Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.
29
votes
6answers
1k views
What is known about the topological structure of spacetime?
General relativity says that spacetime is a Lorentzian 4-manifold $M$ whose metric satisfies Einstein's field equations. I have two questions:
What topological restrictions do Einstein's equations ...
19
votes
2answers
63 views
Kerr Geometry, Separability and Twistors
One of the remarkable properties of the Kerr black hole geometry is that scalar field equations separate and are exactly solvable (reducible to quadrature), even though naively it does not have enough ...
18
votes
7answers
856 views
Why are differential equations for fields in physics of order two?
What is the reason for the observation that across the board fields in physics are generally governed by second order (partial) differential equations?
If someone on the street would flat out ask ...
15
votes
3answers
2k views
What is the physical meaning of the connection and the curvature tensor?
Regarding general relativity:
What is the physical meaning of the Christoffel symbol ($\Gamma^i_{\ jk}$)?
What are the (preferably physical) differences between the Riemann curvature tensor ($R^i_{\ ...
13
votes
4answers
2k views
Are matrices and second rank tensors the same thing?
Tensors are mathematical objects that are needed in physics to define certain quantities. I have a couple of questions regarding them that need to be clarified:
1-Are matrices and second rank tensors ...
13
votes
1answer
926 views
Flux through a Mobius strip
I was sent here from mathoverflow, hoping for a complete answer to this:
===
A friend of mine asked me what is the flux of the electric field (or any vector field like
$$
\vec r=(x,y,z)\mapsto ...
11
votes
5answers
2k views
Laplace operator's interpretation
What is your interpretation of Laplace operator? When evaluating Laplacian of some scalar field at a given point one can get a value. What does this value tell us about the field or it's behaviour in ...
11
votes
1answer
216 views
Covariant derivatives
I need correctly define covariant derivatives on the coset space $G/H$, where a group $G \equiv \{X_i, Y_a\}$ ($X$ and $Y$ are generators) have a subrgroup $H \equiv \{X_i\}$
Lie algebra of $G$ has ...
11
votes
1answer
438 views
Formulation of Transformation optics using a Material Manifold
Dear Community,
recently, Transformation optics celebrates some sort of scientific revival due to its (possible) applications for cloaking, see e.g. Broadband Invisibility by Non-Euclidean Cloaking ...
10
votes
3answers
325 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 ...
10
votes
4answers
319 views
Discontinuities and nondifferentiability in thermodynamics
In physics and engineering sources, calculus-based formalisms - whether differential forms on a manifold, or "differentials" of functions of several variables - are presented as a way of modeling and ...
10
votes
1answer
461 views
Can GR be derived by postulating a maximum force?
This paper was published in a peer review journal, and claims the answer is yes.
http://arxiv.org/abs/physics/0607090
The derivation in the paper seems more like dimensional analysis hand-waving in ...
9
votes
3answers
413 views
Physical and Geometrical interpretation of Differential Forms
I have a doubt about the physical and geometrical interpretation of differential forms. I've been studying differential forms on Spivak's Calculus on Manifolds, but my real intent is to use those ...
9
votes
1answer
191 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 ...
8
votes
4answers
864 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, ...
8
votes
3answers
809 views
Can spacetime be non-orientable?
This question asks what constraints there are on the global topology of spacetime from the Einstein equations. It seems to me the quotient of any global solution can in turn be a global solution. In ...
8
votes
2answers
395 views
Is spacetime simply connected?
As I've stated in a prior question of mine, I am a mathematician with very little knowledge of Physics, and I ask here things I'm curious about/things that will help me learn.
This falls into the ...
8
votes
1answer
489 views
What all is needed to solve for the metric in GR?
Einstein's field equations are:
$R_{ab} - {1 \over 2}g_{ab}\,R + g_{ab} \Lambda = {8 \pi G \over c^4} T_{ab}$
And since the Ricci curvature tensor is "less information" than the Riemann curvature ...
7
votes
6answers
860 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 ...
7
votes
3answers
608 views
Why is the covariant derivative of the metric tensor zero?
I've consulted several books for the explanation of why
$$\nabla _{\mu}g_{\alpha \beta} = 0,$$
and hence derive the relation between metric tensor and affine connection $\Gamma ^{\sigma}_{\mu ...
7
votes
3answers
561 views
What does the dual of a tensor mean (e.g. dual stress tensor in relativistic ED)?
I know what the dual of a vector means (as a map to its field), and I am also
aware of of the definition a dual of a tensor as,
$$F^{*ij} = \frac{1}{2} \epsilon^{ijkl} F_{kl}\tag{1}$$
I just don't ...
7
votes
2answers
163 views
Equivalence of definitions of ADM Mass
ADM Mass is a useful measure of a system. It is often defined (Wald 293)
$$M_{ADM}=\frac{1}{16\pi} \lim_{r \to \infty} \oint_{s_r} (h_{\mu\nu,\mu}-h_{\mu\mu,\nu})N^{\nu} dA$$
Where $s_r$ is two ...
7
votes
4answers
348 views
Hamiltonian and the space-time structure
I'm reading Arnold's "Mathematical Methods of Classical Mechanics" but I failed to find rigorous development for the allowed forms of Hamiltonian.
Space-time structure dictates the form of ...
7
votes
1answer
254 views
Diffeomorphisms, Isometries And General Relativity
Apologies if this question is too naive, but it strikes at the heart of something that's been bothering me for a while.
Under a diffeomorphism $\phi$ we can push forward an arbitrary tensor field $F$ ...
