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

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11
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5answers
569 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 ...
0
votes
0answers
26 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 ...
4
votes
1answer
256 views

How can I mathematically describe the parallel transport in the Roman soldiers example?

I've been trying to understand parallel transport. Many of the descriptions present a mathematical version: $\nabla_V X = 0$. And/or they present an example involving soldiers (usually Roman) ...
7
votes
1answer
92 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)}...
3
votes
1answer
133 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., $S^2, T^2$, ...). ...
2
votes
1answer
131 views

Interesting Hamiltonian System [duplicate]

The definition of a Hamiltonian system I am working with is a triple $(X,\omega, H)$ where $(X,\omega)$ is a symplectic manifold and $H\in C^\infty(X)$ is the Hamiltonian function. I am wondering if ...
3
votes
1answer
482 views

Stokes' theorem in complex coordinates (CFT)

I am studying CFT, where I encounter Stokes' theorem in complex coordinates: $$ \int_R (\partial_zv^z + \partial_{\bar{z}}v^{\bar{z}})dzd\bar{z} = i \int_{\partial R}(v^{z}d\bar{z} - v^{\bar{z}}dz). $$...
4
votes
3answers
80 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) $$ I\colon x^\mu \mapsto \frac{x^\mu}{x^2} $$ ...
2
votes
0answers
65 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 ...
1
vote
0answers
35 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 ...
1
vote
2answers
75 views

General relativity applications other than gravity

Do the Einstein field equations successfully predict/describe physical processes other than gravitational ones?
6
votes
1answer
324 views

Difference Between Algebra of Infinitesimal Conformal Transformations & Conformal Algebra

in Blumenhagen Book on conformal field theory, It is mentioned that the algebra of infinitesimal conformal transformation is different from the conformal algebra and on page 11, conformal algebra is ...
2
votes
0answers
75 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!
4
votes
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\...
3
votes
5answers
560 views

Where does the idea gravity=curvature of spacetime really come from?

I have been searching for quite a while but mostly found the answer: Einstein's genius. Quite unsatisfactory. I know and understand that the idea gravity=curvature of spacetime works. Furthermore I ...
8
votes
1answer
231 views

Homotopy proof of the lack of foliation of the Gödel metric

A common proof of the lack of foliation of the Gödel universe, apparently mostly copy pasted from Hawking and Ellis, goes thusly : A closed timelike curve must cross a spacelike hypersurface ...
-1
votes
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?
3
votes
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" (...
0
votes
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^...
3
votes
1answer
107 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. ...
4
votes
0answers
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}...
7
votes
2answers
312 views

Non-trivial scalar quantity

Is there any scalar quantity made of only the Christoffel symbols, determinant of a metric and tensors, not derivatives? In other words, can we construct a scalar quantity which cannot be written in ...
1
vote
0answers
50 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 ...
3
votes
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 ...
-1
votes
0answers
52 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 ...
0
votes
0answers
51 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
votes
2answers
500 views

Locally flat coordinate and Locally inertial frame

I am having some doubts on myself regarding the above concepts in General Relativity. First, I want to point out how I understand them so far. A male observer follows a timelike worldline ($\gamma$) ...
1
vote
1answer
345 views

What are the mathematical models for force, acceleration and velocity?

In mechanics, the space can be described as a Riemann manifold. Forces, then, can be defined as vector fields of this manifold. Accelerations are linear functions of forces, so they are covector ...
3
votes
1answer
192 views

Local translations in curved spacetime

A global Poincare transformation on a scalar field induces $$\delta(a, \lambda)\phi(x) = [a^{\mu}+\lambda^{\mu\nu}x_{\nu}]\partial_{\mu}\phi(x). \tag{11.46}$$ In curved spacetime we replace $a^{\mu} ...
3
votes
2answers
259 views

Covariant derivative of a covariant tensor wrt superscript

Is it true that when you take the covariant derivative of a covariant tensor, do you always have to do with a subscript? What if you do it wrt a superscript?Does the first term (with the partial ...
1
vote
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 ...
-1
votes
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
2
votes
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 ...
2
votes
2answers
265 views

Timelike Boundary

I was reading in a paper (see 1st paragraph of introduction section in http://arxiv.org/pdf/1510.00709.pdf) that in AdS space, waves can reach the boundary in finite time and, since said boundary is ...
0
votes
1answer
43 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 = \...
0
votes
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?
2
votes
3answers
3k views

Ricci scalar for a diagonal metric tensor

I was wondering if there is a general formula for calculating Ricci scalar for any diagonal $n\times n$ metric tensor?
1
vote
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}=-\...
4
votes
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 $\...
0
votes
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^...
2
votes
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
votes
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?
0
votes
1answer
137 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{...
2
votes
1answer
189 views

If a point r lies in the boundary of the chronological future of another point p, why does the chronological future of r belong to that of p?

I am studying the global causality of the spacetime. Here, I come across a problem. Suppose a point $r\in \partial I^+(p)$. $I^+(p)$ is the chronological future of a different point $p$ in spacetime....
1
vote
1answer
164 views

Generalized spin connection and dreibein in higher spin gravity

I am studying 3D higher spin gravity and I would like to know the mathematical and physical meaning of generalized spin connection and generalized dreibein that appear in this theory. It is well known ...
2
votes
1answer
111 views

Parallel Transported Orthonormal Basis

The following argument results in a conclusion that I find strange, and makes me suspect there is something wrong with the reasoning. First, consider a timelike geodesic $\gamma$ with normalized ...
2
votes
2answers
240 views

Different signatures

I was working out the christoffel symbols, once where the metric that I am using has (+---) signature and another time where it has (-+++) signature because two books had different signatures and I ...