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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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2answers
55 views

Proof that the scalar curvature of a two-dimensional space can be expressed by only one component of the Riemann tensor

So I'm working on a question that asks for a proof that in two-dimensional space, the scalar curvature is given by: $$R = \frac{2R^1{}_{212}}{{g}_{22}}$$ Now, I've been playing around with the ...
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5answers
187 views

Is curved spacetime a real thing or just math

I was curious if the curving of spacetime by mass/energy was actually a real thing or is it just a mathematical construct, a way of visualizing the force of gravity and explaining it and that there is ...
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2answers
112 views

Is there a limitation on the values ​that Einstein tensor $G_{\mu\nu}$ can take?

Is there a limitation on the values ​​that Einstein tensor $G_{\mu\nu}$ can take? For example: Is it always bigger than zero? What is the highest amount that can be taken by it? What is the ...
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1answer
84 views

How spacetime distortion can ever be noticed from inside spacetime itself?

this is a naive question from a non-physicist. It is my understanding that gravitational waves are a deformation of spacetime. However, those are noticeable through, for example, laser interferometry. ...
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1answer
80 views

Is curvature the exterior covariant derivative of the connection?

Let $P\to M$ be a $G$-principal bundle, $G$ a topological group, $\omega$ the connection and $V$ a vector space. We define $d_\omega: \Omega^k_G(P, V)\to\Omega^{k+1}_G(P, V)$ the exterior covariant ...
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1answer
58 views

Conventions in the FRW metric

So the FRW metric is $$ds^2=-c^2dt^2+a(t)^2\left(\frac{dr^2}{1-kr^2}+r^2(d\theta^2+\sin^2\theta d\phi^2)\right). $$ I have seen books mention that $k=1,0,-1$ - depending one of the values listed ...
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0answers
11 views

Can we envision the curvature of a 2d spacetime with the help of a second space dimension near the big bang?

Consider a 2-dimensional spacetime. For a 3-, let alone a 4-dimensional spacetime it's impossible to envision the curvature of spacetime near (at the beginning or just after) the big bang. Is it ...
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39 views

Riemann tensor Contracted with full antisymmetric tensor

I'm not able to show that $\epsilon^{abcd} R_{bcae} = 0$ Note: Properly, I have to show that $\epsilon^{Iabc} R_{abIL} = 0$, where $I,L$ are tetrad index and $a,b,c$ are spacetime index, but it ...
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1answer
60 views

Is space in a rotating frame flat?

Apparently in papers like "Space geometry of rotating platforms: an operational approach" https://arxiv.org/abs/gr-qc/0207104 (page 21) and https://www.amherst.edu/.../view/10267/original/reden05.pdf (...
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1answer
96 views

Bending of space and time

we say massive object bends spacetime, but space is same in all direction, then how do u decide which direction it will bend spacetime, for showing the movement of planet around sun is ok, wat about ...
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1answer
227 views

Make $\pi = 3$ again [closed]

The value of $\pi$, or the circumference divided by the diameter of a circle, is known with absurd precision, but I want it to be 3. The circumference around a black hole outside the Schwarzschild ...
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1answer
64 views

Curvature scalars and singularities

Without resorting to the singularity theorems, can we say that there is a singularity at a particular $r=\textrm{constant}$ if the value of the Ricci and Kretschmann scalars get infinitely large at ...
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0answers
35 views

Is intrinsic curvature of an embedded surface a covariant quantity from the embedding space point of view?

Suppose I have a $(d+1)$-dimensional manifold with metric $g_{\mu\nu}$. In it I have an embedded codimension-$1$ surface, $\Gamma$, with induced metric $\gamma_{ab}$. Is Ricci scalar defined in terms ...
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1answer
69 views

Meaning of $M$ in Schwarzschild metric

In the Schwarzschild metric $$ds^2 = - \left(1-\frac{2M}{r} \right) dt^2 + \frac{dr^2}{1-\frac{2M}{r} }+ r^2 d\Omega^2. $$ Is it safe to call $M$ the mass of the source of curvature? Or should I ...
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2answers
173 views

Is there is a concensus among physicists if spacetime actually curves and if so what is it?

