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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73 views

Visualising/getting inuition for, the manifold resulting from these geodesics?

This is related to another post of mine. I know that the geodesics for flat Euclidean Space are straight lines. But I want to take these curved geodesics and tried to work backwards to determine the ...
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47 views

Discontinuous changes in manifold's curvature?

First of all, there exists a question on PSE which does seem to pertain to my question below, but not exactly. This is one of those questions which requires perhaps an intuitive rather than ...
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Derivative of the quadratic invariant term

I know that the derivative of the contraction of two Ricci tensors with respect to the Riemann tensor must be \begin{equation} \left(\frac{\partial \left(R^{\alpha\beta}R_{\alpha\beta}\right)}{\...
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3answers
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How curvature and field strength are exactly the same?

I am watching this lecture series by Fredric Schuller: "Curvature and torsion on principal bundles - Lec 24 - Frederic Schuller" @minute 34:00. In this part he discusses the Lie algebra valued one ...
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1answer
171 views

Proving that the Berry curvature is always zero

If we have an energy eigenstate $|\psi(\boldsymbol \lambda )\rangle$ which is a function of some external parameter $\boldsymbol \lambda = (a,b)$, then the associated Berry connection is defined to be ...
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2answers
114 views

How small could the universe be?

In the article Will the Universe expand forever?, NASA states: We now know (as of 2013) that the universe is flat with only a 0.4% margin of error. This suggests that the Universe is infinite in ...
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56 views

About 2+1 General Relativity [closed]

I'm currently studying on how to write GR as a CS theory, but i have a problem with one of the basic theorem of this subject. $$ e^\mu_{\ \ a} \epsilon^{\nu \rho \sigma} R_{\ \ \rho \sigma}^a = \det(...
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1answer
223 views

Can a closed universe become open?

My feeling is that no a closed universe can't become open. Under normal evolution according to the Friedman equation the curvature would be more exagerated as time goes on (i.e. a a closed universe ...
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3answers
289 views

What makes the Earth accelerate in a free-falling object's frame of reference?

If I'm an object in free fall near earth, then I'm an inertial frame of reference and I see the earth accelerating towards me with no force acting upon it. What causes that acceleration? The spacetime ...
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1answer
72 views

Spacetime curvature [duplicate]

When large planetary objects moves, it bends spacetime according to the mass this object has. The question is: when large celestial object moves away from say point A, - how does spacetime "knows" ...
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1answer
82 views

Measurable effects of localized negative curvature in spacetime

In the comments to my answer to this worldbuilding.se question, someone mentioned that a curvature sufficient to produce triangles whose angles add up to 179 degrees would produce a large perceived ...
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1answer
92 views

Where does the work come from if tidal forces are stretching elastic objects?

An elastic object e.g. a rubber band will be stretched in accelerated expanding FRW-spacetime during radial free fall in Schwarzschild spacetime by tidal forces due to Ricci- and Weyl-curvature resp.,...
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1answer
72 views

Equations of motion for Lagrangean Density dependent of Curvature tensor

I am trying to find the equations of motion for the following Lagrangean $$\mathscr{L} = \epsilon_{\mu \nu \alpha \beta} R_{\delta \gamma}^{}{}^{\mu \nu} R^{\delta \gamma \alpha \beta}$$ Where R is ...
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1answer
385 views

How do I prove that the Riemann curvature tensor is indeed a tensor? [duplicate]

I know that to prove that something is a tensor I have to show that this thing transforms like a tensor, i.e., like below: $$ R^{'\alpha}{}_{\gamma\phi\lambda} = \partial_{\beta}x^{\alpha'}\partial_{\...
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About tests on the importance of general relativistic effects

I am dealing with the trajectories of charged particles in the vicinity of a Kerr black hole which is inmerse in an asymptotically uniform magnetic field. I pretend to make an estimation of the ...
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2answers
103 views

Volume of an Universe with $k=+1$

I read in the Steven Weinberg’s book “Cosmology”: So far, we have considered only local properties of the spacetime. Now let us look at it in the large. For $k = +1$ space is finite, though like ...
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2answers
134 views

How to get the Ricci tensor of an EM field?

