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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Curvature from change of coordinates

This is from Zee's book on General Relativity. Using $$g_{\mu\nu}=\delta_{\mu\nu}+B_{\mu\nu,\lambda\sigma}x^\lambda x^\sigma+...$$ $$x^\mu=x'^\mu+M^\mu_{\nu\lambda\sigma}x'^\nu x'^\lambda x'^\sigma+.....
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Definition of spacetime in GR

In all the references/textbooks that I have looked at, the precise definition of spacetime is never really clear. By gathering the hypothesis that we need to make, I get the following definition: $$\...
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Locally Flat Understanding

I wanted to make sure that I was definitely understanding the proof of locally flat correctly. I can't see to find a similar proof to the one in the book, so I'm not super sure if my understanding/...
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Doubt on Newman-Janis algorithm for a traversable Wormhole

Recently in a paper $[1]$ the researchers presented a rotating traversable wormhole solution using the famous Newman-Janis Algorithm $[2]$. But something is anoying me. In $[1]$ they presented the ...
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What are the local Lorentz transformations in general relativity?

What is the exact form of local Lorentz transformations (from the point of view of the metric) in a curved spacetime background like in general relativity? It should deviate substantially from ...
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An identity between the d'Alembertian and the covariant derivative

Suppose $f$ is function which depends on $\phi$, $f = f(\phi)$; and $\phi$ is a scalar field. We define $$\square \equiv g^{\mu\nu} \nabla_\mu \nabla_\nu$$ and $$\nabla_\mu f(\phi) \equiv f_{;\nu}$$ ...
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Covariant derivatives in a rank 2 tensor

I was trying to prove that for any second order tensor: $$A^{\mu\nu}_{;\mu\nu}=A^{\mu\nu}_{;\nu\mu}$$ considering the torsion free property and locally flat coordinates. Considering the point where ...
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Conditions on one forms [closed]

I am trying to solve exercise 8.3 from Lightman's problem book, but I don't know where to start to get a sufficient and necessary condition on a field of one forms $\tilde\sigma$ for there to exist a ...
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Is there any way to prove that contrarly to a flat 3D space, a curved 3D space can only be constructed in a 4D manifold?

This question is a result of me trying to understand how this universe can be possibly infinite if it isn't infinitely old. So to compare with an area that is flat it can be constructed both in 2D and ...
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Commutators and coordinate-induced basis (General Relativity)

There is an exercise in MTW that asks to prove that, given two vectors u and v, there exists a coordinate system for which $$ \textbf{u}=\frac{\partial}{\partial x^1} \mbox{ and }\textbf{v}=\frac{\...
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Proof of the differential Bianchi identity

I was trying to prove the differential Bianchi identity by applying the covariant derivatives to each of the Riemann tensor terms $R^{\lambda}_{\sigma\mu\nu;\rho}+R^{\lambda}_{\sigma\nu\rho;\mu}+R^{\...
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Weyl transformation vs diffeomorphism; conformal invariant vs general in/covariant

Background info: My understanding: 1. Weyl transformation is a local rescaling of the metric tensor $$ g_{ab}\rightarrow e^{-2\omega(x)}g_{ab} $$ A theory invariant under this Weyl transformation is ...
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Complex Lie algebra vs Real Lie algebra in Physics [closed]

A Lie algebra is a vector space $\mathfrak{g}$ over some field $F$ together with a binary operation $$\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$$ called the Lie bracket satisfying the following ...
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Sectional curvature to Ricci tensor

We know that to measure the if the n-dimensional manifold is curved or not, we look at the Riemann Tensor. If it is 0, then it is flat otherwise it is curved positively or negatively (which each of ...
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Deforming a nematic line defect to a uniform configuration

In Nakahara section 4.9, "Defects in nematic liquid crystals", it is discussed that the order parameter for a nematic should be the real projective plane $\mathbb{R}P^2$, which has ...
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Weyl connection

Considering the Weyl connection $\Gamma^{\lambda}_{\mu\nu}$ in a torsion-free space, defined by the covariant derivative of the metric tensor and a vector field $A_{\lambda}$: $D_{\lambda}g_{\mu\nu}=...
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Writing the EL equations in the language of differential geometry

I want to explore generalised Noether currents obtained from $q$-form symmetries in an action. The regular way we obtain Noether currents is fairly straightforward: We have a 0-form symmetry $\phi \to ...
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Taking Chart-Coordinate Derivatives of Geodesic Tangents

I've been working to understand cuvature in relation to geodesic deviation. Reading Wald, he uses a family of gedesics parameterized by two coordinates $s,\tau\in\mathbb{R}$ called $\gamma(s,\tau)$, ...
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How did the Lagrangian and Hamiltonian theories of motion inspire the idea that forces should be treated as one-forms instead of vectors?

