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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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About “conserved quantities” in a diffeomorphism-invariant theory by Wald and Zoupas

In this work, Wald & Zoupas developed a framework to define the "conserved quantities" in a diffeomorphism-invariant theory using the covariant phase space formalism. Let $\xi^a$ be a vector ...
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0answers
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deSitter spacetime metric and curvature

I have to compute the metric of an hyperboloid given by $-(X^0)^2+(X^1)^2+(X^2)^2+(X^3)^2=H^{-2}$ in 5D Minkowski spacetime using the following coordinates: $$X^0=H^{-1}\sinh(Ht)\sqrt{1-H^2r^2}$$ $$X^...
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0answers
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Worked examples of Lie derivative [migrated]

I'm trying to find the Lie derivative of a 2 form $\sin(\theta)d\theta d\phi$ with respect to a vector field given in a differential basis and I think the way to go here is to use Cartan's formula but ...
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1answer
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$dS^d$ and $AdS^d$ are conformally equivalent

I have seen in the book by Zee (Einstein Gravity in a Nutshell) the following metric for $dS^4$: $ds^2 = \frac{1}{\cos^2 \tau} \big( - d\tau^2 + d\psi^2 + \sin^2 \psi \, d\Omega_2^2 \big) $, (Eq. IX....
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What is torsion physically in the Einstein-Cartan theory?

In Einsteins theory of gravity the metric gives a unique torsion free connection called the Levi-Civita connection. In the Einstein-Cartan theory we allow any connection compatible with the metric ...
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1answer
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Where does the factor of one half come from in the delta-vector equation involving the Riemann Curvature Tensor?

In Einstein's Theory, A Rigorous Introduction for the Mathematically Untrained, by Grøn and Næss: The change of the covariant components a vector by parallel transport around an indefinitely small ...
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What's the difference between internal spaces and extra dimensions?

From a geometrical perspective, we attach a copy of our gauge groups (e.g. $U(1)$ to each point in spacetime. Since Lie groups are manifolds we can, therefore, imagine that there is a little circle, ...
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How to find Ricci tensor?

I'm trying to find the Ricci tensor in question 3. Here $u=r/R .$ http://imgur.com/gallery/qSAknvz I found the Christoffel symbols but I can't find the Ricci tensors. On the link, there is also my ...
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Are all 4D Ricci flat manifolds locally Euclidean? [closed]

If a 4D manifold with metric signature (++++) is Ricci flat. Does this mean that locally the space is Euclidean? Does this mean that the only difference between these manifolds is there global ...
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Physical interpretation of a singular pde [closed]

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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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
37 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
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Is spacetime wholly a mathematical construct and not a real thing? [closed]

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 $\frac{\cos(\theta)dA}{r^2} = d\omega$ [closed]

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
41 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
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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
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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
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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
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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
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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
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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
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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
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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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1answer
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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
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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
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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
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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
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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
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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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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
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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
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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
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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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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
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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
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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
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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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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
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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
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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
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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
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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
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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
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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
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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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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
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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 ...