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
132 views

How to prove that zero Weyl tensor predicts no deflection of light?

There is Nordstrom theory, which can be given as $$ C_{\mu \nu \alpha \beta} = 0. $$ The solution of Einstein equations for this case is conformally flat metric: $$ g^{\mu \nu} = e^{\epsilon \varphi ...
3
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2answers
467 views

How do gravitons and curved space time work together? [duplicate]

I've heard two different descriptions of gravity, and I'm wondering how they work together. The first is Gravitons: "The three other known forces of nature are mediated by elementary particles: ...
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1answer
196 views

Riemann curvature tensor symmetries confusion

In the context of spacetime, reading Schutz, I'm confused about the symmetries of the Riemann curvature tensor, which I understand are: ...
1
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1answer
148 views

Are there any good references on the “gravitational” curvature of spacetime of a moving mass being distorted due to special relativity?

In this Wikipedia paragraph suggesting an explanation for the phenomenon of inertia, it claims: Another physicist, Vern Smalley, has derived the Lorentz transformation for mass by assuming that ...
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1answer
109 views

Curvature approaching infinity

I assume that all mass-objects curve time-space but the curvature is only measurable with celestial bodies large enough to be significant gravity-wells. What you call "curvature" seems to me to be ...
4
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1answer
479 views

Lie derivative of Riemann tensor along killing vector ( = 0 )

I'm currently learning the mathematical framework for General Relativity, and I'm trying to prove that the Lie derivative of the Riemann curvature tensor is zero along a killing vector. With the ...
5
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2answers
207 views

Exterior (covariant) derivatives and electromagnetism

I'm porting Maxwell's equations to curved spacetime and am having trouble reconciling the tensor and forms treatments. I think the problem boils down to a misunderstanding on my part concerning the ...
4
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3answers
280 views

In the static spacetime, the extrinsic curvature of hypersurface $t=constant$ is zero

How can I prove that in the static spacetime, the extrinsic curvature of hypersurface $t=constant$ is zero? My efforts all are failed. Any hint would be greatly appreciated.
2
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1answer
282 views

The Weyl tensor and gravitational waves

How exactly is the Weyl tensor is connected with information about gravitational waves? And what are physical reasons for that?
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0answers
417 views

How to prove that Weyl tensor is invariant under conformal transformations?

I need to verify that the solution for vanishing Weyl tensor is conformally flat metric $g_{\mu\nu} = e^{2\varphi}\eta_{\mu\nu}$. The most convenient way to show this is to prove that Weyl tensor is ...
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6answers
685 views

Physical meaning of non-trivial solutions of vacuum Einstein's field equations

According to Einstein, the space-time is curved and the origin of the curvature is the presence of matter i.e. the presence of the energy-momentum tensor $T_{ab}$ in Einstein's field equations. If our ...
3
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0answers
208 views

Gauss-Bonnet term in Physics

Given a 4-dimensional compact manifold (torsion free), the Euler characteristic is defined as: $$E_4 ~=~ \int \epsilon_{abcd}R^{ab} \wedge R^{cd}$$ with $R^{ab}$ is the curvature 2-form. Perturb the ...
2
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1answer
275 views

Ricci scalar in Scalar Field in Curved Space-time

I was recently looking at a Lagrangian of a scalar field in curved space-time at http://www.unc.edu/~mgood/research/Carroll_QFT_CS.pdf on page 8. I am not a physicist, and I am currently studying ...
3
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2answers
227 views

The meaning of potential in Bohm-Aharonov experiment

The Bohm-Aharonov experiment involves a magnetic field inside a cylinder which is zero outside that cylinder. Nonetheless it affects the electrons moving outside the cylinder. The explanation for this ...
3
votes
1answer
115 views

What spacetimes satisfy this identity?

What spacetimes satisfy $R^{\mu\nu} R_{\mu\nu} =\alpha R^2$, where $R = g^{\mu\nu}R_{\mu\nu}$ is the Ricci scalar, and $\alpha$ is some constant? A follow-up question: in what spacetimes does ...
5
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1answer
94 views

Metric of following spacetime and refractive index

Let's have metrics $$ ds^{2} = f(\mathbf r)dt^{2} - h(\mathbf r )\delta_{ij}dx^{i}dx^{j}. $$ Hot to show that motion of light in spacetime with this metrics is equal to motion in continuous media with ...
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0answers
33 views

How much extra distance to a CERN event horizon? [duplicate]

How much extra distance would a scientist have to travel to get to the event horizon of a mini black hole if they ever make one?
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1answer
241 views

How much extra distance to an event horizon?

