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

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Problem 1 Chapter 11 Wald

I'm currently trying to solve problem 1, Chapter 11 of Wald, General Relativity. The request is to derive from the condition $$ \tilde\nabla_a \tilde\nabla_b \Omega=0\text{ at }\mathscr I^+, $$ where ...
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52 views

A Calculation in Padmanabhan's Book

I have seen this in Padmanabhan's book. How can I verify this: $$d\Sigma_{mn}=\frac{1}{2!}\epsilon_{mnab}\frac{\partial(x^a,x^b)}{\partial(\theta,\varphi)}d\theta d\varphi=\epsilon_{mn\theta\varphi}r^...
4
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1answer
83 views

Killing tensor in Minkowski space

I'm trying to solve the Killing tensor equation $\nabla_{(a}K_{bc)} = 0$ in Minkowski space. I'd like to generalise the method we use to find Killing tensors in Minkowski space. We can take $\...
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0answers
51 views

Reference for orbifolds in string- and M-theory

A number of orbifold constructions have been studied heavily in string- and M-theory over the years, establishing various dualities between different theories. Can someone point me to a slightly more ...
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2answers
117 views

Is a spacetime of constant positive curvature just a 4-hypersphere?

In discussions of basic cosmological models, I don't see "spacetime of constant positive curvature" described more simply as a "4-hypersphere". What am I missing?
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36 views

Do time and spatial derivative under a 3+1 decomposition commute?

After a certain 3+1 decomposition of the space-time, the derivative of time part and spatial part separate. Let's denote them by $d_t$ and $\partial_\mu$. Here the spatial derivative is covariant but ...
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35 views

Variation with respect to the metric and other tensors

When varying an action with respect to tensors and the metric, I'm afraid I get confused as how to one organizes the Lagrangian and then performs the variation. Take for example, the following example ...
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55 views

Poincare Group (Wald, Chapter 4 Page 59)

In Wald's text on general relativity, he mentions that in special relativity, many different global inertial coordinate systems are possible and can be put into one-to-one correspondence with elements ...
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54 views

Gauss-Weingarten equation

In E Poisson "A relativist tool kit" p.75 it says that the Gauss-Weingarten relation is: $$e^{\alpha}_{a;\beta}e^{\beta}_{b}=\Gamma^{c}_{ab}e^{\alpha}_{c}-\epsilon K_{ab}n^{\alpha}$$ We have the ...
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1answer
41 views

Second covariant derivative, computation problem

I am having a question on the wikipedia article http://en.wikipedia.org/wiki/Second_covariant_derivative Using the notation therein I don't get why $(\nabla_{u}\nabla_{v}w )^a=u^c\nabla_{c}v^b\nabla_{...
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Covariant Derivative of Kronecker Delta

I am reading Carroll's book on GR right now, and I ran into a little trouble in his chapter 3 on curvature. He is establishing the properties of the covariant derivative, and claims that the fact that ...
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81 views

Global Hyperbolicity in spacetime Manifold [closed]

If space time is timelike or null geodesically incomplete but cannot be embedded in a larger spacetime then we say that it has singularity. What does incompleteness means here?
3
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1answer
110 views

Usage of tensors in physics [closed]

As I understand it, tensors are multi-linear maps that map vectors (and dual vectors) to real (or complex) numbers, but I'm hoping to gain some intuition as to why they are useful in physics. Is it ...
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33 views

Commuting of the covariant derivative: Menzel's Mathematical Physics

Menzel defines covariant differentiation as equivalent to partial differentiation with respect to the general coordinates. “To indicate the covariant nature of the differential operator, set $$\frac{\...
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1answer
73 views

Coupling a spinor field to a preexisting scalar field?

So I'm not a physicist, but I'm thinking about a mathematical problem where I think physical insight might be useful. We're working on a Riemannian manifold $(M,g)$ (positive definite metric) with a ...
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30 views

Are general coordinate transformations and diffeomorphisms the same? [duplicate]

Infinitesimal diffeomorphisms $x{}^\mu \rightarrow x{}^\mu + \xi{}^\mu$ (with $\xi{}^\mu \ll 1$) change geometric objects by means of the Lie derivative, that is, $X \rightarrow X + \mathcal{L}_\xi \, ...
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100 views

Scalar Curvature of a Conformally Flat Metric

Suppose that you have a metric $g_{\mu\nu}=\phi^2\eta_{\mu\nu}$ for some function $\phi$. There is a standard formula for what the scalar curvature $R$ looks like in terms of $\phi$, which is given by ...
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100 views

Question on $E_8$ and twistor space [closed]

