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

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Geometric (topological?) structure and consequences of the canonical hamiltonian method

(1) Even though (I think) I understand the concept of a tangent bundle, I have trouble assimilating the idea of the configuration space being one and in relation to what that is the case. How can I ...
16
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6answers
2k views

Does curved spacetime change the volume of the space?

Mass (which can here be considered equivalent to energy) curves spacetime, so a body with mass makes the spacetime around it curved. But we live in 3 spatial dimensions, so this curving could only be ...
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0answers
29 views

What is the minimum math level required to comprehend general relativity? [duplicate]

I am currently working on a B.S. In chemistry and only need to go up to Calculus II, though I like to dabble in physics. What other math classes could I take to gain greater understanding of higher ...
1
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0answers
44 views

Rotating fermion and spin structure on manifold

We know that doing a 2$\pi$ rotation would give a minus sign to wavefunctions of electrons. Since electrons are spin $1/2$ objects. How is this related to the spin structure on the manifold in which ...
2
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0answers
82 views

(1+1)-General Relativity

Goodevening everyone, my question is: What is the interest of studying the (1+1)dimension General Relativity? Can you explain please? Thank's in advance!
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2answers
78 views

General relativity applications other than gravity

Do the Einstein field equations successfully predict/describe physical processes other than gravitational ones?
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1answer
51 views

How to define the distance between two points in a conformal transformed space?

Consider a particular conformal transformation $x^\mu\rightarrow x'^\mu$, and the metric of a flat space transforms in the following way, $$\eta_{\mu\nu}\rightarrow g'_{\mu\nu}=\Lambda^2(x)\eta_{\mu\...
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0answers
41 views

Lagrangian density

I really wonder : Why do we take Lagrangian density as zero for the Stokes theorem in Minkowski-space at infinity? Is there a proof of this situation?
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75 views
+50

Objective time derivative that is no Lie derivative

Summary Led by an interest into the concept of "Material Objectivity", I am asking myself: Are there objective time rates that are not Lie derivatives? The long read I am trying to understand the ...
3
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1answer
81 views

Exotic differentiable structures in physics

When reading a bit on exotic spheres and exotic $\mathbb{R}^4$'s, I came across some papers of Carl H. Brans and Torsten Asselmeyer-Maluga: "Exotic differentiable structures and general relativity" (...
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0answers
22 views

Relation between differentiation of one-form basis and Christoffel Symbols

If I want to covariantly differentiate a one form then I can write: $\nabla_\beta \tilde p = \dfrac{\partial p_\alpha}{\partial x^\beta} \tilde \omega^\alpha + p_\alpha \dfrac{\partial \tilde \omega^...
4
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3answers
81 views

How can I show that inversion is continuously connected to a reflection?

From Ex 3.1 in the TASI lectures on the conformal bootstrap: http://arxiv.org/abs/1602.07982 the problem is the inversion map (with Euclidean signature) $$ I\colon x^\mu \mapsto \frac{x^\mu}{x^2} $$ ...
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52 views

Jet bundles for physicists

In order to make Classical Field Theory rigorous we need the idea of jet bundles. I've seem some books on the subject, but most of them are aimed at mathematicians and tend to go quite deep in the ...
4
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45 views

Is the Weitzenböck connection the only connection with Torsion but without Curvature?

In teleparallel gravity, the (local) connection coefficients of the Weitzenböck connection are given by $$ \Pi^{\beta}{}_{\mu\nu}= h^{\beta}_{i} \partial_{\nu}h^{i}_{\mu} - \Gamma^{\beta}{}_{\mu\nu}...
3
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1answer
110 views

Derivation of Christoffel Symbols

So I am reading a book on relativity & differential geometry and in the text, they gave the Christoffel symbols in terms of the metric and its derivatives, but I wanted to derive it myself. ...
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53 views

contravariant derivative

Initially I understand the covariant derivative, but I really wonder contravariant derivative also. I saw some definitions which use metric tensor and covariant operator to provide contravariant ...
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0answers
52 views

Writing the Yang-Mills topological charge using differential forms

I have a very pedestrian knowledge of differential forms and I am having some trouble in a derivation. The topological charge $Q$ in Yang-Mills theories is supposed to be $$ Q=\int{}q(x)d^4x $$ where $...
3
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2answers
110 views

Proving constant curvature

I'm currently on section 5.1 in Wald's book. He is trying to prove that the cosmological principle implies that space has constant curvature. Given a spacelike hypersurface $\Sigma_t$ for some fixed ...
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1answer
62 views

Covariant derivative [closed]

Hi, Could you explain to me why the subtraction of vector at some point and parallel transported vector is covariant derivative vector. How is it possible
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1answer
44 views

Symmetry group of FLRW metric

$$ g = dt^2 - a^2(t) (dx^2+dy^2+dz^2) = dt^2-a^2(t)(dr^2+r^2d\Omega^2)$$ So this is my metric. What is the symmetry group of it? I think that my Killing vectors are 3 translation vectors: $$K_i = \...
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0answers
40 views

Multidimensional Area and Volume

In 3D the volume is $xyz$, the product of three coordinates. But in $N$ dimension ,how to define area and volume?
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1answer
95 views

Gauge transformation of connection of $\mathcal{O}(n)$

The holomorphic line bundle $\mathcal{O}_X(1)$ over a toric manifold $X$, admits a Hermitian connection, $A^{(1)}$, whose $U(1)$ gauge transformation in a local patch of the base space is $$ A^{(1)}...
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0answers
22 views

Extrinsic curvature for cylinder [migrated]

Suppose we have the following metric describing a cylinder: $$ds^2=ud\rho^2+\rho^2d\phi^2$$ where $u$ is a function of $\rho$. We know the definition of the extrinsic curvature that is, $$K_{ij}=-\...
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1answer
94 views

What is the additional gravitational term from general relativity given by?

Carroll gives the potential energy in general relativity by $$ V(r)=\frac{1}{2}\epsilon-\epsilon\frac{G\,M}{r}+\frac{L^{2}}{2r^{2}}-\frac{G M L^{2}}{r^{3}} $$ My first question is does $V(r)$ have ...
2
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1answer
70 views

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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1answer
51 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
81 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 $\...
2
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0answers
47 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 ...
3
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2answers
115 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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0answers
35 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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0answers
33 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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1answer
54 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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50 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
37 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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3answers
91 views

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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1answer
57 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
109 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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1answer
32 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
71 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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0answers
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 \, ...
3
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0answers
86 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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0answers
95 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 ...
5
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1answer
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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0answers
35 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
69 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 ...
4
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1answer
35 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 ...
2
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2answers
81 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(\...
2
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0answers
56 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 ...
3
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1answer
59 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 (...
4
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0answers
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}\,...