Going off from what others have told me on here, and based on the Wikipedia page for Quantum Gravity, General Relativity can be described mathematically in a way different than the geometrical curved ...
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1answer
98 views

Evidence for zero curvature of universe (besides CMB)

There is a beautiful argument, based on the spherical harmonics of the cosmic microwave background, which calculates the curvature of the universe. Are there any other methods of computing the ...
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1answer
47 views

Riemann curvature tensor components having 3 or 4 distinct components

When, if ever, will we see Riemann curvature tensor (RCT) components having 3 or 4 distinct indices?, like for $R_{txxy}$ or $R_{txyz}$ for ex. How this came about was I that I was reading that ...
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1answer
86 views

Gauge dependence of the Einstein tensor and the Riemann/Ricci curvature tensors in non-linear general relativity

The Einstein field equations are given by (with assuming $\Lambda = 0$), $$ R_{ab} - \frac{1}{2} R g_{ab} = \kappa T_{ab}. $$ The principle of general covariance states that the form of these ...
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2answers
160 views

Does a vacuum solution to the Einstein equation imply flat spacetime?

I have read that a solution to the vacuum Einstein equation has a vanishing Einstein tensor, and therefore a vanishing stress-energy tensor. This means that there is no matter to generate spacetime ...
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0answers
29 views

Is the warping of spacetime proportional to the mass/energy/momentum of an object in GR [duplicate]

Just wondering if the warping of spacetime proportional to the mass/energy/momentum of an object in general relativity?
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1answer
144 views

Proving that a manifold of constant curvature is maximally symmetric

This is a doubt from Jelle Hartong's masters thesis on the geometry of dS spacetime. So basically I know that $$R= \frac{2d}{d-2} \Lambda$$ and $$R_{abcd}= \frac{R}{d(d-1)}(g_{ac}g_{bd}-g_{ad}g_{bc}).$...
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3answers
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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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1answer
66 views

Can we define gravity on Calabi-Yau manifolds?

I have read about applying Hermitian geometry in general relativity in deriving holomorphic gravity. But if we take it some steps further i.e. allowing Kähler manifolds with the Ricci flatness ...
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0answers
43 views

Does Jackson's result for the vector potential of current loop correct?

General form of Maxwell equation is given by $$ \nabla_\mu F^{\mu\nu} = 4\pi J^\nu $$ where $F_{\mu\nu}=\nabla_\mu A_\nu-\nabla_\nu A_\mu$ is the tensor of EM field. Then Maxwell equations can be ...
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1answer
57 views

Is Levi cita tensor an invariant in curved space?

The Minkowski metric and Levi cita tensor is an invariant quantity in Euclidean flat space. But in curved space metric tensor varies. Analogous to it, is the Levi cita tensor varies in any Non-...
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0answers
41 views

Identically null Einstein equations in Schwarzschild spacetime

In deriving Schwarzschild solution one assumes many constraints on the metric, in particular parity invariance (invariance of $g _{\mu \nu}$ under $t \rightarrow-t, \phi \rightarrow-\phi, \theta \...
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0answers
21 views

Einstein field equations in terms of invariants [duplicate]

Is it possible to express Einstein field equations of general relativity in terms of invariants of Riemann tensor and Stress-energy tensor? I suppose field equations should lead to an algebraic ...
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2answers
204 views

Geometric interpretation of the second Bianchi identity?

Assuming a torsion free Christoffel symbol, the covariant derivative can be shown to satisfy the second (differential) Bianchi identity: \begin{equation} [[\nabla_a,\nabla_b],\nabla_c]+[[\nabla_c,\...
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3answers
3k views

Does theoretical physics suggest that gravity is the exchange of gravitons or deformation/bending of spacetime?

Throughout my life, I have always been taught that gravity is a simple force, however now I struggle to see that being strictly true. Hence I wanted to ask what modern theoretical physics suggests ...
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0answers
100 views

Solving the biharmonic equation $\square^2f(\vec{x}) =0$ on a curved space

How to solve the biharmonic equation $\square^2f(\vec{x}) =0$ on a curved space (e.g. $d$ dimensional sphere or hyperboloid)? I was thinking in the following line : I know how to solve $\square f(\...
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0answers
74 views

Spacetime curvature and measurements

From a programming perspective, I've always thought of gravitational influence as a kind of vector field, (crudely drawn) which seems to attribute to the motions of bodies through the field ...
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2answers
71 views

Transform mixed vielbein expressions to $\sqrt{-g}$ times traces in massive gravity?

In the Massive Gravity review by Claudia de Rham the massive gravity action is given by with mass potential in vielbein formulation. Equivalently, the same action can then be described by with I ...
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0answers
23 views

$\rho(t)\gt or \lt \rho_{critical}(t)$ depends upon $k$ for expansion or contraction in cosmology?

