Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.

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6
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1answer
59 views

Does nature of singularity in black hole depend on material that fell in?

Electromagnetic waves have a tracesless stress energy tensor, and therefore if they are the only fields in a region of spacetime, the Ricci curvature scalar $R=0$ according to GR. However $R^{\mu\nu} ...
3
votes
0answers
39 views

Classical toy models of particles with intrinsic spin

Related to my question here (spacetime torsion, the spin tensor, and intrinsic spin in einstein cartan theory), I'd like to be able to put test particles on a manifold with non-zero torsion and see ...
2
votes
3answers
105 views

Why does a bubble take a spherical shape?

I suspect this has something to do with thermodynamics and the isoperimetric inequality and I'm interested in a mathematical derivation of this result.
2
votes
0answers
37 views

New coordinates in terms of old from Jacobian— Software? [duplicate]

Fellow geometry enthusiasts: I am studying a particular geometry and I would like to change from one set of coordinates to a new set that makes the metric unimodular (i.e. it's determinant is one). ...
3
votes
0answers
53 views
+50

Spacetime Torsion, the Spin tensor, and intrinsic spin in Einstein-Cartan theory

In Einstein-Cartan gravity, the action is the usual Einstein-Hilbert action but now the Torsion tensor is allowed to vary as well (in usual GR, it is just set to zero). Variation with respect to the ...
5
votes
1answer
53 views

What's the definition of incompleteness of a coordinate system and a spacetime?

I always see in GR textbooks that some coordinates or some spacetime is incomplete, such as Rindler spacetime and spacetially flat FRW universe with only positive cosmological constant. This ...
5
votes
0answers
49 views

Fermions in Schwarzschild spacetime

To my understanding Geroch proved that on 4-dimensional non-compact manifold a necessary and sufficient condition for a manifold to have a notion of spinors is to be parallelizabe .1 (General ...
0
votes
1answer
68 views

What are great circles of 2-sphere?

What exactly are great circles, and how does one derive them? Given that the Lagrangian is: $$ L =\frac {1}{2}(\dot\theta^2 + \sin^2\theta\dot\phi^2)$$ it was written that the great circles were ...
1
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0answers
53 views

Geodesic equation from Euler - Lagrange

There are several ways to derive the geodesic equation. One of which is the variational method which I seemed to understand it because it was written in great details. Then it was mentioned that the ...
1
vote
0answers
28 views

Parallel transport of a vector on a curve on a sphere [closed]

I want to do a calculation of a parallel transport along a curve on a sphere, but I think I'm lacking a bit the basics of using different coordinate systems and curves there because I never did much ...
1
vote
1answer
44 views

Modeling wall's behaviour

Sorry if the quesiton is inconvenient, but I judged the physics forum would be the best place to go. My house is divided in two parts by a wall, and there's some tree pushing it, so the wall is about ...
5
votes
1answer
66 views

Relation between cohomology and the BRST operator

Given a manifold $M$, we may define the $p$th de Rham cohomology group $H^p(M)$ as the quotient, $$C^p(M) \, / \, Z^p(M)$$ where $C^p$ and $Z^p$ are the groups of closed and exact $p$-forms ...
4
votes
2answers
96 views

Why is the phase space a symplectic manifold rather than a manifold with a metric?

Why does phase space require a symplectic geometry rather than a metric? Is there some scenario where a metric is unable to describe the notion of length in phase space, specifically in relation to ...
2
votes
1answer
85 views

Can these two terms cancel out?

In trying to prove that $$\Gamma_{\mu\nu\lambda} = \eta_{ab}J^a_bJ^b_{\nu\lambda}.$$ The author canceled out while expanding the first equation $$J^a_{\mu\lambda}J^b_\nu$$ with ...
4
votes
0answers
106 views

Level quantization of 7d $SO(N)$ Chern-Simons action

In 3d, one can write down the $SO(N)$ Chern-Simons action to be $$S(A)=\frac{k}{192\pi}\int_{M}\text{Tr}(A d A +\frac{2}{3}A^3),$$ where $A$ is an $SO(N)$ connection. The level quantization can be ...
1
vote
1answer
52 views

Riemann Curvature Tensor Symmetries Proof

I am trying to expand $$\varepsilon^{{abcd}} R_{{abcd}}$$ by using four identities of the Riemann curvature tensor: Symmetry $$R_{{abcd}} = R_{{cdab}}$$ Antisymmetry first pair of indicies ...
1
vote
0answers
64 views

Help with parametrization of surface if I'm given the metric [closed]

I've got a homework question. Consider a 2 dimensional space with metric $$ ds^{2} = \frac{dr^{2}}{1 -\frac{2}{r} } + r^{2}d\theta^{2} .$$ I need to show that this is the induced metric ...
4
votes
1answer
83 views

Conservative Vector Fields

I was always told that to find whether or not a vector field is conservative, see if the curl is zero. I have now been told that just because the curl is zero does not necessarily mean it is ...
2
votes
0answers
70 views

What are Killing spinors?

