# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...