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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1answer
141 views

Curvature and length contraction paradox

I was thinking about the Flat-Earth model and I confronted with the following issue. Consider a sphere and an observer who is leaving the sphere. Using length contraction principle from special ...
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391 views

Can we think of the proof of local flatness theorem as the proof the the Einstein's Equivalence principle? [duplicate]

Sean Carroll states the Einstein's Equivalence Principle (EEP) as "In small enough regions of spacetime, the laws of physics reduce to those of special relativity; it is impossible to detect the ...
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Why gauge-invariant Berry curvature commutator looks like torsion?

The Berry Curvature is defined as (for invariant gauge transformations) $$F_{ij} = [\partial_i, A_j] - [\partial_j,A_i] + [A_i,A_j]$$ The gauge covariance satisfies the transformation $$A_i \...
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Berry Curvature and Curvature Tensor

When the curvature tensor (from Einstein's theory) has a non-zero torsion, it is said to be an antisymmetric tensor in the last two indices composed of the connections of the field. Alternatively, the ...
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260 views

Variation of Gauss Bonnet Invariant

I am trying to do the variation of Gauss Bonnet Invariant, and the Gauss Bonnet Invariant is: $G$=$R^2$+$R_{abcd}$$R^{abcd}$-$4R^{ab}$$R_{ab}$ The variation of $G$ is: $\delta$$G$=$2R\delta$$R$+ $\...
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263 views

Anti de Sitter space

While writing the metric for AdS Space, why are we starting with a five dimensional Flat space and embedding a hyperboloid in it? Does it have to do with the fact that the cosmological constant being ...
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1answer
773 views

Does time dilation cause gravity as explained in this video?

Watch it around 2:00 minutes. https://youtu.be/gcvq1DAM-DE Do objects move closer to Earth because they experience time at different rates, really? Does it make sense? The video also represents the ...
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3answers
464 views

Anti-De Sitter Space

Anti - De Sitter Space is the maximally symmetric solution to field equations with negative cosmological constant. The negative cosmological constant also shows that the spacetime has negative ...
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1answer
171 views

Could there be equivalence between anisotropic space and the presence of a field? [duplicate]

What if we are so used to the curvature of space caused by mass and the range of its effects that we totally ignore the possibility of the existence of "opposite" curvature1, i.e. objects that bend ...
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1answer
102 views

Are the space and time axes of Schwarzschild metric uncurved?

Schwarzschild metric is commonly considered as an expression of curved spacetime: $$ \mathrm ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right) c^2~\mathrm dt^2 + \frac{1}{1 - \frac{2GM}{c^2 r} }~\mathrm dr^...
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3answers
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Instead of warping spacetime, can gravity be represented by locally varying time rates?

Instead of thinking of gravity as mass warping spacetime, could it be thought of as mass warping only time, whereby time would advance at faster rates at locations where more mass is present?
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Extracting energy from curvature

The device Imagine i connect 4 rigid rods of equal length together in a square. This thing has 4 corners, each with a 90° angle. Going around the whole thing i make one full rotation, so 360°. Now i ...
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1answer
250 views

Connecting curvature to the 'failure of squares to close'

In one of the Feynman lectures, Feynman describes curvature as the failure of a square to close. By switching to spacetime, Feynman then claims that the curvature of spacetime is reflected by the ...
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1answer
191 views

What does the Kretschmann scalar really tell us about the geometry of spacetime? [duplicate]

The Kretschmann scalar is one of the measures of spacetime curvature. For flat (Minkowski) spacetime it is zero. The dimensions of the Kretschmann scalar are $[L]^{-4}$. What does that physically ...
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4answers
240 views

How can one understand with an example that Newton's law(s) fail in a curved space?

Is it true that Newton's law is not valid in curved spaces? If yes, how can I understand it and explain to a high school student preferably with an example? I tried to think about the motion of a ...
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348 views

Rindler Space and tensors

How can we immediately see that the Riemann tensor and Ricci tensor in Rindler space are zero? I know that the Rindler metric is given by: $$-ds^2=-a^2x^2dt^2+dx^2+dy^2+dz^2$$ and what I just did ...
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379 views

Connection and curvature using differential forms [duplicate]

I am trying to understand how one would use differential forms to calculate the components of the connection and the curvature tensor given a metric. Can anyone point me to relevant resources for the ...
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1answer
231 views

Is there any additional symmetry in Riemann curvature tensor to tell which components are zero?

