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 status of gauged gravity [duplicate]

The Standard Model of elementary particles is a gauge theory with gauge group $SU(3)\times SU(2)\times U(1)$, which is really a successful theory. We might be able to quantize gravity similarly. ...
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144 views

Level quantization of 7d $SO(N)$ Chern-Simons action

In 3d, one can write down the $SO(N)$ Chern-Simons action to be $$S(A)=\frac{k}{192\pi}\int_{M}\text{Tr}(A d A +\frac{2}{3}A^3),$$ where $A$ is an $SO(N)$ connection. The level quantization can be ...
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195 views

Tricks for Computing Riemann Curvature Tensor with Levi-Civita connection

I am new to differential geometry, so far it seems to me that computing the Riemann tensor tends to be a rather tedious task, I wanted to know whether there are some tricks that I am missing. In ...
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66 views

Why is the Taub-NUT instanton singular at $\theta=\pi$?

Consider the following metric $$ds^2=V(dx+4m(1-\cos\theta)d\phi)^2+\frac{1}{V}(dr+r^2d\theta^2+r^2\sin^2\theta{}d\phi^2),$$ where $$V=1+\frac{4m}{r}.$$ That is the Taub-NUT instanton. I have been ...
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125 views

Lie derivative of Dirac Delta

In the setting of general relativity, I came across a source term of the wave equation of the following form: $$ \frac{1}{\sqrt{q}}\,\delta^{(3)}(p-\gamma(t)) $$ where $p\in M$ is a point in our 4d ...
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79 views

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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142 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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92 views

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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501 views

How to prove that Weyl tensor is invariant under conformal transformations?

I need to verify that the solution for vanishing Weyl tensor is conformally flat metric $g_{\mu\nu} = e^{2\varphi}\eta_{\mu\nu}$. The most convenient way to show this is to prove that Weyl tensor is ...
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53 views

Does heat kernel factorize on product spaces?

I have a doubt regarding whether the trace of the vector heat kernel on a product space factorizes into the corresponding heat kernel traces on each manifold in the product space. I know this holds ...
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96 views

Path Integral on Einstein Cartan Manifold

In condensed matter, crystal with disclination and dislocation has both curvature and torsion. I am looking for a reference in which path integral quantization of Dirac equation on manifold with ...
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254 views

What are endomorphism bundle valued $p$-forms and exterior covariant derivatives and their use in Chern-Simons theory?

Chern-Simons Forms appears in several places in physics for examples, Fractional Quantum Hall Effect, response of Topological Insulator, invariant of knot, electromagnetism in 2+1 space-time, ...
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2answers
192 views

Why do we need non-trivial fibrations?

I am currently reading this paper. I understand how the Bloch sphere $S^2$ is presented as a geometric representation of the observables of a two-state system: $$ \alpha |0\rangle + \beta |1\rangle ...
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3answers
383 views

Why Hausdorff and Paracompact manifold in GR?

What can we say about the transition map if the manifold is a Hausdorff space? Why do we need the manifolds to be Hausdorff and paracompact in General Relativity?
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4answers
133 views

What is the motivation for the definition of a manifold?

In Wald's General Relativity, an $n$-dimensional $C^{\infty}$ manifold $\mathit{M}$ is defined as a set, with subsets $\lbrace{O}_{\alpha}\rbrace$, which satisfies 3 properties. In particular, the ...
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1answer
228 views

Is the apparent lack of (Ricci) curvature in the Schwarzschild metric due to a choice of coordinates?

I've been lightly studying GR lately. Something that has been bothering me has been the lack of (Ricci) curvature produced from the Schwarzschild metric in the few lectures I've watched, as well as ...
3
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2answers
381 views

Which of these two textbook equations of geodesic deviation is correct?

My previous question Geodesic deviation on a 2-sphere - is this the right track? got shot down as “off topic”, so I'm having a second stab at it. Misner et al's Gravitation (p34) gives the geodesic ...
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2answers
416 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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1answer
527 views

Can one raise indices on covariant derivative and products thereof?

