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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17 votes
4 answers
4k views

ALL "forces" as manifestations of properties of space-time

I apologize if this seems like a quack question, but I need some insights by those who know much more than me in Physics. Anyway, the gravitational "force" (not really a force) is a manifestation of ...
0 votes
0 answers
39 views

Can the Ricci curvature tensor vary while we hold fixed the Ricci scalar and the Weyl tensor? (Nordstrom Gravity)

I am reading about Nordstrom's second theory of gravity and have become confused. On the wikipedia page and elsewhere it is said that the theory is completely captured by its equations for the Ricci ...
0 votes
0 answers
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Are there any ways to conceptualize the relationship between gravity and space-time other than curvatures?

This might sound like a random question, but it came to me while I was trying to conceptualize the size of the universe and started thinking of entire galaxies resembling grands of sand floating ...
1 vote
1 answer
48 views

Independent Components of the Riemann Curvature Tensor

I am struggling to understand a general method to calculate the independent components of the Riemann Curvature Tensor (RCT). Firstly, as far as I am aware the number of independent components of the ...
0 votes
4 answers
103 views

How can a triangle have a sum exceeding 180 degrees in a curved space?

I was reading a book to understand the limits of the euclidean space I understand that lines that are parallel in 2d can meet in 3d space like on a sphere but it is hard to imagine or fathom why the ...
0 votes
2 answers
75 views

No torsion with calculating the commutator of the covariant derivatives

For simplicity, I only calculated half of the commutator. I didn't leave everything in components because I'm uncomfortable considering (I previously messed up the indices. The following is the ...
1 vote
1 answer
1k views

Is there a formula to work out how much the fabric of spacetime bends?

From my knowledge, a big mass (planet star etc) can bend the fabric of spacetime. Is there a formula that we can use to work out how much it bends?
0 votes
2 answers
139 views

How do black holes infinitely bend space-time when the bending is mass dependent and not density dependent?

According to Einstein, mass bends the fabric of space-time. And nothing in the universe has infinite mass to infinitely bend space-time. So how do remnants of supermassive stars, i.e black holes ...
0 votes
0 answers
27 views

Can the metric components be determined by the curvature value?

In JT gravity, we are called to solve the equation of motion with respect to the Dilaton field $\Phi$, which read $$R+2=0$$ Since the Ricci scalar $R$ is determined by the metric, then the equation ...
1 vote
1 answer
56 views

Why does trajectory in the space curved by gravity, depend on the speed?

I am sorry about the probably naiive nature of this question (I am a software eng, not a physics student): I (think I) understand the popular curved "trampoline" model of 2-dimensional space,...
1 vote
2 answers
73 views

What is Dirac's reasoning when saying parallel displacement creates vector field with vanishing covariant derivative?

Section 12 of Dirac's book "General Theory of Relativity" is called "The condition for flat space", and he is proving that a space is flat if and only if the curvature tensor $R_{\...
0 votes
0 answers
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Equations of motion in general relativity: Einstein field equations vs geodesic equation

It is said that the equations of motion of a theory are those whose solutions give the coordinates/trajectory of the system. I was wondering ¿which is the correct equation of motion in the theory of ...
2 votes
1 answer
331 views

Variation of the Gauss-Bonnet action and Palatini identity for the purely covariant Riemann tensor

I'm taking the variation of the Gauss-Bonnet action $$\mathcal{L}_{GB} = \frac{1}{2}\left(R^{2} - 4R^{\mu\nu}R_{\mu\nu} + R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}\right)$$ to obtain the equations of ...
1 vote
4 answers
103 views

How to find the double covariant derivative of a general vector?

I have been reading through Carroll's GR textbook and there is a line in the derivation of the Riemann tensor that I do not understand. $$\nabla_\mu \nabla_\nu V^\rho=\partial_\mu(\nabla_\nu V^\rho) - ...
0 votes
2 answers
70 views

Gravitational field of the star changes the paths of light rays, how is this observed during eclipse?

The gravitational field of the star changes the paths of light rays in space-time from what they would have been had the star not been present. I understand that the light cones are bend slightly ...
4 votes
2 answers
562 views

Entropy and the curvature of the Universe

Foreword What I know (and please correct me if I'm stating malarkey): The entropy of the Universe (its description) is contained in Weyl tensor. Einstein's field equations don't directly relate the ...
21 votes
6 answers
4k views

Better explanation of the common general relativity illustration (stretched sheet of fabric)

I've seen many science popularisation documentaries and read few books (obviously not being scientist myself). I am able to process and understand basic ideas behind most of these. However for general ...
0 votes
0 answers
44 views

How does Einstein's spacetime curvature theory for gravity actually makes sense? [duplicate]

I have a lot of confusion understanding the concept that a mass creates a curvature in the space time and other small masses gets attracted towards big masses since they starts falling inside that ...
2 votes
2 answers
761 views

Fourth rank tensor for stress energy

The Weyl tensor equates to the Riemann tensor in a vacuum $$ C_{\mu \nu \eta \lambda} = R_{\mu \nu \eta \lambda} $$ So it makes me wonder about the tensor $$T_{\mu \nu \eta \lambda} = C_{\mu \nu \eta \...
12 votes
4 answers
3k views

How does pressure exist inside a drop of water?

