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

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Curvature Invariants in General Relativity and Singularities

Suppose that I want to check if a given metric is singular or not. I'm interested in curvature singularities, not coordinate singularities, so I can look to scalars made with Ricci, Riemann and Weyl ...
11
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584 views

Classical mechanics: Generating function of lagrangian submanifold

I have a short question regarding the geometrical interpretation of the Hamilton-Jacobi-equation. One has the geometric version of $H \circ dS = E$ as an lagrangian submanifold $L=im(dS)$, which is ...
8
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85 views

Is it known what the necessary and sufficient conditions are for the existence of a “3+1 split” (by means of a foliation) of a (Lorentzian) manifold?

When trying to do physics on a more general pseudo-Riemannian manifold we want to require that there is a foliation of this manifold into three-dimensional subspaces. By this I mean we would like to ...
8
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122 views

Trajectories in Rindler space with zero net time dilation

I've discovered a family of curves in Rindler space that have zero net time dilation. However I struggle to see why this should be so, i.e. what the physical significance of these curves is. My ...
8
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334 views

Extended Born relativity, Nambu 3-form and ternary (n-ary) symmetry

Background: Classical Mechanics is based on the Poincare-Cartan two-form $$\omega_2=dx\wedge dp$$ where $p=\dot{x}$. Quantum mechanics is secretly a subtle modification of this. By the other hand, ...
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108 views

Can some components of metric be Finslerian while the others be Riemannian?

A Finsler metric reduces to a Riemann metric in case it loses its dependence on velocities. Now, my question is this: Can we have a Finsler metric in which some components of the metric have velocity ...
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99 views

Is there a null incomplete spacetime which is spacelike and timelike complete?

Geodesic completeness, the fact we can make the domain of the geodesic parametrized with respect an affine parameter the whole real line, is an important concept in GR. Especially, because the lack of ...
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106 views

Is it possible to have 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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177 views

What exactly is the relationship between the symplectic 2-form and the frequency of leaves of integrable systems in classical mechanics?

In classical mechanics we equip a differential manifold with a closed symplectic 2-form $\omega$. The symplectic leaves of integrable systems also have a unique frequency, in literature denoted ...
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212 views

Why is the Ricci tensor diagonal for isotropic spacetime?

I'm reading Zee's Einstein Gravity in a Nutshell and while calculating the Ricci tensor for FRW spacetime he claims that because the spacelike slices of constant $t$ are rotationally invariant, the ...
5
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159 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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2k 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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216 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 ...
5
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240 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 ...
4
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202 views

How does one determine if a spacetime is globally hyperbolic?

A spacetime $M$ is said to be globally hyperbolic if it is strongly causal and if the sets $J^+(p)\cap J^-(q)$, for all $p,q\in M$, are compact. (For more information, see the Wiki article on causal ...
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120 views

How the Poisson bracket transform when we change coordinates?

I'm studying the book Geometric Mechanics by Darryl D. Holm and there's one exercise in the book I'm not quite getting what has to be done. The same discussion the author makes in the book is made on ...
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65 views

Trajectories in AdS

On page 2 of this paper (http://arxiv.org/abs/1106.6073), Maldacena explains (and has a very nice picture) showing the trajectories that a timelike and null particle would take in AdS space. Of ...
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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 ...
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178 views

Geometric interpretation of quantum Yang-Mills field

In most books\articles review geometric interpretation of classical Yang-Mills field in terms of principal bundle, connections...etc. What are geometric interpretation of quantum Yang-Mills field? ...
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67 views

How many Killing spinors exist on $S^5$?

So, I know that on $S^n$, a spinor of the form $$ \Sigma^\pm = \frac{1 \pm i\gamma^\alpha z_\alpha}{\sqrt{1+z^2}}\Sigma_0$$ where $\Sigma_0$ is a constant spinor, is a Killing spinor on $S^n$ ...
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82 views

Are there relativistic theories with spacetime modelled on $\mathbb C^4$ rather than real Minkowski space $\mathbb R^4$?

