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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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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
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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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1answer
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Casimir Vacuum Energy and the Sakharov Correction for Virtual Particles

There already exists a geometric interpretation in the Casimir equation for vacuum energy since it contains an area: $$\frac{F}{A} = -\frac{d}{dR} \frac{E}{A} = -\frac{\pi^2 \hbar c}{240 R^4}.$$ Can ...
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How gravity really works in 3D and how and massive object attracts another object? [duplicate]

space is 3d then how an massive object creates curvature in 3d an how another object falls into that 3d curvature also if you are answering plese show an image of it.
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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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0answers
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If you chuck a rigid triangle into a curved space, what happens?

As a follow up to How exactly does a real object (e.g. a triangle) behave considering the effects of non-euclidian geometry?, let's say you take a large steel triangle (in a region of space that is ...
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1answer
38 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
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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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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
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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
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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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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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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
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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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$\rho(t)\gt or \lt \rho_{critical}(t)$ 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
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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
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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 ...
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2answers
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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
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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
482 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
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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
46 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
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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
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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
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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
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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
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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
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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
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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 ...
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3answers
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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
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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 ...
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1answer
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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^\...
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Is the Palatini-Lovelock action of order $k$ topological in $2k$ dimensions?

I am interested in Lovelock actions in the metric-affine (or Palatini) formalism. It is well-known that the metric version (starting from the Levi-Civita curvature) of the Lovelock lagrangian of order ...
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1answer
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Do mass and motion affect space-time differently?

Mass is said to create curvatures in space-time thereby creating gravity, yet technically the smallest movements, even on Earth, create gravitational waves. Are there different "types" of disturbances ...
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0answers
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Fiber manifold Ricci flat, physical meaning

In a warped-product spacetime, what a physical meaning we have for Ricci-flat Fiber? I'll explain.. it is well known that a Ricci-flat spacetime means that the cosmological constant need not vanish, ...
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1answer
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Energy spacetime warping

If energy warps spacetime, then does light warp spacetime? And if special relativity says that things near the speed of light increase in relativistic mass, then does light have a relativistic mass? ...
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4answers
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Question on equivalence of acceleration and mass with respect to gravity

Layman’s question here. Let’s say I’m standing on the inside rim of a rotating space station spun at right rate to produce earth-like gravity. Does the spinning warp space time? If so, how can a small ...
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2answers
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Is there a useful way to visualize the symmetries of the relativistic Riemann curvature tensor?

I find it useful to see diagrams such as trees, colored 2D and 3D arrays, etc., which illustrate how terms combine in composite expressions. For example, the following is my visualization of the ...
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1answer
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Phase space as differential manifold

Generally we "draw" phase space as typical coordinate system, where $q$s and $p$s are treated like perpendicular axes. Why do we then regard phase space as generall differential manifold while it ...
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0answers
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How to find the curvature of a surface using directional length dilation?

I've already figured out how to find the curvature of an $f(x,y)$ function at each point. $$K=\frac{f_{xx}f_{yy}-f_{xy}^2}{(1+f_x^2+f_y^2)^2}.$$ Now I want to find out how to calculate curvature ...
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4answers
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Why was pseudo-Euclidean geometry not enough for general relativity?

How would you explain to someone the change that Einstein needed in geometry for his new ideas about gravity and spacetime, what did he seek but could not be described by pseudo-Euclidean geometry? ...
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1answer
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What is the age of a universe with positive, negative, and zero curvature?

I am trying to calculate the age of universes with different curvatures using the Hubble constant and Friedmann equation. What does it mean when we say that the universe started out at equipartition ...
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1answer
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(3+1)D solution to (2+1)D einstein equations?

Imagine a grid in 3D made of pipes smoothed so that it forms one continuous infinite surface. The surface is 2D but it fills 3D space. Like this (at one instant): Could any surface like this be a ...
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1answer
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GR with Torsion: Definition of contorsion

I start doing some computations in manifolds with non vanishing torsion and things are getting a bit confused, basically because of notations and definitions. I understand that in presence of non ...
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1answer
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Does a proton bend spacetime?

Protons have mass and as a result of einstein's field equation dictate that the spacetime is no longer flat. But yet I find in most Quantum Field Theory books the Minkowski flat spacetime metric is ...
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0answers
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Finding the Ricci tensor components for the Schwarzschild metric

I'm trying to use Cartan's method to find the Schwarzschild metric components from Hughston and Tod's book 'An Introduction to General Relativity' (pages 89-90). I'm having problems calculating the ...
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Chern-Simons Gravity term in 3D and equations of motion

In the book "Quantum Gravity in 2+1 dimensions" by Steven Carlip he writes down a possible modification to the Einstein-Hilbert Action in 3d (eq. 1.16 to eq. 1.18) \begin{equation} I_{GCS}=-\frac{1}{...
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What is the physical meaning of the trace-free part of the second fundamental form?

Given a submanifold $X$, the second fundamental form tells you about how the submanifold is embedded in the ambient space (intuitively by measuring how a normal vector field varies from point to point....
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1answer
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Is the gravitational field an illusion, a by-product of geometry? [duplicate]

The principle of general covariance from the Equivalence Principle (EEP) tells us that there is no way in principle to locally distinguish between an inertial acceleration and the effects of a ...
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2answers
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Can matter be described as the result of the curvature of space, instead of vice versa?

Can matter be described as the result of the curvature of space, rather than the curvature of space being the result of matter, and energy being the cause of the curvature of space?