We have the Einstein equations $$G_{\alpha\beta}=\frac{8\pi G}{c^4}T_{\alpha\beta}\\R_{\alpha\beta}-\frac12 g_{\alpha\beta}R=\frac{8\pi G}{c^4}T_{\alpha\beta}$$ I have been asked to show that for ...
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How do we know that bending of light around stars is due to bending of space-time and not diffraction?

One question that popped up during the studies of special and general relativity (which I am forced to take unfortunately) is the following: How do we know that this is due to the bending of space-...
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1answer
96 views

Einstein notation and conventions when raising/lowering indices with the metric

I was trying to find components of the Riemann tensor and it occurred to me that there could be an issue with my notation. For example, if one particular component of the tensor is $$ R^{\...
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2answers
55 views

Regarding 'The Science of Interstellar', space warping section

While reading Thorne's 'The Science of Interstellar', I came across this piece of information: 'Now, the Sun’s equatorial plane divides space into two identical halves, that above the plane and ...
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1answer
102 views

Flat universe and changing energy/matter ratios

Of what I understand, current observations indicate the universe curvature is closely a flat one. This is made possible since adding up the energy densities of regular matter, dark matter and dark ...
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3answers
125 views

Isn't there a contradiction?

In special relativity Einstein used Pythagorean theorem for proving Lorenz transformations. But in general relativity we discovered that space-time has curvature near massive objects, so the geometry ...
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1answer
51 views

For the case where universe is not flat, has it got an extrinsic curvature towards (an)other spatial dimension(s)? [duplicate]

I got the idea that expansion of the universe is not to somewhere, it is just getting stretched of spacetime since a point of singularity. And I know that universe was calculated as flat (which means ...
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1answer
109 views

Non-integer $k$ value in Friedman-Robertson-Walker model?

I understand that $k$ describes positive, negative, or no curvature. However, why can't there be, for example, +0.5 (semi-positive) curvature, etc?
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1answer
198 views

Difference between curvature and Ricci scalar curvature?

I know about curvature by this notation $$\tau=\frac{dt}{ds}$$ the change of tangent vector with respect to arc length $s$ . I also know about Ricci scalar curvature is $$g^{ij}R_{ij}=R$$ I know ...
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2answers
490 views

Does a massive object contract space-time or expand it?

Before asking, I must say that I am not a physicist of any sort, but I do have a strong interest in Relativity that has led me to question my layman’s understanding. So please stay with me on this “...
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54 views

Raising indices of curvature tensor

In spherical polar coordinates, I am looking at the curvature tensor wrt the diagonal metric tensor whose elements are: $$g_{tt}={-u^2}, g_{rr}=\frac{1}{u^2}, g_{\theta\theta}=\frac{1}{r^2}, g_{\phi\...
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2answers
81 views

The invariance of BEC and curved space

I went to a lecture where the speaker explained that people could study space and cosmology and curved space by using BEC states. She tole me that the sound wave (mechanic wave) in BEC was equivalent ...
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Maximally symmetric spacetime metric

I was reading Weinberg's book on Gravitation. In chapter 13 he introduces the metric (eq. (13.3.4)): $$g_{\mu\nu}=C_{\mu\nu}+\frac{K}{1-K C_{\rho\sigma}x^{\rho}x^{\sigma}}C_{\mu\lambda}x^{\lambda}C_{\...
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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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1answer
123 views

Negative mass in Spacetime fabric

In general, a body of some mass in spacetime fabric bend spacetime fabric around it. What will happen if there is a body with negative mass? Does it will make a crest like shape in spacetime fabric?
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1answer
167 views

Entropy and the curvature of the Universe

Foreword What I know (and please correct me if I'm stating malarkey): the entropy of the universe (its description) is contained in Weyl tensor. Einstein's field equations don't directly relate the ...
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1answer
67 views

Do masses interact with each other or not?