What are Killing spinors? How can they be motivated? Are they directly related to Killing vectors and Killing tensors and is there an overarching motivation for all three objects? Any answer is ...
1
vote
0answers
51 views

Why are Lagrangian subspaces called 'Lagrangian'?

I am wondering what the special role of Lagrangian subspaces (or submanifolds) are in mechanics. Do these subspaces have some sort of special property for which we have some sort of `Lagrangian ...
1
vote
0answers
54 views

Equation regarding the Riemann tensor in the Cartan formalism [closed]

I have a problem verifying the following equation (in three dimensions) $$\epsilon_{abc} e^a\wedge R^{bc}=\sqrt{|g|}Rd^3 x$$ where $R$ is the Ricci scalar and $R^{bc}$ is the Ricci curvature ...
3
votes
0answers
54 views

Ricci curvature of embedded spacetime

If I am not mistaken, there is a theorem which states that every Riemannian manifold can be embedded in the $n$-dimensional Euclidean space for some large-enough $n$. Does it also hold for ...
2
votes
2answers
119 views

Momentum vector transformation

I am confused about the way momentum vector transforms in the following case: $$q_k \to q_k'= q_k + \epsilon f_k(q)$$ The Jacobian is thus $\Lambda_{ij} = \frac{\partial q'_i}{\partial q_j} \approx ...
1
vote
1answer
72 views

Derivative chain rule in a triangle, confusing but interesting problem

I asked the question in math.stackexchange. But I think it is better to ask here again. I am new to these sites. Please forgive me if it is not polite. http://math.stackexchange.com/q/921001 You can ...
2
votes
0answers
51 views

What is elliptic genera?

What is elliptic genera in physics? Reading many relevant papers, they just defined elliptic genus as sort of partition function. I try to find useful materials to explain it, but I couldn't find ...
2
votes
0answers
37 views

Mathematical books to become a successful mathematical physicists [duplicate]

My understanding of algebraic topology and Riemannian geometry come from Nakahara's Geometry, Topology, and Physics, which I do not think is sufficient. I am first year PhD student, and I want to do ...
1
vote
1answer
66 views

The Euler-Poincare equation

Can anyone tell me very basically how the Euler-Poincare equation generalises the Euler-Lagrange equation? Further does anyone know if there is an "easy" relationship between the two, i.e. can anyone ...
3
votes
1answer
118 views

Geometric mechanics - Symplecticity

I am just trying to wade through literature on classical mechanics and I really don't know where to start, everything is Fibre bundle this or manifold that, and doesn't really ease you in to the ...
0
votes
1answer
90 views

Geodesic deviation on a unit sphere

Very little interest in the original version of this question so I've rejigged it hoping for a more positive response. I'm trying to use the geodesic deviation ...
1
vote
0answers
28 views

Supergravity solution, metric for the total space, and connection

In supergravity solutions, one sometimes encounters the case where the manifold may be a bundle over some base space, and one has to write down the explicit metric regarding such bundle. I would like ...
1
vote
1answer
92 views

Geodesic curvature and Weyl transformations

The geodesic curvature is given by $$k=\pm t^a n_b\nabla_a t^b,$$ where $t^a$ is a unit vector tangent to the boundary of the string worldsheet and $n_a$ is an outward vector orthogonal to $t^a$. I ...
1
vote
0answers
16 views

Is Hodge star operation can be understood as contraction after tensor product of a $p$-form with the volume element? [migrated]

By defintion, the Hodge star of a $p$-form $\omega_{a_1\cdots a_p}$ on a $n$-dimensional manifold is given by $*\omega_{b_1\cdots b_{n-p}}=\frac{1}{p!}\omega^{a_1\cdots a_p}\epsilon_{a_1\cdots ...
3
votes
2answers
351 views

Which of these two textbook equations of geodesic deviation is correct?

My previous question Geodesic deviation on a 2-sphere - is this the right track? got shot down as “off topic”, so I'm having a second stab at it. Misner et al's Gravitation (p34) gives the geodesic ...
1
vote
0answers
118 views

Textbook disagreement on geodesic deviation on a 2-sphere

Apologies if I have this completely wrong (and for the general long-windedness). I've searched online but can't find anything helpful/relevant. I'm trying to use the geodesic equation ...
2
votes
0answers
55 views

If $S$ is a closed achronal set in a spacetime, any timelike curve starting at a point in $I^+[S]$ and ending at a point in $I^-[S]$ interset $S$?

