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

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How to visualize a sphere bundle?

In the paper ``Gravitational Anomaly Cancellation for M Theory Fivebranes", the authors consider removing a tubular region of radius $\epsilon$ around the M5 brane (in order to make sense of the three ...
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1answer
78 views

Time independent Kerr metric

The Kerr metric expressed in terms of polar coordinates $r,\theta,\phi$, such that $x = r\sin(\theta)\cos(\phi)$, $y = r\sin(\theta)\sin(\phi)$, $z = r\cos(\theta)$. Then the Kerr metric is given as ...
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45 views

Existence of a solution for geodesic differential equations for a singular metric

In order to determine the geodesics, one must solve the following set of differential equations \begin{align} \frac{d^2 x^j}{ds^2} + {j\brace h\,\,k}\frac{dx^h}{ds}\frac{dx^k}{ds} = 0, \end{align} ...
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1answer
134 views

Finding diffeomorphism given vector fields [closed]

Given a vector field how do you find the associated diffeomorphisms? Say I am given a vector field in Minkowski space $$\xi = x \frac{\partial}{\partial t} + t \frac{\partial}{\partial x}.$$ How do ...
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1answer
92 views

What is the sum of the angles of a triangle on Earth orbit?

Gauss went out and measured triangles made up of mountain peaks to show that the angles sum up to 180 degree. However, general relativity leads to non-Euclidian space and I would like to get a better ...
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27 views

Non-time orientable quotient of de Sitter space

Examples of non-time orientable spacetimes are pretty scarce, but it seems the big one is quotients of de Sitter space of the form $dS^n/\pi_1$, where $\pi_1$ is some subgroup of the isometries of de ...
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62 views

Two dimensional spacetime and the Gauss Bonnet theorem

Generally two dimensional spacetimes are deemed to be static, as the Gauss Bonnet theorem implies that the Einstein Hilbert action would be a constant independent of $g$. But as far as I can tell, ...
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2answers
77 views

If curved paths imply that the vehicle is accelerated, how come do we assume that light gets curved whilst its speed is constant?

I don't understand how we can accept these two sentences at the same time: Light speed is constant, therefore experiences no acceleration. On the presence of a gravitation field, light path is ...
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5answers
125 views

Local inertial frame

In general relativity we introduce local inertial frames to be such frames where the laws of special relativity holds. Let $\xi^{\alpha}$ the coordinates in the local inertial frame, so we get ...
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110 views

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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60 views

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 $T_p S$ ...
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74 views

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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36 views

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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0answers
53 views

Trajectories in AdS

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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1answer
70 views

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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30 views

Topics required before studying Tensor Calculus [closed]

Which topics do I need to know before going through tensor calculus?
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1answer
39 views

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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0answers
29 views

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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75 views

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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2answers
63 views

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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0answers
28 views

How many degrees of freedom to diagonalize the metric?

In A. Zee's Einstein Gravity in a Nutshell, he starts with the following expansion of the metric at some point $P$ of a Riemannian manifold, with coordinates $x^\mu$ that have the origin at $P$: $$ ...
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1answer
57 views

How to get null tetrad by metric?

How to get null tetrads ${l^a,n^a,m^a,\overline{m}^a}$ for this metric? This on is from Ryder's book (Introduction to general relativity) page 268 $g^{\mu\nu}=\begin{pmatrix} 0 & \frac{1}{c} ...
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1answer
102 views

Maxwell equations in 2+1 D

I have a problem with the Maxwell equations in (2+1) dimensions using differential form. Following J. Baez "Gauge Fields, Knots and Gravity" page 93 (or any other book), the equations are ...
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0answers
36 views

$S^2$ and monopole [closed]

In mathematics, to describe a sphere $S^2$ we need two coordinate patches which we call the North semi-sphere and South semi-sphere. Between them there is a map which mathematicians called the ...
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1answer
134 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 ...
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32 views

Local Translation of frame field- Geometric Picture

In order to phrase my question I review my geometrical picture of the first order formulation of gravity: Given some d-dimensional manifold $\mathcal{M}$ one constructs the Frame bundle $FM$, a ...
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1answer
99 views

Does $g_{\mu\mu}$ in an expression follow the Einstein summation convention?

Assume that I have the expression for a Christoffel symbol: $$ \Gamma^\mu_{\alpha \beta}=\frac{1}{2}g^{\mu \lambda}(\partial_\alpha g_{\beta \lambda}+\partial_\beta g_{\alpha \lambda} - ...
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2answers
108 views

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?
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3answers
105 views

GR: Pseudo Riemannian or Riemannian?

Is General Relativityy described by Pseudo-Riemannian manifold or Riemannian manifold? I cannot understand the vast difference between the two manifolds. In books, General Relativity is looked as a ...
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1answer
70 views

Are fully raised/lowered versions of Kronecker delta tensors?

I am confused. I have two textbooks contradicting each other, at least, it seems to me so. The first one – "Field theory" by Landau & Lifshitz says that by lowering or raising one index of ...
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0answers
63 views

Constructing Killing tensors from Killing vectors

Background: After reading about Carter constant and symmetries in GR, I became interested in Killing tensors. I tried reading this paper by Alan Barnes, Brian Edgar and Raffaele Rani, discussing ...
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2answers
158 views

Gauge theory for mathematicians?

