# Tagged Questions

In mathematics, topology examines the properties of space (such as connectedness and compactness) that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.

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### The expansion of the universe and its edge

I have understood that the universe is expanding at an accelerated rate. I have also thought, perhaps mistakenly, that this expansion means not that the masses in the universe are moving away from ...
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### If a point r lies in the boundary of the chronological future of another point p, why does the chronological future of r belong to that of p?

I am studying the global causality of the spacetime. Here, I come across a problem. Suppose a point $r\in \partial I^+(p)$. $I^+(p)$ is the chronological future of a different point $p$ in spacetime....
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### How does a position based force work on a torus?

Consider a 2 particle system in $\Bbb R^2$. Let's say they have some force acting between them. For the sake of argument let that be $F=\mathbf r/|\mathbf r|^3$. This is a simple inverse square system ...
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### Yang-Mills potential and principal bundles

In section 2.7.2 of Bertlmann's "Anomalies in quantum field theory", it is stated that since a non-trivial principal bundle (based on a Lie group $G$) does not admit a global section, the Yang-Mills ...
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### Relation between projective representations, connectivity of a group manifold and number of equivalence classes of paths

Projective representations of a multiply-connected group is defined as $$U(g_1)U(g_2)=c(g_1,g_2)U(g_1g_2)$$ where $c(g_1,g_2)$ is phase. Reading various articles, and this old post of mine, it appears ...
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### Lorentz surfaces, conformal metrics and eigenvalues

From what I understand of Lorentz surfaces (spacetimes of dimension 2), it seems that, according to Kulkarni's theorem, two reasonable enough Lorentz surfaces (I am only interested in surfaces with ...
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### A question on the Chern number and the winding number?

Let $\mid \psi(x,y) \rangle$ be a normalized wavefunction living in a $d$-dimensional Hilbert space and depend on two real parameters $(x,y)$ that belong to a closed surface (e.g., $S^2, T^2$, ...). ...
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### Interesting Hamiltonian System [duplicate]

The definition of a Hamiltonian system I am working with is a triple $(X,\omega, H)$ where $(X,\omega)$ is a symplectic manifold and $H\in C^\infty(X)$ is the Hamiltonian function. I am wondering if ...
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### How can I show that inversion is continuously connected to a reflection?

From Ex 3.1 in the TASI lectures on the conformal bootstrap: http://arxiv.org/abs/1602.07982 the problem is the inversion map (with Euclidean signature) $$I\colon x^\mu \mapsto \frac{x^\mu}{x^2}$$ ...
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### Difference Between Algebra of Infinitesimal Conformal Transformations & Conformal Algebra

in Blumenhagen Book on conformal field theory, It is mentioned that the algebra of infinitesimal conformal transformation is different from the conformal algebra and on page 11, conformal algebra is ...
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### Wave equation for odd spacetimes and source terms

It seems to be quite common practice, when solving the wave equation in spacetimes with odd topologies or horizons, to decompose the solution into a sum of the various origins (or destinations) of the ...
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### Homotopy proof of the lack of foliation of the Gödel metric

A common proof of the lack of foliation of the Gödel universe, apparently mostly copy pasted from Hawking and Ellis, goes thusly : A closed timelike curve must cross a spacelike hypersurface ...
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### Validity of topological thermodynamics?

I've been reading some material by R. Kiehn, developing a topological approach to non-equilibrium thermodynamics through Cartan forms, where the fundamental claim is that irreversible processes are ...
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### Causal structure, time orientability and equivalence classes

Quoting from this Wikipedia article, if $(M,g)$ is a Lorentzian manifold then the tangent vectors at each point in the manifold can be classified into three different types. Using a $(+,-,-,-)$ metric ...
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### Geometry inside the event horizon

I'm trying to understand intuitively the geometry as it would look to an observer entering the event horizon of a Schwarszchild black hole. I would appreciate any insights or corrections to the above. ...
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### Can a rotating black hole have a donut-shaped event horizon? [closed]

It is conjectured that a rotating black hole has at its center a ring-shaped singularity. Thus, at the center of the ring-shaped singularity the gravitational field must be zero (similar to ...
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### Equivariant cohomology and Mayer-Vietoris sequence

I'm reading this article upon topological field theory and I'm a bit confused about the way he compute equivariant cohomology of $S^2$ wrt $\mathrm{U}(1)$, i.e. $H^\bullet_{S^1}(S^2)$. You can find ...
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### Constant electric background field from theta-term in QCD?

The theta-term of $1+1$ dimensional QED corresponds to a constant electric background field. Does the theta-term of QCD also induce a constant electric background field, which could be measurable in ...
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### Book covering differential geometry and topology required for physics and applications

I am a physics undergrad, and interested to learn Topology so far as it has use in Physics. Currently I am trying to study Topological solitons but bogged down by some topological concepts. I am not ...
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### Which are the best introductory books for topology, algebraic geometry, differential geometry, manifolds, etc, needed for string theory? [duplicate]

Which are the best introductory books for topology, algebraic geometry, differential geometry, manifolds, etc, needed for string theory?
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### Book recommendations on geometrical methods for physicists (like Topology, Diff. Geometry) [duplicate]

I would like to obtain a book that has to do with geometrical methods/subjects for physicists. When I say geometrical methods/subjects I mean things like Topology, Differential Geometry, Lie ...
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### If we live on the surface of Earth then why Earth images shows maps around it? [closed]

If you visits google map and go to earth we see the image as attached below. My question is if the earth is round like sphere ball and if we live on the surface of this ball (point me if i am ...
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### What do we exactly mean by a “topological object” in physics?

I have been working on topological defects like monopoles, etc. for some time. One think that I have not been able to understand is the physical meaning of the phrase "topological object". I have ...
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### What's the metric of the Standard Non-Time-Orientable Spacetime

If you've read any spacetime topology, you know that spacetime. It is the amazing rotating lightcone identified after half a rotation. And outside of De Sitter space with some identifications, it is ...
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### Moving the endpoints of a wormhole towards each other

Suppose we have a perfectly safe portal/wormhole and we place the two endpoints facing each other so that a person between them would see an endless corridor (with infinite number of herself). What ...
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### Some questions about gauge theory

Let's talk quantum mechanics. I have a charged particle moving on a sphere. It has a wave function $\psi$. At time $t=0$, there is no magnetic flux piercing the sphere. Instantaneously, I introduce a ...
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### Physical meaning of the Morse functions? [closed]

What is the physical correspondence of the Morse functions in a physical system? Currently I am studying Mirror symmetry but I can not get a physical intuition out of it.
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### Resources for algebraic topology in condensed matter physics

I wanted to know if anyone had any good introductions on algebraic topology for the theoretical physicist? I am particularly interested in applications to condensed matter physics, but would be happy ...