Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.
4
votes
1answer
121 views
Kaluza-Klein Christoffel Symbols
I have a question regarding the connection coefficients as they pertain to the following paper: http://www.weylmann.com/kaluza.pdf . When I try to calculate the 4D Christoffel symbols from the 4D part ...
1
vote
2answers
159 views
What is the Riemann curvature tensor contracted with the metric tensor?
Can the Ricci curvature tensor be obtained by a 'double contraction' of the Riemann curvature tensor? For example
$R_{\mu\nu}=g^{\sigma\rho}R_{\sigma\mu\rho\nu}$.
4
votes
1answer
129 views
Do partial derivatives commute on tensors?
For example; is $$\partial_{\rho}\partial_{\sigma}h_{\mu\nu} - \partial_{\sigma}\partial_{\rho}h_{\mu\nu}=0$$ correct?
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
1answer
366 views
Physical significance of Killing vector field along geodesic
Let us denote by $X^i=(1,\vec 0)$ the Killing vector field and by $u^i(s)$ a tangent vector field of a geodesic, where $s$ is some affine parameter.
What physical significance do the scalar quantity ...
1
vote
1answer
106 views
Covariant derivative with upper index
I just need clarification, that is, to see that I'm doing the right thing.
When calculating central charge for certain metric, I need to solve an integral that contains Lie brackets etc. And I have ...
2
votes
1answer
176 views
Difference between $\partial$ and $\nabla$ in general relativity
I read a lot in Road to Reality, so I think I might use some general relativity terms where I should only special ones.
In our lectures we just had $\partial_\mu$ which would have the plain partial ...
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 ...
3
votes
0answers
54 views
Expectation of 2-form field $B_{MN}$ in string theory
In the context of string theory, in particular when we're dealing with a low energy effective action, if we have an effective action of the form:
$$S_{eff} \sim S^{(0)} + \alpha S^{(1)} + (\alpha)^2 ...
2
votes
1answer
72 views
Killing Vectors of BTZ black hole and their calculation in general
I was wondering what are the Killing vectors of BTZ black hole and how to guess them easily? Will it be the same as of AdS? What then will be Killing vectors for AdS-Schwarzschild e.g.?
0
votes
1answer
131 views
How to find the intrinsic covariant derivative component?
How to find the intrinsic covariant derivative component?
In general relativity the elements of the acceleration four-vector are related to the elements of the four-velocity through a covariant ...
7
votes
3answers
603 views
Why is the covariant derivative of the metric tensor zero?
I've consulted several books for the explanation of why
$$\nabla _{\mu}g_{\alpha \beta} = 0,$$
and hence derive the relation between metric tensor and affine connection $\Gamma ^{\sigma}_{\mu ...
1
vote
2answers
204 views
Geometrical interpretation of the Dirac equation
Is there a geometrical intuitive picture behind the Dirac equation, and the gamma matrices that it uses? I know the geometric algebra is a Clifford algebra. Can the properties of geometric algebra, be ...
2
votes
0answers
92 views
Is there a nice way to write Navier-Stokes equations in exterior calculus
I'm considering to study some high-dimensional Navier-Stokes equations. One problem is to do write the viscous equation for vorticity, helicity and other conserved quantities. I think it might be ...
7
votes
1answer
253 views
Diffeomorphisms, Isometries And General Relativity
Apologies if this question is too naive, but it strikes at the heart of something that's been bothering me for a while.
Under a diffeomorphism $\phi$ we can push forward an arbitrary tensor field $F$ ...
2
votes
3answers
181 views
Are the principles of space-time homogeneity and Isotropy independent of one another?
Einstein in deriving the Lorentz transformations, used the principles of space-time homogeneity and Isotropy. Does space-time isotropy follow from space-time homogeneity or are they completely ...
3
votes
3answers
259 views
Could metric expansion create holes, or cavities in the fabric of spacetime?
Is it possible for metric expansion to create holes, or cavities in the fabric of spacetime?
According to the Schwarzschild metric, the metric expansion of space around a black hole goes to infinity ...
2
votes
1answer
70 views
What is physical meaning of $\kappa$ and $R$ in curved space?
What is physical meaning of $\kappa$ and $R$ in curved space?
$$dl^2 = \frac{dr^2}{1 - \kappa\frac{r^2}{R^2}}
+ r^2d\theta^2 + r^2\sin^2\theta d\phi^2$$
0
votes
0answers
87 views
Is there any Calculator capable of calculating and displaying differential geometry? [closed]
Is there any Calculator capable of calculating and displaying differential geometry (display curvature of spacetime)?
$$ds^2~=~g_{ab}dx^adx^b.$$
2
votes
1answer
236 views
Does the universe obey the holographic principle due to Stokes' theorem?
Does the universe obey the holographic principle due to Stokes' theorem?
\begin{equation}
\int\limits_{\partial\Omega}\omega = \int\limits_{\Omega}\mathrm{d}\omega.
\end{equation}
Can this ...
1
vote
1answer
167 views
Tensor Introduction
I have recently started learning about tensors during my course on Special Relativity. I am struggling to gain an intuitive idea for invariant, contravariant and covariant quantities. In my book, ...
-5
votes
1answer
136 views
Differential of square $dw^2$or square of differential$(dw)^2$? [closed]
in Curved space
it seems $dw^2=(dw)^2$ how is it possible!?
$$x^2+y^2+z^2+w^2=\kappa^{-1}R^2,$$
$$dw=w^{-1}(xdx+ydy+zdz),$$
$$\kappa^{-1}R^2-(x^2+y^2+z^2)=w^2,$$
$$dl^2 = dx^2 + dy^2 + dz^2+dw^2,$$
...
3
votes
1answer
317 views
Chern number in condensed matter physics
In mathematics, the Chern number is defined in terms of the Chern class of a manifold. What is the exact definition of Chern number in condensed matter physics, i.e. quantum hall system?
1
vote
2answers
189 views
Example of two linearly independent, nowhere vanishing vector fields in $\mathbb{R}^{2}$
I knew that two linearly independent and nowhere-vanishing vector fields provide a basis for the tangent space at each point in $\mathbb{R}^{2}$.
Is it necessary that these two vector fields commute? ...
1
vote
1answer
150 views
A particular case when Lagrange equation is equivalent to equation of motion on a Riemannian manifold
Suppose a particle is moving on a surface of a sphere,then it contains a holonomic constraint and so the three Cartesian co-ordinates are available with a constraint equation(equation of surface in ...
3
votes
1answer
97 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
196 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
2answers
111 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
votes
1answer
141 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
votes
1answer
224 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 ...
4
votes
0answers
140 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 ...
1
vote
0answers
101 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 ...
2
votes
1answer
205 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 ...
0
votes
1answer
245 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 ...
1
vote
1answer
129 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 ...
1
vote
1answer
66 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?
0
votes
1answer
311 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}} .$$
Can anyone show me how to get this result?
0
votes
1answer
637 views
Killing vector fields
I am facing some problems in understanding what is the importance of a Killing vector field? I will be grateful if anybody provides an answer, or, refer me to some review or books.
2
votes
1answer
124 views
Cosmology with a negative cosmological constant
Based on the Friedmann equation for a universe with only cosmological constant,
$$\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
102 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 ...
1
vote
2answers
93 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 ...
1
vote
2answers
337 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 ...
1
vote
3answers
626 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?
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 ...
1
vote
1answer
104 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
votes
1answer
186 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 ...
7
votes
3answers
559 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
votes
4answers
257 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
268 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 ...
