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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If two hypersurfaces are conformally related and one of them is a Cauchy surface, is the other a Cauchy surface too?

I'm reading Wald's "General Relativity". In appendix D he states that conformal transformations preserve causal structure. Does this mean that they also preserve when a hypersurface is a ...
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Newtonian limit of General Relativity using weak field approximation

I read Geometry, Topology and Physics by Mikio Nakahara. In exercise 7.24 I have to compute the Newtonian potential equation in General Relativity weak field limit : where (7.202) refer to $G_{\mu\nu}...
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Maxwell's action with differential geometry formalism

I'm having problems in showing that the following identity, regarding Maxwell's action, holds true: $$ S_{Maxwell}=-\frac{1}{2}\int F\wedge\star F=-\frac{1}{4}\int \sqrt{-g}\,F_{\mu\nu}F^{\mu\nu}\...
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Is time orientability independent of space orientability?

In the context of Lorentzian manifolds, we can define time orientability and space orientability as follows: Time orientability: A Lorentzian manifold $(M,g_{ab})$ is time orientable if and only if ...
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Derivation of Lovelock gravity field equations from the more general f(Lovelock) theory

I'm trying to derive the Lovelock gravity equations of motion (EOM) from the more general f(Lovelock) theory (see https://doi.org/10.1007/JHEP04(2016)028). In Lovelock gravity $$\nabla^\alpha \frac{\...
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Why is $\frac{d^2x^{\mu}}{d\lambda^2}=0$ not a tensorial equation?

In flat space, the motion of freely falling particles given by the parametrized path $x^\mu(\lambda)$ is given by the geodesic equation $$\frac{d^2x^{\mu}}{d\lambda^2}=0.$$ Why is this not a tensorial ...
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Covariant derivative of spherical harmonics

Given is the metric $\gamma_{jk}$ for the surface of a Sphere $S^2$ with $\gamma_{22}=1,\gamma_{23}=\gamma_{32}=0$ and $\gamma_{33}=\sin^2(\theta)$. The coordinates are $x=$($t,r,\theta,\phi$) and $j$ ...
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How to pick a constraint function for Lagrangian mechanics? [duplicate]

Motivating Example Consider a system which consists of two masses $m_1$ and $m_2$ at positions $x_1$ and $x_2$ respectively joined together by a rigid rod of negligible mass and length $l$ . We have ...
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Derivation of Gauss-Codazzi type equation (Ricci relation)

I am following Padmanabhan's book Gravitation for the particular derivation. The derivation goes as follows, \begin{align} R_{abst}n^t&=\nabla_a\nabla_b n_s-\nabla_b\nabla_a n_s=\nabla_a(-K_{...
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Intensity along null geodesic

Let $\mathcal P$ be a bundle of light with a continous spectrum of frequencies $\nu$ emitting from $x^\mu$ in a static spacetime $g_{\mu\nu}$ (e.g. kerr-schild) in the direction $n^\mu = \frac{p^\mu}{\...
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Christoffel symbols of Poincaré metric using orthonormal tetrad formalism

I want to calculate the Christoffel symbols for the Poincaré metric using the orthonormal tetrad formalism. $$ds^2 = y^{-2}dx^2 + y^{-2}dy^2.$$ I introduce a non coordinate orthonormal basis with one-...
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Riemann curvature tensor is the only tensor that you can write down that has two derivatives of the metric tensor

Is the statement in the main question correct? Can someone send me a proof (link, pdf file etc.)
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What are the analogues of $F_{\mu\nu}$ in General Relativity?

In electromagnetism, the measurable gauge-invariant quantities are the electric and magnetic fields or the six independent components of the field strength tensor $F_{\mu\nu}$. What are the analogues ...
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Why is Divergence of a vector field which is decreasing in magnitude as we move away from origin positive at points other than origin?

divergence of $\frac{\hat{r}}{r}$ is positive, $\frac{\hat{r}}{r^2}$ is zero and $\frac{\hat{r}}{r^3}$ is negative at points other than origin. As I have studied, divergence in 3D space tells whether ...
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Why is the Riemann tensor $C^{\infty}$-multilinear?

According to my textbook on differential geometry, the Riemann tensor $R(\cdot, \cdot)$ is $C^{\infty}$-multilinear. I suppose this means that if $M$ is a manifold, $p \in M$ and $x_1,x_2, y, z \in ...
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What is the significance of the trace of a tensor?

On a Riemannian manifold, the trace $X$ of a tensor $X_{\mu\nu}$ is defined as $$X=g^{\mu\nu}X_{\mu\nu}.$$ In linear algebra, the trace is the sum of the diagonal elements, so a traceless matrix has ...
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Covariant and contravariant vectors with Einstein index notation

I'm trying to "translate" some expressions from tensor notation to Einstein notation and I have some trouble with the formulas for the covariant and contravariant base vectors. How should ...
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What class of spacetimes has an associated temperature?

