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

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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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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
113 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
127 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
197 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
89 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
73 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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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
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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
67 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 ...
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1answer
67 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 ...
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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 ...
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2answers
184 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
81 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
189 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 ...
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1answer
84 views

In field theory, why are some symmetry transformations applied to the field values while other act on the space that the fields are defined on?

My basic understanding is that a field theory consists of symmetry groups, a space $S$ that the symmetry groups act on and of fields defined on that space $S$. In other words, the space $S$ is the ...
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3answers
161 views

Can we say that gravity(indirectly) is responsible for motion of electrons around nucleus? [closed]

From Wikipedia But because general relativity dictates that the presence of electromagnetic fields (or energy/matter in general) induce curvature in spacetime From Wikipedia An ...
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1answer
271 views

Why are the quantum observables defined on opens sets a presheaf and not a sheaf?

In local quantum field theory or AQFT one can mathematically describe over each open set $U$ of a spacetime $M$ the quantum states or observables of the theory. This structure is commonly referred as ...
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23 views

Construct bivariate symmetric (polynomial) Hilbert-Schmidt two-qubit volume functions over the unit square with certain properties

Construct bivariate symmetric polynomials (two-qubit volume functions) f(r,R) = f(R,r) >= 0 over [0,1]^2, with f(1,R) = f(r,1)=0, such that the univariate marginal (integrating over r or R) ...
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39 views

Any good reference on Maslov index (or Morse index)?

Any good reference on Maslov index (or Morse index)? I have some basic knowledge of differential geometry, calculus of variation. So is there any good reference for me?
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0answers
54 views

Foliation of the phase space

Consider an arbitrary classical Hamiltonian system. Given an initial state $(p_0, q_0)$, we can get a solution of the equation of motion, a curve in the phase space. Now the problem is, for a generic ...
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1answer
122 views

Geometric meaning of spin connection

A very short question: Does the spin connection that we encounter in General Relativity $$\omega_{\mu,ab}$$ have a geometric meaning? I am supposing it does because it comes from mathematical terms ...
2
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1answer
85 views

Infinite dimensional manifolds in general relativity [closed]

In GR the concept of a manifold is very useful. However, all of these manifolds are of finite dimension. Is it possible to define a manifold with infinite dimension (ie much like Hilbert space in QM) ...
2
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2answers
183 views

What coordinate systems allows the magnitude of the basis vectors to change with position?

I'm familiar with coordinate systems where the direction of the basis vectors changes with position, but I haven't come across any where the relative magnitude of the basis vectors themselves are ...
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3answers
367 views

Confusion about 1-forms as introduced in “Gravitation” (Kip S. Thorne,…)

In the book Gravitation in chapter 2, paragraph 5, they introduce the concept of 1-forms by thinking about the momentum 4-vector differently. They first introduce the de Broglie 1-form as follows (I ...
3
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1answer
117 views

Hermiticity of Dirac operator in curved spacetime

The Dirac Lagrangian in curved spacetime is usually given by \begin{equation} \mathcal{L} = i\bar{\Psi}\gamma^a e^{\mu}_a(\partial_\mu + \frac{1}{4}\omega_{\mu bc}\gamma^b\gamma^c)\Psi \end{equation} ...
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1answer
87 views

Does $R_{\mu \nu \sigma \rho} R^{\mu \nu \sigma \rho} \propto R$ hold?

For $R_{\mu \nu \sigma \rho}$ the Riemann-tensor and $R$ the Ricci-scalar: Does $R_{\mu \nu \sigma \rho} R^{\mu \nu \sigma \rho} \propto R$ hold? Or is there any way to relate $R$ approximately ...
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1answer
58 views

Interpretation of black hole area

I'm studying properties of Kerr spacetimes and a lot of fuss is made about area of BH. It is defined to be integral of area element on event horizon $r=r_+$, $t=const.$ where $r_+$ is radial ...
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1answer
112 views

Newtonian motion of a particle confined to a smooth surface

Recently, I've been considering a model wherein a lone particle with constant mass m is confined to a surface $F$: $\mathbb{R}^m \to \mathbb{R}^n$ , where $m < n$. I declare this surface to be ...
3
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1answer
271 views

Einstein-Yang-Mills Connections

I am playing around with coupling a classical $SU(2)$ Yang-Mills theory to Einstein's equations. Assuming spherical symmetry, the $SU(2)$ connection can be written \begin{equation} A = ...
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0answers
73 views

Existence of affine parametrization [closed]

This is a question from General Relativity by Wald Chapter 3, problem 5. Given either pseudo-Riemannian or Riemannian metric $g_{ab}$ and manifold $M$. Assume the $\nabla$ is compatible with the ...
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0answers
115 views

Angle sum of triangle in Schwarzschild solution

Curvature of space is often intuitively explained as angles of a triangle not adding up to 180 degrees. My questions concerns that. Suppose you have a perfectly spherical star of uniform density - so ...
6
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1answer
146 views

Where does the “Supersymmetry” in Witten's proof of the Morse inequalities come from?