Can the following be true? $g^{\sigma\rho}\nabla_{\rho}\nabla_{\mu} = \nabla^{\sigma}\nabla_{\mu}$ $g^{\sigma\rho}\nabla_{\nu}\nabla_{\sigma} = \nabla_{\nu}\nabla^{\rho}$ ...
3
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1answer
662 views

Difference between $\partial$ and $\nabla$ in general relativity

I read a lot in Road to Reality, so I think I might use some general relativity terms where I should only special ones. In our lectures we just had $\partial_\mu$ which would have the plain partial ...
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4answers
381 views

Formulation of general relativity

EDIT: I think I can pinpoint my confusion a bit better. Here comes my updated question (I'm not sure what the standard way of doing things is - please let me know if I should delete the old version). ...
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1answer
298 views

Vector potential $A$ on a 2-sphere $S^2$ of radius $R$ with some points removed

I am preparing myself for an exam and I got stuck with the following problem. If I wanted to calculate the vector potential $A$ on a sphere (not off or in), where some points are removed, how would I ...
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1answer
240 views

Etale bundles and sheaves

Before answering, please see our policy on resource recommendation questions. Please try to give substantial answers that detail the style, content, and prerequisites of the book or paper (or ...
3
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2answers
576 views

Geodesic equations

I am having trouble understanding how the following statement (taken from some old notes) is true: For a 2 dimensional space such that $$ds^2=\frac{1}{u^2}(-du^2+dv^2)$$ the timelike geodesics ...
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4answers
982 views

Topology needed for Differential Geometry [duplicate]

I am a physics undergrad, and need to study differential geometry ASAP to supplement my studies on solitons and instantons. How much topology do I need to know. I know some basic concepts reading from ...
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2answers
189 views

Why are totally antisymmetric tensors more useful than totally symmetric tensors?

In an arbitrary number of dimensions, one can naturally define two tensors, Kronecker delta and Levi-Civita epsilon tensor. However, why isn't it advantageous to define some totally symmetric tensor ...
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1answer
121 views

Can geodesics in a Lorentzian manifold change their character?

From a physics perspective, it's pretty easy to see why a a massive particle will be restricted to timelike paths, etc. but does the math guarantee that on its own or do we have to impose it? More ...
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4answers
743 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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4answers
411 views

Complete set of observables in classical mechanics

I'm reading "Symplectic geometry and geometric quantization" by Matthias Blau and he introduces a complete set of observables for the classical case: The functions $q^k$ and $p_l$ form a complete ...
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2answers
606 views

Is there an analogue of configuration space in quantum mechanics?

In classical mechanics coordinates are something a bit secondary. Having a configuration space $Q$ (manifold), coordinates enter as a mapping to $\mathbb R^n$, $q_i : Q \to \mathbb R$. The primary ...
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1answer
202 views

Susy QM and Atiyah-Singer index theorem

Consider maps $t\mapsto x^i(t)$ from circle to some Riemannian (spin) manifold and lagrangian $$ \mathcal L = \frac12 g_{ij}(x) \partial_t x^i \partial_t x^j + \frac12 g_{ij} \psi^j \left(\delta^i_k ...
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2answers
368 views

Space-time Topologies?

When it comes to questions of existence of bounds for PDE's and such, one must often make some assumptions regarding the topology of the space-time to use well known theorems. My question is ...
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1answer
124 views

What is the difference between $\nabla _{\sigma} $ and $ \nabla^{\sigma}$?

What is the difference between: $\nabla _{\sigma} $ and $ \nabla^{\sigma}$? I've been told that the first is the covariant derivative, however I'm just starting a course on spacetime geometry and ...
3
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1answer
105 views

Ricci tensor of direct product of manifolds

Imagine I have a (Lorentzian) manifold with a metric $\left[ {\begin{array}{cc} g_{\mu\nu} &0\\ 0&g_{mn}\\ \end{array} } \right]$ Will the Ricci tensor be also block diagonal ...
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1answer
120 views

What spacetimes satisfy this identity?

