Tagged Questions
2
votes
1answer
43 views
Are there any restrictions on building the topology of spacetime out of the complement of open balls?
I assume that for a Lorentzian manifold (i.e. with Minkowski signature), the analog of an open ball is the interior of a light cone. My question is motivated by the observation that whereas any point ...
2
votes
0answers
43 views
Special case of the hodge decomposition theorem [migrated]
I am trying to prove the following special case of the hodge decomposition theorem in differential geometry for a n component vector field $V_i$ in $\mathbb{R}^n$.
any vector can be written as the ...
7
votes
6answers
259 views
In coordinate-free relativity, how do we define a vector?
Relativity can be developed without coordinates: Laurent 1994 (SR), Winitzski 2007 (GR).
I would normally define a vector by its transformation properties: it's something whose components change ...
9
votes
1answer
213 views
Is there a “covariant derivative” for conformal transformation?
A primary field is defined by its behavior under a conformal transformation $x\rightarrow x'(x)$:
$$\phi(x)\rightarrow\phi'(x')=\left|\frac{\partial x'}{\partial x}\right|^{-h}\phi(x)$$
It's fairly ...
2
votes
1answer
93 views
Energy Functional
I am a graduate student in pure mathematics, during my study on Ricci Flow I faced some functional known as energy functional. For example Einstein-Hilbert functional is called an energy functional, ...
1
vote
0answers
50 views
Geometry for Physics [duplicate]
I am currently a high school student interested in a research career in physics. I have self taught myself single variable calculus and elementary physics upto the level of IPHO . And I am comfortable ...
3
votes
1answer
158 views
Change of coordinates from an arbitrary frame to a locally inertial frame in General Relativity
If I have the following metric:
$$ds^2=(1-2\phi)c^2 dt^2 - (1-2 \phi)(dx^2+dy^2+dz^2)$$
$\phi$ being the gravitational potential with $|\phi| << 1$ everywhere.
How do I find a coordinate ...
5
votes
3answers
244 views
What is a dual / cotangent space?
Dual spaces are home to bras in quantum mechanics; cotangent spaces are home to linear maps in the tensor formalism of general relativity. After taking courses in these two subjects, I've still never ...
10
votes
3answers
351 views
Representing forces as one-forms
First of all, sorry if any of those things are silly or nonsense, I'm just trying to understand better how the concepts of forms, exterior derivative and so on can be used in physics.
This question ...
0
votes
0answers
59 views
Is there a book that discusses General Relativity in terms of Modern Differential Geometry? [duplicate]
All of the physics books that I've seen which discuss General Relativity do so in terms of coordinates - the tensor calculus - even though the naturally relevant entities are invariant under general ...
7
votes
1answer
272 views
Diffeomorphisms, Isometries And General Relativity
Apologies if this question is too naive, but it strikes at the heart of something that's been bothering me for a while.
Under a diffeomorphism $\phi$ we can push forward an arbitrary tensor field $F$ ...
3
votes
1answer
98 views
An issue about the compactness and the existence of CTCs
There is a well known fact that a compact spacetime necessarily contains a closed timelike curve (CTC). Proof can be found in several books on GR (e.g. Hawking, Ellis, Proposition 6.4.2), and in ...
3
votes
2answers
209 views
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
votes
0answers
65 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
votes
0answers
147 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 ...
2
votes
1answer
191 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 ...
3
votes
4answers
262 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 ...
2
votes
2answers
276 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 ...
1
vote
1answer
115 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 ...
3
votes
3answers
375 views
Equations of fluid dynamics and differential geometry
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 ...
1
vote
1answer
70 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 ...
3
votes
4answers
485 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 ...
2
votes
0answers
135 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 & ...
2
votes
1answer
217 views
Vanishing Ricci flow on a curved manifold
If I understand this right the Ricci flow on a compact manifold given by
$\partial g_{\mu \nu} = - 2R_{\mu \nu} + \frac{2}{n}\!R_{\alpha}^{\alpha} \,g_{\mu \nu}$
tends to expand negatively curved ...
6
votes
0answers
251 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 ...
9
votes
1answer
192 views
Can Fermionic symmetries be fully integrated into geometric deformation complexes or symplectic reduction?
How should a geometer think about quotienting out by a Fermionic symmetry? Is this a formal concept? A strictly linear concept? A sheaf theoretic concept?
How does symplectic reduction work with odd ...
6
votes
1answer
305 views
The role of metric in the Wave Equation
The wave equation is often written in the form
$$(\partial^2_t-\Delta)u=0,$$
involving the Laplace-Beltrami operator $\Delta$. However, the Laplace-Beltrami operator $\Delta$ is defined only in the ...
8
votes
4answers
883 views
What does a frame of reference mean in terms of manifolds?
Because of my mathematical background, I've been finding it hard to relate the physics-talk I've been reading, with mathematical objects.
In (say special) relativity, we have a Lorentzian manifold, ...
2
votes
3answers
128 views
Length of a curve in D dimensional euclidean space
In a book I am reading on special relativity, the infinitesimal line element is defined as $dl^2=\delta_{ij}dx^idx^j$ (Einstein summation convention) where $\delta_{ij}$ is the euclidean metric. Next, ...
