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

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Electromagnetism for Mathematician

I am trying to find a book on electromagnetism for mathematician (so it has to be rigorous). Preferably a book that extensively uses Stoke's theorem for Maxwell's equations (unlike other books that on ...
2
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0answers
73 views

A doubt about fuchsian functions in physics?

I'm not sure if this is the right place (or math.stackexchange?) to ask the next What is the difference between fuchsian, theta-fuchsian, and kleinian functions? Please, suggest me an introductory ...
4
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2answers
123 views

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 ...
2
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1answer
208 views

Geometric interpretation of perturbation theory in quantum field theory

I am studying GR right now, and one interesting thing I learned about vectors is that they are defined to have the same properties as derivatives. With this in mind, can I make a differential ...
2
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1answer
687 views

Tiling hexagons on a sphere surface

In attemopt to understand basic principles of non-Euclidean geometry and its relation to physical space, I am reading General Relativity by Ben Crowell. On page 149 there is a discussion of hexagons ...
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0answers
210 views

7 sphere, is there any physical interpretation of exotic spheres?

Basically an exotic sphere is topologically a sphere, but doesn't look like a one. Or more accurately: homeomorphic but not diffeomorphic to the standard Euclidean n-sphere The first exotic ...
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0answers
161 views

Extending General Relativity with Kahler Manifolds?

Standard general relativity is based on Riemannian manifolds. However, the simplest extension of Riemannian manifolds seems to be Kahler manifolds, which have a complex (hermitian) structure, a ...
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1answer
496 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
503 views

Metric tensor under coordinate transformation x to y(x)

Say I have a metric representation $g_{\mu\nu}$ in a coordinate system $x$ and I want to find the representation of the metric in a new set of coordinates $y = y(x)$. I know how to do this if you are ...
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1answer
146 views

Symmetries of spacetime and objects over it

I guess according to mathematical didactic, we first think of spacetime as a set and we reason about elements of its topology and then it's furthermore equipped with a metric. Appearently it is this ...
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1answer
92 views

Is there a formula to work out how much the fabric of spacetime bends?

From my knowledge, a big mass (planet star etc) can bend the fabric of spacetime. Is there a formula that we can use to work out how much it bends?
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1answer
857 views

How to get the gradient potential in polar coordinate

In polar coordinate, $$\nabla U = \frac{\partial U}{\partial r}\hat{\mathbf{r}} + \frac{1}{r}\frac{\partial U}{\partial \theta}\hat{\mathbf{\theta}} .$$ Can anyone show me how to get this result?
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1answer
1k views

Killing vector fields

I am facing some problems in understanding what is the importance of a Killing vector field? I will be grateful if anybody provides an answer, or, refer me to some review or books.
2
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1answer
207 views

Cosmology with a negative cosmological constant

Based on the Friedmann equation for a universe with only cosmological constant, $$\left(\frac{\dot{a}}{a}\right)^2 \sim \Lambda$$ I would expect the scale factor $a(t) \sim e^{-it}$ if $\Lambda < ...
2
votes
2answers
154 views

Diffusion of probability amplitudes

Let's say I have a probability amplitude $\psi:\Sigma\rightarrow\mathbb{C}$ for some domain $\Sigma$ (so, $\psi$ satisfies $\int_\Sigma |\psi|^2=1$). Is there a way to use $\psi$ as initial ...
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2answers
127 views

Similarity of probability amplitude functions

Let's say I have two probability amplitude functions given by $\psi_1$ and $\psi_2$. That is, $\psi_i:\Sigma\rightarrow\mathbb{C}$ for some domain $\Sigma$ with $\int_\Sigma|\psi_i|^2=1$ for ...
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2answers
686 views

Metric tensor and its inverse

Is it always allowed to represent the metric tensor $g_{\mu \nu}$ in General Relativity as a $4\times 4$ matrix? If the last one is represented for example with a $4\times 4$ matrix ...
4
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3answers
2k views

What is a Killing vector field?

I recently read a post in physics.stackexchange that used the term "Killing vector". What is a Killing vector/Killing vector field?
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2answers
268 views

Mathematical probabilistic interepretation of probability amplitude

As a warning, I come from an "applied math" background with next to no knowledge of physics. That said, here's my question: I'm looking at the possibility of using probability amplitude functions to ...
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1answer
120 views

What is the curvature scalar $\Psi_{4}$?

What is the curvature scalar $\Psi_{4}$? Is it related to the scalar curvature $R$? What does its real and imaginary parts represent?
2
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1answer
251 views

Killing vectors for SO(3) (rotational) symmetry

I am reading a paper$^1$ by Manton and Gibbons on the dynamics of BPS monopoles. In this, they write the Atiyah-Hitchin metric for a two-monopole system. The first part is for the one monopole moduli ...
9
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3answers
1k views

What does the dual of a tensor mean (e.g. dual stress tensor in relativistic ED)?

I know what the dual of a vector means (as a map to its field), and I am also aware of of the definition a dual of a tensor as, $$F^{*ij} = \frac{1}{2} \epsilon^{ijkl} F_{kl}\tag{1}$$ I just don't ...
3
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4answers
341 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 ...
3
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2answers
406 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
138 views

what is wrong with the following argument about stokes law in compact universes?

