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 ...
10
votes
3answers
322 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 ...
5
votes
1answer
434 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 ...
8
votes
2answers
393 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 ...
7
votes
6answers
853 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 ...
8
votes
3answers
807 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 ...
18
votes
7answers
854 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 ...
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 ...
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 ...
4
votes
3answers
566 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 ...
0
votes
0answers
87 views
Is there any Calculator capable of calculating and displaying differential geometry? [closed]
Is there any Calculator capable of calculating and displaying differential geometry (display curvature of spacetime)?
$$ds^2~=~g_{ab}dx^adx^b.$$
11
votes
1answer
430 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 ...
5
votes
3answers
217 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
1answer
167 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 ...
10
votes
4answers
318 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 ...
6
votes
1answer
257 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
$$H(\sigma) = -\sum_{i = 1}^N J_i\sigma_i\sigma_{i + 1} - h\sum_{i = ...
4
votes
3answers
182 views
What are some mechanics examples with a globally non-generic symplecic structure?
In the framework of statistical mechanics, in books and lectures when the fundamentals are stated, i.e. phase space, Hamiltons equation, the density etc., phase space seems usually be assumed to be ...
1
vote
1answer
120 views
Interpreting Vector fields as Derivations on Physics
I have a subtle doubt about the physical interpretation of the mathematical definition of vector field as a derivation. In basic physics we understand a vector quantity as a quantity that needs more ...
6
votes
2answers
231 views
Mathematical probabilistic interepretation of probability amplitude
As a warning, I come from an "applied math" background with next to no knowledge of physics. That said, here's my question:
I'm looking at the possibility of using probability amplitude functions to ...
5
votes
2answers
278 views
Conical spacetime of cosmic string
Inspired by: Angular deficit
The 2+1 spacetime is easier for me to visualize, so let's use that here. (so I guess the cosmic string is now just a 'point' in space, but a 'line' in spacetime) Edward ...
4
votes
1answer
95 views
Flat space metrics
This question concerns the metric of a flat space:
$$ds^2=dr^2+cr^2\,\,d\theta^2$$ where $c$ is a constant. Why is it necessary to set $c=1$ to avoid singularities and to restrict $r\ge 0$?
Thanks.
4
votes
1answer
283 views
Is 4-volume element a scalar or a pseudoscalar in special relativity?
In general relativity 4-volume element $\mathrm{d}^4 x = \mathrm{d} x^0\mathrm{d} x^1 \mathrm{d} x^2\mathrm{d} x^3$ is clearly a pseudoscalar (or scalar density) of weight 1 since it transforms as ...
3
votes
2answers
97 views
Electric Field One-Form
I know for instance that we can interpret the electric field as the one-form that given a vector gives the change in potential in the direction of the vector, however I'm very unsure about how to ...
1
vote
2answers
91 views
Ricci tensor for a 3-sphere without Math packets
Let's have the metric for a 3-sphere:
$$
dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right).
$$
I tried to calculate Riemann or Ricci tensor's ...
1
vote
3answers
620 views
What is a Killing vector field?
I recently read a post in physics.stackexchange that used the term "Killing vector". What is a Killing vector/Killing vector field?
1
vote
3answers
166 views
Where 2 comes from in formula for Schwarzschild radius?
In general theory of relativity I've seen several times this factor:
$$(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 ...
0
votes
1answer
629 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.