It is very well known that the Schwarzschild metric carries a temperature inversely proportional to the mass. Is there a much wider class of spacetimes that has temperatures associated with it? What ...
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3answers
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Is the matrix $[g_{\mu\nu}]$ of metric tensor a linear operator?

Starting in an arbitrary coordinate system with basis vectors $\textbf{e}_\mu$ and metric components $g_{\mu\nu}$, we can always diagonalize the matrix $[g_{\mu\nu}]$ so that the components of the ...
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Physical meaning of geodesic equation $p^\lambda \nabla_\lambda p^\mu=0$

In Sean Carroll's GR book pg. 109, it was said that the geodesic equation for timelike paths can be written in terms of the four momentum $p^\mu=mU^\mu=m\frac{dx^\mu}{d\tau}$: $$p^\lambda \nabla_\...
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Why have we not found an interior Kerr solution?

The Schwarzschild interior solution was found not so long after the exterior solution was found. I understand that Kerr solution is significantly more complicated and there are more conditions at the ...
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Raising/lowering indices in linearized GR

In linearized general relativity, we have the unperturbed metric and the perturbed metric. In all textbook treatments, they say that they are going to raise and lower indices with the unperturbed ...
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Is the Poincaré gauge theory a real gauge theory in the mathematical sense?

When studying Poincaré gauge theory using Milutin Blagojevich's book on "Gravitation and gauge symmetries" we find an interesting line of thought. But to get that I need to set some ...
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Why is the Kaluza-Klein ansatz the natural choice?

In standard Kaluza-Klein theory we choose a parametrisation for the 5-dimensional metric: $$d\hat{s}^2 \equiv \hat{g}_{ab} dx^a dx^b = g_{\mu\nu}dx^\mu dx^\nu + \phi^2(dz + A_\mu dx^\mu)^2 $$ where $...
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Where can I read about the broader implications of diffeomorphism invariance?

I have had 2 semesters of GR. Diffeomorphism invariance was stated as this very important thing although all that was explained was that it is required for physics to be the same in all "...
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Vanishing torsion condition preserved when doing a connexion transformation

I have a simple question but I can't wrap my head around it. First let's consider a Riemannian manifold with a metric $g$ with an affin Connection $ \Gamma $ If we want to construct a Levi-Civita ...
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Euler characteristic of an Einstein-Cartan manifold [migrated]

In the Einstien-Cartan theory, the Euler characteristic of the manifold $M$ is given by $$\chi(M) = \int F_{ab}\wedge\star F^{ab}$$ where $F_{ab} = d\omega_{ab}+(\omega\wedge\omega)_{ab}$, $\star F_{...
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Coordinates vs. parametrization of a worldsheet

In introductory string theory, the worldsheet is described (e.g. Tong, Polchinski) as a surface $X^\mu(\tau,\sigma)$ in Minkowski spacetime indexed by two parameters: $(\tau,\sigma)$. Now, I initially ...
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Why is the projection tensor defined the following way?

In Spacetime and Geometry, Carroll(2019) and more generally in physics the projection tensor/operator for a hypersurface $\Sigma$ is defined by $$P_{\mu\nu} = g_{\mu\nu} - \sigma n_{\mu} n_{\nu},$$ ...
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Formula for curl in polar coordinates using covariant differentiation

For the plane in polar coordinates $(r,\theta)$ with metric $$ds^2=dr^2+r^2d\theta^2,$$ the curl on a vector field $v^a\partial_a$ is given by the rank-2 antisymmetric tensor $\nabla_av_b-\nabla_bv_a$....
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GHY term in loop quantum gravity

Does the GHY (Gibbons-Hawking-York) boundary term play any role in loop quantum gravity? If it does where and how does it show up? (I am more interested in 4D)
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In what context can pullbacks be interpreted as the dot product betwen vector fields [migrated]

Context / intuition I'll use Picard theorem to state the reasoning behind the question. The question itself is not related to this theorem. In $\mathbb R^2$ with coordinates $(x,u)$. Consider $\theta=...
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Local Rindler coordinates

According to arXiv:1607.07506, in equation (3.10), any Riemannian manifold can be locally endowed with a local Rindler coordinate, such that the metric is locally given by $$d s^{2}=d r^{2}-r^{2} d \...
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Important identity for covariant derivatives involving extrinsic curvature