I'm looking for a textbook or set of lecture notes on gauge theory for mathematicians that assumes only minimal background in physics. I'd prefer a text that uses more sophisticated mathematical ...
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45 views

A Lie derivative $\mathcal{L}_{\alpha^A}$ with respect to a spinor $\alpha^A$?

Suppose we work with Minkowski flat space $M$ (just to make things easy). If $\textbf X$ is a Killing vector field it is possible to define the Lie derivative of an spinor $\alpha^A$ with respect to ...
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1answer
211 views

Derivation of one-form/vector equation in Carroll confusion

I don't understand the derivation of Equation 2.14$$\mathrm{d}f\left(\frac{d}{d\lambda}\right)=\frac{df}{d\lambda} \tag{2.14}$$ in Carroll's Lecture Notes on General Relativity ...
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2answers
68 views

Field theory where fields are differential forms, other than electromagnetism [closed]

I am looking for a few examples of field theories (classical or quantum) that can be formulated taking the fields to be differential forms at least of degree 1 (not counting 0-forms) excluiding ...
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2answers
82 views

Maintaining symmetry? [closed]

Minkowski metric is found to be $$ds^2=-dt^2+dr^2+r^2d\Omega^2$$ where $d\Omega^2$ is the metric on a unit two-sphere. Why should we keep track of the $d\Omega^2$ so that spherical symmetry holds ...
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2answers
74 views

Textbook for mathematical Lagrangian mechanics [duplicate]

I'm looking for a textbook or online notes or a review article etc on a rigerous formulation of Lagrangian mechanics. I'm well aware of the book by Arnold but I would like something to accompany it. ...
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67 views

Deriving the adiabatic relation for an ideal gas from a differential geometry point of view

Recall $Q = dU + W$ (First Thermodynamic law, energy conservation). If $Q=0$ (for adiabatic process, either adiabatic expansion or contraction), and supposing $W = pdV$, then $0 = dU + pdV$. Either ...
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2answers
102 views

How strong would the Earth's “magnet” be if it was the size of a fridge magnet?

If you shrunk the "magnetic part" from inside the Earth down to a fridge magnet size, how strong would it be in Gauss? This is not a home work question. I just watched this video which included the ...
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39 views

Is the coordinate transformation of an object the same of the action of a group on this same object?

I am having troubles in understanding frame transformations in physics from the mathematical point of view. What I understand for a coordinate transformation is just a function to one chart to another ...
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0answers
61 views

Transformation of Christoffel symbols [closed]

Friends I have little problem with transformations:) In General relativity is Christoffel symbol of second kind defined as: $$ \Gamma^{l}_{ij}=g^{lk}\left(\frac{\partial g_{ki}}{\partial ...
5
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1answer
133 views

How to measure Torsion and Non-metricity?

In General Relativity, we most often work with the Levi-Civita connection (metric and torsion-free). What kind of experiment can we make to be sure that our physical space-time indeed is torsion-free ...
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1answer
66 views

Is a local inertial frame of reference a Lorentz frame?

I started reading "Gravitation" (the big black book with the apple) and in the first chapter it is said that a local inertial frame of reference rocks. A little later it is said that Lorentz frames ...
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2answers
175 views

What is a Christoffel symbol?

What is a Christoffel symbol? I often see that Christoffel symbols describe gravitational field and at other times that they describe gravitational accelerations. Then, on some blogs and forums, ...
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3answers
119 views

Can space and time separately be curved?

How can I imagine curved time, if it is not a part of four dimensional spacetime? Similarly for space. What are the measurable, observable consequences of these two phenomena in a laboratory or in ...
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2answers
89 views

How does a metric of the form $\mathrm{d}z \mathrm{d}\bar z$ work, if $z$ and $\bar z$ are not independent? [duplicate]

My question is motivated by 2D CFT where one works in "complex coordinates". The question is the following: Suppose I am in 2D flat Euclidean space, i.e. $$\mathrm{d}s^2 = \mathrm{d}x^2 + ...
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1answer
178 views

Tetrad formalism vs coordinate formalism example

Sources I have been reading Chapter 11 and 25 of Andrew Hamilton's amazing notes which has some material on tetrad formalism in general relativity (formulating GR in coordinate-free fashion). ...
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4answers
183 views

Do $\vec r$ and $d \vec r$ have the same direction?

One question is bugging me for a long time but I couldn't make out anything nor could my friends. Here it goes: We know, $\vec r$ is regarded as the position vector. So we can say, $$\vec r \cdot\vec ...
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66 views

Canonical Momentum Conjugate vs. Momentum

I stumbled upon this while reading about Legendre Transforms today. So consider an n-particle system. The Lagrangian is a function of $ q_i$'s and $\dot q_i$'s. If you consider the manifold $M$ where ...
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1answer
66 views

Clarification about some steps in the derivation of the Lie derivative (mechanics)

First of all, this question may seem to be undefined, because I'm not sure how to connect this (to me) newly introduced concept with the abstract notion of the Lie derivative. I'm not even sure if I ...