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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168 views

Is gravity a force given that it derives from curved massless space-time?

One of the answers to a similar question regarding gravity concluded that gravity is an "observed effect" of the curvature of space-time. I read this (and other answers) to imply that gravity results ...
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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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What is the curvature of an empty universe?

My calculations tell me an empty universe has hyperbolic curvature. Is this correct? If it is, can anyone help me understand why this is intuitively?
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Curvature and length contraction paradox

I was thinking about the Flat-Earth model and I confronted with the following issue. Consider a sphere and an observer who is leaving the sphere. Using length contraction principle from special ...
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37 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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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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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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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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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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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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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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36 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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Parametric and covariant expressions for the acceleration vector

I am reading S. Neil Rasband book about Classical Dynamics. In the first chapter, there are two different forms of the acceleration: What he calls the "intrinsic". Given a trajectory with parameter $...
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How is Gravity created in opposite to centrifugal force?

Wikipedia points out that Gravity is: most accurately described by the general theory of relativity (proposed by Albert Einstein in 1915) which describes gravity not as a force, but as a ...
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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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What does the metric condition $\nabla_\rho g_{\mu\nu}=0$ in General Relativity intuitively mean for an observer measuring distances?

In General Relativity, the following condition hold: $\nabla_\rho g_{\mu\nu}=0$, where $g_{\mu\nu}$ is the metric of spacetime which has to do with measuring distances and angles and $\nabla$ is the ...
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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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1answer
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Expanding a summation of covariant derivatives

I hope this is not a silly question but I am trying to understand how this part of the equation works: $$ \nabla_{\lambda} \left( \nabla_{\mu}(R_{\nu \lambda}) + \nabla_{\nu}(R_{\mu \lambda}) \right) ...
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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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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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Gravity: Why Do things fall to Earth? [duplicate]

If gravity is in reality spacetime geometry why when I drop an object on the surface of the Earth does it fall to the ground? Does spacetime push it?
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1answer
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General Relativity in a Differential Geometry concept

I know that in $\mathbb{R}^2$ we can define the curvature of a parametrized curve $\textbf{x}(t)=\bigl( x(t), y(t)\bigr)$ as $$ \kappa(t) = \dfrac{\text{det}(\textbf{x}',\textbf{x}'')}{||\textbf{x}'||^...
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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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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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1answer
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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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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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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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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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60 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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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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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
120 views

What is the age of a universe with positive, negative, and zero curvature?

I am trying to calculate the age of universes with different curvatures using the Hubble constant and Friedmann equation. What does it mean when we say that the universe started out at equipartition ...
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Conformally Flat metric

There is a theorem in D'Inverno Introducing Einstein's relativity which is as follows. "Any two dimensional Riemannian manifold is conformally flat". What does this mean? Does is mean that any ...
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2answers
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How does one determine black hole mass based on the curvature of a photon?

How much mass would a black hole need to create a Schwarzschild radius that would trap a photon, whereby the photon would (to an outside observer) be continually curved 0.004km/s at the horizon? (...
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1answer
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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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Vanishing of the Ricci tensor in higher spacetime dimensions

I understand how, if the Riemann tensor is 0 in all its components, since we construct the Ricci tensor by contracting the Riemann, Ricci tensor would be 0 in all components as well. I've read that ...
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Why is the hypersphere not seriously considered by cosmologists as the best model for the overall shape of the universe?

Cosmologists seem to not seriously consider the hypersphere as the best model for the universe even though they mention it as a candidate from time to time. If you look closely, it seems to be a very ...
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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
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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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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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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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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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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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1answer
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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
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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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1answer
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Is the Palatini-Lovelock action of order $k$ topological in $2k$ dimensions?

I am interested in Lovelock actions in the metric-affine (or Palatini) formalism. It is well-known that the metric version (starting from the Levi-Civita curvature) of the Lovelock lagrangian of order ...
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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 ...
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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 ...