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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 Calabi-Yau manifold.

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Spinning top moving in curved spacetime

If I have a spinning top in empty space, it would take work to change the orientation of the angular momentum vector of the top. Suppose I throw a spinning top in flat space such that the direction of ...
Matrix23's user avatar
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24 views

An exercise from Jackson's Classical Electrodynamics [closed]

Use Gauss'theorem to prove that at the surface of a curved conductor the normal derivative of electric field is given by $$\dfrac{1}{E}\dfrac{\partial E}{\partial n}=-\bigg(\dfrac{1}{R_1}+\dfrac{1}{...
Adam Darx's user avatar
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22 views

How does the Ricci tensor describe the changing separation of two airplanes flying from the equator? Conceptually understanding the Ricci tensor

I'm trying to understand the concept of the Ricci tensor and its physical implications using a concrete example involving two airplanes. Suppose two airplanes start at the equator, separated by a ...
bananenheld's user avatar
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-3 votes
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Does Mass Actually Displace Space-Time, or does Mass only Distort it?

1. Question Given the plethora of space-time illustrations, there is a sense that space-time is actually being displaced by mass, (planets). But on its face, this doesn't really make sense because ...
elika kohen's user avatar
-1 votes
0 answers
47 views

Probabilistic curvature of spacetime [duplicate]

I was wondering since matter tells space-time how to curve, and since matter is probabilistic in position (say hydrogen atom) is the curvature also probabilistic? black holes slowly shrink by ...
Mantu Das's user avatar
0 votes
1 answer
44 views

Spatial Curvature of Universe at recombination vs now

From my understanding, we use the CMB data to measure the spatial curvature of the universe today. Why is it the value for today if the CMB data reflects the universe at recombination (380K years ...
KaraboMadisa's user avatar
0 votes
4 answers
156 views

Are there closed simply connected 2D manifolds that do not require a third dimension?

In the context of cosmology, space is commonly described as potentially having a global curvature that can be positive, zero, or negative. A common way that textbooks describe positive curvature is by ...
scottduhnam's user avatar
1 vote
2 answers
48 views

Viable values for the $K$ parameter in the FLRW metric

The FLWR metric is sometimes given as $$c^2 d\tau^2 = c^2 dt^2 - \frac{a(t)^2}{(1-KX^2)} dX^2. $$ I am not interested in the tangential motion so I set $d \Omega = 0$ although it is of interest in ...
KDP's user avatar
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1 vote
2 answers
42 views

Cone vs. small circle parallel transport

I'm having trouble reconcile the following two seemingly contradicting conclusions (in 2d space): A cone is flat, because you can unfold it and it's a flat 2d surface. A cone as shown in the picture ...
Cosmo's user avatar
  • 313
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49 views

Extrinsic Curvature Calculation on the Sphere

Given the following 2+1 dimensional metric: $$ds^{2}=2k\left(dr^{2}+\left(1-\frac{2\sin\left(\chi\right)\sin\left(\chi-\psi\right)}{\Delta}\right)d\theta^{2}\right)-\frac{2\cos\left(\chi\right)\cos\...
Daniel Vainshtein's user avatar
1 vote
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Derivation of Einstein-Cartan (EC) action for parametrized connection $A$ & introduction of torsion

I have some trouble with one missing step when I want to get the teleparallel action from general EC theory, which I am not fully understanding. The starting form of action (3-Dimensional) is: $$ S_{...
StarPlatinumZaWardo's user avatar
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72 views

Where is the mass in a Black Hole without a "central" curvature singularity?

Not all black holes have a curvature singularity at their center (an example). But in principle, I thought that the curvature singularity was a direct result of the fact that the mass is concentrated ...
Aleph12345's user avatar
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40 views

Extrinsic Curvature in a conformally-flat spacetime that is also asymptotically-flat spacetime

I would appreciate if someone can confirm or correct my understanding of extrinsic-curvature (as in the ADM 3+1 decomposition of spacetime) when dealing with a conformally-flat spacetime. (I updated ...
AmnonJW's user avatar
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3 answers
76 views

Does Matter Cause Curvature or Vice-Versa [closed]

From the way explanations about gravity-acceleration-curvature equivalence are usually phrased here or elsewhere, it would appear many or most think that matter causes space-time curvature. I cannot ...
Prototypist's user avatar
1 vote
3 answers
86 views

Does (covariant) divergence-freeness of the stress-energy tensor ${T^{\mu\nu}}_{;\nu}=0$ follow from the Bianchi identity?