On page-5 of this paper1 by E. Minguzzi titled "A geometrical introduction to screw theory", he writes: Who adopts this point of view argues that it should also be adopted for forces in ...
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Spinor covariant derivative conventions

The covariant derivative of a spinor $\psi$ is given by $$ \nabla_\mu \psi = \partial_\mu \psi + \Omega_\mu \psi $$ where $\Omega_\mu$ is the spin connection. In equation (7.227) of Geometry, Topology ...
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Why isn't $g^{\mu \nu}\partial_{\nu}$ equal to $\partial^{\mu}$ in General Relativity?

The question is pretty short. I have been told that unlike for flat spacetime, in curved spacetime we cannot write $g^{\mu \nu}\partial_{\nu}f=\partial^{\mu}f$ With $f$ an scalar function, but I don't ...
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Could spacetimes have singular manifolds?

Let's take a spacetime as a pair $(M,g)$ where $M$ is the manifold and $g$ the metric. I've seen that there exist a generalization of manifolds. This generalization consist in accept singularities in ...
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How can a 2-sphere exist in Euclidean 3-space?

I don't know if this is a simple question to answer however, I have trouble understanding how a spherical object (such as a planet) with positive curvature can exist in Euclidean 3-space with no ...
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How momentum is dual to the velocity vector at a point on a differentiable manifold?

The tangent space $T_pM$ which is a real vector space on a point $p$ of a differentiable manifold $M$, has a cotangent bundle $T_p^*M$ at $p \in M$, such that for any $v \in T_pM$ and for any $w \in ...
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Weyl Tensor Theorems

Is there any quick way of proving that for a conformal metric of the form: $g^{}_{\mu\nu}=\Omega^2\eta^{}_{\mu\nu}$ where the $\eta^{}_{\mu\nu}$ is the usual Minkowski metric, the Weyl tensor vanishes ...
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Extrinsic Description of spacetime curvature, is it possible? How many dimensions would we need? [duplicate]

Imagine we had an extra spatial dimension $f$ and so an overall spacetime interval: $ds^2=-dt^2+dx^2+dy^2+dz^2+df^2\tag{1}$ Now let's say that we're restricted to a 4-dimensional "plane" in ...
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Christoffel Symbols Different Representation?

I was given the $\mathbb{R^2}$ metric in polar coordinates, as follows: $$ ds^2=dr^2+r^2d\theta^2 $$ In this context we denote $e_1=\partial_r=(\cos(\theta), \sin(\theta))$, $e_2=\partial_{\theta}=(-r\...
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Explicit expression of gradient, laplacian, divergence and curl using covariant derivatives

For my course in General Relativity I am given the problem to find the expressions for the gradient, laplacian, divergence and curl in spherical coordinates using covariant derivatives. I have tried ...
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The role of diffeomorphism in physics

Suppose space time is the manifold $M $ isomorphic $ \mathbb{R^4}$ with the metric $-\eta_{00}=\eta_{11}=\eta_{22}=\eta_{33}=1$ in the Cartesian coordinates $\Psi(p)=(x^0,x^1,x^2,x^3)$ for $p \in M $ ....
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Generalisation of Equilateral Triangle in curved space [migrated]

It is known that in flat Euclidean Space the internal angles of a triangle sum to $180^{\circ}$ but if we draw a similar triangle in curved space then the sum of interior angles will be more or less ...
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Variation of d'Alembertian operator

I am working in a higher derivative quantum gravity theory, and I'm having trouble with the variation of d'Alembertian operator. Suppose we have the following action: \begin{equation} \mathcal S[g]=\...
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Do separation vectors exist in general relativity?

Deriving the equation of geodesic deviation one looks at two test masses on positions $x^\mu$ und $\tilde{x}^\mu$ and defines the separation vector $\boldsymbol{\chi}$ as $$\tilde{x}^\mu=x^\mu+\chi^\...
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Quantum Geometric Tensor and Berry Connection [closed]

The Quantum Geometric Tensor is given by $$ \begin{split} Q_{\mu\nu}=\langle\partial_{\mu}\psi|\partial_{\nu}\psi\rangle-\langle\partial_{\mu}\psi|\psi\rangle\langle\psi|\partial_{\nu}\psi\rangle \end{...
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Horizontal vector subsepace of electromagnetic connection in Minkowskian spacetime

For Minkowskian space-time $M$, the principal bundle for electromagnetism is $(M \times U(1), M, proj_{1}, U(1))$. I imagine there is a global gauge potential (since I can choose a global section, ...
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How Would Projectile Motion Be Described in Non-Euclidean Space?