The Kahler $4$ form constructed from two-forms $\{\alpha, \beta\} \in H^2(M,\mathbb Z)$, and $M$ a $4$-manifold, is induced by $\alpha\wedge\beta$ with the map $H^2(M, \mathbb Z)\otimes H^2(M, \mathbb ...
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70 views

Signature of $f: \Lambda^2(\mathbb{R}^4) \times \Lambda^2(\mathbb{R}^4) \to \mathbb{R}$, $f(\omega, \omega') = \omega \wedge \omega'$ [closed]

Define$$f: \Lambda^2(\mathbb{R}^4) \times \Lambda^2(\mathbb{R}^4) \to \Lambda^4(\mathbb{R}^4) \cong \mathbb{R}, \quad f(\omega, \omega') = \omega \wedge \omega'.$$ What is the signature of $f$? ...
3
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36 views

Induced metric is a scalar for transformation from $x\to x'$? (Poisson E.A p.62)

I have a (simple) question about the induced metric $h_{ab}$. In Poisson E.A. (a relativist toolkit) it says in p. 62 that the induced metric $$h_{ab}=g_{{\alpha}{\beta}} \frac{\partial x^{\alpha}}{\...
3
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1answer
73 views

How do you actually use the geodesic equation?

The geodesic equation used in general relativity is the following: $$ {d^2 x^\mu \over ds^2} =- \Gamma^\mu {}_{\alpha \beta}{d x^\alpha \over ds}{d x^\beta \over ds}. $$ It states that the ...
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1answer
37 views

Lie Derivative of Kahler 2-form

Suppose there is a Killing vector $k$ on a Kahler manifold $M$. By definition, $k$ generates isometries of the metric. That is, $L_kg=0$, where $L$ is the Lie derivative. At the same time, there is a ...
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2answers
82 views

Lie derivative for a covariant derivative of vector

I would like to calculate the $\mathcal{L}_\xi(\nabla_a K^b)$ for the case where $\mathcal{L}_\xi(K^b)=0$ The only Idea that I have is that $$\mathcal{L}_\xi(\nabla_a K^b)=\mathcal{L}_\xi(\...
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57 views

Determinants in path integrals in gauge theories and geometry

I know that in the formalism of path integral it is easy to show how determinants, corresponding to gauge fixing condition and FP ghosts, appear. But there is strict explanation of these determinants ...
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64 views

Torsion in kerr black holes

In General Relativity, we generally assume that the derivative operator is torsion-free, i.e., second covariant derivatives commute on functions. However, in Kerr black holes, spacetime is dragged (...
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39 views

Schwarzschild metric, speed of ball as measured by observer who catches the ball, just before ball is caught? [closed]

Inspired by this question here. The Schwarzschild metric, describing the exterior gravitational field of a planet of mass $M$ and radius $R$, is given by$$ds^2 = -(1 - 2M/r)\,dt^2 + (1 - 2M/r)^{-1}\,...
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17 views

Lapse Function and Shift Vector in Minkowski and de Sitter

I'd like to find the lapse function and shift vector in 1+1 Minkowski as well as 1+1 de Sitter (flat foliation) for a region foliated this way: The $y$-axis represents time while the x-axis ...
5
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1answer
263 views

Killing tensor and Riemann tensor identity

I know that if we have a Killing vector then it's straightforward to show the identity: $$\nabla_a \nabla_b K_c = R_{cba}^k K_d$$ I'm now trying to show the following identity for a $(0,2)$ Killing ...
4
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1answer
119 views

Covariant derivative of a covariant derivative

I'm trying to find the covariant derivative of a covariant derivative, i.e. $\nabla_a (\nabla_b V_c)$. This is something I've taken for granted a lot in calculations, namely I though that by the ...
5
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1answer
58 views

Schwarzschild metric, acceleration of ball before it's dropped [duplicate]

The Schwarzschild metric, describing the exterior gravitational field of a planet of mass $M$ and radius $R$, is given by$$ds^2 = -(1 - 2M/r)\,dt^2 + (1 - 2M/r)^{-1}\,dr^2 + r^2(d\theta^2 + \sin^2\...
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1answer
56 views

Why does the 'Jacobian of at least one combination of $n$ functions shall be different from zero'?

I've started reading The Variational Principles of Mechanics by Cornelius Lanczos; here is the concerned excerpt from p. 11: The generalized coordinates $q_1,q_2,\ldots, q_n$ may or may not have a ...
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1answer
52 views

Wald's Equation 3.3.6

I have an issue with Eq. 3.3.6 of Wald's General Relativity. There he would like to prove that for Gaussian normal coordinates, the geodesic tangent field remains orthogonal to all coordinate basis ...
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57 views

Flux with or without sources

I want to check whether my understanding about flux is correct. So let me consider a $d$-dimensional spacetime manifold $M$ and a $(p+2)$-form flux (or field strength) $F$. Then there are two ...
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How can you tell if spherical-like coordinates are locally flat across the origin?