Consider pressure inside a drop of water. I have seen a formula for it where $$\text{Pressure Inside}=\text{Pressure Outside}+2\frac{S}{R}.$$ I want to know how pressure exists inside the liquid. Is ...
1 vote
1 answer
90 views

Limit of gravity

Is there any limit to the bending of spacetime due to gravity? I have been reading about wormholes and how they bend spacetime and connect two systems. But if there is no limit to gravity, we can ...
0 votes
2 answers
133 views

Why does curvature reduce the inward attractive force of molecules?

This question concerns the first 2 paragraphs of this text. I am struggling to visualize why molecules on the flat surface experience more net inward force than those on a curved surface. If we have 2 ...
1 vote
1 answer
190 views

Issue calculating the scalar curvature for static, spherically symmetric spacetime

I was trying to follow the Cartan method to find the curvature as outlined here, in the case of a static, spherically symmetric spacetime. It all seemed to work fine, and for a metric of $$ds^2 = -a^...
0 votes
0 answers
40 views

What shape does a rope make when compressed / coiled up?

Let's say we have a rope placed on the line $y=0$ on a flat table in the $(x,y)$ plane. We place one hand on the rope at $x=0$ and $x=l$ and then move our hands together along the $x$-axis so that ...
0 votes
0 answers
82 views

Mathematical description of higher spin gauge theories

Ordinary Yang-Mills gauge theory giving spin 1 gauge bosons can be mathematically described by connection 1-forms and curvature of principal bundles. I wonder, what the proper mathematical description ...
1 vote
1 answer
62 views

Does 2-form curvature $\Omega \in \Omega^2(P,\mathfrak{g})$ represent a physical quantity in gauge theory?

In gauge theory, all measurable physical quantities remain invariant under a gauge transformation. I have always seen that the curvature 2-form $\Omega \in \Omega^2(P,\mathfrak{g})$ associated to a ...
3 votes
1 answer
96 views

How many photons exist in another dimension spaces?

As I understand, there are 2 types of photons in our (3+1) space with photon helicity $\pm 1$. How many photons exist in another spaces like (2 + 1) or (1 + 1)? Can we apply the same for gravitons?
0 votes
0 answers
26 views

What is the correct type of the Berry curvature?

I am studying Berry curvature for a specific material and faced different types of the Berry curvature formula. Some papers use only valence eigenstates (u1) like this $$i*(<(∂U1/∂kx)| (∂U1/∂ky)>...
1 vote
1 answer
215 views

Variation of the square of the Weyl tensor

After reading a bit about Conformal gravity, I came across the Lagrangian of the form: $$L=C_{abcd}C^{abcd} \sqrt{-g}$$ Where $C$ is the Weyl tensor, I am interested in finding the field equations ...
2 votes
0 answers
42 views

Equation of motion in conformal gravity theory?

In conformal gravity theory, the action is given by $$L=\int \sqrt{-g}C^{abcd} C_{abcd} d^4x=\int \sqrt{-g}(R^{ab}R_{ab}- \frac{1}{3}R^2)d^4 x.$$ However, the variation of the first term $\int \sqrt{-...
0 votes
3 answers
124 views

What is spacetime like according to general relativity? [duplicate]

It is often said that matter curves space (or rather spacetime) in general relativity. But why should matter curve space one way and not the other way? So it seems like a metaphor, I guess. I read ...
7 votes
3 answers
608 views

Why does the Weyl tensor not show up in the Einstein field equations?

In the Einstein field equations, the only tensor that shows up is the Ricci tensor and the metric tensor, together with the Ricci scalar. The Weyl tensor though is a tensor that is a part of the ...
0 votes
1 answer
99 views

Flat space between colliding black holes

When 2 black holes approach each other, they both bend space in an opposite direction. There must always be a flat space between 2 colliding black holes. However, I heard that they actually merge, ...
10 votes
2 answers
512 views

How will two equal discs, rotating with equal but opposite angular velocities and put on top of each other affect the spacetime "surrounding" them?

In this article the Ehrenfest paradox is described. You can read in it that, according to Einstein's General Relativity (to which this paradox contributed), the spacetime around a rotating disc is non-...
3 votes
0 answers
84 views

Can a CTC contaning spacetime be purely electric?

