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Questions tagged [differential-geometry]

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

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Physical interpretation of a singular pde [on hold]

Can you kindly have a look at the following question attached in the link below: https://math.stackexchange.com/questions/3058490/physical-interpretation-of-a-singular-pde?noredirect=1#...
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0answers
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Advanced and Retarded Coordinates for Hawking Radiation

I am trying to solve the Klein Gordon equation near the horizon of a black hole. For a more accurate result I am using the Vaidya Metric in which the mass is a function of retarded time u. When ...
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1answer
34 views

Is it possible understand Berry curvature as Gaussian curvature in some limit?

I would like to understand the Berry curvature and the Chern number from mathematical geometry-topology. I understand that in electronic QHE, there is a map from $k^2$ to a vector space where the ...
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11answers
9k views

Is spacetime wholly a mathematical construct and not a real thing? [on hold]

Speaking of what I understood, spacetime is three dimensions of space and one of time. Now, if we look at general relativity, spacetime is generally reckoned as a 'fabric'. So my question is, whether ...
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Derivation of $\frac{\cos(\theta)dA}{r^2} = d\omega$ [on hold]

Note: I asked the same question on the mathematics stackexchange, but was advised to ask it here. It also seems to arise quite often in physics. I've been looking for a (formal) derivation of the ...
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1answer
39 views

Schwarzschild geometry for negative masses

Imagine that in a part of our universe there exists negative masses $M=-|M|<0$. The metric around this object -- say a black hole -- will be of the form $$ ds^2 = -\Big(1+\frac{2|M|}{r}\Big)dt^2 + ...
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1answer
44 views

Will I be destroying spherical symmetry if I write the mass of the body as a function of time?

Will I be destroying spherical symmetry if I write the mass of the body as a function of time? If yes, then how can I write a metric for a body with mass as a function of time?
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2answers
92 views

Non-static spherical symmetry spacetime

The Schwarzschild solution is a static spherically symmetric metric. But I wanted to know that how would the space-time interval look in a Non-Static case. I tried to work it out and got $$ds²= Bdt² -...
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0answers
41 views

Kaluza-Klein approach and Gauss-Cadazzi approach

Can you tell me the difference or physical application of Kaluza Klein approach and Gauss Codazzi approach? In Kaluza Klein theory, 5 dimensional theory can be dimensional reduced to 4 dimensional ...
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1answer
29 views

Confusion regarding the transformation law under diffeomorphism

While revisiting the transformation laws - under a diffeomorphism - of the partial derivative of a vector field I got confused by the following result. So, for a vector field $A$ the usual ...
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1answer
121 views

Spacetime has an infinite number of choices for differentiability. Coincidence?

Spacetime can be modelled using a four-dimensional topological manifold. Say we denote the manifold using $(M, \mathcal{O})$ where $\dim M =d$. The structure $(M,\mathcal{O})$ is not sufficient for ...
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0answers
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Question about tensor integration

A (covariant)$(0,2)$-tensor can be written as: $$T = \sum_{\mu}\sum_{\nu}t_{\mu \nu} e^{\mu}\bar{\otimes}e^{\nu}\tag{1}$$ with the particular basis vectors $e^{\mu}\bar{\otimes}e^{\nu}$ that spans ...
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1answer
63 views

Line element 1-form

It was pointed out that dual vectors of a manifold, and hence differential 1-forms, are not dependent on the metric (Intuition behind dual vectors ('Bongs of a bell' does not help)). But doesn'...
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3answers
49 views

Ricci Scalar as Curvature

So I understand that the Ricci scalar represents the curvature of the space. Since any manifold can be considered locally flat, is Ricci scalar always zero locally for any manifold? On one hand it ...
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3answers
92 views

Intuition behind dual vectors ('Bongs of a bell' does not help)

Similar to the post here (How to visualize the gradient as a one-form?), I'm wondering about an intuition behind dual vectors and differential forms (and the link in that answer to Thorne's notes is ...
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0answers
21 views

Obtaining Brans Dicke theory scalar (wave) equation

I have trouble with obtaining d'Alambert equation for scalar field in Brans-Dicke gravity (http://www.scholarpedia.org/article/Jordan-Brans-Dicke_Theory). B-D gravity langrangian density is given by: ...
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1answer
56 views

Derivation of equation for geodesic deviation

I am trying to figure out the calculation which leads to the geodesic deviation on this site. So far I understood all steps until (14.7) and managed to show that (14.6) = (14.7), namely $$ \ddot\xi^\...
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2answers
79 views

Isometry group on a coset manifold

In ''Einstein Gravity in a Nutshell'' Zee says ''On a coset manifold $G/H$, the isometry group is evidently just $G$'' when discussing the relation between the Killing vector fields and Lie ...
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1answer
79 views

How to “resolve a state” with respect to a spacelike hypersurface in Minkowski Spacetime QFT?

