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

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Fourier Transform on a Riemannian Manifold

The question is quite simple: What would be the definition of Fourier Transform (and it's inverse) on a Riemannian Manifold? I've found that a similar question has been asked at Mathematics.SE but ...
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110 views

Help with the understanding of boundary conditions on $AdS_3$

So I am trying to reproduce results in this article, precisely the 3rd chapter 'Virasoro algebra for AdS$_3$'. I have the metric in this form: ...
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Fermions in Schwarzschild spacetime

To my understanding Geroch proved that on 4-dimensional non-compact manifold a necessary and sufficient condition for a manifold to have a notion of spinors is to be parallelizabe .1 (General ...
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73 views

How is foliation of manifolds' theory useful in General Relativity?

I am interested on getting some hints on how Foliations Theory of Manifolds can be used fruitfully on General Relativity. I just started my Ph.D on Mathematics this semester focusing on studying ...
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81 views

Intuitively what's the relationship between forces and connections?

In Einstein's General Relativity we relate the effects of gravity with the curvature of the Levi-Civita connection on the spacetime manifold. Also, when we get the electromagnetic tensor $F = dA$ ...
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135 views

Heat kernel expansion for entanglement entropy

Can somebody please let me know where I can find a reference for calculating heat kernel coefficients on a manifold with conical singularities? I am trying to compute the entanglement entropy for ...
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194 views

An introductory resource for learning AdS space

Can someone please point me to introductory resources about the geometry of Anti DeSitter Space ? What are some examples of other spaces used in theoretical physics ?.I'm learning Differential ...
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141 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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435 views

Does the universe obey the holographic principle due to Stokes' theorem?

Does the universe obey the holographic principle due to Stokes' theorem? \begin{equation} \int\limits_{\partial\Omega}\omega = \int\limits_{\Omega}\mathrm{d}\omega. \end{equation} Can this ...
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610 views

Where do I start with Non-Euclidean Geometry?

I've been trying to grok General Relativity for a while now, and I've been having some trouble. Many physics textbooks gloss over the subject with an "it's too advanced for this medium", and many ...
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91 views

Conservative Vector Fields

I was always told that to find whether or not a vector field is conservative, see if the curl is zero. I have now been told that just because the curl is zero does not necessarily mean it is ...
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268 views

In the static spacetime, the extrinsic curvature of hypersurface $t=constant$ is zero

How can I prove that in the static spacetime, the extrinsic curvature of hypersurface $t=constant$ is zero? My efforts all are failed. Any hint would be greatly appreciated.
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485 views

How do we know the geodesic is a minimum?

The geodesic equation is derived from the Euler-Lagrange equation, which (as I understand it) is a necessary but not sufficient condition to ensure that the geodesic is a minimum. The introductory GR ...
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Why is the phase space a symplectic manifold rather than a manifold with a metric?

Why does phase space require a symplectic geometry rather than a metric? Is there some scenario where a metric is unable to describe the notion of length in phase space, specifically in relation to ...
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94 views

Aside from experimental evidence, is there any reason to model space as Euclidean?

Obviously experiment is the end-all-be-all of any science, but I'm curious if there's any a priori reason to model space as Euclidean three-space (from a pre-relativity viewpoint, of course; I'm ...
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243 views

Dimensions of strings in string theory

In the above image taken from wikipedia, at the string level the strings have been shown as some loops, the article in wikipedia says that in string theory the particles at lower level are broken ...
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217 views

Metric tensor in special and general relativity

I'm having trouble understanding the metric tensor in general relativity. What I've understood so far has come from my course lecture notes used in conjunction with "The Road to Reality" by Roger ...
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201 views

Why do we require manifolds to be a topological space?

Roughly speaking, we define a manifold $M$ to be covered by a set of charts $\{(U_i , \varphi_i)\}$ such that locally the $n$-dimensional manifolds looks like $\mathbb{R}^n$. One of the conditions is ...
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259 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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671 views

Christoffel symbols and affine connection

What is the difference between the "affine connection" (S. Weinberg, Cosmology) and "Christoffel symbols?"
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684 views

Stokes' theorem in GR

I read this formula in Sean Carroll's book of GR: $$\int_{\Sigma}\nabla_{\mu}V^{\mu}\sqrt{g}d^nx~=~\int_{\partial\Sigma}n_{\mu}V^{\mu}\sqrt{\gamma}d^{n-1}x$$ where n is the 4-vector orthogonal to ...
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182 views

What makes a coordinate curved?

Bear with me while I try to explain exactly what the question is. The question Can a curvature in time (and not space) cause acceleration? is imagining a coordinate system in which the curvature is ...
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228 views

Do partial derivatives commute on tensors?

For example; is $$\partial_{\rho}\partial_{\sigma}h_{\mu\nu} - \partial_{\sigma}\partial_{\rho}h_{\mu\nu}=0$$ correct?
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Real, non-constant scalar field with special properties in class of 4-dimensional spacetimes

David Deutsch (Oxford University) asked the following question which I think is an interesting one: In what class of 4-dimensional spacetimes does there exist a real, non-constant scalar field φ with ...
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980 views

Covariant derivative and Leibniz rule

I read the Wikipedia page about the covariant derivative, my main problem is in this part: http://en.wikipedia.org/wiki/Covariant_derivative#Coordinate_description Some of the formulas seem to lead ...
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74 views

Two expressions for topological instanton number

I have begun to study instantons and I have the following difficulty: $\newcommand{tr}{\operatorname{Tr}}$ I am considering theory with $SU(2)$ gauge group: $S=\frac{1}{2g^{2}}\int \tr ...
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73 views

What's the point of a cobasis?