From friedmann equation $$1=\frac{\rho(t)}{\rho_c(t)}-\frac{k}{a^2H^2},$$$$\dot a(t)=+-\sqrt\frac{k}{\frac{\rho(t)}{\rho_c(t)}-1}$$ for $k\gt0$,$$\rho(t)\gt\rho_c(t)$$ and for $k\lt0$,$$\rho(t)\lt\...
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1answer
162 views

Einstein-Hilbert action and Lagrangian density for vacuum Ricci scalar

From the action, $$\int L\,\mathrm dt=\int R \sqrt{|g|}\,\mathrm d^4x,$$ why is the Lagrangian density for the gravitational field replaced by the Ricci scalar, which yield field equations in vacuum $$...
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0answers
71 views

Possible extra term in the Gauss-Bonnet Action

Is it possible to add a term like $\epsilon_{\alpha\beta\gamma\delta}R^{\alpha\beta}_{\enspace\mu\nu}R^{\mu\nu\gamma\delta}$ to the Gauss-Bonnet action in higher dimensional theories of gravity? Or ...
2
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2answers
200 views

How does density affect gravity?

Say we have two masses, mass A and mass B. These two masses are identical in every dimension. The only difference is the density. Do they not curve the same amount of space-time, and if not, why?
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0answers
27 views

Is a magnetic force caused by a curvature of something? [duplicate]

If gravity is not a force, but a manifestation of spacetime curvature, what about other forces? What about magnetic force (or Lorenz force)? Is it not a force, but a manifestation of the curvature of ...
2
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2answers
606 views

Einstein and Riemann curvature tensor

Riemann curvature tensor is dirrectly related to a path dependence of parallel transport. I read that Einstein first thought of this tensor to be the one that goes into his field equation but it didn'...
2
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1answer
57 views

Instructions for mapping the independent Riemann coefficients to the redundant Riemann coefficients

Introduction: I have been developing a General Relativity utility for working out the stress tensor coefficients for a given metric and all the related Riemannian coefficients which build up to it: ...
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1answer
56 views

Is the space-time curvature linearly additive?

Could someone please show using equations if space-time curvature due to two bodies being linearly additive or not in general.
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1answer
54 views

Where does the factor of one half come from in the delta-vector equation involving the Riemann Curvature Tensor?

In Einstein's Theory, A Rigorous Introduction for the Mathematically Untrained, by Grøn and Næss: The change of the covariant components a vector by parallel transport around an indefinitely small ...
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0answers
107 views

How to find Ricci tensor?

I'm trying to find the Ricci tensor in question 3. Here $u=r/R .$ http://imgur.com/gallery/qSAknvz I found the Christoffel symbols but I can't find the Ricci tensors. On the link, there is also my ...
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1answer
98 views

Is it possible understand Berry curvature as Gaussian curvature in some limit?

I would like to understand the Berry curvature and the Chern number from mathematical geometry-topology. I understand that in electronic QHE, there is a map from $k^2$ to a vector space where the ...
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11answers
11k views

Is spacetime wholly a mathematical construct and not a real thing? [closed]

Speaking of what I understood, spacetime is three dimensions of space and one of time. Now, if we look at general relativity, spacetime is generally reckoned as a 'fabric'. So my question is, whether ...
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1answer
70 views

Relationship between Energy density and Curvature

I don't know GR so while answering the question so keep in mind that. In the Friedmann Equations, is energy density has an effect on curvature or vice versa? Or they are separate things and they don'...
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0answers
26 views

Why Empty universe have to obey the Negative Curvature? [duplicate]

For empty universe it seems to me that we can have two solutions. $$H^2=\frac {8\pi G\epsilon} {3c^2}-\frac {\kappa c^2} {R^2a^2(t)}$$ For an empty universe when we set $\epsilon=0$ we get $$H^2=\...
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3answers
367 views

Ricci Scalar as Curvature

So I understand that the Ricci scalar represents the curvature of the space. Since any manifold can be considered locally flat, is Ricci scalar always zero locally for any manifold? On one hand it ...
2
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3answers
131 views

Is spacetime-curvature relative?

Velocity is relative, which means kinetic energy is. Since, according to general relativity, energy bends spacetime around it, wouldn't this mean observers moving in different inertial frames measure ...
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0answers
27 views

Propagation of gravitaional waves near black holes [duplicate]

As we know near black holes light gets strongly deflected. And if the gravity of the black hole is strong enough, the light can move in circles around the black hole. How the gravitational wave ...
0
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1answer
129 views

Derivation of equation for geodesic deviation

I am trying to figure out the calculation which leads to the geodesic deviation on this site. So far I understood all steps until (14.7) and managed to show that (14.6) = (14.7), namely $$ \ddot\xi^\...