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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Conserved quantities with charge and Killing's vectors
I'm trying to solve the following problem:
A particle with electric charge e moves with 4-velocity $U_{\alpha}$ in a spacetime with metric $g_{αβ}$ in the presence of a vector potential $A_µ$. The ...
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Conformal covariant derivative of a scalar
Suppose that I have a metric $g_{ij}$ with covariant derivative $\nabla_{j}$ and another metric, $\gamma_{ij}$, with covariant derivative $D_{j}$ that is conformally related to $g_{ij}$ as $\gamma_{ij}...
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Problem with understanding contravariant component transform
I am reading Susskinds book General Relativity: The theoretical minimum and I got a bit stuck on the transformation rule of contravariant components. The book defines the components of a vector $(V^{’}...
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Nambu-Goto action and the World-Sheet Area
I am studying string theory from the book "String theory and M-theory", written by Becker, Becker and Schwartz. My question is:
We are told that the Nambu-Goto action is simply the one that ...
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Metric Tensor times its inverse (non-zero curvature)
so I am quite confused regarding the spatial metric tensor $g_{ij}$. If I have $g_{ij}g^{ij}$ I essentially get the trace of the metric tensor $g$ right? Or, do I get $\delta^i_i = 3$ instead?
The ...
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Confusion regarding Riemann Tensor and Ricci Tensor
Ricci Tensor is the contraction of the Riemann Tensor. Even if all the components of the Ricci Tensor is zero, that doesn't mean that the spacetime is flat. If all the components of the Riemann Tensor ...
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How to transform a partial derivative to a directional derivative with respect to some affine parameter?
Suppose an affine parameter $\lambda$ is defined along a null geodesic with $dx^\mu/d\lambda=k^\mu$. How could I write the partial derivative $\partial f/\partial x^\mu$ by using $df/d\lambda$? If $k^\...
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Lie group generators - exponential map
In the framework of Quantum Physics, I have to explain to some of my colleagues what is a Lie group, a Lie algebra and their connections with the exponential map. This is mainly to make them ...
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How can I calculate the solid angle of a planet? [closed]
I've read several articles about solid angles recently and have one question now. They can be calculating by using the following formula: $\Omega = \frac{A}{r²}$
A stand for the area and r for the ...
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HaMiDeW coefficients - recursive calculation of the coincidence limits
In his book Aspects of Quantum Field Theory in Curved Spacetime Stephen Fulling calculates the coincidence limit $[a_1]$ and gives an idea of how $[a_n]$ with $2 ≤ n$ can be found recursively.
Since ...
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What is the $r$ coordinate in a $\mathbb{S}^3$ FLRW spactime?
I'm having trouble understanding what the $r$ reduced-circumference coordinate really is in a 3-sphere $\mathbb{S}^3$ context.
Let's start with the unit 3-sphere metric in hyperspherical $(\psi, \...
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About Lorentz transformations
Is this definition of Lorentz transformations correct?
Consider 3+1 dimensional space-time manifold $M$. Let $v,u$ are two vectors of the vector bundle and $g$ be a metric on $M$.
Now we can define ...
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What diffeomorphism does the Hamiltonian constraint generate?
Consider the Hamiltonian constraint $\mathcal H(x)$ in the ADM formalism. What diffeomorphism does this generate?
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Partial derivative of Christoffel symbols
I have faced one issue regarding the partial derivative of the Christoffel symbols. $\Gamma$'s themselves are not tensors. But if we take the difference between two $\Gamma$'s at the same point, then ...
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Lie algebra of a Lie group
I'm going through some notes on group theory for physics. After introducing the concept of Lie group and Lie algebra the writer makes the connection between the two.
Let $G$ be a Lie group of ...
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Relation between Lorentz transformations in QFT and GR [duplicate]
I often have difficulty expressing certain doubts because I am not (not even my self, yes) fully aware of what's going on that bothers me, so forgive me if the question isn't the clearest.
I noticed ...
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Christoffel symbol and partial derivative of the metric
It may seem like a dumb question but I'm trying to solve a problem involving coordinate transformations on the Christoffel symbol and to solve it they do the product rule
$$\partial_\alpha g_{\beta ' \...