What spacetimes satisfy $R^{\mu\nu} R_{\mu\nu} =\alpha R^2$, where $R = g^{\mu\nu}R_{\mu\nu}$ is the Ricci scalar, and $\alpha$ is some constant? A follow-up question: in what spacetimes does ...
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2answers
555 views

Orthochronous Lorentz transformations are time-preserving and $SL(2,\mathbb{R})$

Let's consider the psuedosphere/hyperboloid in $\mathbb{R}^{1,2}$ given by $$x^2+y^2-z^2=-R^2.$$ We know that the Lorentz group $$O(1,2)=\{ A \in Mat(3,\mathbb{R}): A^tGA=G \},$$ where ...
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3answers
860 views

Equations of fluid dynamics and differential geometry [closed]

Where can I look for equations of fluid motion written in terms of nifty things from differential geometry like exterior derivative, Hodge dual, musical isomorphism? Preferably both with and without ...
3
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2answers
193 views

Correct tetrad index notation

There seems to be some different conventions on the indexes of the tetrad. I am wondering which is the standard, which is correct, and which is an abuse of notation. In Sean Carroll's notes and in ...
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1answer
188 views

Geometric mechanics - Symplecticity

I am just trying to wade through literature on classical mechanics and I really don't know where to start, everything is Fibre bundle this or manifold that, and doesn't really ease you in to the ...
3
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1answer
129 views

Ricci flat compact manifold with $U(1)\times{}SU(2)\times{}SU(3)$ isometry group?

As the title says, is it possible to have a Riemannian Ricci flat compact manifold with $U(1)\times{}SU(2)\times{}SU(3) $ isometry group?
3
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1answer
189 views

Geodesics in AdS3

I'm having some trouble doing an easy computation with the AdS space. I'm considering $\text{AdS}_3$ space with the Poincaré coordinates, so the metric reads $$ds^2 = \frac{R^2}{z^2}(dz^2 - dt^2 + ...
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3answers
145 views

Linear independence of the Covariant Derivative

What's the easiest way to show that the covariant derivative $\nabla U^{\mu}$ is linearly independent to $U^{\mu}$, which is a vector? I mean I'm assuming they are since I'm proving the second ...
3
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2answers
222 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 ...
3
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1answer
73 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
940 views

'Easy way' of finding out the Killing vector fields?

Is there a way for calculating the Killing vector fields of a given metric in a quick way? Sure I can guess looking at the metric at the symmetries, and then guess some of them, but, for instance, in ...
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2answers
260 views

Differential Forms and Densities

I've heard that differential forms are related to densities, however I'm still a little confused about that. I thought on the case of charge density and I came to that: let $U\subset\mathbb{R}^3$ be a ...
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2answers
780 views

Metric coefficients in rotating coordinates

Let $(t,x,y,z)$ be the standard coordinates on $\mathbb{R}^4$ and consider the Minkowski metric $$ds^2 = -dt^2+dx^2+dy^2+dz^2.$$ I am trying to compute the metric coefficients under the change of ...
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1answer
845 views

What bends fabric of space-time?

I know that mass can bend fabric of space-time, which causes gravity by making an object curve around a planet or star but is there anything else that can bend it? Other energy sources, forces ...
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1answer
63 views

What is the physical cause that circulation on a closed surface is zero?

This is quoted from Feynman's Lectures: We would like to see what happens when the loop shrinks down to a point, so that surface boundary disappears - the surface becomes closed. Now, if the ...
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2answers
71 views

Symplectic structure and isomorphisms

In his book Mathematical Methods of Classical Mechanics, V.I. Arnold writes To each vector $\xi$, tangent to a symplectic manifold $(M^{2n},\omega^2)$ at the point $\mathbf{x}$, we associate a ...