# 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 Calabi-Yau manifold.

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### Can 1D beings in 1+1D determine if they are in a curved universe?

For hypothetical beings living in a the surface of a sphere, it is possible to determine if their world is Euclidean or curved and closed by making triangles and measuring the sum of the angles (...
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### Could the curvature of spacetime, as in general relativity, result from the interaction of quantum fields?

If both the general and special theories of relativity deal with space as spacetime, then the special theory of relativity deals with spacetime as flat, and the general theory of relativity deals with ...
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### How are the Riemann tensor inside a mass-energy distribution and the Weyl tensor outside it connected?

As we all know, the Riemann tensor inside a mass-energy distribution is made up of the Ricci trace part and a traceless Weyl part which is the part which exist outside of the distribution. How are ...
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### Why are distances to event horizons linear with mass when gravitational effects fall off as $1/r^2$?

Black holes' gravitational effects fall off as $1/r^2$, but their event horizon grows linearly with increasing mass.  $R$ (event horizon) grows the same rate as $M$ (mass of black hole).  Okay lets ...
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### What should be the shape of spacetime to repel opposite masses from one another?

Spin-2 gauge particles like gravitons cause attraction. Attraction in general relativity is accompanied by an appropriate spacetime. Now if we have positive and negative mass, the gravitons cause ...
1 vote
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### What is the meaning of $\nabla _{\mu}\nabla _{\nu}\phi(r)$ in general relativity?

I know the covariant derivative of a tensor is $$\nabla_{\mu} V_{\nu}=\partial_\mu V_\nu-\Gamma_{\mu\nu}^{\lambda}V_{\lambda}$$ Now I want to obtain $\nabla_{\mu}\nabla_{\nu}\Phi(x)$ where $\Phi(x)$...
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### How long does flat spacetime take to become curved? [duplicate]

Just as a thought experiment, given that you start with a flat spacetime metric, $\eta _{\mu \nu }$, and then you take $$\eta _{\mu \nu } \to g _{\mu \nu }$$ Where $g _{\mu \nu }$ is some general ...
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### Is space — as opposed to space-time — curved by a gravitating mass?

Or is the question in the title fundamentally wrong? We label each point in space-time with four coordinate values, one of which typically is suggestively called $t$ for time. This made me think that ...
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### Would gravity pulling person toward Earth change if velocity of Earth changes? [duplicate]

So I was reading the Albert Einstein's theory of how gravity works. From my understanding, the more mass an object has, the more space-time around it it bends. All objects travels in completely ...
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### Is the curvature of spacetime zero, in the center of a Kerr black hole?

In a Kerr black hole the singularity in the center has become a ringularity. Roughly said a singularity that is not point-like but circle-like. The curvature blows up not at the exact center of the ...
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### How can a graviton curve space(time) if it only couples to particles in spacetime?

In the theory of quantum gravity (insofar it's incomplete version exists) gravitons couple to particles like photons or other gauge fields. Unlike the other gauge fields, which are vector-like or ...
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### Visualizing PNDs in Petrov classification of Weyl Curvature

In 3+1 dimensional space-times, we can decompose Weyl curvature as $D(2,0)$ and $D(0,2)$ irreducible representations of $SL(2,\mathbb{C})$. Further, if such space-time admits spin-bundle, we can map ...
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### Proof of second Bianchi identity

I have a problem. I thought it was second Bianchi identity at first, but it's not. I have checked again and this formula is not wrong. How to prove it? $$R^n{}_{ikl;m}+R^n{}_{ikm;l}+R^n{}_{ilm;k}=0$$
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### Why doesn’t horizon distance move exactly proportional to the height of the observer?

For instance if someone is 8 inches above the surface of the Earth, they can see approximately 1 mile to the horizon. However, if someone is viewing the horizon at an eye level of 5’5 they can only ...
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### Symmetries of Riemann tensor

Is there a way to show that the symmetries of Riemann tensor are preserved even if the indices are raised or lowered in general. I know how to do it individually for each symmetry but am not sure how ...
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### Why is the flatness problem called the "flatness" problem? What is its connection to geometry?

My understanding of the flatness problem is that it says that if we leave out dark energy and inflation, then the density parameter $\Omega(t)$ tends to $\infty$ or $0$ unless we have $\Omega(t) = 1$ ...
1 vote
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### Does the spacetime curvature in the vicinity of a massive body increase, decrease or remain unchanged with the increasing velocity of an observer?

Does the spacetime curvature in the vicinity of a massive body such as the sun increase, decrease or remain unchanged with respect to an observer's increasing velocity relative to that massive body?
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### Looking back to Earth via general relativity [duplicate]

Based on Eddington experiment proving the effect of gravity on light, would it be conceivable that some light emitted from our sun bounce back on earth and continued in the universe. During this ...
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### How can an observer observe the metric of spacetime?

I don't mean how can we measure the metric in practice. I only mean in principle. Suppose you are an omnipresent being, no experimental limitations. What measurements do you need to measure the metric ...
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### Stress tensor trace anomaly in two dimensions

I'm trying to calculate the expectation value of the stress tensor in 2D following the book "Quantum fields in curved space" (Birrell and Davies). In 2D the divergent contribution to the one-...
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### Geometry: path length in atmosphere ("round" Earth)

I'm having trouble obtaining this formula. I'll paste the text from the book: Considering the curvature of the Earth ($R$ is the Earth radius) and a non-vertical direction (zenith angle $θ$), the ...
1 vote
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### Is the solar system sitting in the centre of curved spacetime and if so, are we viewing the rest of the universe from inside that "bubble"?

I read an article about a huge bubble being discovered in which the solar system sits bang in the middle. It got me thinking about the curvature of spacetime. The bubble was created by several ...
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### Why does gravity seem to have two natures (force or warping of space and time)?

In classical mechanics, gravity is regarded as a force but in general relativity it's a warping of space and time in presence of mass. Are these two definitions the same? Or is this a duality nature ...
624 views

### Tidal force in equivalence principle

The inertial frame of reference in a gravitational field is defined locally, but couldn't a sufficiently sensitive instrument detect tidal forces in the gravitational field and thus make the frame in ...
1 vote
Let's assume a theoretical spherically symmetric metric which leads to a finite volume with infinite extension. The metric is characterized by \mathrm{d}s^2=-B\,c^2 \mathrm{d}t^2+A\,\mathrm{d}r^2+r^...