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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Which finite-dimensional representations of the Lorentz group do $p$-forms correspond to?
On the Wikipedia article about the representation theory of the Lorentz group, the finite-dimensional representations $(1,0)$ and $(0,1)$ are referred to as "$2$-form" representations. On ...
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Precise definition of a string worldsheet as a manifold in string theory
I've spent some time studying some definition in smooth manifolds theory in order to give a proper definition of a worldsheet in classical string theory at least. My attempt is the following:
...
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Can the Lagrangian density of vacuum Maxwell equation be written into tensor contraction without a basis?
The Lagrangian density of the Maxwell equations in vacuum is
$$
\mathcal{L} = - \frac{1}{4} F_{\mu\nu}F^{\mu\nu} . \tag{1}
$$
My question is, $F$ is a tensor, namely
$$
F = \frac{1}{2}F_{\mu\nu} dx^{\...
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A problem about the detailed derivation in Einstein's paper
My question is from chapter #18 "The impulse-energy law for matter as a consequence of the field equations", from the derivation of equation above (57). He says that we can get (57) by ...
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Scalar curvature of a 2-sphere via the Ricci tensor
Using the usual coordinates on a 2-sphere of radius $r$, I get the metric tensor $g_{\mu\nu}=\text{diag}(r^2, r^2\sin^2\theta)$ and so $g^{\mu\nu}=\text{diag}(1/r^2,1/r^2\sin^2\theta)$.
Hence the only ...
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Motivation for $dF=0$?
Recently I have been studying differential geometry, and in the book I’m studying Maxwell’s Equations are derived using the 2-form $$F=E\cdot dr\wedge dt+B\cdot d\sigma.$$ They then state that $$dF=0.$...
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Double Hodge star proof [closed]
I'm having some trouble understanding the proof of acting the Hodge star operator twice on some p-form. Specifically, I don't understand how they went from the first line to the second one. The rest ...
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A question from Einstein's original paper on general relativity [closed]
I'm working on Einstein's original paper on general relativity (1916). I have a problem on its derivation.
I can't understand the process from (52) to (53), how is it derived? Is there something that ...
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Question on index notation
I am working my way through Carroll's text on GR and am having trouble understanding what it means when an index is inside/outside parentheses. For example, in his discussion of geodesic deviation, ...
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Alternate derivation of the covariant derivative of a contravariant vector
In Dirac's “General Theory of Relativity”, he derives the covariant derivative of a contravariant vector (his Eq. (10.7)):
$$ A^\mu_{: \sigma} = A^\mu_{, \sigma} + \Gamma^\mu_{\alpha \sigma}A^\alpha $$...
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Difference between position of indexes in Kronecker delta symbol [duplicate]
I am studying the Dirac gamma matrices and have encountered the Kronecker delta $\delta_{ij}$ That I am accustomed to. However, I have also come across a different form, $\delta_{\mu}^{\nu} $.
This ...
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Hawking & Ellis: typo on page 16?
On page 16 of The Large Scale Structure of Space-Time (1973) by Hawking and Ellis, they describe the basics of tangent spaces. This line appears near the top of the page:
Thus the tangent vectors at $...
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What prevents the line element being Minkowskian in the vicinity of a point mass?
This is probably a naive question and I'm missing something really simple. The Schwarzschild solution has been constructed in consideration of the following requirements:
The field equations $ \frac{\...
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About the correctness of a Christoffel-symbol-related equivalence
Disclaimer: This question had been asked 10 months ago in Mathematics SX, but it is still unanswered there.
In "Introduction to Tensor Analysis and the Calculus of Moving Surfaces" of ...
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Why can we put these conditions on coordinates of worldsheet?
https://www.asc.ohio-state.edu/mathur.16/classicalstring.pdf
At first, I write some notations I need here.
$I=[0,1]$, $M$ means $(1,3)$ Minkowski space, smooth map $X:I\times I\to M$ is timelike ...
2
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1
answer
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Existence of a Trapped Surface to the Existence of a Black Hole
Does the existence of a trapped surface in a region of space (not necessarily either of the vacuum or symmetric spacetime) indicates (theoretically) the existence of a "black hole" there? ...
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Showing that the contravariant base vector transforms as a vector [closed]
I wanna show that $Z^a$ is indeed a contravariant vector in the same way I showed that $Z_i$ is indeed a covariant vector (see attached image).This is how I define $Z^a$ : $Z^a = \frac{\partial y^a}{\...