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Berry curvature in cartesian coordinates
I am trying to find the Berry curvature in Cartesian coordinates for the spin-1/2 Hamiltonian given by
\begin{equation}
\hat{H} = \sum_{i}d_i \hat\sigma_i
\end{equation}
Where $\mathbf{d} = \mathbf{d(...
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In the fibre bundle description of gauge theories, what is actually the difference between dimension of spacetime and dimensionality of the bundle? [closed]
Please pardon me if my words are not rigorous mathematically, but I hope you understand what i mean.
In the fibre bundle description of gauge theories, what is actually the difference between ...
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Dipolar field in terms of demag. tensor
Assuming a potential as
$$\Phi=\frac{1}{4\pi}\oint_S \frac{\vec M(\vec r')\cdot\hat n}{|\vec r - \vec r'|}da' \tag{1}\label{1}$$
where the integral can be written as
$$\int_V \nabla' \left(\frac{\vec ...
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The Poisson bracket for a Dirac field on an Eddington-Finkelstein background metric
I am interested in deriving the anti-commutation relations of a Dirac field for a $(1+1)$D Eddington-Finkelstein metric given by
$$ ds^2 = f(r) dt^2 - 2 dt dr, \quad f(r) = 1 - \frac{r_s}{r}.$$
where $...
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Inverse of the covariant derivative
Given the covariant derivative of some tensor, for the sake of this example a covariant vector:
$$\nabla_\mu A_\nu$$
Is there a well-defined inverse operation on the covariant derivative such that it ...
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Does Curved Edge exist for a smooth infinitely long right circular Cylinder? [migrated]
This question is in the continuation of this question. As it is cleared from the comments of the respective question that an infinitely long cylinder which is also a right circular, is a smooth $3$D ...
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Vector and dual vector in Newman-Penrose formalism
I am confused about the vector and dual vector in the tetrad formalism. Start from Schwarzschild metric
$$\mathrm{d} s^2=\left(1-\frac{2m}{r}\right)dt^2 - \left(1-\frac{2m}{r}\right)^{-1}\mathrm{d} r^...
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Why does $(\partial_\mu g(x))g(x)^{-1} \in T_e G$?
I'm reading Folland's book about QFT and at chapter nine he states:
How must the Lagrangian be modified in order to remain invariant under this larger group of transformations?
As before, the point ...
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Misner space in Hawking & Ellis book
I am studying the Taub-NUT space presented on page 170, of Hawking & Ellis' book "The large scale structure of the universe". There are some aspects that are not clear to me about the ...
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What is the three-parameter family of time translation Killing field in Minkowski spacetime?
In section 5.1 of Wald's Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, there are two paragraphs before the discussion of Unruh effect as follows.
Let us reconsider the ...
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How to get rid of the affine parameter in geodesic equation?
I encounter a problem that require me to calculate the geodesic of
$$ds^2=\frac{dx^2+dz^2}{z^2}$$ with the endpoint $(x_L,0),(x_R,0)$. I get the answer $\ddot{x}-\frac{\dot{x}\dot{z}}{z}=0$ and $\...
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Combining two spacetimes of different curvature
This is based on a question by someone else on this site:
Combining metric tensors/curvature tensors
They asked:
Consider a particle which causes a metric $g_{\mu\nu}$
on an otherwise Minkowski ...
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How to determine the electric and magnetic fields from $*F=q\sin \theta d\theta \wedge d\phi$?
I have the 2-form $*F=q\sin \theta d\theta \wedge d\phi$, how can I determine the eletric and magnetic fields from that?
I have tried wrtting F in the vector potential form for them finding the ...
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What did happened with the 2 in equation (3.82) to (3.83)?
I have tried to do the same computing by the definition of the Christoffel symbols by the metric and the result is the same that the image, but i don understand what occur with the number 2. Why did ...
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How to derive the energy tensor invariantly?
On a (pseudo-)Riemannian manifold $(M, g)$ I can define the following action for any $\phi \in C ^{\infty}(M)$:
$$
\mathcal{S}(\phi) = \int_M g(\text{grad }\phi, \text{grad }\phi)
\mathrm{d} V.
$$
...
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Gravity from a reduction of the structure group of a frame bundle $FX$ to a Lorentz group $SO(1,3)$
According to https://en.wikipedia.org/wiki/World_manifold, gravity can be understood as follows:
In accordance with the geometric Equivalence Principle, a world manifold possesses a Lorentzian ...