It is known that $ R^{\alpha}_{\beta \gamma \delta} = -R^{\alpha}_{\beta \delta \gamma} $. I.e it's skew-symmetric in its last two indices. So if the last two indices are the same one can just say by ...
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5answers
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Are the Christoffel symbols all zero in gravity-free space?

I was looking at the geodesic equation, $$\ddot{x}^\mu + \Gamma^\mu{}_{\nu\rho} \dot{x}^\nu \dot{x}^\rho = 0, $$ and thinking about how to identify gravity-free spaces by looking at the Christoffel ...
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Test-particles motion in general space-time with *torsion* : geodesics or auto-parallel curves?

Consider a general curved space-time with torsion. In the standard Einstein-Cartan-Kibble-Sciama theory (ECKS or ECSK), torsion is non-dynamical and doesn't propagates in free space. But a more ...
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3answers
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Does knowing the curvature of a patch of space-time determine all of space-time?

I think I read somewhere that knowing the curvature on a 3d slice of space-time is enough to determine the curvature of all 4D space-time everywhere? Sort of like analytic continuation? Is this true?...
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1answer
308 views

Different formulation of Riemann tensor

I want to write the Riemann tensor in two dimensions as $$R_{abcd}=\frac{R}{2}(g_{ac}g_{bd}-g_{ad}g_{bc})$$ Now I know that the Ricci scalar is defined as $$R={R^{a}}_{a}=g^{ab}R_{ba}=g^{ab}R_{ab}$...
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111 views

Is $(\Box \nabla_\nu-\nabla_\nu\Box)\phi=0$?

In Eq. 15 of https://arxiv.org/abs/1104.0819 the author has a term that looks like $(\Box \nabla_\nu-\nabla_\nu\Box)\phi$. Since for a scalar quantity $\nabla_\nu\phi = \partial _\nu\phi$ and $\Box\...
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6answers
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Can we interpret the Einstein field equations to mean that stress-energy *is* the curvature of spacetime?

What do I mean? There are two kind of equalities, or two ways to interpret an equality. Take for example the ideal gas law $$PV = Nk_BT$$ We all know what this equation means: when you calculate ...
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3answers
367 views

How did Einstein relate energy and the curvature of spacetime?

My understanding of the mathematics of General Relativity is quite limited. So I am not expecting any mathematically rigorous answers, but what I do understand is that energy curves spacetime, which ...
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1answer
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The curvature of our universe?

When someone says that the universe is flat it should be infinite. But, this contradicts the Big bang because infinity can never be reached from below. If the universe is in fact infinite then shouldn'...
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1answer
155 views

Where is the pole of distorted spacetime fabric? [closed]

Recently scientist have proved that gravitational waves travel through spacetime fabric and any mass object distorts the space fabric because of which we fill gravitational pull. My question is if the ...
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3answers
96 views

When a massive object warps the space around it, does the amount of space expand?

I may have watched those physics shows for too long, but the warping of space is usually described as a surface which bends. Whenever that happens, the surface they are showing increases in surface ...
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1answer
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Cosmological constant in GR can induce vector field tachyons?

A recent question on Light dispersion in gravitational theories, led to an implication that confuses me. It appears that the field equation for a massless vector field travelling in a background with ...
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4answers
2k views

Why do we say “Spacetime Curvature is Gravity”?

Although "Spacetime Curvature is Gravity" is the first statement that is told to an infant by the "popular science", I believe that really really misrepresents what GR has to say. I am posting this ...
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1answer
96 views

Extract Weyl curvature spinor

Eq. (27) in http://arxiv.org/abs/1110.2662 says I can construct the Weyl spinor according to $$\Psi_{ABCD} = \frac 14 C{}_{\mu\nu\lambda\rho} \left( \sigma^\mu \right){}_A{}^{\dot A} \left( \sigma^\...
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2answers
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Light dispersion in gravitational theories

GR predicts no Ricci curvature in vacuum (or at least when we can ignore the cosmological constant). Would theories that violate this lead to observable light dispersion in solar system tests of ...
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2answers
698 views

