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

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What is the meaning of “background” in General Relativity?

I was reading a book on Topological defects of the very early universe as an example of the fundamental groups, and they say that "in an expanding homogeneous and isotropic universe, the background ...
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Is it possible to build up holography in a closed manifold, i.e., in a manifold with a mathematical boundary?

I was wondering about the AdS/CFT correspondence basics. It is constructed on the idea of conformal compactification, in which a open manifold $M$ is homeomorphic related to a closed one $N$ through a ...
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1answer
114 views

$\mathcal{N}=2$ supersymmetry and $SU(2)$ holonomy

I was reading this Phys.SE question. I was unable to understand how an $SU(2)$ holonomy would produce $\mathcal{N}=2$ in four dimensions. Could anyone shed some light on this?
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90 views

Curvature based derivation of Schwarzchild Metric

I'm a third year maths undergrad and I'm trying to find (and follow) a curvature based derivation of the Schwarzchild metric, if there exists such a proof?
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1answer
92 views

What is the neatest way to describe a “non-autonomous” (lagrangian) system?

The configuration space of a system of particles $(m_i,x_i)$, $i=1,\dots,n$, subject to constraints $$\Phi (x)=0,\qquad \Phi\colon \mathbb R^{3n}\to \mathbb R ^{3n-k},\qquad x=(x_1,...,x_n),$$ if the ...
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163 views

Partial Differentiation of a Tensor

I have doubts in the statement that the partial or ordinary differentiation of tensor is not a tensor. The argument for this is that the partial differentiation of the tensor involves evaluating the ...
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102 views

Lie derivative of a scalar and PDE

I am reading about differential geometry, and in particular the Lie derivative and its relation to (relativistic) hydrodynamics. In particular, I was wondering if, given two scalar functions ...
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196 views

Riemannian manifold: Integration by parts of Lie derivative

How can we apply integration by parts to the Lie derivative? Background: In the Hamiltonian formulation of general relativity, we have the momentum constraint (using abstract index notation, $D_a$ is ...
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241 views

Must every isometry have an associated Killing vector?

I understand that the flows of Killing vector fields are isometries, and that one-parameter groups of isometries have an associated Killing vector which generates them, but are your Killing vectors ...
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87 views

How to implement wedge product and form in quantum mechanics?

For example use p-form for p identical fermion system? The Maxwell equations can be rewritten as a single function using forms plus an intrinsic equation showing the Faraday tensor is a two form. ...
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102 views

Further explanation of the Penrose Conjecture

I'm currently a third year maths undergrad, writing a dissertation on the application of minimal surfaces in space. I have recently come across the Penrose Conjecture that the mass of a spacetime is: ...
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31 views

How to express “curvature scalars” in terms of "discrete curvature values $\kappa_n$?

We know from MTW [1] and Synge [2] how, for participants who were (pairwise) rigid to each other, it may be determined whether or not they were straight to each other, plane to each other, or ...
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284 views

Curvature of Spacetime

I have been exploring for some time both the Special and General Relativity, hoping to glean at least a conceptual grasp of their basic tenets. In reading the book "Gravitation" by Misner, Thorne and ...
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56 views

Bending a flat sheet

I am interested in the following problem: starting from a flat sheet, once can bend it into a (sector) of a cylinder isometrically. If further (orthogonal to the first plane) curvature is induced, the ...
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54 views

Contracting Indices in General relativity [duplicate]

I was reading a book about general relativity and I came across these two equations $$ \begin{align} \mathrm{g}^{\mu\nu}_{,\rho}+ \mathrm{g}^{\sigma\nu}{\Gamma}^{\mu}_{\sigma\rho}+ ...
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2answers
180 views

Bracket Notation on Tensor Indices

I know about the () symmetrisation and [] anti-symmetrisation brackets on tensor indices so long as they appear on their own, such as : $$V_{[\alpha \beta ]}=\frac{1}{2}\left ( V_{\alpha \beta ...
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311 views