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Is Newton-Cartan theory really equivalent to Newton's theory of gravity?
It is often said that Newton-Cartan theory is a reformulation or perhaps a generalization of Newton's theory of gravity, and it is said that (given certain conditions/assumptions) the two theories are ...
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Mermin-Wagner theorem for curved spaces?
i'm searching for the Mermin-Wagner theorem formulation for negative curvature surfaces. Something like that exist?
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How do I understand the Hodge $⋆$ operator in Yang-Mills Lagrangian?
The gauge-invariant part in Yang-Mills Lagrangian is
$$
\mathcal{L}_{\text{gauge}} = -\frac{1}{2}TrF_{\mu\nu}F^{\mu\nu} = -\frac{1}{4}F_{\mu\nu}^aF^{a, \mu\nu}.
$$
Sometimes I see the lagrangian ...
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Non-surjective coordinate transformation
Is coordinated transformation which is not surjective (so it's inverse remove some region on spacetime) allowed in general relativity?
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Classification of Penrose Diagrams
I am new to the concept of Penrose Diagrams and I am curious, does there exist a classification of Penrose Diagrams? Maybe simply the conformal diagram is the manifold mod the group action, $M/SO(3)$ (...
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Is there a natural coordinate system in general relativity?
After reviewing some different coordinates systems that describe Schwarzschild spacetime (such as Gullstrand–Painlevé coordinates) it seems like we can always make a coordinates transformation very ...
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How to derive connection Lie algebra valued one-form on the frame bundle if given the pulled back of it on the physical space?
I am following this YouTube lecture by Schuller where he finds the appropriate formalism for the quantum mechanics in the physical curved space.
Everything makes sense to me but at the very end I see ...
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Derive Laplacian in polar coordinates using covariant derivative [migrated]
In cartesian coordinates since the metric components $g^{ab}$ are constant, we know that $\partial_c = \nabla_c$ where the RHS is the covariant derivative.
So we can write the laplacian in cartesian ...
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Parallelism in General Relativity
I have a small technical question.
When two 4-vector fields $A^{\mu}$ and $B^{\mu}$ are said to be parallel in the context of general relativity, do we mean that $A^{\mu}=\alpha(x^{\nu})B^{\mu}$ where ...
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Notational meaning of $\nabla_{\lambda}V^{\rho}$ and $\nabla_{\mu}\nabla_{\nu}V^{\rho}$
This question is related to Reconciling different expressions for Riemann curvature tensor, but it's different since it asks for some notational clarification arising out of calculations I did. To not ...
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Understanding how the geometric understanding of SR comes from its classic postulates
My background is in mathematics, and I am well versed in differential geometry and general relativity. I have not, however, studied special relativity as much, and frankly view Minkowski space as a ...
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What's the geometrical meaning of the Einstein Tensor? [duplicate]
What is the geometrical meaning of the Einstein tensor $G_{\mu \nu}$ ?
I know the field equations and all of that standard stuff.
I know that the tensor $R = g^{\mu \nu}R_{\mu \nu}$ quantifies the ...
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Simplest way to see that the Euler density is topological
I’m looking for a simple way to see that the integrated Euler density $\sqrt{g} \, E_{2n}$ is topological (i.e. metric-independent in general even dimension $2n$). I can see it it must be true in two ...
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Sorts of relativistic effect around black holes [closed]
There are many effects around black holes. In particular it is possible to study the motion of geodesics, calculate tidal tensors, lense-thirring effetcs and so on.
So, beyond tidal effects, geodesic ...
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Calculations with tensors give two different results from seemingly equivalent paths:
$\require{cancel}$
I want to calculate some operators in the context of Metric-Affine-Gravity: specifically spinors. I will work in tangent space, where all greek ($\mu,\nu,..$) indices are cast into ...
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Exterior derivatives Leibniz rule
I want to prove Sean Carroll's "spacetime and geometry"'s eq.(2.78):
$$
\mathrm{d}(\omega \wedge \eta)=(\mathrm{d} \omega) \wedge \eta+(-1)^p \omega \wedge(\mathrm{d} \eta) \tag{2.78}
$$
...
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On finding geodesics in general relativity
In the following, as usual, a spacetime is a pseudo-Riemannian manifold $M$ with metric $g$ compatible with the Levi-Civita connection $\nabla$. If an affinely-parametrized curve is a geodesic, then ...
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