Using partial derivatives for Maxwell's equations in curved spacetime

I am having trouble understanding when/why we can sometimes use partial derivatives in place of covariant derivatives for electrodynamics in a curved spacetime. And how to interpret / intuitively ...
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Why is the Ricci scalar the only independent scalar constructed from products of the metric and its first and second derivatives? [duplicate]

In Sean Carroll's book, last paragraph of page 160, this statement is found: "The Riemann tensor is of course made from the second derivatives of the metric, and we argued earlier that the only ...
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1answer
74 views

path of light pulse: from EM in static background to massless particle in curved space-time

Is there a way to 'derive' the equation of massless particle geodesics in curved space-time starting from the field equations of electromagnetism in curved space-time? I'm imagining something along ...
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361 views

Has the curvature of spacetime been measured at the human scale?

The curvature of spacetime has been observed many times from the deflection of light around massive astronomical objects. But has it been observed around small objects in a lab? In the Cavendish ...
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Relation of $\dfrac{\partial T}{\partial q^\kappa}$ with the Curvature of the Configuration Space

$T$ being the kinetic energy and {${q^\kappa}$} being the generalized coordinates, in Chapter $1$, Page $20$, Classical Mechanics, Goldstein claims that ...speaking in the language of differential ...
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2answers
74 views

What would the skies look like in a small, positively-curved universe?

In the positively curved universe going straight would bring you back where you came from. If such a universe were very small, not expanding, and there was only one solar system, we should be able ...
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1answer
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Conformally Flat metric

There is a theorem in D'Inverno Introducing Einstein's relativity which is as follows. "Any two dimensional Riemannian manifold is conformally flat". What does this mean? Does is mean that any ...
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3answers
248 views

If all matter were removed from space-time, what would be remaining?

I would like to gain an understanding of the nature of the characteristics of space, as described in Einstein's theories. I appreciate that an answer using the language of mathematics is probably most ...
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1answer
312 views

How can particles account for the curvature of spacetime?

Classical General Relativity rests on the idea that what we call gravity actually is one property of spacetime itself. The matter distribution determines the metric by means of the Einstein field ...
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1answer
81 views

Is there any proof of warped space? [closed]

If space is warped around the sun, wouldn't the sun act as a gravity lens? Other than the theory of GR, is there any actual proof of warped space? If the earth orbits the sun due to warped space - ...
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1answer
980 views

Is de Sitter space with non-zero curvature an acceptable model for the universe?

On wikipedia I can find that a de Sitter-space has maximal symmetry and a constant curvature. Recently the interest of de Sitter spaces has increased as it could serve as a model for the universe, an ...
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1answer
332 views

How does density relate to the curvature of space time in this scenario?

Assume there is a spaceship floating between two much larger and more massive spherical bodies m1 and m2. The two bodies m1 and m2 have equivalent mass, while the body m1 is much more dense than m2. ...
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2answers
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What is the Fabric of Spacetime? [duplicate]

I always get the analogy of the pretend sun on a piece of cloth pulling everything down such as the image attached below: But What I never really understood is what IS that piece of cloth. Like is it ...
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229 views

What is the geometry of light cones if space is curved/non-Euclidean?

In light cone diagrams, the plane corresponding to the present is always the Euclidean one, but what if space is curved? Now, I've also seen diagrams where spacetime is supposed to be regarded as ...
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2answers
549 views

Why does a flat metric imply coordinates?

When in a completely flat spacetime, a metric $\eta_{\mu\nu} $ implies that in a stationary reference frame, you are dealing with three cartesian space coordinates, and one time coordinate. On a ...
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6answers
2k views

How can mass bend spacetime, if there's nothing to bend?

The Michelson-Morley experiment proved there was no aether, nothing that light moves through in space. Yet in GR, mass bends spacetime so that light travels in arcs around large masses. How can ...
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1answer
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Will the flatness of spacetime decrease with the expansion of the Universe?

My question specifically, how has the curvature of the universe remained just about flat topologically as the universe has expanded and thus if the expansion of the universe is accelerating then ...
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2answers
646 views

Indices of the Riemann tensor

Recently I have been using a video guide/explanation type thing on YouTube to gain a better understanding of the Riemann tensor and it's links to the Ricci tensor: https://youtu.be/sBDGPIuJ-HA . I ...