In general relativity, with spherical-like coordinates in a radial gauge, I have a metric that looks like: $$-g_{tt}\mathrm{d}t^2 + g_{rr}\mathrm{d}r^2 + r^2(\mathrm{d}\theta^2 + \sin^2\theta\ \...
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1answer
44 views

Problem books for concept building in applications of Riemannian and other geometries to mechanics

As a student of physics I have learned solving Euler equations for rigid bodies by solving examples and exercises in self-contained books rather than understanding the proofs of Euler equations (I ...
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Why is the Einstein Static Universe represented as an infinite cylinder when it seems like only half a cylinder?

The Einstein static universe metric is $$ds^2=-dt^2 + d\chi^2 + \sin(\chi)^2d\Omega^2$$ where $-\infty<t<\infty$ , $0<\chi<\pi$ and $d\Omega^2$ is the metric on a $S^2$. It describes the ...
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What is basic tensor algebra in teleparallel equivalent of general relativity?

Teleparallel gravity represents a viable alternative to general relativity where gravitation comes from torsion rather that curvature. The theory is based on a new modified connection, and the ...
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173 views

A question on the Chern number and the winding number?

Let $\mid \psi(x,y) \rangle$ be a normalized wavefunction living in a $d$-dimensional Hilbert space and depend on two real parameters $(x,y)$ that belong to a closed surface (e.g., $S^2, T^2$, ...). ...
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31 views

AdS boundary global vs Poincare'

Is the global boundary of AdS the same of the boundary written in Poincare' coordinates?
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28 views

Volume form of the AdS_{4} Space

Regarding the unit radius $AdS_{4}$ space, the metric in global coordinates, is given by: $$ds^{2}_{AdS_{4}}=\frac{1}{\cos^{2}{\rho}}[dt^{2}-d\rho^{2}-\sin^{2}\rho d\Omega_{2}^{2}]$$ where $$d\...
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1answer
54 views

Anomalies and determinant bundle curvature

I heard that anomalies and curvature of determinant bundle are related. Namely, curvature of determinant bundle is related to Chern-Simons form (which are involved in description of gauge anomalies). ...
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2answers
87 views

Straight line null geodesics in Minkowski, De Sitter and Schwarzschild

I'm trying to understand which part of the following metric determines whether photons travel on a "straight" line (thinking of $(t,r,\theta,\phi)$ as a flat background), the metric I'm considering is:...
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45 views

Tensors Contracting Indices

I'm pretty confused regarding the components of a tensor once you take its trace (or contraction). I'll use $B\in T_2^1(V)$ to be specific. Let $V$ be an $n$-dimensional vector space with basis $\{E_i\...
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1answer
70 views

Particle on $S^1$ and $U(1)$-principal bundle

I have a question arisen from a simple QM problem: let consider a boson on $S^1$ minimally coupled with a constant gauge field $A$. Taking the stationary Schrödinger (S) or Klein-Gordon (KG) equation ...
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1answer
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Relationship between fracture of solid plates and its curvature

I do some school projects about fracture of glass, my assumption is this phenomenon related to the curvature of these plates, specifically, fracture starts from the point which is having “maximum” ...
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129 views

Two Robertson-Walker observers, at what time will a light signal be received?

Here is a question I have that is inspired by this question here. The spacetime metric of a radiation-filled, spatially flat ($k = 0$) Robertson-Walker universe is given by$$ds^2 = - dT^2 + T[dx^2 + ...
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1answer
158 views

Torsion-free, symmetric connection and non-coordinate basis

The torsion tensor is defined as (Hawking p.34) \begin{equation} \mathbf{T}(\mathbf{X},\mathbf{Y}) = \nabla_{\mathbf{X}}\mathbf{Y} - \nabla_{\mathbf{Y}}\mathbf{X} - [\mathbf{X},\mathbf{Y}]. \end{...
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1answer
54 views

About periodicity of coordinates given a metric

If I am given a metric how do I decide which coordinate is periodic? Eg. can I look at metric in plane polar coordinates and tell that θ direction is periodic. Also How do I calculate the period of ...
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1answer
53 views

Difference between Fermi and Riemann normal coordinates

What is the difference between Fermi normal coordinates and Riemann normal coordinates? Which one of them is related to the vanishing of the Christoffel symbols?
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1answer
66 views

From states in string theory to differential forms

This question is probably really easy and isn't usually discussed in textbooks. So suppose when we construct states in string theory we obtain something like $\left|- \right\rangle \otimes \left|+ \...