Does anybody know of references to theories where relativity & spacetime is modelled on a (complex/Kähler) manifold which is locally diffeomorphic to $\mathbb C^4$ rather than $\mathbb R^4$, hence ...
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184 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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319 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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103 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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174 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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84 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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215 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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120 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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895 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 ...
4
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64 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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101 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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313 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, ...
3
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42 views

A problem with ADM mass in the derivation of 1st law of black hole thermodynamics

The definition of ADM mass is $$M=\frac{1}{16\pi}\lim_{r\rightarrow\infty}\int \left(\frac{\partial h_{\mu\nu}}{\partial x^\mu}-\frac{\partial h_{\mu\mu}}{\partial x^\nu} \right)N^\nu dA$$ according ...
3
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74 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 ...
3
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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 ...
3
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81 views

Projector and delta function on a cycle $\Sigma$ of a manifold $\mathcal{M}_6$

In the paper ``Hierarchies from Fluxes in String Compactifications'' by Giddings, Kachru and Polchinski, the following example is considered for a localized source that may have negative tension (my ...
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77 views

Branes wrapping curves in M-theory. What does it mean?

What does it mean that a M5-branes wraps a holomorphic curve in M-theory? In specific a lot of Vafa's paper involve various branes (not only M5) wrapping some cycles. What does this really mean ...
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317 views

Maxwell's equation in curved spacetime - how come? And experimental evidence?

I'm trying to understand the generalization of Maxwell's equations to curved spacetime. In FLAT (Minkowski) SPACETIME: If we define the "four-potential" as $$\ ...
3
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124 views

Chocolate dynamics

Now I have found a possible model on how to describe chocolate when it is chewed. It has to do with geometrical transformations when a curve $\gamma$ intersects a manifold $M$. The chocolate is ...
3
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100 views

Time functions in general relativity

In my general relativity notes a function $f$ is called time function, if $\nabla f$ is time-like past-pointing. Say that we are in Schwarzschild spacetime and I want to check if $f=t$ is a time ...
3
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83 views

Classical toy models of particles with intrinsic spin

Related to my question here (spacetime torsion, the spin tensor, and intrinsic spin in einstein cartan theory), I'd like to be able to put test particles on a manifold with non-zero torsion and see ...
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100 views

What are Killing spinors?

What are Killing spinors? How can they be motivated? Are they directly related to Killing vectors and Killing tensors and is there an overarching motivation for all three objects? Any answer is ...
3
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117 views

Ricci curvature of embedded spacetime

If I am not mistaken, there is a theorem which states that every Riemannian manifold can be embedded in the $n$-dimensional Euclidean space for some large-enough $n$. Does it also hold for ...
3
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0answers
90 views

Is the “Force” of Gravity Simply Hamilton's Principle on a Curved Spacetime?

It's my understanding that General Relativity abstracts away the concept of gravity as a force, and instead describes it as a feature of spacetime by which massive objects cause curvature. Then it ...
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99 views

Avoiding Pseudo-tensors when addressing global conservation of energy in GR

Discussions about global conservation of energy in GR often invoke the use of the stress-energy-momentum pseudo-tensor to offer up a sort of generalization of the concept of energy defined in a way ...
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72 views

Asymtotically flat spacetime applicable for spacetimes which are not diffeomorphic to $\mathbb{R}^4$

I wanted to investigate changes on a compact 4-manifold $M$. More specifically it is the K3-surface. I follow a paper by Asselmeyer-Maluga from 2012. The idea there was to make sure that the manifold ...
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55 views

Scalar product of torsional forms - how are the standard identities modified?

It is known that for any smooth, orientable, compact manifold $X$ without boundary and $\alpha \in \Omega^{r}(X), \beta \in \Omega^{r-1}(X)$ it holds \begin{equation} (d\beta,\alpha)= (\beta, ...
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70 views

Rigid rectangle in Schwarzschild

Say I build a perfect rectangle. Side lengths $l_1$ and $l_2$ and perfect right angles. I am on earth and the metric is given by the Schwarzschild metric. Setting $dt=0$ leads to the spatial ...
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204 views

Computing the Einstein tensor for a spherically symmetrical metric using the tetrad formalism

I am having some trouble understanding how to use the tetrad formalism. I will start with what I have so far, my question will be after that. I begin with the metric $$ \text{d}s^2 = e^{2a} \text{ ...
3
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126 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 ...