Contracting Indices

Does anyone know how to get from (1) to (2) in the system $$ \begin{align} \mathrm{g}^{\mu\nu}_{,\rho}+ \mathrm{g}^{\sigma\nu}{{\Gamma}}^{\mu}_{\sigma\rho}+ ...
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1answer
58 views

Unable to resolve 2 equivalent geodesic equations

A free particle moves along geodesics, one form being \begin{split} \ddot x^\mu &= -\Gamma^{\mu}_{\sigma \rho} \dot x^\sigma \dot x^\rho \\ &= -\frac{1}{2}g^{\mu \nu}(\partial_\sigma g_{\rho ...
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1answer
81 views

Invariants of Connection Form

I am somewhat going out "on a limb" here, since I am much more grounded in the physics side of things than I am in mathematics. Nonetheless, I am wondering if someone is able to comment on the ...
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167 views

Vanishing of Weyl Tensor Contraction

Within the context of Einstein space-times, we know that the contraction of the Weyl tensor across a set of indices always vanishes, like so : $$C{^{\alpha }}_{\mu \alpha \nu }=0$$ From a purely ...
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175 views

Geodesics equations via variational principle

I would like to recover the (timelike) geodesics equations via the variational principle of the following action: $$ \mathcal{S}[x] = -m \int d\tau = -m \int \sqrt{-g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}} $$ ...
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1answer
133 views

Symmetry, Transformations and non-linear transformations

I am a physics student. My mathematical background is quite weak. I just want to know the similarities (if there are any) or differences between coordinate transformation of two kinds : Rotation of ...
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190 views

Understanding Calculus Notation in Physics

I have just started a first-year calculus-based physics course about electromagnetism and waves. I am having trouble understanding what calculus notation means in the context of physics. Here is a ...
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1answer
105 views

Sign of $dr$ in Schwarzschild geodesics

There is an equation that relates energy $E$, angular momentum $L$ and other constants and variables to find $\left(\frac{dr}{d\tau}\right)^2$ in a plane. ...
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59 views

Does the covariant derivative of a derivative of the metric vanish?

The title holds the main question. For a little more backgroud: In a calculation I keep coming across terms of the form $$ D_M \partial_N e_P^A \overset{?}{=} 0$$ where $e_P^A$ is the vielbein and ...
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63 views

Riemann normal chart and special relativity

When you pick Riemann normal coordinates at a point, then the Christoffel symbols vanish and the metric is flat, but the Riemann curvature tensor does not necessarily vanish. Since Einstein said that ...
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70 views

Finding the components of the Riemannian tensor given the components of a metric

I am looking at a manifold of dimension $n$ (And I am considering a local co-ordinates system $x_1,x_2,\ldots x_n$) and the metric defined by the components $g_{ij} = \frac{\delta_{ij}}{x_1^2}$. I'm ...
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In which field theories with fermions do string- and fivebrane structures not come up?

A year ago, username @Greg Graviton asked in a thread here about the Spin group as covering of the spatial rotations. A subquestion was: What other groups, even larger than SU(2) are there that ...
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79 views

Geometric quantization in Kepler problem in hydrogen atom

Why in the usual geometric quantization calculation the dimensions of eigenspaces is wrong (we can see this obstacle for Kepler problem in hydrogen atom). Here is a refference see
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1answer
123 views

Are conformal, Killing and homothetic vector fields the same in pseudo-riemannian manifolds?

I work in the Lorentzian manifolds, more generally in pseudo Riemannian manifolds and applications to general relativity. I know the definitions of conformal, Killing and homothetic vector fields in ...
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Timelike Loop Spaces as Projective Null Twistor Spaces

Let $\mathcal{M}$ be a spacetime, and let $\Omega\mathcal{M}$ denote the loop space of the spacetime. My idea is that the set of all closed timelike curves of $\mathcal{M}$ forms the projective null ...
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1answer
88 views

How can I show non-Abelian CS term is a total derivative?