How much extra distance would I have to travel through space to get from Earth to a stellar mass event horizon? (compared to the same point in space without a black hole)
5
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0answers
118 views

Euclidean black hole extrinsic curvature

I have read that the extrinsic curvature at the horizon of a euclidean black hole is zero? Does anybody know how this can be shown?
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2answers
873 views

What's the idea behind the Riemann curvature tensor?

The Riemann curvature tensor can be expressed using the Christoffel symbols like this: $R^m{}_{jkl} = \partial_k\Gamma^m{}_{lj} - \partial_l\Gamma^m{}_{kj} + \Gamma^m{}_{ki}\Gamma^i{}_{lj} ...
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2answers
263 views

Vanishing of the Ricci tensor in higher spacetime dimensions

I understand how, if the Riemann tensor is 0 in all its components, since we construct the Ricci tensor by contracting the Riemann, Ricci tensor would be 0 in all components as well. I've read that ...
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3answers
339 views

Is it true to say *Space time curvature* $\Leftrightarrow$ *Matter*

Is it true to say Space time curvature and Matter are just the same thing, part of the same coin and that therefore Space time curvature $\Leftrightarrow$ Matter? In other words is Space time ...
3
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1answer
2k views

Riemann tensor in 2d and 3d

Ok so I seem to be missing something here. I know that the number of independent coefficients of the Riemann tensor is $\frac{1}{12} n^2 (n^2-1)$, which means in 2d it's 1 (i.e. Riemann tensor given ...
4
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1answer
130 views

Curvature of spacetime: pincushion distortion?

This may be an elementary question, but if gravity causes a curvature in spacetime, then why isn't everything distorted when looking down on earth, or up at the moon? Shouldn't there be a pincushion ...
2
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2answers
282 views

How are the Weyl & Riemann curvature tensors related to the stress energy tensor in GR?

Einstein's vacuum equations, that is without matter, allows the possibility of curvature without matter. For instance, we may consider gravitational waves. The question is: Is there some link ...
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2answers
66 views

Expand metric $g_{ij}$ about flat space

I expand metric $g_{ij}$ about flat space $\delta_{ij}$ $$g_{ij}=\delta_{ij}+h_{ij}$$ where $|h_{ij}|\ll 1$. I would like to find $R_{ij}$, to linear order, in terms of $h_{ij}$, but I dont know ...
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0answers
44 views

Is the space-time deformation only a way to say how gravity works? [duplicate]

Here we have the classical picture of the deformation of the space-time: https://blogs.stsci.edu/livio/files/2012/06/spacetime.jpg And I would to know if this representation is only a way to say how ...
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2answers
392 views

Is the tide on Earth caused by curvature of spacetime

The tide on Earth appears absolutely whenever the moon is overhead. Is that tide caused by spacetime, re-curvature in space or attraction gravity?
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3answers
243 views

A thought experiment on vision and curved spacetime

What follows is a long self-made example to deal with my conceptual issues of visualizing curved spacetime. Imagine an observer floating somewhere in space. He feels no strain on his body, ...
6
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1answer
152 views

Resultant curvature tensor from the Casimir Effect

I've often seen the Casimir effect cited as a source of negative energy/exotic matter with regards to ideas like the Alcubierre drive. The articles then go on to note that the energy required by the ...
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1answer
230 views

Metric tensor in General Relativity or otherwise [closed]

What is the metric tensor? How can this be a covariant and contravariant tensor, or a mixed tensor, by raising and lowering indices? How it relates to distance function (metric) and angles? How ...
2
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1answer
139 views

How would it be to live in a very small universe, let´s say 20x20 square meters?

Let´s consider a curved universe that is very small, say 20 square meters and not expanding. If you stood at the middle of this tiny universe and looked forward, you wouldn´t see any walls, since it ...
4
votes
1answer
148 views

Components of the Ricci Tensor

Is there any interpretation of what each of the components of the Ricci tensor corresponds to? For example, for the stress-energy tensor, $T_{00}$ corresponds to energy density, $T_{0i}$ is the ...
7
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1answer
334 views

Hodge star operator on curvature?

I've a question regarding the Hodge star operator. I'm completely new to the notion of exterior derivatives and wedge products. I had to teach it to myself over the past couple of days, so I hope my ...
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3answers
1k views

How energy curves spacetime?