Consider usual free QFT in Minkowski spacetime. For simplicity let us consider a real scalar field $\phi$. Usually quantization is performed with respect to one inertial reference frame. This is is ...
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2answers
71 views

General Relativity: Exchanging a field with its infinitesimal components on metric tensor

On a youtube video about Einstein's field equations, the author writes the following equation (https://youtu.be/foRPKAKZWx8?t=1078): $$d\phi=\sum_{n} \frac{\partial \phi}{\partial x^n} dx^n\tag{1}$$ ...
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2answers
142 views

I almost got the Schwarzschild solution [closed]

I was trying to derive the Schwarzschild metric, but i got the following solution: $$ds^2=(C_1-\frac{1}{r})dt^2-\frac{1}{(C_1-\frac{1}{r})}dr^2-r^2d\Omega^2.$$ The correct place for the $C_1$ should ...
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1answer
52 views

Question about Lagrange method and line element

Consider the following line element: $$ds^{2} = K(x,y,z,t)(-dt^2+dx^2)+M(x,y,z,t)dxdt+dy^2+dz^2$$ Then the lagragian method give to us the lagrangian from line element: $$\mathcal{L}^2 = K(x,y,z,t)(...
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0answers
19 views

Geodesic curve definition [duplicate]

Do we have a choice in defining the covariant derivative by the use of a set of coefficient functions(Christoffel gammas)? If so, could we then say that these coefficient functions need not to ...
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1answer
90 views

Is there a useful way to visualize the symmetries of the relativistic Riemann curvature tensor?

I find it useful to see diagrams such as trees, colored 2D and 3D arrays, etc., which illustrate how terms combine in composite expressions. For example, the following is my visualization of the ...
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1answer
54 views

Do we need really need tetrads? OR: Does the No-Go theorem for spinors in curved space only apply to a linear connection?

When I first learned General Relativity, the tetrad formalism was introduced with near simultaneity. I was immediately taught that, to utilize spinors in any way, I had to formulate a local ...
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1answer
52 views

Phase space as differential manifold

Generally we "draw" phase space as typical coordinate system, where $q$s and $p$s are treated like perpendicular axes. Why do we then regard phase space as generall differential manifold while it ...
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0answers
26 views

How to find the curvature of a surface using directional length dilation?

I've already figured out how to find the curvature of an $f(x,y)$ function at each point. $$K=\frac{f_{xx}f_{yy}-f_{xy}^2}{(1+f_x^2+f_y^2)^2}.$$ Now I want to find out how to calculate curvature ...
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1answer
63 views

Radius of Star, The Schwarzschild metric and Black Holes

From Section 9.1, in General Relativity by Woodhouse: For a normal star, the Schwartzchild radius is well inside the star itself. As it is not in the vacuum region of space-time, the Ricci tensor ...
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1answer
61 views

When can we raise lower indices on “nontensors” as described in Dirac's book *General Theory of Relativity*?

This is a follow on to my previous question: Dirac's book *General Theory of Relativity*: Doesn't this show the partial derivative of the metric tensor is zero? I should not have made that ...
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2answers
84 views

Is there a good treatment of “familiar” physics using exterior calculus, AKA differential forms?

By familiar physics, I mean the physics of things I can reach out and touch. In other words, neither relativity nor analytical dynamics, etc. After re-reading chapter 4 of MTW's Gravitation yet ...
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0answers
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Causal structure of spacetime. A pathological example of spacetime?

Consider a spacetime $(M,g)$ and $p \in M$. It is known that locally, the boundary of the chronological future of p is just the set of points along (future-directed) null geodesics starting at p. ...
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1answer
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Hamiltonian as differential manifold

I know that phase space $(q^i, p^i)$ can be treated as manifold. But for me defining hamiltonian as a function also leads to new manifold $(q^i, p^i, H(q^i,p^i))$. Like map from open set $(q^i, p^i) \...
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1answer
61 views

Deriving the equation relating the metric and the coordinates to the proper time in general relativity

If $x^a = x^a(\tau)$ is the worldline of a particle in general motion, then $$V^a = \frac{dx^a}{d\tau}$$ is a four-vector field along the worldline. If $$g_{ab}V^aV^b = g_{ab}\dot{x}^a\dot{x}^b = ...
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1answer
88 views

Dirac's book *General Theory of Relativity*: Doesn't this show the partial derivative of the metric tensor is zero?