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How do you decompose a general tensor into a sum of outer products?
Dirac's book "General Theory of Relativity" says on p. 2 that a general rank-2 tensor can be written as a sum of outer products:
$$ T^{\mu\nu} = A^\mu B^\nu + A'^\mu B'^\nu + A''^\mu B''^\nu ...
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Why we need hypersurface to solve initial value problem in general relativity? [duplicate]
Almost all in general relativity book to tackle the initial value problem there we need the concept of hypersurface and another concept like future or past domain of dependence.
My question is what is ...
2
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1
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Ultrastatic spacetime and cosmological constant
A spacetime $(\mathcal{M},g)$ is called "ultrastatic", if it admits a set of coordinates such that
$$g=-\mathrm{d}t^{2}+h$$
where $h$ is a Riemannian metric, which does not depend on time. ...
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Vectors and One-forms in Cylindrical Coordinates and the Angular Momentum
The angular component of the velocity of a particle in cylindrical coordinates has different units if we consider the vector component $v^{\phi}$ or the one-form component $v_{\phi}$:
$$ v^{\phi} = \...
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Help with geometric view of conjugate momenta and Legendre transformation
I'm familiar with the ''coordinate view'' of Lagrangian and Hamiltonian mechanics where if $\pmb{q}=(q^1,\dots, q^n)\in\mathbb{R}^n$ are any $n$ generalized coordinates and $L(\pmb{q},\dot{\pmb{q}})$ ...
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What is the formal criteria that the spacetime is curved?
We suppose we have three scenarios.
We are far away from mass and energy in a spot in the universe. We put in free movement a small object $m$, for example, an apple. At the same time, we send a ...
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Static Spacetime = no cosmological constant?
I stumbled over a strange result, which cannot be true: In the (3+1)-formulation of general relativity, one considers a metric of the type
$$g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}=(-\alpha^{2}+\...
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Gabriel's horn and General Relativity
Is there anything in GR that involves Gabriel's Horn?
This idea came to me when I met Flamm's paraboloid. If we take Schwarzschild metric at constant time and $\theta=\pi/2$, we get
$$ds^2=\left(1-\...
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What do Hawking/Ellis mean exactly by "non-rotating families of geodesics"?
In The Large Scale Structure of Space-Time, Hawking and Ellis refer twice (page 4, page 78) to non-rotating families of geodesics.
I don't know what that means. Is a rotating geodesic one that ...
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What is the physical meaning of non-commuting tetrads?
I'm reading about the tetrad formalism in GR and one main difference between the coordinate and the tetrad frame is that coordinate derivatives commute $\partial_\mu \partial_\nu = \partial_\nu \...
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What does the differential $d\Sigma_{ab}$ means when integrating over a two-surface?
In the paper $[1]$, Bardeen integrated an identity between Killing vectors and the Ricci tensor. I'll reproduce the calculation and explain my question in the following.
Consider then the identity,
$$...
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Are spinors intrinsically nonlocal?
I would prefer a purely classical answer since I don't think quantum mechanics (quantum field theory etc.) are necessary to answer this question and such answers will likely complicate matters. If you ...
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Observational effects of torsion in general relativity
Torsion is usually described as a rotation of the tangent vector along the geodesic, like the image below from Wikipedia:
Does this mean that if you add torsion, and you have an elevator falling ...
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Gauge Covariant Derivative of Gauge Field
Given a Gauge Theory with the covariant derivative defined as:
$${(D_\mu)^A}_B={\left(\delta \partial_\mu-igA^c_\mu (\mathcal{T}_c)\right)^A}_B={\delta^A}_B \partial_\mu-igA^c_\mu {(\mathcal{T}_c)^A}...
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Is energy "equal" to the curvature of spacetime?
When you are solving the Einstein field equations (EFE), you basically have to input a stress–energy tensor and solve for the metric.
$$
R_{\mu \nu} - \frac{1}{2}R g_{\mu \nu} = 8 \pi T_{\mu \nu}
$$
...
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(Carroll) In $\mathbb R^d$, what is the induced volume element on a $n$-dimensional submanifold?
I am following Carroll's book on general relativity [1]. In Eq. D.35, he states that the components of
the induced volume element $\hat \epsilon_{\mu_1...\mu_{n-1}}$ on a $(d-1)$-dimensional ...
