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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4answers
653 views

How to prove that a flat spacetime admits Minkowski coordinates?

How should I prove the following in general relativity? A flat spacetime can be covered by Minkowski coordinate neighborhoods. A flat spacetime with the trivial topology can be covered by a global ...
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2answers
382 views

What does it mean that spacetime is curved? [closed]

I am not sure whether this is a physics question, a maths question or even a linguistics question; please forgive me if I have chosen the wrong platform. I am trying to understand what it really means ...
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2answers
84 views

Physical differences between our universe and strongly negatively curved one

Interestingly, the idea of a negatively curved universe isn't entirely science fiction. In particular if decrease a single physical constant (known as the density parameter) to be less than one, the ...
3
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1answer
101 views

How does electromagnetism work in a negatively curved universe?

Let's say that there was a negatively curved universe (in particular $\Omega < 1$). I assume that means the universe would be like Hyperbolic space. In our universe, electromagnetism obeys an ...
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2answers
152 views

Is Flatness Problem still a problem since the observable universe is flat but the whole universe can be curved on much larger scale?

When we say the universe is flat, we mean just the observable universe. So that doesn’t mean the shape of the whole universe is also flat (ie. it can be curved on much larger scale) Is Flatness ...
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1answer
121 views

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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1answer
176 views

Infinite gravity

We know that gravity works at an infinite distance. By Einstein's theory of gravity it is the result of space time curvature. So can we say that the curvature is infinitely long? How can this be even ...
12
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1answer
203 views

Nontrivial example of a spacetime for which we need the real definition of asymptotic flatness?

Asymptotic flatness basically means that you can apply a conformal transformation to your spacetime so that it becomes compact, and it admits a boundary having the same causal structure as the ...
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1answer
569 views

What does asymptotically flat solution mean?

Can somebody explain what does it mean when a solution is "asymptotically flat"? like the schwarzschild metric which is asymptotically flat solution to vacuum Einstein equations.
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2answers
476 views

Forms of the extrinsic curvature tensor

I am having trouble showing a relation in Carroll's GR book in his appendices. He defines the extrinsic curvature tensor as $$K_{\mu\nu}\equiv\frac{1}{2}\mathcal{L}_nP_{\,u\nu},$$ where $$\qquad P_{\...
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0answers
59 views

Can a 2D universe be closed?

Consider a universe as a curved 1D line looped onto itself. The second dimension is time. On one hand, this line is easy to visualize as a circle embedded in a flat 2D plane. However, there is only ...
3
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1answer
141 views

Could LIGO have detected R'lyeh if it existed?

In Possible Bubbles of Spacetime Curvature in the South Pacific, that R'lyeh might be exist within a bubble of spacetime. (It appears to be a joke paper since some of the references are fictional, but ...
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2answers
122 views

A problem with how curved spacetime explains gravity [duplicate]

I've read quite a few online explanations of how curved spacetime is the reason objects are drawn to each other, and that gravity is an illusion. Most of them follow the same path: explain what ...
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1answer
283 views

Does CFT in AdS/CFT live in flat spacetime?

As the title says, does CFT in AdS/CFT live in flat spacetime, or is it only approximately flat?
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2answers
160 views

Curved Space and Great Circles in General Relativity

I have been studying general relativity by B. Schutz. It says locally straight lines on a sphere does not remain parallel, as great circles do intersect on a sphere. Below is a picture where two ...
2
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1answer
194 views

1+1D curved spacetime diagram example

This is a very basic question about General Relativity. I haven't take any GR course yet. Suppose a flat spacetime with one space direction and one time direction, as follows: Now add a mass at rest ...
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1answer
344 views

2D Liouville Stress-Energy tensor

I am working on 2D Liouville field theory and trying to follow mostly Harold Erbin's note on 2d quantum gravity and Liouville theory. I have a really simple question: One consider the Euclidean ...
2
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2answers
104 views

Can I state that a spacetime is homogeneous and isotropic iff $\nabla_\mu R = 0$?