We know through General Relativity (GR) that matter curves spacetime (ST) like a "ball curves a trampoline" but then how energy curves spacetime? Is it just like matter curvature of ST?
7
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1answer
319 views

Maxwell's Equations in curved spacetime

I know that we can write Maxwell's equations in the covariant form, and this covariant form can be considered as a generalization of these equations in curved spacetime if we replace ordinary ...
3
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1answer
225 views

Deriving the transformation under Weyl rescaling in Polchinski eq. (1.2.31)

I have another question in Polchinski's string theory book volume 1, namely how to derive Eq. (1.2.32)? $$(-\gamma')^{1/2} R'=(-\gamma)^{1/2} (R-2 \nabla^2 \omega) \tag{1.2.32}$$ I have awared his ...
13
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1answer
458 views

What are the local covariant tensors one can form from the metric?

Normally in differential geometry, we assume that the only way to produce a tensorial quantity by differentiation is to (1) start with a tensor, and then (2) apply a covariant derivative (not a plain ...
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2answers
260 views

Triple-right triangle experiment: what's the minimum distance?

Among the other ways, one way to prove the Earth is round is the triple-right triangle. The idea is simple: Starting from point A you move in a straight line for a certain distance. At point B, ...
2
votes
1answer
145 views

How can I express the Riemann tensor of the 4-metric in terms of quantities derived from the 3-metric and the normal to it?

I want an expression for the Riemann tensor of the four metric in terms of extrinsic curvature, normal, lie derivative of the normal, etc. The first Einstein-Codacci eq. gives the Riemann tensor of ...
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1answer
223 views

Curvature tensor of 2-sphere using exterior differential forms (tetrads)

$ds^2= r^2 (d\theta^2 + \sin^2{\theta}d\phi^2)$ The following is the tetrad basis $e^{\theta}=r d{\theta} \,\,\,\,\,\,\,\,\,\, e^{\phi}=r \sin{\theta} d{\phi}$ Hence, $de^{\theta}=0 ...
3
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1answer
594 views

Problem with calculating the curvature tensor of the $n$ dimensional sphere

I am calculating the Riemann curvature tensor, Ricci curvature tensor, and Ricci scalar of the $n$ sphere $$x_0^2 + x_1^2 + ....+x_n^2=R^2,$$ whose metric is $$ds^2=R^2(d\phi_1^2 + \sin{\phi_1}^2 ...
3
votes
2answers
121 views

5D Ricci Curvature

As part of a hw problem for a class, we're supposed to be deriving the equivalence given in equation 2.3 of this paper http://arxiv.org/abs/1107.5563. I was wondering if there is some special ...
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1answer
207 views

Does the actual curvature of spacetime hold energy?

My understanding of GR is that curvature of spacetime reflects the density of energy-matter. Does the curvature itself have energy? Or if energy is assigned to curvature it simply reflects the energy ...
3
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0answers
125 views

Curvature and spacetime

Suppose that it is given that the Riemann curvature tensor in a special kind of spacetime of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a ...
6
votes
3answers
631 views

Why Can We Observe Space Curvature / Warping At All?

I don't understand why we are able to see and measure curvature / warping of space at all. Space as I understand it determines distances between objects, so if space were "compressed" or warped, ...
2
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1answer
231 views

Parallel transport of a vector along a closed curve in curvilinear coordinates

There is an expression indicating the change of the vector parallel translation along a closed infinitesimal curve in curvilinear coordinates (one way of introducing curvature tensor): $$ \Delta A_{k} ...
2
votes
2answers
451 views

Ricci tensor for a 3-sphere without Math packets

Let's have the metric for a 3-sphere: $$ dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right). $$ I tried to calculate Riemann or Ricci tensor's ...
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0answers
562 views

How to calculate Riemann and Ricci tensors for a sphere? [closed]

Let's have the metric for a sphere: $$ dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right). $$ I tried to calculate Riemann or Ricci tensor's ...
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0answers
51 views

How to prove the derive the expression for space part of Riemann tensor for homogeneous and isotropic space-time?

It's not a homework!! For spheric, hyperbolic and flat case $$ dl^{2} = R^{2}\left(d \psi^{2} + sin^{2}(\psi )(d \theta^{2} + sin^{2}(\theta )d \varphi^{2})\right), $$ $$ dl^{2} = R^{2}\left(d ...