Questions tagged [curvature]

Use this for questions pertaining to curvature of manifolds. Does not need to be specific to general relativity, but also for curvature of e.g. a [tag:calabi-yau] manifold.

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Do virtual photons follow spacetime curvature?

I have read this question: https://link.springer.com/chapter/10.1007%2F978-3-319-13443-7_26 The electric field lines from a point charge — and the rays of light when the charge is replaced by a ...
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48 views

What is the logic connection between these two statements?

What is the connection between these two statements: the berry curvature change sign under time-reversal operation If the system has the time-reversal symmetry, then berry curvature is odd in k. ...
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Likelihood, posterior, prior interpretation and credibility/confidence_level with bayesian/frequentist approaches [closed]

I try to understand the following article : testing general relativity from curvature and energy contents at cosmological scale I don't understand the title of figure 1 : where it is indicated ...
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What is the skin depth of space?

It is known that space behaves like a 3D rubber sheet, when mass is present, and bends. Gravity can be explained by the curvature of this bending. Water forms a highly flat surface that also behave ...
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Globally constant vector field in a curved spacetime

Is it possible to define a globally constant vector field in a curved spacetime, that is a vector field for which the covariant derivative vanishes along every world line? The vector field $V^{\mu}=0$ ...
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How come the universe is considered flat if zero point energy is infinite?

If quantum field theory calculates that the vacuum energy is infinite and Einstein's theory of gravity implies this energy should produce a curvature of space-time then why shouldn't the universe be ...
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Do the Christoffel symbols $\Gamma_{rn}^w\partial_sV_w = \Gamma_{sn}^w\partial_rV_w$?

In lecture 3 (about 97 min into the lecture) of Leonard Susskind's general relativity course, he suggests finding the Riemann curvature tensor in terms of the Christoffel symbols as an exercise. I ...
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Numerical Calculation of Berry Curvature

I am trying to calculate some berry curvature (BC) in a 2D lattice and I have some things I am getting lost with. In the 2D lattice, we set up the eigenvalue problem $H|u_1\rangle = \epsilon_i|u_i\...
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Contracting Riemann Tensor Troubles

It has been several years since I looked at General relativity, and I am trying to brush up on it because it was always interesting and I am in need of it for my research. Specifically, I am looking ...
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53 views

A priori knowledge of the components of the Ricci tensor

Source: Thomas Moore's A General Relativity Workbook In Moore's "diagonal metric worksheet" he doesn't explain his process of determining the "only possible non zero components" of the Ricco tensor, ...
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Ricci scalar in terms of vierbein and spin connection

I have been trying to derive the following form for the Ricci scalar in terms of vierbein and spin connection $$R=(e^{\mu a}e^{\nu b}-e^{\mu b}e^{\nu a})(\partial_\mu \omega_{\nu ab}+\omega_{\mu a}^{\...
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Deriving Kretschmann scalar for Schwarzschild solution

I'm trying to derive kretschmann scalar for schwarzschild solution, which is \begin{equation} K=\frac{48M^{2}}{r^{6}} \end{equation} I know I have to compute $R_{abcd}R^{abcd}$, but it seems like an ...
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How does spatial curvature apply to the planets' orbits?

We all know that in the presence of large, massive objects, spacetime is positively curved, more so the more massive it is. This means that the path of an object without any forces on it is a straight ...
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Prove that colliding celestial bodies produce perturbations in the fabric of space-time

So I was recently asked this question by one of my professors, it really is confusing for me at the moment since I only have a basic grasp of and the ideas Einstein proposed. P.S: I was wondering if ...
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How is spacetime warped by a massive object?

I was going through this question (Why don't planets have Circular orbits?) related to planetary orbits. In the accepted answer it is stated that orbits are actually conic sections. Given this ...
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Confusion regarding Ricci Scalar

Source: Thomas Moore's A General Relativity Workbook Equation 1: $R= g^{\mu\nu}R_{\mu\nu} = R^\nu{}_\nu$ Equation 2: $R^{\mu\nu}=g^{\mu\beta}g^{\nu\sigma}R_\beta\sigma$ Question: Does $R$ also ...
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Does curvature affect the change in the acceleration/deceleration of the Universe?