I'm working through Chap. $30$ of Dirac's "GTR" where he develops the "comprehensive action principle". He makes a very slick and mathematically elegant argument to show that the ...
Khun Chang's user avatar
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2 answers
70 views

Homogeneous and Isotropic But not Maximally Symmetric Space

Is this statement correct: "In a homogeneous and Isotropic space the sectional curvature is constant, while in a maximally symmetric space the Riemann Curvature Tensor is covariantly constant in ...
Nayeem1's user avatar
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4 votes
3 answers
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Constant curvature on a sphere?

$ds^2 = \frac{1}{1- r^2}dr^2 + r^2d\theta^2$ denotes a 2d spherical surface and it should have a constant curvature. The Riemann curvature tensor components are linear in their all 3 inputs. Since the ...
Nayeem1's user avatar
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5 votes
3 answers
240 views

How did Einstein figure out mass (and hence energy) bends spacetime?

I can understand that once I fix the velocity of light at $c$, there is a relative variation in space-time based on special relativity (inertial frame of reference). It's not clear to me how Einstein ...
iVenky's user avatar
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1 answer
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Geodesic variation and the (Riemann) curvature tensor -- what about uniform gravitational fields?

BACKGROUND: The equation of geodesic variation is in almost every GR book. Suppose $x(s)$ and $x(s)+\epsilon(s)$ are two nearby geodesics at $x$, with $\epsilon(s)$ small, and also $d\epsilon(s)/ds$ ...
Khun Chang's user avatar
2 votes
2 answers
917 views

Theoretically, can perfectly flat space exist in the universe?

According to general relativity, mass and energy cause the curvature of space. To have perfectly flat space, there must be a completely empty vacuum state with no mass or energy. Theoretically, is it ...
NOH WHIREA's user avatar
2 votes
3 answers
148 views

Motivation for pure Yang-Mills Lagrangian

The Lagrangian for pure Yang-Mills theory is given by $$-\frac14 F^{a\mu\nu}F^a_{\mu\nu} \tag{1}$$ where $$F^a_{\mu\nu} = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + gf^{abc}A^b_\mu A_\nu^c.\tag{2}$$...
CBBAM's user avatar
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3 votes
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Physical interpretation of the square of the Ricci tensor

I know that the Ricci scalar $\mathcal{R}=g^{\mu \nu} R_{\mu \nu}$ intutively measures the local "curvature radius" at each point in a manifold (if the manifold can be approximated by by an ...
Bairrao's user avatar
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2 votes
1 answer
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Understanding Wormholes Geometrically

Is the folding sheet analogy really that good for understanding what a wormhole is? After all, space-time curvature doesn't require any ambient space (it's intrinsic), as such a picture would suggest. ...
user345249's user avatar
26 votes
10 answers
13k views

How do black holes move if they are just regions in spacetime?

If black holes are just regions of spacetime, how can black holes even move? When matter moves through spacetime, it bends the spacetime around it, but if black holes are just regions of spacetime, ...
Rick Gennings's user avatar
0 votes
1 answer
67 views

If space has a positive curvature, is the expansion of the universe caused by time, not "dark energy"? [closed]

Ok, I will assume that space has a positive curvature, where space is the "surface" of this sphere, and time is the radius from the center, so the universe is a 4D hypersphere. Under these ...
Rick Gennings's user avatar
1 vote
0 answers
48 views

How can you use gravity while trying to model gravity? [duplicate]

So consider the usual pop-science spacetime model, a bowling ball on a trampoline. Apparently, the ball should sink into the trampoline, causing a dip in the fabric which causes nearby objects to fall ...
stickynotememo's user avatar
2 votes
0 answers
35 views