How can I demonstrate this identity? $$\nabla_a(K^{ab}-\lambda Kg^{ab})=\Gamma^b_{ac}(K^{ac}-\lambda K g^{ac})$$ where $\lambda$ is a parameter, $K_{ij}$ is extrinsic curvature in ADM formalism and $...
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1answer
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Intrinsic definition of canonical vector field on tangent bundle

I have been trying to solve the following question's last part (iv): on giving an intrinsic definition to the vector field whose local definition is that in part (iii). I believe this is called the ...
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About the existenece and uniqueness of time-like vector fields in a time orientable manifold

I'm reading Wald's book "General Relativity" and I'm having a bit of trouble understanding his proof on the following lemma: Let $(\mathcal{M}, g_{ab})$ be time orientable. Then, there ...
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Ricci scalar of AdS in $D$ spacetime dimensions from structure equations

Starting from the AdS metric in $D$ spacetime dimensions in Poincare coordinates $ds^2 = \frac{R^2}{(x^3)^2}\eta_{\mu\nu}dx^\mu dx^\nu$ (R here is the AdS radius), I would like to compute the ...
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1answer
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Transformation law for the Levi-Civita symbol under a change of basis

I'm trying to prove that the Levi-Civita symbol $\epsilon_{i_1 ... i_n}$ is a tensor density of weight $w=-1$. For this purpose, it has to be shown that the transformation law for the components of ...
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1answer
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Definition of normal 4-vector of hypersurface in Poisson

In Poisson book on GR (toolkit) ch-$3$ normalized normal vector on a hypersurface ($\Phi(x)=0$) is defined as $$n_{\alpha}=\frac{\epsilon\Phi_{,\alpha}}{|\Phi_{,\mu}\Phi^{,\mu}|^{1/2}}$$ where $\...
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Transforming absolute derivative of separation four-vectors in GR

Moore (A General Relativity Workbook) claims that the following implication holds, where $\boldsymbol{n}$ is the separation four-vector between two objects at a given proper time: $$ \frac{d\...
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What does the derivative of tangent means? [closed]

While studying the circular motion I had to find the derivative of a tangent so I thought what the derivative of a tangent could probably mean since the derivative of position gives velocity. Or think ...
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2answers
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What exactly is a "smooth surface" in geometrical optics?

Often in elementary optics, one considers so-called "smooth surfaces" (this is to be viewed in contrast with the idea of a "rough surface"), with plane mirrors being a prototypical ...
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Does intrinsic curvature in a higher dimension mean that the lower dimensions also exhibit curvature?

If our universe has intrinsic curvature in a higher dimension, would that mean the 3 dimensions that we live in would be curved? and if so would the lower dimensions exhibit intrinsic or extrinsic ...
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Why does Christoffel symbol $\Gamma^0_{01}$ not vanish for weak gravitational field in my calculation?

The metric for a static weak gravitational potential $\Phi(\vec x)$ is in first order $g_{00}=1+2\Phi$ $g_{ij}=-(1-2\Phi$) with all other components zero. When I calculate Christoffel symbol $\Gamma^\...
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1answer
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Why would geodesics depend on velocity in general relativity?

In general relativity, it's the curvature of spacetime that gives the effect of gravity, due to objects following geodesics. What I learnt about geodesics is that they are like straight lines in ...
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2answers
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Dependence of entropy on topology?

In black hole thermodynamics, there is a nontrivial dependence of thermodynamical entropy on the area of the event horizon which is a geometric object. I want to know how this may be intuitively ...
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Intrinsic and extrinsic curvature

Does having intrinsic curvature always mean that there is extrinsic curvature? Can you have one without the other? Is it possible to have one without the other all the time? And is it hypothetically ...
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Can the spin connection defind on an orthonormal frame be non-metric compatible?

In the (holonomic) coordinate frame {$\partial_\mu$} with the affine connection $\Gamma^\alpha_{~\beta \gamma}$ and a generic metric $g_{\mu \nu}$, the manifold can acquire both torsion $T_{\alpha \...
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Change of coordinates in Kerr-Newman (KN) spacetime and partial derivatives

I'm currently studying the Kerr-Newman (KN) spacetime, and I was trying to reproduce a result but it seems that the change of coordinates is throwing me off. In standard Boyer-Lindquist coordinates, (...
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Starting with 1-form potential $A$, derive the relation $F_{\mu\nu}^{\nu}=A_{\nu,\mu}^{,\nu}-A_{\mu,\nu}^{,\nu}=4\pi J_\mu$ [closed]

First the problem in detail should be, "Starting with 1-form potential $\mathbf{A}$, derive the relation $F_{\mu\nu}^{\;\;\;,\nu}=A_{\nu,\mu}^{\;\;\;\;,\nu}-A_{\mu,\nu}^{\;\;\;\;,\nu}=4\pi J_\mu$...

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