If a spacetime is homogenous and isotropic can I say that $\nabla_\mu R =0$? I was reading this paper https://arxiv.org/abs/astro-ph/0610483 and, I think that is the justification for the authors ...
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3answers
119 views

A star with its radius not much larger than its Schwarzschild radius

I was asked about the question today: suppose that you are observing from afar a spherically symmetric star of mass $M$. Its radius $R$ is $\textbf{not}$ much larger than its Schwarzschild radius. ...
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3answers
485 views

The commutation of partial derivatives in curved spacetime

While following a lecture series on General Relativity, an argument was presented that in order the spacetime to be flat, a vector parallel transported along two different paths should yield the same ...
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2answers
361 views

Intuition for why mass and energy curve the space-time fabric and for why this relationship is linear?

The force of gravity does not exist I understand(-ish) that, following general relativity, an apple falls onto earth, not because there is a force pulling earth and the apple toward each other but ...
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2answers
223 views

Riemann tensor and the non-commutativity of parallel transports

Riemann curvature s denoted by $R^a_{bcd}$. If you try to attempt to parallel transport a tangent vector $\psi$ along an infinitesimal parallelogram given by two tangent vectors along non-parallel ...
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1answer
105 views

Contribution of the metric tensor towards the Riemann curvature tensor

If I specify a metric tensor in terms of a choice of bases in the co-tangent space at a given point in a manifold, how much information does the metric tell me about the curvature tensor at that point?...
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1answer
57 views

Estimating the curvature tensor of spacetime using a rotating body

In curved space, you can define the Riemann curvature tensor at a point. In Riemann normal coordinates with that point as origin, the Riemann curvature tensor can be expressed as depending only on the ...
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2answers
58 views

The central point is singular or not?

Let take the metric \begin{eqnarray} \mathrm ds^2 = -f^2(r) dt^2 + g^2(r)dr^2 + r^2~\mathrm d\Omega^2 \end{eqnarray} with $f^2(r) = 1 -ar^3$ and $g^2(r) =\frac{b}{1-ar^3}$. Is the central point $r=...
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2answers
155 views

How will two equal discs, rotating with equal but opposite angular velocities and put on top of each other affect the spacetime “surrounding” them?

In this article the Ehrenhaft paradox is described. You can read in it that, according to Einstein's General Relativity (to which this paradox contributed), the spacetime around a rotating disc is non-...
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0answers
153 views

Scalar curvature of warped-product manifolds - intuition

Let $(M, g) = (N_1, g_1) \times f(N_2, g_2)$ be an Einstein warped-product manifold, with metric $g=g_1+f^2g_2$. What does it mean if you choose the scalar curvature, of its base-manifold $(N_1, g_1)$...
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1answer
139 views

Assuming a fixed total mass, will the spacetime geometry outside a spherical mass distribution depend on the shape (of the distribution)?

Consider two independent spheres of equal masses but of different radius and in different spacetimes. The first sphere is less dense than the second one, i.e., it has a larger radius. For example, if ...
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1answer
312 views

Silly question about Schwarzschild Solution

The following question is based upon elementary concepts of General Relativity, and Tensor Calculus The condition for a space-time manifold to be flat is: $$R^{a}_{bcd} \equiv 0 $$ I.e., the ...
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1answer
134 views

Is a 4 dimensional spherical universe possible with flat curvature?

I'm trying to understand this snippet from Wikipedia, in particular the section I've emphasized: The curvature of the universe places constraints on the topology. If the spatial geometry is ...
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2answers
214 views

How do we know that universe is flat?

Last measures by WMAP indicates that the universe is flat with a 0.04 margin of error. What does that exactly mean? Is it almost flat but with a soft curvature that could be negative or positive?
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2answers
517 views

Can the fabric of space-time be contoured into hills instead of just wells?

Einstein's general theory of relativity states that gravity is the distortion of space-time into gravity wells. In order to illustrate this, a flat plane is used to represent undistorted space-time ...
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1answer
139 views

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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0answers
376 views

Can we think of the proof of local flatness theorem as the proof the the Einstein's Equivalence principle? [duplicate]

Sean Carroll states the Einstein's Equivalence Principle (EEP) as "In small enough regions of spacetime, the laws of physics reduce to those of special relativity; it is impossible to detect the ...
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0answers
160 views

Why gauge-invariant Berry curvature commutator looks like torsion?

