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

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Geometric interpretation of zero metric

What is the geometric interpretation of a metric vanishing at a point on a Riemannian manifold? What does it look like locally in the case of a surface?
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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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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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A problem with ADM mass in the derivation of 1st law of black hole thermodynamics

The definition of ADM mass is $M=\frac{1}{16\pi}\lim_{r\rightarrow\infty}\int(\frac{\partial h_{\mu\nu}}{\partial x^\mu}-\frac{\partial h_{\mu\mu}}{\partial x^\nu})N^\nu dA$ according to Wald. ...
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1answer
75 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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46 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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34 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
111 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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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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43 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
92 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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43 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 ...
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2answers
58 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
79 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
66 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$$\vec{e_{r}}=\frac{\partial x}{\partial r}\vec{e_{x}}+\frac{\partial y}{\partial r}\vec{e_{y}.}$$ ...
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1answer
76 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
96 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?
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What does coordinate invariance mean?

I would like to really understand what the mathematical as well as Physical meaning of coordinate invariance is. I have pretended to know what this means, but upon thinking a little harder today, I am ...
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2answers
75 views

What is the metric tensor for?

I am wondering how to use the metric tensor, in practice? I read the book and done the exercises in A student's guide to vectors and tensors by Dan Fleisch. The concept of a tensor and their ...
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1answer
97 views

Einstein tensor of a gravitational source

In section 4.4 of gravitational radiation chapter in Wald's general relativity, eq.4.4.49 shows the far-field generated by a variable mass quadrupole: $$ \gamma_{\mu \nu}(t,r)=\frac{2}{3R} \frac{d^2 ...
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28 views

Weyl Transformations and Group actions [migrated]

I have the following question. Let $(M,g_{ab})$ be a Riemannian manifold $M$ with metric $g$, and with an action of a Lie group $G$. Moreover, the Riemannian metric $g_{ab}$ is taken to be invariant ...
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1answer
93 views

Derivation of Schwarzschild metric using the full machinery of differential geometry [closed]

How would one derive the Schwarzschild metric using the full machinery of differential geometry, using the component approach as little as possible? Something along these lines: Begin with a manifold ...
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1answer
27 views

Inverse gauge transformation in general relativity [closed]

Can someone explain to me how (8.21) follows from (8.20). The Picture comes from A first course in general relativity (Schutz). Thanks and regards, Jens Wagemaker
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1answer
101 views

Why pseudo-Riemannian metric cannot define a topology?

It is not clear for me why a positive definite metric is necessary to define a topology as noted in some textbooks like the one by Carroll. Does this imply that in cosmology, say through FLRW metric, ...
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Symmetry of the Polyakov action?

Let us look at the Polyakov action for a string moving in a spacetime with metric $g_{\mu \nu}(X)$:$$S_P = -{1\over{4\pi \alpha'}} \int d^2 \sigma \sqrt{-\gamma} \gamma^{ab} \partial_a X^\mu ...
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1answer
61 views

Is an event formally a 4-vector? [duplicate]

An event is a 4D point in spacetime. At every point in spacetime there is a tangent space. 4-vectors live in the tangent space. One can contract two events using a metric tensor. Is there a process ...
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1answer
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Killing vector and one-form [closed]

p. 21 in this paper (http://arxiv.org/abs/0704.0247) $V$ is Killing vector, where $V^2 = −4b\bar{b}$, which means it is timelike Killing vector. The authors say: From $V^2 = −4|b|^2$ and $V = ...
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0answers
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Squashed spheres in general dimension

The point of this question is to help me find references regarding squashed spheres in general dimension. I am interested in the general theory of squashing for arbitrary dimension. All of the ...
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1answer
64 views

Variation of a tensor

Let a change of coordinates be given by $x^{\mu}\to x^{\mu '}=x^{\mu}+\varepsilon \xi^{\mu}(x)$ with epsilon a small quantity. Given a tensor $T$ we define $\delta T:=T'(x)-T(x)$. I guess this means ...
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2answers
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Killing field in Minkowski space-time

If we look at the killing equation for a vector field $X$ in $\mathbb{R}^{(p,q)}$ (or on an open subset thereof) in coordinates with constant diagonal pseudo-metric we get: ...
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3answers
131 views

What is the physical meaning of the Levi-Civita connection?

I'm taking a course in General Relativity and I have studied the fundamental theorem of Riemannian geometry: Let $M$ be a manifold with metric $g$. Then exists an unique torsion-free connection ...
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1answer
66 views

What is the relationship between a brane, a manifold, and a space?