Take a time-oriented Lorentzian manifold $(M, g)$ where $M$ is a topological 4-manifold and $g$ a Lorenzian metric. Assume such a spacetime contains a CTC. Since the manifold is time-oriented, one can ...
0 votes
0 answers
24 views

Extrinsic curvature of constant time hypersurfaces in Minkowski

Along the geodesic of a stationary observer in Minkowski spacetime we have the following tangent vector $$t^\mu = (1,0,0,0)$$ We have that hypersurfaces of constant time along this are just 3D ...
6 votes
1 answer
123 views

Raychauduri Equation in "The Large Scale Structure Of Space-Time" book by Hawking & Ellis

Previously to Raychauduri equation, Hawking-Ellis obtain equation (4.25) (pg. 84) namely $$ \frac{d \theta_{\alpha\beta}}{ds} = - R_{\alpha 4 \beta 4} - \omega_{\alpha\gamma} \omega_{\gamma\beta} ...
4 votes
4 answers
769 views

Why do we call the Riemann curvature tensor the curvature of spacetime rather than the curvature tensor of its tangent bundle?

I was studying the mathematical description of gauge theories (in terms of bundle, connection, curvature,...) and something bothers me in the terminology when I compare it with general relativity. In ...
3 votes
0 answers
35 views

Why are tidal forces conformally invariant?

Tidal forces are encoded in the Weyl-tensor $C^\mu_{\nu\lambda\sigma}$. It is well-known that the Weyl-tensor is invariant under conformal transformations: $g'_{\mu\nu}(x) = \Omega(x)g_{\mu\nu}(x)$. ...
1 vote
1 answer
252 views

Curvature in space time during Big Bang and present scenario

Space time in the presence of masses is curved. But during the time of Big Bang it's presumed that all the matter in this universe was at a single point, so it must have been super dense and had very ...
12 votes
10 answers
7k views

Why doesn't planet Earth expand if I accelerate upwards when standing on its surface?

According to General Relativity I am being accelerated upwards by planet earth while writing this question. But a curious person on the the other side of the planet relative to me would have the same ...
1 vote
1 answer
61 views

Does the variation of $I$ yield Bach tensor?

For $$I_1=\int \sqrt{-g}C_{abcd}C^{abcd}d^4x,$$ where $C_{abcd}$ is the Weyl tensor. If we neglect the Gauss-Bonnet term this can be reduced to $$I_2=2\int \sqrt{-g}(R^{ab}R_{ab}-\frac {1}{3} R^2)d^4x....
3 votes
1 answer
645 views

Does the Casimir effect only occur between flat plates?

What happens to the strength of the Casimir effect when the Casimir plates are curved instead of being completely flat. Does this have an effect on the negative vacuum pressure at different points ...
0 votes
1 answer
196 views

How we find the contorsion tensor?

I know that the formula for contorsion tensor is $$K^{\mu\nu}_a=\frac12({T_a}^{\mu\nu}+T^{\nu\mu}_a-T^{\mu\nu}_a)$$ I want to know how I can find ${T_a}^{\mu\nu}$. What kind of contraction do I follow ...
3 votes
1 answer
872 views

Understanding Riemann Curvature Tensor in Misner, Thorne and Wheeler's Gravitation

I'm trying to understand section 11.4 of Misner, Thorne and Wheeler's Gravitation textbook, which explains how the output of the Riemann Curvature Tensor $Riemann(...,A,u,v)$ is a vector describing ...
0 votes
1 answer
148 views

Is there any physical interpretation of the curvature in electromagnetism?

Electromagnetism can be modeled as a $U(1)$-principal bundle over Minkowski spacetime. The strength of the electromagnetic field is given by the 2-tensor $F_{\mu\nu}$. In differential geometry this is ...
1 vote
0 answers
64 views

Definition of asymptotically flat spacetime

Following the definition in Wald's book on general relativity, in page 276 asymptotically flat spacetimes are defined using conformal isometry with conformal factor $Ω$. Then one of the requirements ...
1 vote
0 answers
86 views

How does a curvature in time equate to Newtonian gravity? [duplicate]

I often read that a curvature in time (the rate at which clocks tick) near a massive object, is considered to be the source of Newtonian gravity. This got me wondering, does General Relativity use the ...
2 votes
2 answers
159 views

Calculating proper volume in the Alcubierre spacetime

I'm trying to calculate the proper volume of a portion of the alcubierre spacetime to see how it compares to the euclidean volume element. As I understand it, the proper volume element in cartesian ...
0 votes
1 answer
78 views

Cosmological perturbation theory and relationship to Taylor series?

In cosmological perturbation theory, it's hard to find papers that would expose the general principle to perturb physical quantities (metric, fluid pressure and density, speed...) up to the $n$th ...

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