The Berry Curvature is defined as (for invariant gauge transformations) $$F_{ij} = [\partial_i, A_j] - [\partial_j,A_i] + [A_i,A_j]$$ The gauge covariance satisfies the transformation $$A_i \...
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1answer
486 views

Berry Curvature and Curvature Tensor

When the curvature tensor (from Einstein's theory) has a non-zero torsion, it is said to be an antisymmetric tensor in the last two indices composed of the connections of the field. Alternatively, the ...
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1answer
255 views

Variation of Gauss Bonnet Invariant

I am trying to do the variation of Gauss Bonnet Invariant, and the Gauss Bonnet Invariant is: $G$=$R^2$+$R_{abcd}$$R^{abcd}$-$4R^{ab}$$R_{ab}$ The variation of $G$ is: $\delta$$G$=$2R\delta$$R$+ $\...
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1answer
255 views

Anti de Sitter space

While writing the metric for AdS Space, why are we starting with a five dimensional Flat space and embedding a hyperboloid in it? Does it have to do with the fact that the cosmological constant being ...
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1answer
748 views

Does time dilation cause gravity as explained in this video?

Watch it around 2:00 minutes. https://youtu.be/gcvq1DAM-DE Do objects move closer to Earth because they experience time at different rates, really? Does it make sense? The video also represents the ...
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3answers
450 views

Anti-De Sitter Space

Anti - De Sitter Space is the maximally symmetric solution to field equations with negative cosmological constant. The negative cosmological constant also shows that the spacetime has negative ...
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1answer
169 views

Could there be equivalence between anisotropic space and the presence of a field? [duplicate]

What if we are so used to the curvature of space caused by mass and the range of its effects that we totally ignore the possibility of the existence of "opposite" curvature1, i.e. objects that bend ...
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1answer
98 views

Are the space and time axes of Schwarzschild metric uncurved?

Schwarzschild metric is commonly considered as an expression of curved spacetime: $$ \mathrm ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right) c^2~\mathrm dt^2 + \frac{1}{1 - \frac{2GM}{c^2 r} }~\mathrm dr^...
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3answers
123 views

Instead of warping spacetime, can gravity be represented by locally varying time rates?

Instead of thinking of gravity as mass warping spacetime, could it be thought of as mass warping only time, whereby time would advance at faster rates at locations where more mass is present?
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0answers
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Extracting energy from curvature

The device Imagine i connect 4 rigid rods of equal length together in a square. This thing has 4 corners, each with a 90° angle. Going around the whole thing i make one full rotation, so 360°. Now i ...
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1answer
248 views

Connecting curvature to the 'failure of squares to close'

In one of the Feynman lectures, Feynman describes curvature as the failure of a square to close. By switching to spacetime, Feynman then claims that the curvature of spacetime is reflected by the ...
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1answer
186 views

What does the Kretschmann scalar really tell us about the geometry of spacetime? [duplicate]

The Kretschmann scalar is one of the measures of spacetime curvature. For flat (Minkowski) spacetime it is zero. The dimensions of the Kretschmann scalar are $[L]^{-4}$. What does that physically ...
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4answers
238 views

How can one understand with an example that Newton's law(s) fail in a curved space?

Is it true that Newton's law is not valid in curved spaces? If yes, how can I understand it and explain to a high school student preferably with an example? I tried to think about the motion of a ...
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1answer
340 views

Rindler Space and tensors

How can we immediately see that the Riemann tensor and Ricci tensor in Rindler space are zero? I know that the Rindler metric is given by: $$-ds^2=-a^2x^2dt^2+dx^2+dy^2+dz^2$$ and what I just did ...
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1answer
372 views

Connection and curvature using differential forms [duplicate]

I am trying to understand how one would use differential forms to calculate the components of the connection and the curvature tensor given a metric. Can anyone point me to relevant resources for the ...
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1answer
226 views

Is there any additional symmetry in Riemann curvature tensor to tell which components are zero?

It is known that $ R^{\alpha}_{\beta \gamma \delta} = -R^{\alpha}_{\beta \delta \gamma} $. I.e it's skew-symmetric in its last two indices. So if the last two indices are the same one can just say by ...