I've read many ways to define manifold; one way is to define it as a type of mathematical space (a type of topological space to be exact). All of the definitions that I've seen for brane, on the ...
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Quantization of KK Theory

I know that electromagnetism is force via curvature in a U(1)-bundle. I am now trying to literally visualize this, and write down equations that make this manifest. KK (Kaluza-Klein) theory is the ...
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3answers
118 views

Is there any physical interpretation for $\nabla\cdot(\nabla \times F)=0$?

It is well known that the divergence of the curl is always 0. Mathematically I understand why this happens ($d^2=0$ where $d$ is the exterior derivative) but today I was wondering what is the physical ...
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0answers
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Proper time and asymptotic flatness

I'm trying to understand the concept of asymptotic flatness in general relativity, and came up with the following question: If the proper time $\tau$ is infinite for a timelike geodesic, does it mean ...
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Is there any reason (other than convenience) to assume the universe is paracompact?

In this discussion on MathOverflow, it is mentioned that the universe, being a Riemannian manifold, must be paracompact. But is there any reason to assume the universe is globally 'small enough'? In ...
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Variation of Bazanski Lagrangian

The Bazanski Lagrangian is defined as $$ L=g_{\alpha \beta }U^{\alpha }\frac{D\psi ^{\beta }}{Ds} $$ and $$ U^{\alpha }=\frac{\mathrm{d} x^{\alpha }}{\mathrm{d} s} $$ $x^{\alpha }$ is the ...
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1answer
323 views

What, to a physicist, are instantons and the Donaldson invariants?

I study gauge theory from a mathematical perspective. To me, one of the most fundamental ideas is the notion of an instanton on a 4-manifold. To be precise, I have a Riemannian 4-manifold and a ...
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1answer
55 views

Possible inconsistency of mixed index tensor notation

I am posting this here, because in my experience, this sort of thing exists in physics-related works only. Given a local frame $\{e_{(i)}\}$ on some $n$-dimensional manifold $M$, and given a local ...
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1answer
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Local Coordinate Expressions for Lie Derivatives

I'm currently working through the math chapters of Norbert Straumann's book on General Relativity. I have trouble understanding the coordinate expression of the Lie derivative of a basis vector. The ...
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1answer
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Equation of a torus

In the recent paper http://arxiv.org/abs/1509.03612, page 37. They say that a torus can be described by the equation $$y^2=x(z-x)(1-x)$$ where $x$ is a coordinate on the base $\mathbb{P}_1$. Could ...
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4answers
136 views

Which tensor describes curvature in 4D spacetime?

I heard these two statements which don't work together (in my mind): In 4D spacetime the curvature is encoded within the Riemann tensor. He holds all the information about curvature in spacetime. ...
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1answer
138 views

Null geodesic equations

If one is constrained to the $xt$ plane, one can define the intersection with that plane of the null hypersurfaces originating at some point $P$ as $$ g_{tt} \frac{d P^t}{d \lambda}\frac{d P^t}{d ...
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1answer
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Variations of actions of (lie algebra valued) differential forms

I have always found it a bit difficult to understand the variation of an action written in differential form language. For example, take the action $$\int tr A\wedge A\wedge A$$ where $A=A_\mu ...
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1answer
43 views

Notation: tetrad indices

I am trying to understand the meaning of upper and lower indices as used in the Newman-Penrose formalism. The tetrad is $\lbrace l^{a},n^{a},m^{a},\overline{m}^{a}\rbrace$, where the upper index ...
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Magnetic monopole using differential forms

I'm trying to understand the different variations of the Maxwell's equations using differential forms. The Maxwell's equations are $$dF=0\\ *d*F=J$$ where $F$ is the electromagnetic tensor ($F=dA$) ...
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Geodesic equation (free particle)

How to find a coordinate system whose geodesic equation does not have the "Christoffel symbol" term? (i.e. free particle - generalized Newton's second law.)
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How the Poisson bracket transform when we change coordinates?

I'm studying the book Geometric Mechanics by Darryl D. Holm and there's one exercise in the book I'm not quite getting what has to be done. The same discussion the author makes in the book is made on ...
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1answer
72 views

Are all maximally symmetric spacetimes constant curvature spacetimes?

A $d$ dimensional maximally symmetric spacetime is a spacetime with the maximum allowed number of Killing vectors. This number is $\frac{d(d+1)}{2}$. Constant curvature spacetimes are spacetimes ...
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Global angular forms

In the study of anomaly inflow due to an M5 brane (see for instance the paper by Freed, Harvey, Minasian and Moore), one regularizes the $\delta$ function near a source as $$\delta^{(5)} \rightarrow ...