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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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32 views

Does Electrostatic potential energy bend Space-time? [duplicate]

Okay, there are various questions. First, "matter and energy bends space-time" does this mean any form of energy can bend space-time? Does theory of relativity assume that there is no other form of ...
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2answers
79 views

If gravity can be thought of as masses leaving dents on a spacetime 'sheet', what is holding up that sheet? [duplicate]

If the force of gravity can be thought of as masses leaving dents on a sheet of spacetime, what is holding up that sheet?
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On the Computation of Gibbons-Hawking-York Boundary Term

The Gibbons-Hawking-York (GHY) boundary term is given by $$S_{GH}=\frac{1}{8 \pi G}\int_{\partial M}\sqrt{|\gamma|}K,$$ where $\gamma_{ij}$ is the boundary induced metric, and $K$ is the trace of the ...
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45 views

Can a flat space have nonzero torsion?

I know that in general a curved space can have torsion or be torsion-free, however, can torsion exist in a flat space? I'm guessing that it cannot for the reason that torsion is the ...
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0answers
36 views

Weyl- Squared Lagrangians

I'm studying conformal gravity theories, in particular I read that if we take $L=\sqrt{g}C_{abcd}C^{abcd}$ where $C$ is the Weyl tensor the theory we get is not unitary. What does it means unitary at ...
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What is the best comparison to understand how space is formed without resorting to the design of the temporary space blanket? [closed]

For Einstein, space is like a blanket that curves according to the mass of the bodies, so objects should move up, sideways, and down very little, according to the mass of the bodies. This ...
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31 views

Variation of the Ricci tensor “squared” and antisymmetrization of the derivatives

I'm dealing with some extension of GR, with action: $S=\int d^4x\Big[\sqrt{-g} f(R,R_{\mu\nu}R^{\mu\nu})$ Varying this action gives: $\delta S=\int d^4x\Big[\delta\sqrt{-g} f(R,R_{\mu\nu}R^{\mu\nu})...
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1answer
55 views

Positive local spatial curvature of the universe implies that the universe is compact (i.e. finite)?

I quote from the Wikipedia page about the shape of the universe: If the spatial geometry [of the universe] is spherical, i.e., possess positive curvature, the topology is compact. I'm trying to ...
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1answer
64 views

Geometrical interpretation of curvature invariants

Consider a Riemannian manifold. It is possible to describe it by curvature invariants. Now, is there any geometrical description (intuition) for simple invariants such as scalar curvature, Ricci ...
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1answer
43 views

Counting independent components of the Riemann curvature tensor

In 4D spacetime, we may choose a locally inertial frame at point P, that is we always have a transformation such that $g_{{\mu'}{\nu'}}(P) = \eta_{{\mu'}{\nu'}}$ and its first derivatives vanish. ...
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Black hole singularity with zero curvature (do you feel weightless)?

As described by GR, at the center of a black hole, may lie a gravitational singularity, where the spacetime curvature becomes infinite. https://en.wikipedia.org/wiki/Black_hole Though, by the ...
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1answer
126 views

Can a straight rod exist next to or inside a black hole?

A black hole is defined as a part of spacetime where gravity is so strong, that spacetime curvature reaches extreme levels. Not even light can escape. https://en.wikipedia.org/wiki/Black_hole Now as ...
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2answers
81 views

Pertubation of Riemann tensor in a general curved space-time

It is a direct and simple question. I am fully developing the perturbation of Einstein Field Equations, and I need to calculate the perturbation of the Riemann tensor. However the background metric is ...
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1answer
61 views

What does bent in space-time means exactly? How does mass of an object affect space and time? [duplicate]

I don't understand how does of mass an object for example say earth causes distortion in space and time. I am just new to this field so it is difficult imagine this phenomenon.
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What is the definition of the “characteristic radius”?

