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

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172 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
108 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
137 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
372 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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0answers
48 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
221 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 (http://ned.ipac....
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2answers
90 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
83 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
101 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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0answers
79 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
349 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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0answers
45 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
67 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 x^{j}}+\...
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1answer
173 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
133 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
332 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
139 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
101 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 + \mathrm{d}y^...
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1answer
419 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
212 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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0answers
94 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
75 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 ...
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2answers
354 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 ...
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1answer
56 views

What happens if locally manifold is seen as an Euclidean space? [closed]

I have been trying to understand the definition of a manifold and I have found out that the most common definition can be paraphrased as: A manifold is a space that has a complex "topology" globally ...
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1answer
81 views

The Ricci tensor and its relation to volume

From Wikipedia's entry on Ricci tensor, In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a geodesic ball in ...
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2answers
81 views

Configuration manifold of a rigid body

As I know, a rigid body is a set of $N$ particles in three-dimensional space subject to the following constraint: if $b_1,\dots,b_N\in \mathbb{R}^3$ are the initial positions of the particles and if $...
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1answer
72 views

Einstein space - proper definition [closed]

Excuse, this is my first question at this forum, I try to be clear and short. What is the exact definition of Einstein spaces? It's enough to say $$ G^{\mu\nu}_{;\mu}=0~? $$ Where $$ G_{\mu\nu}=R_{\...
4
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1answer
176 views

Gradient one-form [duplicate]

I am trying to understand what gradient one-form means actually. In the book that I'm following (A first course on General Relativity by Schutz) it's told that gradient is a one-form and it's ...
3
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0answers
58 views

What kind of math do I need got general relativity? [duplicate]

I'm 15 this year and have a passion in physics What kind of math do I need to tackle general relativity? Also what year in uni do we learn about general relativity?
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0answers
58 views

Chern-Simons function

The Chern-Simons function on the space of connections, mod the gauge transformations, on a 3-manifold can be defined by an integral. I study mathematics as profession, so I want to know what is the ...
2
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2answers
95 views

Where does the 3-velocity live?

Imagine a four-dimensional affine space $\mathcal{M}$ with the standard metric $\eta = \text{diag} (1,-1,-1,-1)$. Let $\mathcal{C}$ be a worldline of a point particle parametrized by an affine ...
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1answer
53 views

Euler density of two-dimensional manifolds

I am asking this question after reading this post: What is Euler Density?. For a two dimensional manifold, the Euler density is given by: \begin{equation} E_2=2R_{1212} \end{equation} (note that $...
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1answer
136 views

Schwarzschild metric: Change in coordinates corresponds to change in object?

I have been reading about the Schwarzschild metric in the book "General Relativity: An Introduction for Physicists" by Hobson, Efstathiou and Lasenby and it appears to say something counter intuitive. ...
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2answers
202 views

What equation (/solution) predicts the existence of black holes?

Where does our theoretical prediction of the existence of black holes come from? If it is (as I am guessing) from the Einstein Field Equations, which solution predicts it and why?
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1answer
83 views

Classical Field Theory Using Geometry

I would like to know if there are good introductory courses on Classical Field Theory taught in a differential geometry approach yet one doesn't need a background in those mathematical subjects but ...
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0answers
82 views

Tangent Vectors as Infinitesimal Displacements

I'm reading Wald's General Relativity, and I'm stuck on something that is stated very early on in the book. For an abstract manifold $M$, he goes through the usual definition of a tangent vector at $p\...
4
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1answer
77 views

How to obtain initial conditions to image Kerr black hole?

I'm reading Gravitational Lensing by Spinning Black Holes in Astrophysics, and in the Movie Interstellar to make a raytracer code to image Kerr black holes. The paper introduces a Fiducial Observer ...
4
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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) ...
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0answers
45 views

Helical killing vector

A killing vector X is defined as a vector field that satisfies the relation $$\mathcal{L}_X g_{\mu\nu}=0.$$ which basically means that if one were to transport the metric along this vector, there ...
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0answers
80 views

Riemannian generalization/adaption of the Hubbard–Stratonovich transformation

I'd like to write the Hubbard–Stratonovich (HS) transformation of a scalar function on a Riemannian manifold. This transformation is quite simple in Euclidean space. One can consider it as a Fourier ...
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0answers
41 views

Diiference between three squashed sphere and three sphere and number of susy

First I know that three sphere $S^3$ and squashed sphere $S_b^3$ \begin{align} S_{b}^{3} = \begin{array} & R^2 \times S_{r}, \quad r=b, \quad b\rightarrow 0 \\ R^2 \times S_{r}, \quad r=\frac{1}{b}...
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0answers
40 views

Is there a well defined way to shift Cauchy surfaces back and forth?

Let $S$ be a Cauchy surface on a four-dimensional connected Lorentzian manifold. Define $\Gamma(S)=\{\gamma:\mathbb{R}\to M\mid \gamma(0)\in S, \gamma$ is a unit speed geodesic passing orthogonally ...
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0answers
139 views

What is the degrees of freedom of metric tensor?

As $g_{\mu\nu}$ can be taken to be symmetric, it contains 10 functions of spacetime in 4 dimensions. But, why we call these 10 functions as the degrees of freedom of the metric while they are the ...
2
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1answer
142 views

Intuition about Momentum Maps

I'm studying Classical Mechanics and there is one object that appeared recently on the book I'm not being able to get a physical intuition about it. The mathematical definition goes as follows: Let $...
2
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1answer
212 views

Differentiation of a vector with respect to a vector

Does differentiation of a vector with respect to a vector make any sense? Even if it makes sense, how does it make any physical meaning? I mean what is the physical interpretation?
2
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1answer
109 views

Derivation of geodesic deviation equation from two neighbouring geodesics

I'm stuck trying to follow Foster and Nightingale's derivation of the geodesic equation from two neighbouring geodesics $x^{a}\left(u\right)$ and $\tilde{x}^{a}\left(u\right)$ joined by a ...
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1answer
79 views

How to approach proofs in Electricity and Magnitism that involve integrals?

I have read through both Franklin and Jackson's Electromagnetism books and I am able to understand the different proofs involving integrals but when I try to re-derive them on my own later I am always ...
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53 views

How would you describe what the affine parameter is in layman's terms? [duplicate]

I've been trying to learn it from other sites, but I'm not well-versed enough in mathematics to understand.
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0answers
35 views

Null geodesic equation with affine parameters [duplicate]

A photon's geodesic equation is defined by re-parameterizing the geodesic equation to some parameter other than proper time. This is done because $ds=0$ for the photon. Again if we use affine ...
2
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1answer
102 views

Classical conformal invariance

So I am trying to understand classical conformal invariance. So we move gently from general coordinate invariance to Weyl invariance to conformal invariance, and now they start out with this thing ...