See the bold text for my question. This question regards Dirac's General Theory of Relativity page 13. In his demonstration that the length of a vector is unchanged by parallel displacement Dirac ...
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4answers
202 views

Why was pseudo-Euclidean geometry not enough for general relativity?

How would you explain to someone the change that Einstein needed in geometry for his new ideas about gravity and spacetime, what did he seek but could not be described by pseudo-Euclidean geometry? ...
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1answer
64 views

2D global conformal transformations and the $z= \frac1w$ argument

For instance in Blumenhagen's CFT, there is a standard argument which determines that globally defined conformal transformations on the Riemann sphere where $$l_n = -z^{n+1} \partial_z$$ is an ...
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0answers
38 views

Derivatives in Poincare' gauge theory

I have been reading the lectures: http://www.damtp.cam.ac.uk/research/gr/members/gibbons/gwgPartIII_Supergravity.pdf about Poincare' gauge theory. The Poincare' group is considered as semidirect ...
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1answer
84 views

Tensorality of the Lie Derivative and $i([X,Y]) = [L(X),i(Y)]$

I'm trying to understand the equation following (2.15) on p.9 of Blau's Symplectic Geometry and Geometric Quantization. For two vector fields $X,Y$ on a symplectic manifold $M$ we are told one ...
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3answers
3k views

Does time speed up or slow down near a black hole?

The Schwarzchild geometry is defined as $$ds^2=-\left(1-\frac{2GM}{r} \right)dt^2+\left(1-\frac{2GM}{r} \right)^{-1}dr^2+r^2(d\theta^2+\sin^2(\theta) d\phi^2)$$ Lets examine what happens close to ...
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0answers
32 views

How to define an Operator Product Expansion (OPE) on arbitrary Riemann surface for a CFT?

Whenever we define the OPE of a 2D CFT, we do so (at least in the literature that I have come across) on the complex plane. Similarly, the commutation relations between conformal generators $L_n$ and ...
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0answers
12 views

Geometric phase of an optical fiber or the space curve

Hi everyone I am trying to understand the geometric phase along an optical optical fiber. I am a geometer and I am almost ok with the geometric part but I have some diffuculties to extract physical ...
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0answers
23 views

Regarding the shortest path conneting two points on a sphere of radius $R$ [migrated]

Consider a path on the surface of the sphere given by a function $\phi(\theta)$. The line element along the path $\phi$ is given by $$ \mathrm{d}s=R\mathrm{d}\theta\sqrt{1+\left (\frac{ \mathrm{d} \...
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2answers
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Spin statistics from the fundamental group of $SO(D)$

I read the answer to this question and am very intrigued by its simple and elegant explanation of the emergence of anyon, boson & fermion statistics. @Trimok basically says: In a space-time ...
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1answer
78 views

GR with Torsion: Definition of contorsion

I start doing some computations in manifolds with non vanishing torsion and things are getting a bit confused, basically because of notations and definitions. I understand that in presence of non ...
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1answer
78 views

Rigorous definition of generalized coordinates

In Goldstein's classical mechanics and in many other books I haven't seen a rigorous definition of generalized coordinates. In a system of $N$ particles described by $\textbf{r}_1, \dots, \textbf{r}...
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0answers
37 views

How to indicate this formula with nabla operator? [migrated]

I found this in the fluid mechanics: $F(u)=\frac{\partial u_x}{\partial x}\frac{\partial u_y}{\partial y}+\frac{\partial u_x}{\partial x}\frac{\partial u_z}{\partial z}+\frac{\partial u_y}{\partial y}\...
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0answers
53 views

Finding the Ricci tensor components for the Schwarzschild metric

I'm trying to use Cartan's method to find the Schwarzschild metric components from Hughston and Tod's book 'An Introduction to General Relativity' (pages 89-90). I'm having problems calculating the ...
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1answer
57 views

Ambiguous curl for a vector field

I'm trying to work out how to explain why the curl/divergence for the following image is ambiguous without giving the explicit vector function for the following: E has an ambiguous divergence and I ...
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0answers
40 views

Tetrad/vierbein for Eddington-Finkelstein coordinates

I have trouble with obtaining vierbein (and orthonormal frame components) given by $\eta_{(a)(b)} = e_{(a)}^{\mu}e_{(b)}^{\nu}g_{\mu\nu} $ with tetrads $e_{(i)}= e_{(i)}^{\mu}\partial_\mu$ and ...
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0answers
23 views

What is the Newman-Penrose (NP) form of Rarita-Schwinger (RS) equation?

Does anyone knows how to write down the Rarita-Schwinger equation in Newman-Penrose formalism?