Partial integration of the Gibbons-Hawking-York boundary term

In https://arxiv.org/abs/1402.6334 on page 16 in their Eq. (5.21), they go from the equation $$S_G+S_\chi\approx\int d^2x\sqrt{-g}XR+\int dt\sqrt{-\gamma}XK$$ to the equation $$=\int d^2x \sqrt{-g}\...
mp62442's user avatar
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1 vote
0 answers
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Einstein's equation of gravitation field [duplicate]

I'm looking for the reason why there is the number eight $8$ at the r.h.s. of EI: $$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\frac{8\pi G}{c^2}T_{\mu\nu}.$$ My attempt was to take the limit of this equation ...
user2925716's user avatar
2 votes
0 answers
40 views

Calculating Gaussian Curvature for metric

I am trying to calculate Gaussian curvature of an optical metric $$ d \sigma^2=\frac{r\left(\omega_{\infty}^2-\omega_e^2\right)+2 m \omega_e^2}{(r-2 m) \omega_{\infty}^2}\left(\frac{d r^2}{1-\frac{2 m}...
sabir ali's user avatar
4 votes
5 answers
262 views

How is Gravity, assuming only General Relativity, *not* like Centrifugal Force?

It is common to state that "Gravity is not a force" due to its interpretation as a curvature effect in general relativity. By this, is it right to say that gravity is a fictitious force due ...
Anthony Khodanian's user avatar
1 vote
0 answers
58 views

How to use parallel propagator explain path dependence derived from a metric field?

In Sean Caroll's Spacetime and geometry. He gives a method to write down a solution to the parallel transport equations. If a vector $V(x)$ being transported down a path $l$, The vector field is ...
Jianbingshao's user avatar
3 votes
2 answers
505 views

Is the mass of curved space, additional mass?

According to Einstein, mass, say in the form of matter, curves space. It is the curvature of space that gives rise to gravity. Now I have heard there is an energy associated with the curvature of ...
John Hobson's user avatar
-7 votes
1 answer
138 views

So just because gravity "merely" bends space and isn't "really" a force at a distance - isn't it still a thing at a distance? [closed]

As a preamble, just for clarity as far as I can remember (I was awfully drunk) I have a degree in physics, math and comp sci: my point is "here's a probably stupid question at the level of person ...
Fattie's user avatar
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-1 votes
1 answer
53 views

If an area in 2D cannot be curved and finite is the same regarding the space of our pressumed 3D universe?

Is the sentence in the title right that our universe is infinite? And if so does it mean that stars are not evenly distributed along our universe but they all move from a populated centre to a fairly ...
Krešimir Bradvica's user avatar
2 votes
2 answers
158 views

Alternative definition of the Ricci Scalar

I came across this definition of the Ricci Scalar on its Spanish Wikipedia page: $$R=-g^{\mu\nu}\left(\Gamma_{\mu\nu}^{\lambda} \Gamma_{\lambda\sigma}^{\sigma} - \Gamma_{\mu\sigma}^{\lambda}\Gamma_{\...
Stargazer's user avatar
4 votes
0 answers
102 views

Which Lorentzian metrics are comformally equivalent to some Einstein metric or some Ricci flat metric?

I know that given a conformally flat Lorentzian metric, I can implement a Weyl transformation, $$g_\text{ab}\mapsto\bar{g}_\text{ab}=\Omega^2 g_\text{ab}$$ to flatten it as $\bar{g}_\text{ab}=\eta_\...
Daniel Grimmer's user avatar
0 votes
1 answer
68 views

Selecting Indices for the Riemann Tensor

How do I know when computing the Riemann Tensor (in two dimensional) which indices to select? Consider the Riemann Tensor $R^a_{bcd}$ how do I know what values to take for $a$? As an example, consider ...
missyclarke1998's user avatar
19 votes
5 answers
14k views

Is quantum gravity research implying that gravity is actually a force and not spacetime curvature according to GR?

