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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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25 views

How to interprete this singularity?

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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50 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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0answers
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
38 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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35 views

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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1answer
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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
48 views

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
90 views

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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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
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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
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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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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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General relativity gravity and curvature, Why light doesn't accelerate? [duplicate]

When we are talking about curvature, in fact, we are talking about the stretching and compressing and bending and warping of space-time loom. My question is, if the presence of a mass produces a ...
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2answers
156 views

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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2answers
86 views

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
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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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1answer
52 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
45 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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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
62 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
56 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
144 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
89 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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1answer
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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
771 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
109 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
87 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
81 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
201 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
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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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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
95 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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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
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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
79 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
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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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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 ...