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Killing vectors [closed]
I have an assignment:
For a metric $g_{\mu\nu}$ with everywhere timelike Killing vector $K^\mu$, a free particle with $p^\mu$ and mass $m$ show that its conserved energy $E=-p_\mu K^\mu$ is bound ...
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What comprises of a 'sensible' coordinate transformation?
I am doing the course on general relativity at my university and have been struggling with covariant and contravariant vectors. I understand that components of contravariant vectors transform in a way ...
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Is a stationary spacetime automatically globally hyperbolic?
Is a stationary spacetime automatically globally hyperbolic? Can one construct a Cauchy-Surface by the existence of a global timelike Killing Vector field?
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Is spacetime isomorphic to a metric space?
I know that spacetime, as described by General Relativity (GR), is a pseudo-Riemannian manifold. The label "pseudo" is due to the fact that the metric of spacetime entails not only positive ...
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How to derive the Kerr killing vector?
The Kerr metric have two killing vectors:
$$t^{\mu} \equiv (k_{t})^{\mu} = (1,0,0,0)\hspace{5mm} \mathrm{and}\hspace{5mm} \phi^{\mu} \equiv (k_{\phi})^{\mu} = (0,0,0,1). \tag{1}$$
In general, it is ...
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Implications of Parameter Choice in Geodesic Equation
Is there a difference, conceptually speaking, between solving the geodesic equations using $\lambda$ as an arbitrary parameter vs substituting a coordinate from the metric in it's place?
For instance, ...
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How to vary a metric
I am doing some GR computations and I want to rescale a metric, this is straightforward becuase I can just rescale the coordinate in question: $x \to x_{\text{rescaled}} = kx$ where k is just some ...
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Metric in dilatation transformation of massless scalar field
The lagrangian density of the massless real scalar field is
\begin{align}
L = \frac{1}{2}\eta^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi = \frac{1}{2}\partial_\mu\Phi\partial^\mu\Phi.
\end{align}
I want ...
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Confusion about convention for curvature tensor
I am a little bit confused about the convention of the curvature tensor. The books of Wald and Misner/Deser/Wheeler seem to have the same conventions, i.e. the indices of the Riemann curvature tensor ...
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How does the fiber bundle perspective on geometric phase lead to a certain connection one-form?
I'm trying to understand why the relevant connection one-form when calculating geometric phase in quantum systems is
$$\mathcal{A}_\psi(X):=i \text{Im}\langle \psi | X\rangle.$$
Set-up: I'll set the ...
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151
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Riemann curvature tensor in flat space
Can Riemann curvature tensor be non-zero in flat space
if the 3rd term (Lie bracket term) is non zero,
if the 3rd term is zero?
I was experimenting with random vector in flat space (Minkowoski ...
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The inhomogeneous Maxwell equation ${*}\mathrm d{*}F=J^\flat$ is only true for the signature $(+---)$?
In short, we would expect that considering the Riemannian manifold $(M,g)$ and the Riemannian manifold $(M,-g)$ leads to the same Maxwell equations, i.e. we would expect that
$\newcommand{\imult}{\...
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Sufficient and necessary condition for being a black hole energy-momentum tensor
Is there any necessary and sufficient mathematical condition(s) so that a (general) energy-momentum tensor can possess an assemblage of black holes? Or in other words, if I'm given a general energy ...
- 628
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1
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162
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Uniqueness of the diagonal form of metric
For Schwarzschild solution, if we use the coordinates ($t$,$r$,$\theta$,$\phi$). the metric in these coordinates are diagonal, my question is, is there exist another set of coordinates ($t^{'}$,$r^{'}$...
- 391
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Thermodynamic Potential Minimization
My textbook states that at equilibrium thermodynamic potentials are minimized. I am having trouble understanding how this minimization work and how to visualize it. For example, the Helmholtz free ...
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63
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Newtonian Gravity from curved space?
Imagine you have the arc-length of a curve, in spherical, coordinates:
$$
s = \int_{\mathcal C}{d\tau \; \sqrt{f(r)^2 \left (\frac{dr}{d \tau} \right )^2 + r^2 \left (\frac{d \theta}{d \tau} \right )^...
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Connection between covariant derivative operators upon conformal compactification
I'm having trouble determining the connection between two covariant derivative operators. These are: the one associated with the original space-time (and thus with the metric $ \tilde{g}_{ab}$) and ...