Where does the "Supersymmetry" in Witten's proof of the Morse inequalities (original paper and outline of proof for mathematicians) come from? Hopefully someone can provide an intuitive understanding? ...
3
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1answer
185 views

Maxwell's equations in integral form using differential geometry

So I've been trying to convert from Maxwell's equations in terms of differential forms to the integral versions of Maxwell's equations that we know from vector calculus. We have, in vector calculus ...
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1answer
91 views

Covariant derivative commutator on spinors [closed]

What is this object $[\nabla_{\mu},\nabla_{\nu}]\epsilon$ in terms of curvature tensor $R_{\mu\nu}$? Where $\nabla_{\mu}$ is the covariant derivative on a four sphere and $\epsilon$ is spinor. PS: I ...
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1answer
44 views

Elementary question about distributive property of variation operator on an exterior product

I am trying to work out the equations of motion of a 11-dimensional supergravity action $$S = \frac{1}{2\kappa^2}\left(\gamma\int d^{11}x\sqrt{|g|}\mathcal{R} - \frac{\alpha}{2}\int G \wedge \star G ...
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1answer
96 views

Is EM interpreted in a principal or vector bundle?

I've read in a few places that EM is a $U(1)$-principal bundle; but is this correct? Isn't it rather an associated vector bundle using the adjoint representation of $U(1)$?
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2answers
341 views

How does one write Newtons 2nd Law using the language of forms?

Newton's second law says that $F=ma$. Supposing that the force is conservative and can thus be expressed in terms of a potential $V$ we have that $F=-dV$. We have that $V$, being a function, can ...
3
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2answers
94 views

Telling if geometry is curved without Riemann tensor

If you're only interested in telling whether a certain geometry is flat or curved, and you do not need to know in which way it is curved, do you still need the Riemann tensor? When I try to visualize ...
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0answers
80 views

Normal of a null surface and null junction conditions in general relativity

I am trying to use the null junction formalism in general relativity (as explained in eg http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.43.3763&rep=rep1&type=pdf, "Junctions and thin ...
3
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1answer
225 views

What are the equations of motion for the scalar field in the tetrad formalism?

The action of a massless scalar field in curved spacetime is given by: \begin{equation} S(\phi)=\int d^{4}x \sqrt{-g}\left(g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}\right) \end{equation} Now the action can ...
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0answers
147 views

Master's Thesis in General Relativity [closed]

just throwing a query out to the Physics community. I'm about to embark on a master's in Gravitation, Cosmology and General Relativity and was looking for possible subjects to start researching. My ...
3
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1answer
70 views

Gradient in the Frenet-Serret coordinate

I was simply thinking that the gradient in the Frenet-Serret coordinate at a particular point is similar to the gradient in the Cartesian coordinate. I simply assumed that Frenet space is an ...
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3answers
1k views

Does spacetime position not form a four-vector?

When one starts learning about physics, vectors are presented as mathematical quantities in space which have a direction and a magnitude. This geometric point of view has encoded in it the idea that ...
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1answer
458 views

Variation of Christoffel symbol and Lie derivative

I've also asked this question on Math Overflow; I hope that asking in two separate fora is not a solecism. Under an infinitesimal diffeomorphism the Riemann metric changes by the Lie derivative $$ ...
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0answers
26 views

The null geodesic for given geodesic [duplicate]

What is null geodesic equation for the static and spherically symmetric line element in $$ds^{2}=-K^{2}dt^{2}+\frac{dr^{2}}{K^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta{d\phi^{2}})$$ where ...
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0answers
73 views

What is the null geodesic equation? [duplicate]

What is null geodesic equation for the static and spherically symmetric line element in $$ds^{2}=-K^{2}dt^{2}+\frac{dr^{2}}{K^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta{d\phi^{2}})$$ where ...