I've been learning about tensor analysis, and things have been going well so far, but I'm a bit stuck when it comes to the idea of a cobasis (by which I mean the reciprocal basis; not sure which term ...
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82 views

Question about exterior derivatives

I know from Carroll that the integration in GR is basically a mapping from n-form to the real number. And it's given that $$d^nx=dx^0\wedge\ldots\wedge ...
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116 views

“WLOG” re Schwarzschild geodesics

Why, when studying geodesics in the Schwarzschild metric, one can WLOG set $$\theta=\frac{\pi}{2}$$ to be equatorial? I assume it is so because when digging around the internet, most references seem ...
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141 views

Wick Rotation in Curved space

So over time I have learned to do exhaustive searches before asking things here. Wick rotations are cool if you are trying to work in qft and make statements about the thermodynamics of some physical ...
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173 views

Computing Curvature via Cartan Formalism

Given a metric $g_{\mu \nu}$, one can select an orthonormal basis $\omega^{\hat{a}}$ such that, $$ds^2= \omega^{\hat{t}}\otimes\omega^{\hat{t}} - \omega^{\hat{x}} \otimes \omega^{\hat{x}} - ...$$ By ...
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115 views

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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Proof that higher genus surface admits a metric of negative Ricci scalar everywhere

In the Green, Schwarz and Witten Superstring Theory textbook, the paragraph below equation 3.3.15 says, For genus greater than one, it can be shown that the surface admits a metric of everywhere ...
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226 views

What are the generators of spherical symmetry?

The title says it all. I think this should be a pretty simple question but I just couldn't find the answer. Ok -- I'll give a bit more context to my question. I'm encountering this in the context of ...
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793 views

What's the idea behind the Riemann curvature tensor?

The Riemann curvature tensor can be expressed using the Christoffel symbols like this: $R^m{}_{jkl} = \partial_k\Gamma^m{}_{lj} - \partial_l\Gamma^m{}_{kj} + \Gamma^m{}_{ki}\Gamma^i{}_{lj} ...
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386 views

Could metric expansion create holes, or cavities in the fabric of spacetime?

Is it possible for metric expansion to create holes, or cavities in the fabric of spacetime? According to the Schwarzschild metric, the metric expansion of space around a black hole goes to infinity ...
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157 views

How should one interpret the de Sitter slicings?

When 'constructing' the usual de Sitter space in $\mathcal{M^5}$ by invoking the contraint $-X^{2}_{0} +X^{2}_{1} +X^{2}_{2} +X^{2}_{3} + X^{2}_{4} = \alpha^2$ we quickly see that we end up with a ...
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281 views

Relativistic space-time geometry

What subject (suggest book titles, etc.) should I study to get a clear grasping of hypersurfaces, 2-surfaces, and integration on them, mostly in special relativity (I'm not messing with general ...
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107 views

The relationship between Lorentz Lie algebra and curvature

Here I transfered the question from the comment The relationship between spin and spinor curvature How $\mathcal{R}_{ab} = \frac{1}{4}R_{abst}\gamma^s \gamma^t$ is from $\Psi \mapsto \Psi + ...
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95 views

The properity of $\mathbb{R}^4$ that has infinitely many differential structures is related to Yang-Mills field?

I heard a saying that $\mathbb{R}^4$ having infinitely many differential structures which are not diffeomorphic to each other has a relationship with Yang-Mills field. Does anyone can explain it, and ...
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121 views

What is the notion of a spatial angle in general relativity?

Is there a notion of spatial angles in general relativity? Example: The world line of a photon is given by $x^{\mu}(\lambda)$. Suppose it flies into my lab where I have a mirror. I align the mirror ...
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154 views

Hilbert action's invariance under general coordinate changes

In an article, when considering invariance of the Hilbert action under a general coordinate change this formula appears for how the metric changes ...
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106 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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92 views

Metric of a manifold foliated by maximally symmetric submanifold

I am reading the last chapter (Schwarzchild solution and Black Holes) of Sean Caroll's GR notes (http://arxiv.org/abs/gr-qc/9712019). While talking about spherical symmetry, he says how the ...
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451 views

How does (or can) SR/GR extend to phase space or symplectic manifolds?

I'm asking this question because of an article in New Scientist about a recent preprint by a group including Lee Smolin. I haven't taken the time to comprehend the paper completely. My knowledge of ...
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209 views

Is there a simple way to define/solve for a null cone for a general spacetime geometry?

I'm wondering if there's any simple way to define and solve for a null cone for a general spacetime geometry in $n+1$ dimensions, given its vertex $p^\mu$. I can't seem to find a simple way to do it ...
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119 views

Geometric meaning of parallel transport

The definition of parallel transport of a vector $v^b$ along a curve $C$ with tangent field $\it{t}^a$ is given by Wald's GR as $$t^a \nabla_a v^b = 0$$ Is it correct to think of $\nabla_a v^b$ as ...
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Why is $D$ a $2$-form and $E$ a $1$-form?

Usually in electrostatics we start by introducing the vector field $\mathbf{E}$ representing the electric field due to some charge distribution. Later when we study fields in materials we consider the ...
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83 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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429 views

Lie derivative of Riemann tensor along killing vector ( = 0 )

I'm currently learning the mathematical framework for General Relativity, and I'm trying to prove that the Lie derivative of the Riemann curvature tensor is zero along a killing vector. With the ...