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 ...
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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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Notation for the metric of $\rm dS_4$ and/or $\rm AdS_4$

4D de Sitter and anti-de Sitter spaces may have their metrics inferred from the induced metric on a hyperboloid embedded in 5D Minkowski space: $$ -( x^0)^2+( x^1)^2+( x^2)^2+( x^3)^2+( x^4)^2=\pm \...
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How (if) can we connect a 2D "throat" piece of a wormhole to two hyperbolic 2D manifolds?

This question wad closed on the mathematics site, as it lacked clarity. So I try my luck here. My question is cosmology-inspired. Imagine two 2D hyperbolic manifolds. I connect them by a manifold like ...
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5 answers
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Where is the Lorentz signature enforced in general relativity?

I'm trying to understand general relativity. Where in the field equations is it enforced that the metric will take on the (+---) form in some basis at each point? Some thoughts I've had: It's baked ...
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Why doesn't light 'orbit' Earth? (why doesn't light follow a geodesic?) [duplicate]

I learned that things like the international space station or the moon are not "pulled" by the gravity of the earth but instead are following a geodesic (a straight line in curved spacetime)....
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How do clocks, at rest wrt to us, in a saddle-shaped 3D space tick relative to one another?

Clocks placed in a 3D flat space all tick at the same rate, if we look at them when we are at rest wrt to them. On a positively curved space around a mass, the clocks show different paces if we are at ...
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Curvature due to changes in surface tension

What happens to curvature of a surface when the surface tension starts to decrease?
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How does curvature of our eye affect our perception of the world? [closed]

The front part of the eye which can allow light to enter is a bit curved, so shouldn't this cause us to see a curved distorted version of reality when it is really not there? Is there any way to ...
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Cosmological constant divergence [closed]

The cosmological constant is measured to be of the order of $10^{120} $ smaller than the value which has been calculated from quantum mechanics. As far as I know, usually this divergence is attributed ...
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Is the volume in general relativity independent or dependent on the coordinates?

The volume in curved space is calculated as: $$V=4 \pi\int_{\Omega}r^2\sqrt{g_{rr}} d\Omega$$ Is this volume dependent or independent from the chosen coordinates? As I understand it should be ...
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Infinite volume gain of the whole universe due to curvature of spacetime from a finite mass?

The volume in curved space of a spherically symmetric metric calculates as $$V_{prop}=4 \pi \int_{0}^{R} r^2\sqrt{g_{rr}(r,\alpha)}dr$$ where $g_{rr}(r,\alpha)$ is the rr - component of the metric ...
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Equation of motion in quadratic gravity

I am going through the paper https://arxiv.org/abs/1502.01028 which considers the quadratic gravity with the action \begin{align} S = \int d^4x \sqrt{-g} (R - \alpha C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\...
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Could time be a secondary effect due to curvature of space?

In general relativity, four-dimensional spacetime is considered and curvature is calculated for spacetime, not only space alone. However, looking deeper into the equations, many sources of symmetry ...
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What is the general dependence of volume change from mass change due to general relativity?

The volume excess of earth due to general relativity in comparison to euclidean space has been calculated to $$ \Delta V = \frac{ G M \pi R^2}{5 c^2} = 113 km^3 $$ (this is done in this physics....
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What is the component of gravity other than curvature?

This question started long, but it summarizes as follows: there is light there is mass These are 'many' things for the purpose of a single force, gravity. Yet there are 'many' components of gravity: ...
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What is the Ricci curvature tensor in a region of sparse matter?

In empty space, where the energy-momentum tensor $T_{\mu\nu} = 0$, any solution to the Einstein field equation has Ricci tensor $R_{\mu\nu} = 0$. So what does a solution look like in a region of space ...
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Has there been any experiment trying to measure the Ricci Curvature of the universe?

One of the major component of the Einstein equation is the Ricci Curvature. As of now, I understand it mathematically as some sort of trace of the Riemann Tensor, and geometrically as the factor by ...
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How do we figure out what is the right geometry of space?

In page-319 of Visual Differential Geometry, the following is written: When we speak of a solution to Einstein's equation, we mean a geometry of space time (defined by it's metric) that satisfies the ...
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Conformal cyclic cosmology, black hole and Weyl curvature

Roger penrose in his theory "conformal cyclic cosmology" states that after a google there would be a giant black hole. Does black hole have Weyl curvature? If it does, then we know that ...
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The Derivation of the Schwarzschild Solution

I went to this site to find the solution. However, I have a few questions about where these equations come from. In the category of Assumptions and Notations, what equation gives you $\partial_tg_{\...
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Are tensors constructed such that one forms "act" on some complex vector field?

I have some confusion understanding the motivation in constructing tensors (or tensor fields). On a differentiable manifold $\mathcal{M}$ consider a vector field $X$. At any point $p\in \mathcal{M}$, ...
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How to calculate the Kretschmann scalar in 3+1 decomposition?

Does anyone know a source for calculating the Kretschmann scalar and/or 4D Riemann tensor using the variables of the 3+1 decomposition? I need to monitor a curvature invariant and calculating it ...
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Where does gravity come from?

First, in my childhood, I learned that gravity is a force. Later, I learned that gravity is a property of spacetime. If gravity is a property of spacetime, then why is it one of the fundamental forces....
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Do Einstein field equations only relate local spacetime curvature to local energy-momentum of matter?

Do Einstein field equations only relate local spacetime curvature to local energy-momentum of matter? If so, can we extend Einstein field equations globally relating global spacetime curvature to ...
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2 votes
1 answer
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What is the significance of commutator relationships in physics, e.g. $qp-pq = i \hbar$, $R(X, Y)Z = \nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z$, etc?

Quantum mechanics has the commutator relationship: $$qp-pq = i \hbar$$ In relativity the Riemann tensor is a measure of how much covariant derivatives along a path commute. $$ R(X, Y)Z = \nabla_X\...
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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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4 votes
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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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3 answers
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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 $\Omega(t) = 1$ ...
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1 answer
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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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8 votes
1 answer
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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 ...
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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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10 votes
4 answers
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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 ...
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8 votes
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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 ...
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How is electric charge acting as a negative mass in general relativity?

In a NASA script on negative masses, there was the following statement: How is it understandable that, there, charge "acts negative on mass"? In general relativity, all kinds of energy ...
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How to calculate the stress-energy-momentum tensor of a field that leads to finite volume with infinite extension? [duplicate]

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^...
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