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

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98 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 ...
3
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2answers
269 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
42 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
66 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 ...
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1answer
163 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
114 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
278 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
124 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
99 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 + ...
4
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1answer
329 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). ...
4
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4answers
209 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
84 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
71 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 ...
2
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2answers
234 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
55 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
74 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
78 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
68 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 $$ ...
4
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1answer
152 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
55 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
55 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
52 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 ...
3
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1answer
122 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
188 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
78 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 ...
3
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0answers
77 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 ...
4
votes
1answer
73 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
228 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
44 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
76 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
36 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 ...
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0answers
36 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
123 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
135 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
201 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
99 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 ...
0
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1answer
75 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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0answers
51 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
votes
1answer
98 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 ...
3
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0answers
69 views

Classical Statistical thermodynamics phase space and residue $h$

In classical statistical mechanics we have to divide the partition function by a factor of $1/h^n$. In almost every calculation of a real quantity this cancels out and is thought to be a remnant of ...
1
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1answer
68 views

Meaning of “physical” and “gravitational” metrics

I've recently been reading some notes (following a paper by J.D. Bekenstein, titled "The Relation between Physical and Gravitational Geometry": http://arxiv.org/abs/gr-qc/9211017) on alternative ...
2
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0answers
65 views

How is the Routhian of classical mechanics defined?

The Hamiltonian is a function on the cotangent bundle to a configuration manifold $H:T^*M\rightarrow \mathbb R$. The Lagrangian is a function on the tangent bundle to the configuration manifold ...
3
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2answers
185 views

General relativity without curvature?

Is there a reformulation of general relativity without curved space time, just with fields (like classical E&M)? Edit: removed the part about E&M with curvature (multiple posts).
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0answers
40 views

Diameter of manifold with negative curvature

Are there any results (papers/books) on this problem? I am working on a finite dimensional Riemannian manifold which has a negative curvature almost everywhere. But I do not know if such kind of ...
2
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0answers
50 views

Diameter of the space of unitary operation manifold for quantum computation?

I am considering the unitary operation manifold for quantum computation. In order to examine the computational complexity of an algorithm using n qubits, we need to define the complexity of a certain ...
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1answer
82 views

Why closed in the definition of a symplectic structure?

Why do we want the 2-form $\omega $ to be closed? What if it is not?
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1answer
191 views

New “oscillator basis” of gamma matrices?

It was mentioned in http://kclpure.kcl.ac.uk/portal/files/12371620/Studentthesis-Mehmet_Akyol_2013.pdf page 28, a new concept "oscillator basis" or more precisely the author defines gamma matrices of ...