Suppose I'm modelling a Universe with non-zero curvature, filled with matter, radiation and dark energy (further described by quintessence). The appropriate Friedmann equation would be of the form: $$...
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Some kind of slower time principle [duplicate]

I'm always trying to find underlying principles, like that the force is always directed toward a (locally) lower potential energy and alot of stuff like that. Recently I've begun to gain some layman ...
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Curvature of a two-dimensional hyperboloid

I have a question about an exercise in Misner, Thorne, and Wheeler's Gravitation. On page 334, exercise 14.1. "Curvature of a two-dimensional manifold hyperboloid" says: Compute the curvature of ...
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Schwarzschild and Einstein Field Equation

In his paper Über das Gravitationsfeld eines Massenpunktes nach der Einsteinsche Theorie Schwarzschild used this equation (Paper eq. (4)): $$\frac{\partial}{\partial x_\alpha}\Gamma^\alpha{}_{\mu\nu}+...
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GR visualization

I'm watching some GR lectures by Schuller (more or less rushing through them so bear with my ignorance here please) in Lecture 10: Metric Manifolds. He's talking about geodesics in a manifold with a ...
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Derivation of Raychaudhuri equation - Trace

In Wald (Wald: General Relativity on page 218, equation 9.2.10) is stated that $$v^c∇_cB_{ab}=−B^c_bB_{ac}+R^d_{cba}v^cv_d $$ and to continue in order for the equation to be derived one needs to take ...
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General theory of relativity questions [duplicate]

I do not understand how this theory (the theory of general relativity) would work because gravity is what makes things heavy and weight is what bends things. All of the proof makes it hard to question ...
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Ricci Curvature Tensor in a static gravitational field (non-relativistic)

Pg 171 of "Tensors, Relativity and Cosmology" The non-relativistic limit of the metric in a static gravitational field is defined as $$ds^2=\left(1+\frac{2 \phi}{c^2}\right)(dx^0)^2+g_{\alpha \beta}...
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Using symmetry of Riemann tensor to vanish components

The Riemann tensor is skew-symmetric in its first and last pair of indices, i.e., \begin{align} R_{abcd} = -R_{abdc} = -R_{bacd} \end{align} Can I simply use this to say that, for example, the ...
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Origin of the Inverse Mean Curvature Flow (IMCF)

I was reading The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality by Gerhard Huisken and Tom Ilmanen and it is stated on page $7$ that Geroch [30] introduced the inverse mean ...
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A doubt on this paper regarding Ricci flow on space-time

https://arxiv.org/abs/1812.06239 In this paper,the authors use ricci flow to construct Lifshitz spaces. But it is known that ricci flow is limited by Riemannian manifold, which has a positive metric....
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Space-time bending and compression [closed]

Is microscopic relativity correlated with macroscopic relativity? "Small" versus "large" as it applies to relativity may not abide by the the same assumptions using current theories. Thoughts?
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The idea behind the geometrical description of a closed curved space

This question is about the idea of how to describe a curved space, I'm not asking for formulas because I've never studied this topic before. Let's imagine an ant on a sphere, it will use two ...
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What is the CFT dual of the stress tensor in the bulk?

I am new to AdS/CFT. I know that the dual of the bulk metric is the CFT stress tensor but what about the dual of the bulk stress tensor? I mean in principle one can extrapolate whatever bulk fields to ...
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Einstein equations in the spherically symmetric, static case

This question is not about the solutions but much rather about the equations we write in GR for a spherically symmetric, static vacuum 4D spacetime. The Einstein equations are $$G_{\mu\nu}=0\;\;\;\...
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153 views

Uniqueness constraint(s) on spacetime

What additional constraint(s), if any, must be used with the gravitational field equations $$R_{\mu\nu}=\kappa \left( T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu} \right)$$ to uniquely determine the Christoffel ...
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Lowering index of Riemann tensor