I want to understand what is wrong with the following argument: in a topologically compact spacetime, a closed 3D boundary separates the spacetime in two connected components, because of this ...
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3answers
230 views

Where 2 comes from in formula for Schwarzschild radius?

In general theory of relativity I've seen several times this factor: $$(1-\frac{2GM}{rc^2}),$$ e.g. in the Schwarzschild metric for a black hole, but I still don't know in this factor where 2 comes ...
3
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3answers
590 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 ...
4
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1answer
851 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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1answer
106 views

What is the physical intepretation of harmonic coordinates?

When I see harmonic coordinates used somewhere, what should my association be? Is there some general use or need to consider the harmonic cooridnate condition? I don't really see what's ...
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5answers
1k views

What does symplecticity imply?

Symplectic systems are a common object of studies in classical physics and nonlinearity sciences. At first I assumed it was just another way of saying Hamiltonian, but I also heard it in the context ...
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4answers
466 views

Hamiltonian and the space-time structure

I'm reading Arnold's "Mathematical Methods of Classical Mechanics" but I failed to find rigorous development for the allowed forms of Hamiltonian. Space-time structure dictates the form of ...
7
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7answers
1k views

Introduction to differential forms in thermodynamics

I've studied differential geometry just enough to be confident with differential forms. Now I want to see application of this formalism in thermodynamics. I'm looking for a small reference, to learn ...
5
votes
1answer
550 views

Is 4-volume element a scalar or a pseudoscalar in special relativity?

In general relativity 4-volume element $\mathrm{d}^4 x = \mathrm{d} x^0\mathrm{d} x^1 \mathrm{d} x^2\mathrm{d} x^3$ is clearly a pseudoscalar (or scalar density) of weight 1 since it transforms as ...
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1answer
78 views

Smooth trajectory on a smooth manifold

Physicists talk about a smooth trajectory of a particle on a smooth manifold and they label it as q(t) where q_1(t)....q_n(t) are component functions coming from the homeomorphism. I don't see how we ...
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3answers
230 views

What are some mechanics examples with a globally non-generic symplecic structure?

In the framework of statistical mechanics, in books and lectures when the fundamentals are stated, i.e. phase space, Hamiltons equation, the density etc., phase space seems usually be assumed to be ...
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6answers
1k views

What is a tensor?

I have a pretty good knowledge of physics but couldn't understand what a tensor is. I just couldn't understand it, and the wiki page is very hard to understand as well. Can someone refer me to a good ...
6
votes
1answer
842 views

Maxwell's equations in microscopic and macroscopic forms, and quantization

The macroscopic Maxwell's equations can be put in terms of differential forms as $$\mathrm{d}\mathrm{F}=0,\quad\delta \mathrm{D}=j\implies \delta j=0,\quad \mathrm{D}=\mathrm{F}+\mathrm{P}.$$ ...
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4answers
320 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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0answers
431 views

de Sitter and anti de Sitter metric

Is the following correct for the distance $d$ from the origin $(0,0)$ to point $(t,x)$ in the 2-dimensional de-Sitter and anti de-Sitter spaces? Here, $t$ is time and the distance may be called the ...
4
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1answer
145 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 ...
3
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3answers
281 views

How to connect Einstein's Special Relativity(SR) with General Relativity(GR)?

How Einstein's SR becomes GR? $$ds^2=dr^2-c^2dt^2,$$ $$ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}.$$ When the $s$ is constant $ds^2=0$, isn't it true? How to connect Einstein's SR with GR? What is the ...
3
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4answers
629 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 ...
6
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1answer
161 views

If a fundamental theory exibits e.g. a mirror symmetry, in what sense it the underlying geometry real?

Are the more recently discovered symmetries in string theory such that the theories based on mirroring geometries are absolutely the same from an observable point of view? I have mirror symmetry ...
3
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0answers
81 views

Relating the deformation of Calabi-Yau metrics and the conformal quantum field theories

(v2) As I read e.g. in this question, the nice holonomy group features of Calabi-Yau manifolds are valuable regarding supersymmetry (I suspect because it's a symmetry involving the target manifold, ...
6
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1answer
853 views

What is the stress energy tensor?

I'm trying to understand the Einstein Field equation equipped only with training in Riemannian geometry. My question is very simple although I cant extract the answer from the wikipedia page: Is the ...
2
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0answers
167 views

What are the topics of string theory that are comprehensible with only a mathematical background on Manifolds and Algebraic Topology?

What are the topics of string theory that are comprehensible with only a mathematical background on manifolds and algebraic topology? Also, I have read only the first four chapters in Peskin & ...
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0answers
174 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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2answers
533 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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3answers
1k views

Why do objects follow geodesics in spacetime?

Trying to teach myself general relativity. I sort of understand the derivation of the geodesic equation ...
2
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1answer
212 views

A question regarding particle trajectories in the symplectic manifold formalism

How to solve a free particle on a 2-sphere using symplectic manifold formalism of classical mechanics ? Is there a way to get coriolis effect directly, without going into Newton mechanics? And is ...