6
votes
3answers
130 views
From Manifold to Manifold?
Tensor equations are supposed to stay invariant in form wrt coordinate transformations where the metric is preserved. It is important to take note of the fact that invariance in form of the tensor ...
6
votes
2answers
129 views
First and second fundamental forms
I'm writing notes about the 3+1 formalism in general relativity, for myself. Inevitably I came across the notions of first and second fundamental forms. Mathematically, it is clear how these objects ...
6
votes
5answers
546 views
What does symplecticity imply?
Symplectic systems are a common object of studies in classical physics and nonlinearity sciences.
At first I assumed it was just another way of saying Hamiltonian, but I also heard it in the context ...
6
votes
2answers
266 views
Dirac equation in curved space-time
I have seen the Dirac equation in curved space-time written as $$[i\bar{\gamma}^{\mu}\frac{\partial}{\partial x^{\mu}}-i\bar{\gamma}^{\mu}\Gamma_{\mu}-m]\psi=0 $$
This ...
6
votes
5answers
178 views
In coordinate-free relativity, how do we define a vector?
Relativity can be developed without coordinates: Laurent 1994 (SR), Winitzski 2007 (GR).
I would normally define a vector by its transformation properties: it's something whose components change ...
6
votes
1answer
532 views
What is the stress energy tensor?
I'm trying to understand the Einstein Field equation equipped only with training in Riemannian geometry. My question is very simple although I cant extract the answer from the wikipedia page:
Is the ...
6
votes
2answers
231 views
Mathematical probabilistic interepretation of probability amplitude
As a warning, I come from an "applied math" background with next to no knowledge of physics. That said, here's my question:
I'm looking at the possibility of using probability amplitude functions to ...
6
votes
1answer
273 views
Diffeomorphisms and boundary conditions
I am trying to find out how did the authors in this paper (arXiv:0809.4266) found out the general form of the diffeomorphism which preserve the boundary conditions in the same paper.
I found this ...
6
votes
1answer
146 views
If a fundamental theory exibits e.g. a mirror symmetry, in what sense it the underlying geometry real?
Are the more recently discovered symmetries in string theory such that the theories based on mirroring geometries are absolutely the same from an observable point of view?
I have mirror symmetry ...
6
votes
1answer
43 views
How does a geodesic equation on an n-manifold deal with singularities?
My general premise is that I want to investigate the transformations between two distinct sets of vertices on n-dimensional manifolds and then find applications to theoretical physics by:
...
6
votes
1answer
258 views
What is the information geometry of 1D Ising model for a complex magnetic field?
Consider the one-dimensional Ising model with constant magnetic field and node-dependent interaction on a finite lattice, given by
$$H(\sigma) = -\sum_{i = 1}^N J_i\sigma_i\sigma_{i + 1} - h\sum_{i = ...
6
votes
0answers
246 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 ...
6
votes
1answer
303 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 ...
5
votes
5answers
2k views
How do I calculate the perturbations to the metric determinant?
I am trying to calculate sqrt(-g) in terms of a background metric and metric perturbations, to second order in the perturbations. I know how to expand tensors that depend on the metric, but I don't ...
5
votes
2answers
401 views
Why do objects follow geodesics in spacetime?
Trying to teach myself general relativity. I sort of understand the derivation of the geodesic equation ...
5
votes
3answers
221 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 ...
5
votes
2answers
541 views
Visualizing Ricci Tensor
By definition Ricci Tensor is a Tensor formed by contracting two indices of Riemann Tensor. Riemann Tensor can be visualized in terms of a curve, a vector is moving and orientation of the initial and ...
5
votes
2answers
280 views
Conical spacetime of cosmic string
Inspired by: Angular deficit
The 2+1 spacetime is easier for me to visualize, so let's use that here. (so I guess the cosmic string is now just a 'point' in space, but a 'line' in spacetime) Edward ...
5
votes
2answers
288 views
Sewing together flat spacetime pieces = flat spacetime?
I'm trying to imagine the geometry "operations" here:
Angular deficit
and
Conical spacetime of cosmic string
If we sew flat spacetime pieces together, what is the requirement for the sewing to not ...
5
votes
1answer
200 views
Is GR vacuum equation unique?
The title question would be too long if I tried to specify it clearly. So let me be more clear. Consider the class of theories having the following properties:
The langrangian density is only ...
5
votes
1answer
436 views
Angular deficit
If one starts with a flat piece of paper, removes a wedge, and tapes the paper together, you get a cone. The angle of the removed wedge is called the "angular deficit".
Now if this is done in 3 ...
5
votes
1answer
171 views
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 ...
5
votes
1answer
75 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:
...
4
votes
3answers
567 views
Why is the symplectic manifold version of Hamiltonian mechanics used in Newtonian mechanics?
Books such as Mathematical methods of classical mechanics describe an approach to classical (Newtonian/Galilean) mechanics where Hamiltonian mechanics turn into a theory of symplectic forms on ...
4
votes
2answers
174 views
Does Kaluza-Klein Theory Require an Additional Scalar Field?
I've seen the Kaluza-Klein metric presented in two different ways., cf. Refs. 1 and 2.
In one, there is a constant as well as an additional scalar field introduced:
$$\tilde{g}_{AB}=\begin{pmatrix}
...
4
votes
2answers
334 views
Christoffel symbols and affine connection
What is the difference between the "affine connection" (S. Weinberg, Cosmology) and "Christoffel symbols?"