I am all the time reading that gravity is actually the curvature of spacetime according to general relativity (GR) established theory and not a force, like the known three fundamental forces of nature,...
Markoul11's user avatar
  • 4,170
0 votes
3 answers
133 views

The Curvature of Electric Field Lines

I have been practicing many questions regarding electrical field lines. However, I can't seem to understand when electrical field lines remain straight and when they start to curve. Does it depend on ...
improvement dude's user avatar
1 vote
0 answers
35 views

Timelike normal vector becomes null

I have a metric given by \begin{equation} ds^2 = \frac{e^{2 A(z)}}{z^2} \left(-g(z) dt^2 + \frac{dz^2}{g(z)} + dx^2 + dx^2_1 + dx^2_2 \right) \end{equation} where $A(z) = -a \ln(b z^2 + 1)$ and $g(z)$ ...
mathemania's user avatar
1 vote
0 answers
45 views

A covariant derivative computation in General Relativity [duplicate]

I am trying to compute $\nabla^\mu\nabla^\nu R_{\mu\nu}$. I proceed as follows: \begin{align} \nabla^\mu\nabla^\nu R_{\mu\nu}&=g^{\mu\rho}g^{\nu\lambda}\nabla_\rho\nabla_\lambda R_{\mu\nu} \\ &...
vyali's user avatar
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1 vote
0 answers
86 views

Regarding Derivation of Einstein Field Equations

In most sources I come across that try to justify the Einstein Field Equations outside the context of Einstein-Hilbert action, the argument goes mostly as follows: In analogy with the Poisson equation ...
ksnad's user avatar
  • 73
0 votes
1 answer
59 views

Kretschmann Scalar for the FLRW Metric

I am trying to understand the Wikipedia definition of the Kretschmann scalar for a cosmological solution. The metric is given by the standard FLRW metric \begin{equation*} ds^2 = -dt^2 +a^2(t)\left(\...
LolloBoldo's user avatar
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1 vote
1 answer
65 views

Ricci tensor in locally Lorentz frame

In the book A Relativist's Toolkit by Eric Poisson, section 4.1.4, page $123$, it is written that in a local Lorentz frame at a point $P$: $$\delta R_{\alpha \beta} \stackrel{*}{=} \delta\left(\Gamma^\...
darkphysics's user avatar
1 vote
1 answer
57 views

Does the Weyl tensor amount to tidal effects of gravity?

The Ricci tensor, for the spacetime surrounding the Earth, is zero, so the spacetime around the Earth is Ricci-flat. The Riemann tensor though is not zero since spacetime certainly is curved. This ...
Il Guercio's user avatar
2 votes
1 answer
61 views

Derivation of Wald's general relativity equation 7.5.8 ; conformal transformation of Riemann tensor

I am trying to derive some nice properties of conformal transformation of Riemann tensor. I found some formulas on appendix G on Carroll and appendix D in Wald, and recognize their starting point is ...
phy_math's user avatar
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1 vote
2 answers
115 views

Is the curvature so extreme at the event horizon, that you could see curved laser beams?

I have read this: Because the spacetime curvature at the horizon is so great that there is no light-like world line the extends beyond the horizon. Why does time stop in black holes? If the ...
Árpád Szendrei's user avatar
1 vote
0 answers
16 views

How to find out the relation between liquid lens curvature and changing the focal point? [duplicate]

I have a question regarding liquid lens. how much we have to change the curvature to get an image shift of 30mm?
Marjan Shojaei's user avatar
4 votes
0 answers
216 views

Does the term $d ( \omega_{ab} \wedge \theta^a \wedge \theta^b )$ have any significance?

If $\omega_{ab}$ is the spin connection 1-form, and $\theta^a$ are the tetrad 1-forms, then one has the equality \begin{equation} \int \, d ( \epsilon_{abcd} \omega^{ab} \wedge \theta^c \wedge \theta^...
user1379857's user avatar
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2 votes
2 answers
111 views

Why does the warping of spacetime make objects move closer together?

I understand why the warping of spacetime affects moving objects, but why would it affect stationary ones if it even does? Would two completely stationary objects not move closer together because they ...
Hunter Sherring's user avatar

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