I want to show:$$ Tr\left (F\tilde{F} \right )=\partial_{\mu}K^{\mu }=\partial_{\mu}\left (\varepsilon _{\mu \nu \rho \sigma }Tr\left ( F_{\nu \varrho }A_{\sigma }-\frac{2}{3}A_{\nu }A_{\rho ...
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63 views

Compatibility conditions of spinors and Riemannian Metrics

I came across an interesting article by Montesinos (J. Geom. Phys. 2 (1985), no. 2, 145–153.). In it, he finds that spin structures (as lifts of $SO(4)$) are not compatible with all Riemannian metrics ...
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Kähler and complex manifolds

I was woundering if anyone knows any good references about Kähler and complex manifolds? I'm studying supergravity theories and for the simplest $\mathcal{N}=1$ supergravity we'll get these. Now in ...
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267 views

What is the geometrical interpretation of Ricci tensor?

In differential geometry and general relativity space is said to be flat if the Riemann tensor $R=0$. If the Ricci tensor on manifold $M$ is zero, it doesn't mean that the manifold itself is flat. So ...
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130 views

Schwarzschild geodesics

I've found on Wikipedia that energy $E$ and angular momentum $L$ of a particle are conserved quantities in Schwarzschild metric. It's written: $$L=mr^2 \frac {d\phi} {d\tau},$$ ...
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174 views

Lorentzian and Einstein Manifold

I am studying for my Bachelor thesis (in Mathematics). I and my advisor agreed on the Penrose-Hawking singularity theorems. My question is: 1) Which mathematical background should I focus on ...
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1answer
63 views

What is the need to consider a singular spacetime?

To have a consistent superstring theory (which is to avoid the conformal anomaly on the worldsheet CFT) we are forced to build our theory on the critical dimension $n=10$. However, the Standard ...
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1answer
129 views

Where is a closed form also exact?

I'm not very familiar with exterior derivatives. I've some trouble following argument (which is a part of a proof that if the Riemann tensor vanishes, $R^{\,\rho}_{\;\,\sigma \mu \nu}=0$, iff there ...
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4answers
376 views

General relativity in terms of differential forms

Is there a formulation of general relativity in terms of differential forms instead of tensors with indexs and subindexs? If yes, where can I find it and what are the advantages of each method? If ...
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A question about polarization in quantum mechanics

We start our question we a definition A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if \ $P$ is Lagrangian P involutive dim$P\cap\bar ...
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What is the physical meaning of the Eddington - Finkelstein metric?

I want to see a some physical process (experimental) that could explain the many transformations of coordinates into this mathematical procedure. (really two transformations, but i think that is a ...
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95 views

Geometric quantization AND nuclear physics

Classical mechanics has a natural mathematical setting in symplectic geometry and it may be asked if the same is true for quantum mechanics. Geometric quantization is one formalization of the notion ...
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1answer
91 views

Warped AdS geometry

I am having difficulty of finding more basic information on warped geometries. All the standard textbooks are not covering it. In the wiki article it's only said that warped geometry is the one which ...
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General relativity and global aspects [duplicate]

The theory of general relativity tells me something about the global structure of space-time, eg simply connected ?
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1answer
80 views

how affine connection follows from Two derivative operator

IN wald's GR book in chapter 3 This is stated behind the definition of affine connection : First He showed that if we have two derivative operator $\nabla_a , \tilde\nabla_a$ (both of which ...
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1answer
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Extent of coordinate freedom to set metric components along a spacetime path

If we describe spacetime with a Lorentzian manifold, it is always possible to choose a coordinate system such that at any particular point $x^\alpha$, the components of the metric are: $$ ...
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Prereqs for The Geometry of Physics by Frankel [duplicate]

I'm interested in giving The Geometry of Physics a read, and I was wondering what the mathematical and (more importantly) physical prerequisites are. My background is a bit stronger on the ...
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The geodesic line on Poincare half plane

I was calculating the geodesic lines on Poincare half plane but I found I somehow missed a parameter. It would be really helpful if someone could help me find out where my mistake is. My calculation ...
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How to prove that zero Weyl tensor predicts no deflection of light?

There is Nordstrom theory, which can be given as $$ C_{\mu \nu \alpha \beta} = 0. $$ The solution of Einstein equations for this case is conformally flat metric: $$ g^{\mu \nu} = e^{\epsilon \varphi ...