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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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General Relativity and cosmology [duplicate]

What is the physical meaning of Ricci scalar is a covariantly constant?
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1answer
94 views

Witten Index of Riemannian Manifold

Consider a system on a Riemannian manifold with the Lagrangian $$L = \frac{1}{2}g_{IJ} \dot{\phi}^I \dot{\phi}^J + \frac{i}{2}g_{IJ}(\overline{\psi}^I D_t \psi^J - D_t \overline{\psi}^I \psi^J) - \...
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1answer
105 views

Einstein Tensor doesn't vanish [on hold]

Hello my question is how is it possible that the Einstein tensor doesn't vanish in the Schwarzschild metric. For example $G_{11}= \frac{1}{2} g_{11} g^{22} g^{33} \partial_2 \partial_2g_{33}$ is one ...
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Curvature created by an object near Earth via energy-stress tensor

From Misner... who uses the convention of (-, +, +, +) for the metric $g^{\mu\nu}$, with the electromagnetic stress energy tensor being(pg.141): $$T^{\mu \nu} = \frac{1}{\mu_0}(F^{\mu \alpha}F^{\nu}{...
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Why do we visualize space time as trampoline surface? [duplicate]

It's a silly question probably. But the thing is that what is the fascinating about fabric surface. I am so sorry for this silly question in advance. I read a lot articles regarding my question. But ...
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Writing the curvature tensor in terms of spin connection

I am trying to know what are the missing steps in deriving the following expression for the Riemann curvature tensor in terms of the spin connection $\omega$ and the triad $e_\mu^j$. Where the ...
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1answer
74 views

Whats the quickest way to compute the Ricci tensor?

I have been going through exam papers and often they ask us to calculate ricci tensor components and affine connections from a given metric. They seem to take far too long for the time you are ...
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Does the Einstein field equation uniquely determines the topology of spacetime? [duplicate]

I am trying to understand whether the Einstein field equation uniquely determines the topology of spacetime. As far as I know, given a metric, we can always find the induced topology. However, I was ...
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2answers
893 views

Does kinetic energy warp spacetime?

My interpretation of GR leads me to think that energy (namely kinetic) also adds to the curvature of space-time. Which, has raised a thought experiment. If a $10000$ kg ship closely passed a $1$ kg ...
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2answers
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Finding the dimensions of a spacetime given the Riemann tensor

The question is: For a spacetime the Riemann tensor is given below: $$R_{\mu \nu \rho \sigma} = \frac{R}{6} (g_{\mu \rho} g_{\nu \sigma} - g_{\mu \sigma} g_{ \nu \rho} )$$ What is the dimension of ...
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2answers
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Graviton and spacetime

General Relativity and the concept of curved spacetime replacing the "force" of gravitation is really beautiful, and I thought one could probably find similar descriptions of other forces like ...
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4answers
79 views

Curved space-time and metric tensor

I'm studying about curved spaces and I read that a manifold is flat if there a coordinate system such that the metric tensor is constant everywhere. Then I also read that when the space-time tensor ...
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2answers
41 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
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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
105 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
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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
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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
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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
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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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0answers
36 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
56 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
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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
222 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
50 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
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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
53 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
165 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
84 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
34 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
58 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
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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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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
93 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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1answer
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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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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
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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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0answers
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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
144 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
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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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0answers
56 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
63 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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$\rho(t)\gt or \lt \rho_{critical}(t)$ 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
93 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
61 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 ...
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2answers
114 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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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
537 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'...