Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.
6
votes
3answers
133 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 ...
1
vote
0answers
81 views
Self-organizing maps
I'm currently interested in this subject but all I can see is about neural networks and I'm more interested on the Theoretical point of view: "how can a system (Lagrangian/Hamiltonian) alter it's ...
6
votes
0answers
247 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 ...
11
votes
1answer
218 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 ...
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 ...
3
votes
1answer
262 views
How does (or can) SR/GR extend to phase space or symplectic manifolds?
I'm asking this question because of an article in New Scientist about a recent preprint by a group including Lee Smolin. I haven't taken the time to comprehend the paper completely. My knowledge of ...
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 ...
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 ...
8
votes
4answers
867 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
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 ...
4
votes
1answer
155 views
Is there a simple way to define/solve for a null cone for a general spacetime geometry?
I'm wondering if there's any simple way to define and solve for a null cone for a general spacetime geometry in $n+1$ dimensions, given its vertex $p^\mu$. I can't seem to find a simple way to do it ...
2
votes
3answers
125 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, ...
4
votes
2answers
337 views
Christoffel symbols and affine connection
What is the difference between the "affine connection" (S. Weinberg, Cosmology) and "Christoffel symbols?"
3
votes
3answers
241 views
Singularity-free stationary electro-vacuum solution
Let's say we have a spherically symmetric fluid:
$$
T^{\alpha \beta} = \begin{bmatrix}
\rho & 0 & 0 & 0 \\
0 & p & 0 & 0 \\
0 & 0 & p & 0 \\
0 ...
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 ...
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 ...
4
votes
1answer
227 views
Relativistic space-time geometry
What subject (suggest book titles, etc.) should I study to get a clear grasping of hypersurfaces, 2-surfaces, and integration on them, mostly in special relativity (I'm not messing with general ...
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 ...
4
votes
3answers
569 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 ...
3
votes
0answers
272 views
I lost a factor of two in the electromagnetic field tensor
I apologize for this simple question, but I lost a factor of 2 and can't find it anymore, so now I'm looking on the internet, perhaps one of you has some information about its whereabouts. :-)
...
5
votes
1answer
437 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 ...
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 ...
8
votes
1answer
490 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 ...
5
votes
1answer
201 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
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 ...
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 ...
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 ...
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_{\ ...
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 ...
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 ...
