note that \begin{align} G^{\mu}_{\phantom{\mu} ...
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33 views

Identity $ \epsilon_{abcd} R^{cd}_{\phantom{cd}mn} = \epsilon_{mncd} R^{cd}_{\phantom{cd}ab}$ in vacuum

starting from \begin{align} \epsilon_{\rho\lambda\xi \kappa} R^{\xi \kappa}_{\phantom{ab} \sigma\tau} + \epsilon_{\rho\sigma \xi \kappa} R^{\xi \kappa}_{\phantom{ab} \tau \lambda} + \epsilon_{\rho ...
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1answer
43 views

In the orthonormal tetrad method, what is the relation between basis one forms and commutation coefficients?

If $\omega_i$ are dual basis one forms corresponding to an orthonormal tetrad basis $e_i$, and given that the commutation coefficients $C_{ij}^k$ are defined by \begin{equation} [e_i,e_j]=C_{ij}^k ...
5
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1answer
115 views

Non-Euclidean mechanics; is it useful?

Special relativity has the following single-particle Lagrangian: $$S = \int_{t_0}^{t_f}\sqrt {\langle \mathrm d\vec{s},\mathrm d\vec{s}\rangle}.$$ Clearly it is based on Euclidean norms; it is in ...
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85 views

Is it known what the necessary and sufficient conditions are for the existence of a “3+1 split” (by means of a foliation) of a (Lorentzian) manifold?

When trying to do physics on a more general pseudo-Riemannian manifold we want to require that there is a foliation of this manifold into three-dimensional subspaces. By this I mean we would like to ...
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0answers
42 views

A problem with ADM mass in the derivation of 1st law of black hole thermodynamics

The definition of ADM mass is $$M=\frac{1}{16\pi}\lim_{r\rightarrow\infty}\int \left(\frac{\partial h_{\mu\nu}}{\partial x^\mu}-\frac{\partial h_{\mu\mu}}{\partial x^\nu} \right)N^\nu dA$$ according ...
2
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1answer
80 views

Components of dual vectors

(This is a close retelling of Wald, problem 2.4b. Not for homework; just curiosity and an increasingly alarming suspicion that I've never actually understood anything.) Let $Y_1 ... Y_n$ be a ...
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1answer
63 views

Divergence of inverse of metric tensor

I know that the Levi-civita connection preserves the metric tensor. Is the divergence of the inverse of metric tensor zero, too?! I'm not so familiar with the divergence of the second ranked tensor. ...
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1answer
43 views

Metric components transformation under change of coordinates

I have been studying Lie derivatives and some applications. While searching the web I found a refence with the following statement: For a general Riemannian manifold $M$, take a tangent vector field ...
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1answer
116 views

What does $L^2(S^1,\mu_H)$ mean?

It's a Hilbert space, $\mu_H$ stands for the Haar measure on $U(1)$, but what does $S^1$ mean? I found it in one of my quantum mechanics books which approaches from a very 'mathematical' way.
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0answers
63 views

Are all spacetimes locally conformally flat?

No, is the answer. However, I am confused. Let $M$ be a (2+1) Lorentzian manifold (for simplicity) . Then the line element is given by : $ds^{2}=g_{\mu\nu}dx^\mu dx^\nu=−N^2 dt^2 + γ^{ij} (dx^i + ...
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49 views

Does nature really follow the heat equation?

I think the heat equation says that the first derivative of temperature with respect to time in a stationary solid varies as the negative of the second derivative of temperature with respect to ...
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2answers
120 views

Chern-Simons theory

The Chern-Simons 3-form is given by $\omega_3={\rm Tr} \left[ A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right]$ where $A$ is a connection one-form in the adjoint representation of a non-Abelian gauge ...
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1answer
59 views

Transformation matrices for basis and coordinate transformation in non-orthonormal coordinates

The transformation matrices for covariant and contravariant vectors are different but in orthonormal coordinate system numerical values in matrices turn out to be same although in mathematical proof ...
3
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2answers
67 views

Computational advantages of various notations for electromagnetism [closed]

Most undergraduate electromagnetism classes and textbooks use vector notation to describe Maxwell's equations. However, there are other notations like differential geometry and geometric calculus ...
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1answer
145 views

Intuitive meaning of Globally Hyperbolic

I am been studying differential geometry and spacetime and I keep coming across the term globally hyperbolic. I am having a hard time coming up with an intuitive understanding of this idea. What is an ...
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2answers
79 views

Two ways of writing coordinate basis vectors confusion

In Schutz's A First Course in General Relativity (p122) he derives the polar coordinate basis vector$$\vec{e_{r}}=\frac{\partial x}{\partial r}\vec{e_{x}}+\frac{\partial y}{\partial r}\vec{e_{y}.}$$ ...
3
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1answer
204 views

How should Christoffel symbols be written (in LaTeX)? [closed]

I'm writing a summary of a lecture on relativity, and we've recently introduced the Christoffel symbols. It seems that the upstairs indices are the "leftmost" and the downstairs indices are somewhat ...
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2answers
100 views

A manifold question: Why smooth functions and what is a Jacobian?

My question is what does a Jacobian have to do with the change of coordinates (coordinate transformation). Why do we care about this notion to start with? Also, why should it be non-singular?