# Tagged Questions

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

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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 $... 2answers 117 views ### Killing field in Minkowski space-time If we look at the killing equation for a vector field$X$in$\mathbb{R}^{(p,q)}$(or on an open subset thereof) in coordinates with constant diagonal pseudo-metric we get: $$X_{\mu,\nu}+X_{\nu,\mu}=... 3answers 201 views ### What is the physical meaning of the Levi-Civita connection? I'm taking a course in General Relativity and I have studied the fundamental theorem of Riemannian geometry: Let M be a manifold with metric g. Then exists an unique torsion-free connection \... 1answer 101 views ### What is the relationship between a brane, a manifold, and a space? I've read many ways to define manifold; one way is to define it as a type of mathematical space (a type of topological space to be exact). All of the definitions that I've seen for brane, on the ... 0answers 41 views ### Quantization of KK Theory I know that electromagnetism is force via curvature in a U(1)-bundle. I am now trying to literally visualize this, and write down equations that make this manifest. KK (Kaluza-Klein) theory is the ... 3answers 141 views ### Is there any physical interpretation for \nabla\cdot(\nabla \times F)=0? It is well known that the divergence of the curl is always 0. Mathematically I understand why this happens (d^2=0 where d is the exterior derivative) but today I was wondering what is the physical ... 0answers 49 views ### Proper time and asymptotic flatness I'm trying to understand the concept of asymptotic flatness in general relativity, and came up with the following question: If the proper time \tau is infinite for a timelike geodesic, does it mean ... 0answers 28 views ### Is there any reason (other than convenience) to assume the universe is paracompact? In this discussion on MathOverflow, it is mentioned that the universe, being a Riemannian manifold, must be paracompact. But is there any reason to assume the universe is globally 'small enough'? In ... 0answers 45 views ### Variation of Bazanski Lagrangian The Bazanski Lagrangian is defined as$$ L=g_{\alpha \beta }U^{\alpha }\frac{D\psi ^{\beta }}{Ds} $$and$$ U^{\alpha }=\frac{\mathrm{d} x^{\alpha }}{\mathrm{d} s} $$x^{\alpha } is the ... 1answer 414 views ### What, to a physicist, are instantons and the Donaldson invariants? I study gauge theory from a mathematical perspective. To me, one of the most fundamental ideas is the notion of an instanton on a 4-manifold. To be precise, I have a Riemannian 4-manifold and a ... 1answer 67 views ### Possible inconsistency of mixed index tensor notation I am posting this here, because in my experience, this sort of thing exists in physics-related works only. Given a local frame \{e_{(i)}\} on some n-dimensional manifold M, and given a local ... 1answer 50 views ### Local Coordinate Expressions for Lie Derivatives I'm currently working through the math chapters of Norbert Straumann's book on General Relativity. I have trouble understanding the coordinate expression of the Lie derivative of a basis vector. The ... 1answer 1k views ### Equation of a torus In the recent paper http://arxiv.org/abs/1509.03612, page 37. They say that a torus can be described by the equation$$y^2=x(z-x)(1-x)$$where x is a coordinate on the base \mathbb{P}_1. Could ... 4answers 183 views ### Which tensor describes curvature in 4D spacetime? I heard these two statements which don't work together (in my mind): In 4D spacetime the curvature is encoded within the Riemann tensor. He holds all the information about curvature in spacetime. ... 1answer 187 views ### Null geodesic equations If one is constrained to the xt plane, one can define the intersection with that plane of the null hypersurfaces originating at some point P as$$ g_{tt} \frac{d P^t}{d \lambda}\frac{d P^t}{d \... 1answer 91 views ### 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 dx^\...
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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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### 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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### Cones with deficit angle 2$\pi$ and euler characteristics

I've managed to confuse myself with cones and deficit angles. Let's consider a conical defect in 2 dimensions. So the metric is the usual one in polar coordinates, $$ds^2 = dr^2 + r^2 d\phi^2,$$ ...
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### Integral curves in null hypersurfaces [closed]

Let be $(M^{n+1},g)$ a spacetime (Lorentz manifold, connexe and time-oriented), $n\ge 2$, and $S\subset M$ a null hypersurface (codim $S=1$ and the restriction of $g$ to each tangent space $T_p S$ is ...
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### What are the implications of integrating the Poisson bracket? [closed]

Reading Ref. 1 I admit I am a little lost in some places. I was hoping that someone in this area could explain the basic premise of integrating the Lie bracket and further is it connected to nlab's ...
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### How does the expanding of null hypersurface orthogonal geodesic congruence imply a particular result?

Sorry that I do not know how to summarize my problem in the title. First, please go to the website here (free access, even though it looks otherwise) to download the paper done by R. Sashs on ...
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On page 2 of this paper (http://arxiv.org/abs/1106.6073), Maldacena explains (and has a very nice picture) showing the trajectories that a timelike and null particle would take in AdS space. Of ...
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### Operational Definition of Reference Frame in General Relativity

Most treatments of GR begin with the assumption that spacetime is a pseudo-Riemannian manifold (or, sometimes, that it is a more general manifold). But this entails quite a few tacit assumptions about ...
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### Expanding the Ricci tensor by summing over indices

I had an attempt at deriving the Schwarzschild metric. This is a 4-dimensional problem where the indices are being summed from 0 to 3. I got up to the part where I calculate the Ricci tensor which is ...
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### How to derive the relation satisfied by “gravitational magnetic field” from an equation of the Weyl tensor?

Let us call the spacetime $M$ with a metric $g_{ab}$. There is a unit spacelike vector field $\eta^a$ orthogonal to a hypersurface. So that we can define the so-called gravitational electric and ...
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### Noether's theorem clarification

There is wonderful explanation of the derivation of Noether's theorem here: Noether's current expression in Peskin and Schroeder However, I would like to dig a little deeper because I am still ...
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### Local Lorentz transformations

If $\gamma^m$ denotes a tangent space gamma matrix, and $\gamma^\mu$ denotes a curved space gamma matrix, then they are related by $$\gamma^\mu(x) = \gamma^m e_{m}^{\mu}(x)$$ where $e_{m}^{\mu}(x)$ ...
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### What does naturally mean here?

We often cross the sentence "Kahler geometry emerges naturally in sugra". I have always wondered what does this mean; actually what does naturally mean in that sentence?