Upon solving exercises regarding relativity, I have run into the problem below. The inverse square radius of curvature of spacetime is of orer the tidal field, $R^{-2} \approx \nabla^2 \phi$ where $\...
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Gravity in a spacetime with 2 indistinguishable dimensions, with all spacetime directions equivalent

A spacetime with 2 indistinguishable dimensions and all spacetime directions equivalent would have the signature (++) meaning that there would be no difference between spacelike and timelike ...
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0answers
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How many different ways can Riemann-Christoffel Curvature Tensor can be derived? [closed]

In today's Relativity and Gravitation class, my prof was discussing about Riemann-Christoffel Tensor and he derived it. But in the end he told that there are many ways one can derive the Riemann ...
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5answers
764 views

Intuitive methods for representation of Cartesian Coordinates in terms of Spherical Coordinates as basis [closed]

I was going through Griffith's Electrodynamics and came upon an example, where he used that, $$\cos\theta \ \hat{r} - \sin\theta \ \hat{\theta} = \hat{z} $$ Now I admit I was confused for a while ...
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1answer
106 views

Equivalence principle doubt

There is something about Einstein Equivalence Principle that I don't quite get. This is my reasoning: Equivalence principle $\rightarrow$ locally, acceleration is equivalent to a gravitational field ...
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1answer
98 views

How to translate this equation into physicist's notation? [closed]

I asked this in math stackexchange but no one has answered there so I ask here. How to translate this equation into physicist's notation, i.e. tensors with indices? $$\left\langle R_{N}\left(u,v\...
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1answer
86 views

Spacetime curvature is relative?

I have the following conceptual doubt. These are my assumptions: 1) The geometry of spacetime is the same for all observers, regardless their motion 2) All motion is relative (both uniform and not ...
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1answer
52 views

Euclidean view in curved manifold

Let's suppose I am an ant who lives in a 2D curved space. Locally the world seems 2d-euclidean to me, but it is not if I consider a large portion of space. Now let's consider a human being who lives ...
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1answer
78 views

Pseudo-Riemannian 2D manifold (visualize time curvature)

My goal is to visualize somehow the curvature of time, as opposed to the curvature of space. I know that we generally talk about spacetime curvature altogether; however, the fact that spacetime has ...
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2answers
197 views

Why does nobody ever consider the possibility that the universe is not smooth?

Disclaimer: I'm not an astronomer, physicist, mathematician, etc. so this is a question from a complete newbie. One of the greatest mysteries of our age is "where is the dark matter?" The universe ...
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0answers
87 views

Weak Solutions to the Einstein Equation across a Junction

Consider the principle part, i.e., the part which contains the highest derivatives of the metric (which is the $2^{nd}$ derivative) is $$\mathcal{P}\{R_{ab}\}=\frac{1}{2}g^{cd}\left(\partial_{a}\...
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1answer
45 views

Does cosmological expansion change the gravitational field around a massive body?

Do expanding universe affect the curvature of spacetime? If so, does Einstein's field equation account for the change in curvature (gravitational field) around massive object in the expanding universe?...
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1answer
50 views

Intrinsic curvature calculation

Gauss theorem egregium says that it is possible for the inhabitants of a 2d surface to calculate the surface curvature without knowing that it is embedded in a 3d euclidean space, simply calculating ...
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0answers
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Curved space relative motion

Suppose I live in a 2d curved space. I can describe intrinsically the motion of a particle which moves in my 2d manifold using time "t" as a parameter. Now suppose there is a second observer (let's ...
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1answer
93 views

Spacetime curvature replaces acceleration?

In my understanding, not only mass but any kind of energy/ force bends spacetime. So is it correct to say that every object in the world moves along geodesic? If the object is submit to a force, it ...
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0answers
27 views

Relative kinematics curved space

I know that in a general curved manifold it is possible to describe parametric curves. Then, the length of the curve will be a indipendent from the coordinate system used --> the metric tensor ...
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1answer
23 views

Angular velocity in curved space (2d manifold)

In 3d Euclidean geometry, the velocity of any point of a rigid body is given by the cross product between its angular velocity and the position vector which links the instantaneous rotation center to ...
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1answer
77 views

Why spacetime fabric don't tears due to mass of heavy black hole?

In GR, All objects create curvatures in the space-time fabric. Why space-time fabric doesn't tear due to the mass of a heavy black hole? What is it made of?
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1answer
75 views

Why is the coordinate basis never locally orthonormal in curved spacetime

In Carrolls GR book “Spacetime and Geometry” he comments that “This is not a situation we can define away; on a curved manifold, a co- ordinate basis will never be orthonormal throughout a ...
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65 views

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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1answer
54 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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0answers
39 views

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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2answers
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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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2answers
64 views

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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0answers
62 views

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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0answers
82 views

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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1answer
141 views

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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2answers
60 views

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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1answer
58 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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1answer
55 views

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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1answer
81 views

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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2answers
59 views

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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2answers
87 views

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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2answers
50 views

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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1answer
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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: $$...