Suppose $S$ is an achronal set in a spacetime $M$. And $S$ is closed. At the same time, any null geodesic of $M$ intersects $S$. Then, why does any timelike curve from $I^+[S]$ to $I^-[S]$ intersect ...
3
votes
1answer
73 views

Christoffel symbol

For two nearby points in General Theory of Relativity. The change in the vector components when parallel transported is given by Now, since the parallel transport change must depend on the path ...
4
votes
0answers
102 views

Tricks for Computing Riemann Curvature Tensor with Levi-Civita connection

I am new to differential geometry, so far it seems to me that computing the Riemann tensor tends to be a rather tedious task, I wanted to know whether there are some tricks that I am missing. In ...
2
votes
2answers
68 views

Definition of derivative operator on a manifold

I'm hoping to understand the motivation for certain parts of the definition of a derivative operator $\nabla$ on a manifold $M$. In Wald's General Relativity, two clauses of the definition are: ...
4
votes
1answer
139 views

Wick Rotation in Curved space

So over time I have learned to do exhaustive searches before asking things here. Wick rotations are cool if you are trying to work in qft and make statements about the thermodynamics of some physical ...
3
votes
4answers
113 views

What is the motivation for the definition of a manifold?

In Wald's General Relativity, an $n$-dimensional $C^{\infty}$ manifold $\mathit{M}$ is defined as a set, with subsets $\lbrace{O}_{\alpha}\rbrace$, which satisfies 3 properties. In particular, the ...
10
votes
4answers
266 views

Is force a contravariant vector or a covariant vector (or either)?

I don't understand whether something physical, like velocity for example, has a single correct classification as either a contravariant vector or a covariant vector. I have seen texts indicate that ...
2
votes
0answers
45 views

Why can any pair of master coordinates be used to calculate a nonlinear mode of a nonlinear dynamical system?

This is a question I have been asking myself for some time since the following technique is often used in the nonlinear dynamics community, but never managed to get an answer why it could be applied. ...
2
votes
0answers
47 views

Deformation of light-cone

In the paper The geometry of free fall and light propagation by Ehlers and his colleagues (Gen. Relativ. Gravit. 44 no. 6, pp. 1587–1609 (2012)), when the authors introduce the differentiable ...
2
votes
0answers
20 views

Using Polyakov-Alvarez Anomaly Formula [closed]

Take $\Sigma=\mathbb{D}$ to be the unit disk with metric $g=\frac{4}{(1+|z|^2)^2}\,|dz|^2$. If $\phi$ is a nice enough function on $\mathbb{D}$, then I want to compute $$\int_{\partial \Sigma} k_g ...
1
vote
1answer
43 views

If a point r lies in the boundary of the chronological future of another point p, why does the chronological future of r belong to that of p?

I am studying the global causality of the spacetime. Here, I come across a problem. Suppose a point $r\in \partial I^+(p)$. $I^+(p)$ is the chronological future of a different point $p$ in ...
2
votes
1answer
39 views

How does a spatial covariant derivative act on tensors that are not purely spatial?

I have a possibly dumb question on ADM formalism. Starting with a metric in ADM form \begin{equation} ds^2 = -N^2dt^2 + q_{ij}(dx^i + N^idt)(dx^j + N^jdt) \end{equation} where $i,j$ only run over the ...
1
vote
0answers
66 views

How to test that a flat metric represents a global three-torus geometry

When introducing Robertson-Walker metrics, Carroll's suggests that we consider our spacetime to be $R \times \Sigma$, where $R$ represents the time direction and $\Sigma$ is a maximally symmetric ...
1
vote
0answers
53 views

Riemann curvature of a unit sphere

The Riemann curvature of a unit sphere is shown in many textbooks to be sine-squared theta where theta is the azimuthal angle of spherical co-ordinates. But what is the significance of the angle and ...
-2
votes
1answer
62 views

Length in polar coordinates

Say we are in 3 dimensions and use $(-++)$. If we have the metric $$ds^2=-dt^2+dr^2+r^2df^2(t),$$ then what is the third coordinate if the first two were $t$ and $r$? $$X^iX_i=-t^2+r^2+?$$
2
votes
3answers
100 views

Configuration manifolds and constraints

In Classical Mechanics there's this notion of configuration manifold. Although I've heard about that a lot and although I often use that concept, I'm not sure I really understand them well because ...