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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77 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 ...
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1answer
63 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 ...
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66 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, ...
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1answer
78 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? ...
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4answers
89 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 ...
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2answers
170 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 ...
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1answer
141 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 ...
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43 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 ...
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4answers
295 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? ...
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1answer
120 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 ...
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1answer
72 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 ...
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1answer
157 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 ...
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1answer
144 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 ...
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126 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 ...
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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}{...
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56 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....
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1answer
122 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 ...
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2answers
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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?
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217 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 ...
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0answers
50 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 ...
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1answer
82 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 ...
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1answer
129 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 ...
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45 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 ...
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2answers
164 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 ...
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2answers
286 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, $$T_{\mu\nu} = (\rho + P)u_{\mu}u_{\nu} - Pg_{\mu\nu}.$$ ...
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1answer
83 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 ...
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1answer
389 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. ...
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1answer
137 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?
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1answer
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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 ...
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3answers
134 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, ...
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0answers
88 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, ...
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1answer
41 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 ...
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1answer
122 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 ...
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0answers
89 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)...
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2answers
125 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.
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3answers
105 views

According to general relativity, why are two objects at rest attracted to each other? [duplicate]

I'm trying to understand gravity in General Relativity and I'm having some questions. I can understand that an object in orbit around another more massive object is free falling and simply following a ...
148
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25answers
21k views

Simple check for the global shape of the Earth

I have been on a date recently, and everything went fine until the moment the girl has told me that the Earth is flat. After realizing she was not trolling me, and trying to provide her with a couple ...
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1answer
61 views

Can we gauge away curvature in supergravity?

In locally supersymmetric theories we can make a supertransformation which rotates one field into another. With enough supersymmetries could one not make a supertransformation at each point which made ...
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1answer
341 views

Interpretation of Ricci rotation coefficients in tetrad formalism

Given an orthonormal frame (tetrad in 4 dimensions, vielbein more generally) $\{(e_{\mu})^{a}\}$ with $g(e_{\mu}, e_{\nu}) = \eta_{\mu\nu}$, the Ricci rotation coefficients are defined as $$\omega_{\...
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1answer
66 views

Does bending the direction of light generate space curvature?

When light passes a mass, the light path is bent due to the curvature of space caused by the object mass. Is there an inverse effect of bending a light path to cause an effect of generating a level of ...
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0answers
63 views

Contracted Bianchi Identity for FLRW Metric

I am trying to verify that the Contracted Bianchi Identity $\nabla_\mu G^{\mu\nu}=0$ holds for the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric, $$g_{00}=-1,g_{i0}=0,g_{ij}=a^2(t)\delta_{ij}$$ ...
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1answer
189 views

How would General Relativity be different if we assumed Galilean instead of Lorentz transformations?

If we assume a universe where Galilean transformations are the correct transformations between inertial reference frames, would GR be any different ? Gravitational and inertial mass would still be ...
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1answer
96 views

Can a curvature singularity (i.e. BH), as defined in terms of geodesic incompleteness, actually exist in nature?

A curvature invariant is a scalar representation of curvature derived from a curvature tensor. The classic example is the Kretschmann scalar derived from the Riemann curvature, where $K=R_{μνλρ}R^{...
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1answer
235 views

Why do we say Minkowski space is flat when hyperbolic geometry has negative Gaussian curvature?

I often hear that spacetime corresponds to a hyperbolic geometry. The distance metric is not Euclidean, it is hyperbolic, and Lorentz transformations correspond to hyperbolic rotations. Hyperbolic ...
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2answers
140 views

Why is the right hand side of the Equation of Geodesic Deviation 0?

I'm reading through several primers on GR and I keep seeing the same thing over and over again with no explanation: why is the right hand side of this equation zero: $$\frac {D^2\xi^a}{Du^2}+R^{\alpha}...
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2answers
61 views

Perception of curvature of human eye [closed]

How are human eyes able to detect the different curvatures of surface ? Basically how are the human eyes able to differentiate between a plane surface and a convex or a concave surface?
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1answer
85 views

Are the actual singularities with geodesic incompleteness the same as curvature singularities?

Types of singularities include curvature singularities and conical singularities. So, for a curvature singularity(black hole) with geodesic incompleteness, is it the same as a physical singularity? If ...
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1answer
84 views

General Theory of Relativity [duplicate]

Why is General Theory of Relativity not applicable in the singularity of Black Hole?
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2answers
154 views

Distortion of space and time

What will happen to the distorted space and time around a mass when it is converted into energy? Will it back to its original configuration (0 gravity)? Or space time oscillates? Or anything else
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3answers
168 views

Is gravity a force given that it derives from curved massless space-time?

One of the answers to a similar question regarding gravity concluded that gravity is an "observed effect" of the curvature of space-time. I read this (and other answers) to imply that gravity results ...