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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39 views

Not sure why the geodesic derivation equation involved second ordinary derivative

My question is related to this link (18:10) https://www.youtube.com/watch?v=oQZTYt_Pxcc&list=PLJHszsWbB6hpk5h8lSfBkVrpjsqvUGTCx&index=28 Recently i have watched a video about volume ...
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Can you put the Riemann Tensor in block diagonal form?

I'm following the notes by Freed about the Dirac Operator. In section 5.4, equation (5.4.25-27), he makes the following claim about the Dirac operator. In different notation than he is using, he is ...
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1answer
70 views

Riemann curvature tensor and parallel transport

I'm taking a course on general relativity and I'm confused about something in my course notes. First there is an explanation about a curved 2D plane where it is shown how curvature can be defined by ...
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1answer
59 views

Making tidal forces “disappear” via coordinate transformations

I'm watching this introductory lecture on general relativity by Leonard Susskind (see 31:50 onwards for the relevant part) [Video title: General Relativity Lecture 1, channel: Stanford]. To give ...
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1answer
63 views

Is a brachistochrone a straight line in curved space?

Please bear with me, and don't get upset if i have lack in knowledge about spacetime. Brachistochrone: Given two points A and B in a vertical plane, what is the curve traced out by a point acted ...
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35 views

Dirac Delta Function and Schwarzschild Singularity [duplicate]

Pardon my naive question. I recently found out about Dirac Delta function. It is interesting to note that Schwarzschild singularity gives the infinity values of the General relativity field equations ...
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1answer
46 views

Does the fabric of spacetime go through objects? [duplicate]

I've always seen spacetime shown with a ball on top of some cloth that curves around it, and I don't really understand it, since it really only has 2 dimensions and there's some sort of external ...
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51 views

General covariance and the curvature scalar

I'm not asking what general covariance means. I am asking 2 questions: Technically what follows from it is that as long as you don't make a mathematically erroneous transformation, you are allowed ...
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43 views

Taylor’s theorem on curved surfaces

Consider a curved surface, say a sphere whose metric can be defined. How does one expand a function on the curved surface using Taylor’s theorem? The taylor expansion of the function about $r’$ is ...
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58 views

Is Earth flat according to Einstein? [duplicate]

I didnt understand how to apply general relativity to Earth. If there is no such a thing as gravity but spacetime bending. Can someone just show, send something to read how gravity works in Earth ...
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38 views

Kretschmann scalar is discontinuous

I am studying this paper: "Non-singular rotating black hole with a time delay in the center" by T. de Lorenzo et al. In this paper authors calculate a new metric for a regular, rotating black hole. ...
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1answer
65 views

How to interprete this singularity? [closed]

I am calculating the Kretschmann scalar for the Schwartzchild metric. This is the graphic I get: Where $R$ is the radial coordinate and $x=\cos(\theta)$. So, there is the singularity at $R=0$ as it ...
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2answers
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Determining geometry/topology from a Line Element

Is it possible given a line element, to determine its geometry? For example whether the line element $ds^2$ of a 2D surface corresponds to $\mathbb{R}^2$ or $S^2$ geometry?
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Part of a bigger question about spacetime geometry

This is a two-part question. First: I'm imagining that a laser with infinite coherence length is used to shine light toward a black hole from far away. A mirror is placed a few Schwarzchild radii ...
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52 views

Curvature and Symmetries of spacetime

Is there any relation between symmetries of spacetime and the curvature invariants? For example is spherical symmetric spacetimes, necessarily have positive curvature? Could we define any spherical ...
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18 views

How can the Milne model of the Universe have curvature? [duplicate]

The title says it all. The Milne model of the universe is a model where the universe is empty, there is no matter at all, yet the spatial curvature is different from 0. I'd expect that a spacetime ...
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1answer
39 views

maximally symmetric spacetime

An empty spacetime has zero or constant Ricci Scalar (depending on the cosmological constant). Is there a theorem which guarantees that such a spacetime should be Minkowski or dS/AdS? In other words, ...
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Arguments for a vanishing Riemann Tensor

Consider the following metric: $$ds^{2} = -(1+2\Phi(x,y,z))dt^{2}+dx^{2}+dy^{2}+dz^{2} \tag{1}$$ Where, in fact, $\Phi$ is the Newtonian Potential. Consider the Riemann Tensor: $$\mathrm{R}_{abcd} ...
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Are there any CMB-independant probes of the curvature of the Universe?

There is a preprint today (think it also appeared in Nature Astronomy on Nov 4) which argues that a $\Lambda{\rm CDM}+\Omega_k$ model with negative $\Omega_k$ fits the Planck Legacy 2018 CMB data ...
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1answer
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Mathematical relations of Gravitational waves and the Metric Tensor $T$

Ok so as we all know that Spacetime Curvature has Geometric Disturbances which are mathematically called Gravitational Waves. But the question I am asking is that why the Coordinative value of the ...
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1answer
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Deriving $\nabla_\mu \nabla_\sigma \mathcal{K}^\rho=R^\rho_{\sigma\mu\nu}\mathcal{K}^\nu$

I want to derive this equation from Carroll's book. $$\nabla_\mu \nabla_\sigma \mathcal{K}^\rho=R^\rho_{\sigma\mu\nu}\mathcal{K}^\nu$$ We know that $\mathcal{K}^\nu$ is a killing vector and ...
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1answer
40 views

The contracted Riemann tensor in vacuum

When we say the background geometry satisfies Einstein’s equations in the vacuum does that mean that $R_{\mu\nu}=0$? I'm positive that not everything is zero in the equations since we have the ...
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2answers
37 views

Product rule of variations

I am deriving the Einstein equation using the Einstein-Hilbert action: It is obvious that the variation in the Riemann Tensor is calculated from a variational product rule. What is not obvious to ...
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35 views

Accelerated Planet

I'm confused about this matter. If I had a planet sitting still in space-time, would I be bending space-time the same as if this planet was being accelerated in the space-time? Wouldn't there be ...
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3answers
439 views

Can we see the curvature of a surface?

After reading the Feynman lectures' (chapter 42, Vol.2) , it had me thinking if it is by any way possible to measure the curvature of a surface (think, surface of earth) just by observing the nature ...
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1answer
49 views

Roger Penrose's conformal cyclic cosmology (CCC)

Does the Weyl curvature tensor $C$ of the black hole singularity in the conformal cyclic cosmology diverge to infinity unlike the Big Bang (C = 0)?
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2answers
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Einstein's initial clue that spacetime is curved [closed]

I did General Relatively years ago at Uni. I have revised a lot of the maths demo Dirac''s book. It is incredible the leap in thought to noting from the Bianchi identities that the curvature term's on ...
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Can artificial force curve spacetime?

By artificial force, I mean a physical force applied by us onto an object which sets it in an accelerated motion (& not a natural force like gravity). eg: hitting a ball. Excuse me if the ...
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1answer
33 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
86 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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35 views

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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1answer
55 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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50 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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37 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
64 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
65 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
61 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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1answer
149 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
103 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
779 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
111 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
99 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
88 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
85 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
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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 ...