If I and a friend found ourselves in a world described by spherical geometry (as simulated in this linked video https://youtu.be/yY9GAyJtuJ0 ) how would the kinematics equations need to be augmented ...
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Heat kernel: DeWitt iterative procedure

The DeWitt ansatz for the heat kernel is given by $$K(t ; x, y ; D)=(4 \pi t)^{-n / 2} \Delta_{V V M}^{1 / 2}(x, y) \exp \left(-\frac{\sigma(x, y)}{2 t}\right) \Xi(t ; x, y ; D)$$ where $\sigma$ is ...
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Prove $\displaystyle{\not}{\nabla}^2=\nabla_{\mu} \nabla^{\mu}-R/4$ [closed]

I am trying to show $\displaystyle{\not}{\nabla}^2=\nabla_{\mu} \nabla^{\mu}+R/4$ where $R$ is Ricci scalar. $\nabla_{\mu}$ is covariant derivative for spinor: \begin{equation} \nabla_{\mu}=\partial_{...
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Definition of parallel transport

The definition of parallel transport is $t^iD_i u^j=0$, where $\vec{t}$ is the tangent vector to the curve and $\vec{u}$ is the vector being parallel transported along the curve. In flat space, using ...
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Transverse metric being 2-dimensional in null case

In Wald section 9.2 page 221 he says that We turn our attention; now , to null geodesic congruences. Again, we parameterize the geodesics by an affine parameter $\lambda$, but , unlike the timelike ...
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Cauchy Surfaces inside Schwarzschild Black Hole

Context: Working in a globally hyperbolic spacetime we know by definition that we can foliated our spacetime $\mathcal M$ with disjoint cauchy surfaces $\Sigma_\alpha$ $\left(\therefore \bigcup \...
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Does spacetime topology have importance in physics?

Generally in textbooks they represent spacetime as $(M,\nabla,g,t)$ where $M$ is a Lorentzian manifold,$\nabla$ a torsion-free connection,$g$ a metric and $t$ a time orientation. But they do not talk ...
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What does a universe with a boundary look like?

Physically, what would it look like if we lived in a universe with a boundary at finite distance?
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Alternative formula for the affine connection in a new coordinate basis

In Hobsons's General Relativity: An Introduction for Physicists, pg. 64, he gave two different expressions for the affine connection $\Gamma'^a_{bc}$ in a transformed coordinate basis $x'^a$ (the ...
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Isn't the following addition wrong on manifold as done in Frankel book?

In ch-$4$ when calculating expression of Lie derivative using Hadamard's Lemma before $(4.4)$ Frankel's do following manipulation: $$\lim_{t\rightarrow0}\frac{\textbf{Y}_{\phi_tx}(f)-\textbf{Y}_x(f)}{...
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Conformal vector fields in $m$-dimensional Euclidean manifold

A vector field $X=X^\mu\partial_\mu\in\mathfrak{X}(M)$, where $M$ is a (pseudo-)riemannian manifold with a generic metric tensor $g_{\mu\nu}$, is a conformal Killing vector field if the conformal ...
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What motivates defining vectors as first order differential operators?

I have read some introductions to geometrical ideas and tensors and physics and what some of them do (see, for example, Frankel's Geometry of Physics) is define a vector as a first order differential ...
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Does it make sense to take an infinitesimal volume of shape other than a cube?

The question clearer: Is the infinitesimal cube the absolute smallest infinitesimal volume? (Sorry if people thought that it meant: "Is it possible and is it done in daily life to use anything ...
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1answer
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Meaning and application of the connection coefficients (Christoffel symbols)

I know that in polar coordinates, it is $\frac{\partial \,{{\mathbf{e}}_{r}}}{\partial \theta }={{\mathbf{e}}_{\theta }}$ and $\frac{\partial \,{{\mathbf{e}}_{\theta }}}{\partial \theta }=-{{\mathbf{e}...
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Conformal Transformation and angle preserving [duplicate]

For general co-ordinate transformation $V^\mu$ transforms as $V^\mu \rightarrow V'^\mu= \large\frac{\partial x'^\mu}{\partial x^\alpha}\ V^\alpha $ therefore the inner product between two vectors ...
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From Reissner-Nordstrom to Dilatonic Gravity and Einstein Equation

I am doing some calculations of this paper: https://arxiv.org/abs/1711.08482 and in particular I am having trouble with 2 parts: Dimensional Reduction (16): I have managed to get equation (16) from ...

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