I'm trying to undertand the lowering of index of Riemann curvature tensor, but I'm not sure what I have to do. I know that $R_{ebcd} = g_{ea}{R^a}_{bcd}$. But let's say I have the coordinates ($t,r,\...
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What did Einstein mean by-matter curves and warps spacetime? [duplicate]

In general relativity one keeps hearing that. Please tell what is really means and its connection to reality we see around us?
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Proof of Schur's Theorem

On Pg. 123 of Schaum's Tensor Calculus: At an isotropic point of $R^n$ the Riemannian curvature is given by $$K=\frac{R_{abcd}}{g_{ac}g_{bd}-g_{ad}g_{bc}}=\frac{R_{abcd}}{G_{abcd}}$$ for any ...
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Proof of first Bianchi identity

The proof is often simplified by using the following theorem: "If the metric tensor $(g_{ij})$ is positive definite, then, at the origin of a Riemannian coordinate system $(y^i)$, all $\partial g_{...
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Indices of the Riemann Tensor of the first kind

When establishing the identity $V^i_{,kl}-V^i_{,lk}=-R^i_{tkl}V^t$ (, denotes covariant differentiation), one of the steps involves raising one of the indices of the Riemann Tensor of the first kind . ...
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How to prove the absence of naked singularities?

Suppose we have a static, spherically symmetric solutions of Einstein equations coupled to a certain matter source, and we are able to show that the scalars obtained by the stress energy tensor on ...
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Is there a known mechanism for mass-energy distorting spacetime?

I’ve been really interested in learning about the mechanisms behind physical phenomena that go beyond just learning to manipulate the equations and give a physical intuition about HOW something ...
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Is it mathematically impossible to incorporate the space curvature into the equations of motion and gravity? [duplicate]

Obviously, I haven't studied GR, I know no more than common knowledge. However, I'm wondering, is it impossible to develop a mathematical model based on flat space, in which the new equations of ...
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Torsion and Curvature in general relativity

in Landau's book, field theory, the famous physicist stated, "Because of equivalence principle, there should be a 'Galileo' frame, in which the Christoffel symbols should be zero, thus the torsion ...
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Could matter be simply spatial geometry, and nothing else? [closed]

As it will become evident by my question, I have no degree in physics or math. Question: could matter be nothing more than the curvature of space? In that case, different atoms and different ...
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Is there a definition for a *geometric entropy*?

In statistical mechanics, entropy of a system is usually defined as a measure of the system's micro-state randomness, or as an averaged "surprise" of its micro-state: \begin{equation}\tag{1} S_{\text{...
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Help with calculating the Ricci tensor for the PPN formalism

I'm trying to follow the calculation done by Will in his book Theory and experiment in gravitational physics, and I was hoping for some help in calculating the Ricci tensor components in Section 5.2 (...
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General Relativity and cosmology [duplicate]

What is the physical meaning of Ricci scalar is a covariantly constant?
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Accelerated frame vs gravity frame

Can accelerated frame change curvature of space as gravity does? Can there accelerated frame be pure inertial frame?
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Witten Index of Riemannian Manifold

Consider a system on a Riemannian manifold with the Lagrangian $$L = \frac{1}{2}g_{IJ} \dot{\phi}^I \dot{\phi}^J + \frac{i}{2}g_{IJ}(\overline{\psi}^I D_t \psi^J - D_t \overline{\psi}^I \psi^J) - \...
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Curvature created by an object near Earth via energy-stress tensor

From Misner... who uses the convention of (-, +, +, +) for the metric $g^{\mu\nu}$, with the electromagnetic stress energy tensor being(pg.141): $$T^{\mu \nu} = \frac{1}{\mu_0}(F^{\mu \alpha}F^{\nu}{...
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Why do we visualize space time as trampoline surface? [duplicate]

It's a silly question probably. But the thing is that what is the fascinating about fabric surface. I am so sorry for this silly question in advance. I read a lot articles regarding my question. But ...
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86 views

Whats the quickest way to compute the Ricci tensor?

I have been going through exam papers and often they ask us to calculate ricci tensor components and affine connections from a given metric. They seem to take far too long for the time you are ...