Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

2
votes
1answer
34 views

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: ...
0
votes
1answer
39 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.
1
vote
1answer
47 views

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 ...
1
vote
0answers
47 views

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 ...
-1
votes
0answers
38 views

Are all 4D Ricci flat manifolds locally Euclidean? [closed]

If a 4D manifold with metric signature (++++) is Ricci flat. Does this mean that locally the space is Euclidean? Does this mean that the only difference between these manifolds is there global ...
1
vote
1answer
37 views

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 ...
48
votes
11answers
10k views

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 ...
-2
votes
0answers
17 views

Mass bending spacetime: what is the mechanism? [duplicate]

What is the mechanism that causes mass to bend spacetime? Merely inferring that it does is not an answer. How it does it is the answer.
0
votes
1answer
23 views

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'...
0
votes
0answers
23 views

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=\...
0
votes
3answers
50 views

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 ...
2
votes
3answers
89 views

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 ...
0
votes
0answers
21 views

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 ...
0
votes
1answer
57 views

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^\...
1
vote
0answers
42 views

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 ...
-2
votes
0answers
38 views

3D positively curved space [migrated]

If we consider 2D euclidean surface consists of infinite concentric circles and 3D euclidean surface consists of infinite concentric spheres. If 2D surface is positively curved the radius of the ...
1
vote
1answer
57 views

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 ...
1
vote
0answers
52 views

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, ...
1
vote
1answer
63 views

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? ...
0
votes
4answers
71 views

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 ...
4
votes
1answer
90 views

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 ...
1
vote
1answer
52 views

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 ...
0
votes
0answers
26 views

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 ...
3
votes
4answers
205 views

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? ...
1
vote
1answer
68 views

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 ...
2
votes
1answer
65 views

(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 ...
6
votes
1answer
78 views

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 ...
2
votes
1answer
120 views

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 ...
1
vote
0answers
58 views

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 ...
3
votes
0answers
44 views

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}{...
0
votes
0answers
39 views

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....
0
votes
1answer
101 views

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 ...
14
votes
2answers
1k views

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?
1
vote
0answers
68 views

Proving the first Bianchi identity only from the other three Riemann curvature tensor identities [closed]

Given that $R_{abcd}=-R_{bacd}$, $R_{abcd}=-R_{abdc}$ and $R_{abcd}=R_{cdab}$ can I prove that $R_{abcd}+R_{acdb}+R_{adbc}=0$ without using the definition of the Riemann curvature tensor? Are the ...
1
vote
0answers
46 views

Einstein equation outside of the source

When solving Einstein equation outside the source it is assumed that we have $R=0$ where $R$ is Ricci tensor. But if we have as a source Earth and for example, a black hole, how can this equation ...
1
vote
1answer
46 views

Berry phase covariant derivative

I have been studying some simple examples of the covariant derivative for 2D surfaces and the way that it is constructed is by taking the usual derivative in the 3D Euclidean space at a point $p$ on ...
3
votes
1answer
90 views

Is this geometric argument enough to show that special relativity assumes flat spacetime?

I am preparing myself to teach a class about special relativity a few weeks from now. To make sure they'll understand that spacetime must be flat for special relativity to work, I came up with the ...
1
vote
0answers
25 views

Show the conformal transformation of the components of the Schouten tensor at the Spatial Infinity in an asymptotically flat spacetime

In Ashtekar & Hansen, the authors discussed a unified treatment of null and spatial infinity in general relativity. In Section 5.D., they derived the relation (20). I failed to reproduce it. Let ...
-1
votes
2answers
80 views

Why do we say the three-dimensional space is flat (in Physics)? [closed]

This is quote from Hawking's book: The surface of the Earth is what is called a two-dimensional space. That is, you can move on the surface of the Earth in two directions at right angles to each ...
0
votes
2answers
115 views

Einstein GR and metric signature

Let us take the einstein Equation $R_{\mu\nu} -\frac{1}{2}g_{\mu\nu}R = T_{\mu\nu}$. I'm just ignoring all the constants. For a perfect fluid, $$T_{\mu\nu} = (\rho + P)u_{\mu}u_{\nu} - Pg_{\mu\nu}.$$ ...
1
vote
1answer
74 views

A question about the expression of Riemann tensor in Landau & Lifshitz

I was reading Landau & Lifshitz "The Classical Theory of Fields" and there is a expression at the beginning of section 92-Properties of the curvature tensor I don't understand. The author ...
3
votes
1answer
135 views

Commute covariant derivatives of spinors

Consider a spinor field $\psi$ on a general smooth Lorentzian manifold. Let $\Sigma_{ab}$ be a matrix representation of the Lorentz group, and let Greek/Latin letters represent world/Lorentz indices. ...
0
votes
1answer
101 views

Is the spacetime curvature of the Earth the reason for the orbiting of the Moon around the Earth?

The planets revolve around the Sun due to its spacetime curvature of gravity. Does the same apply to the satellites of the planets?
3
votes
1answer
81 views

Physical meaning of curvature in relativity [closed]

I understand space is not a rigid structure which actually bends (like a metal bar or rubber sheet) so "curvature" due to energy momentum pressure and stress (stress energy tensor) is?? This is were I ...
1
vote
3answers
102 views

Doubt about the vacua equations of General Relativity

I'm facing a quite annoying conceptual problem concerning the Einstein Field Equations (EFE) in so called "vacuum". This problem is both physical and mathematical. So, in a elementary point of view, ...
1
vote
0answers
44 views

Riemann curvature in orthonormal frame and Lorentz transformations

I have problem with understading how Riemann tensor in orthonormal frame transforms using Lorentz transformation of frames. I was reading Morris Thorne paper from 1988 (American Journal of Physics 56, ...
1
vote
1answer
33 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 ...
2
votes
1answer
85 views

Relation between curvature in orthonormal basis and in “standard” metric form

Im familiar with both formulations of GR - standard with metric and connection coefficients and that based on orthonormal frames and differential forms (Cartan's structure eqns) in solving Einstein's ...
1
vote
0answers
59 views

Mass-Density relation in General Relativity

Suppose one has a static and spherically symmetric spacetime with line element defined by: $$ds^{2}=-c^{2}e^{\nu(r)}dt^{2}+e^{-\nu(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),$$ where $\nu(r)...
1
vote
1answer
71 views

Attraction of matter in curved spacetime

Is there still going be a force between them (converging space which makes the two bodies meet together at a point)if both of them are absolute rest with respect any frame of reference.