I want to know whether there is any interaction between masses due to gravity. To illustrate my point suppose two masses are in space. They will get attracted to each other. But is this interaction ...
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2answers
186 views

Transforming to Riemann Normal Coordinates

I'm trying to do an ostensibly straight forward exercise in Zee's GR book, but am failing to understand the basic steps involved. Here's the suggestion on page 89: "Suppose you were given a space ...
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141 views

How does QFT in curved spacetime work when state is in momentum space?

I am not really sure how this Fock space picture in QFT would port to curved spacetime, but in QFT, you can form a state that has particles of some momentum, with their spacetime information spread ...
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92 views

Silly conceptual doubt about Einstein Field Equations (EFE)

When we study basic General Relativity, the interior solution of Schwarzschild spacetime sometimes are skipped. In order to determine the $\kappa = \displaystyle \frac{8\pi G}{c^{4}}$ constant of EFE ...
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1answer
116 views

Revisiting the flatness problem of the FRW Universe

The flatness problem in a nutshell One of the Friedman equation is given by $$ H^2\equiv\Big(\frac{\dot{a}}{a}\Big)^2=\frac{8\pi G}{3}\sum\limits_{i}\rho_i-\frac{k}{a^2}.\tag{1} $$ In terms of the ...
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2answers
264 views

Intuition for Interchange Symmetry of the Riemann Tensor

Is there an intuitive/geometric picture for the interchange symmetry of the Riemann tensor? I have seen plenty of algebraic derivations, but would like to understand if the symmetry expresses ...
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1answer
103 views

How would you calculate the age of a non-flat universe?

I understand that you can calculate the age of the universe by taking the inverse of Hubble's constant, but this assumes a flat universe. What happens if $k≠0$ and the universe's geometry is either ...
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1answer
285 views

Why is the Ricci scalar non-zero in this case?

The Einstein equations can be written as (1): $$R_{ab}-\frac{1}{2}Rg_{ab} = -8\pi GT_{ab}$$ or by contracting the above equation with the metric tensor and resubstituting: (2) $$R_{ab}=8\pi G(\frac{1}...
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What (& why) exactly does a massive body curve/bend in General Relativity?

So, in General Relativity a massive body bends, curves the spacetime continuum... But WHAT exactly is it that "thing" that gets curved? What exactly is empty space? Does it have a real, physical ...
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2answers
263 views

Would it be possible for a human to simply “step” through a traversable wormhole?

(For the purposes of a science-fiction story where the author is attempting to have as much basis in fact as possible) if one were to create or make use of a Morris-Thorne or similar traversable ...
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Einstein field equations in empty space, question about non-zero curvature

After reading parts of Chapter 8 in Hobson, 'General Relativity: An introduction for Physicists,' I have a question regarding the observation on page 184 regarding the gravitational field equations in ...
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1answer
306 views

What is the difference between Kretschmann and Ricci scalar curvature? [duplicate]

When you solve the vacuum field equations of gravity $ R_{ab} = 0$ you come to the conclusion that the spacetime is flat in terms of scalar curvature when you have chosen a suitable metric tensor. ...
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1answer
139 views

Significance of Kretschmann scalar to flat spaces?

If you are given a spacetime embedded with a particular metric tensor that satisfies the vacuum field equations of general relativity, how do you confirm that you aren’t simply dealing with a ...
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1answer
291 views

Technically, what is a spacetime singularity? [duplicate]

In popular science books and articles, one often finds that the BigBang is a singularity of spacetime, and it is expected to be solved by a successful theory of Quantum gravity. Technically what is a ...
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2answers
438 views

Why do objects “fall” along spacetime geodesic lines?

I'm working on a paper that also addresses the topic of general relativity (among other topics). The most common answer I get to the question above (why do objects fall) is that the objects are not ...
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1answer
282 views

Ricci tensor derivation in Kaluza-Klein theory

I've been trying to follow the article Kaluza-Klein for Kids where the author derives the lagrangian density in the Kaluza-Klein theory. He takes scalar function $\Phi =1$, then he uses the "ansatz" ...
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1answer
44 views

Vertical scale of spherometer

Does the vertical scale of the spherometer have any use? I only depend on the circular scale of the spherometer.
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651 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 ...