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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2answers
260 views

asymptotic curvature of the universe and correlation with local curvature

There is not-so-rough evidence that at very large scale the universe is flat. However we see everywhere that there are local lumps of matter with positive curvature. So i have several questions ...
3
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1answer
110 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 ...
3
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1answer
496 views

What bends fabric of space-time?

I know that mass can bend fabric of space-time, which causes gravity by making an object curve around a planet or star but is there anything else that can bend it? Other energy sources, forces ...
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1answer
160 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 ...
3
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1answer
452 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
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1answer
246 views

Curvature, Omega, the Flatness problem, and the evolving shape of the universe

I'm a little confused by this: http://en.wikipedia.org/wiki/Flatness_problem Which seems to imply the universe is more curved now than it was soon after the Big Bang. Look at the graph on the right ...
3
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1answer
208 views

Material strain from spacetime curvature

Let's say that you moved an object made of rigid materials into a place with extreme tidal forces. Materials have a modulus of elasticity and a yield strength. Does the corresponding 3D geometric ...
3
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1answer
344 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 ...
3
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2answers
193 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
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3answers
286 views

Why geometrically four acceleration is a curvature vector of a world line? And what is proper acceleration?

Why geometrically four acceleration is a curvature vector of a world line? Geometrically, four-acceleration is a curvature vector of a world line. Therefore, the magnitude of the ...
3
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1answer
236 views

A question about surface tension of membranes and their curvature

I'm reading a review about membranes properties and I have reach a section about fluid membranes. The section discuss the principal curvatures ($c_1, c_2$) and the spontaneous curvatures ($c_0$). ...
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0answers
40 views

Rigid rectangle in Schwarzschild

Say I build a perfect rectangle. Side lengths $l_1$ and $l_2$ and perfect right angles. I am on earth and the metric is given by the Schwarzschild metric. Setting $dt=0$ leads to the spatial ...
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0answers
59 views

Computing the Einstein tensor for a spherically symmetrical metric using the tetrad formalism

I am having some trouble understanding how to use the tetrad formalism. I will start with what I have so far, my question will be after that. I begin with the metric $$ \text{d}s^2 = e^{2a} \text{ ...
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0answers
103 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 ...
3
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0answers
109 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 ...
2
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1answer
307 views

Difference between $\partial$ and $\nabla$ in general relativity

I read a lot in Road to Reality, so I think I might use some general relativity terms where I should only special ones. In our lectures we just had $\partial_\mu$ which would have the plain partial ...
2
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2answers
191 views

What is the curvature of the universe?

What is currently the most plausible model of the universe regarding curvature, positive, negative or flat? (I'm sorry if the answer is already out there, but I just can't seem to find it...)
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1answer
202 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 ...
2
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2answers
182 views

Curved space or curved spacetime?

As I understand it, you can have time + flat space = curved spacetime. So, when one is trying to emphasise that there is a curvature to the space, is it more technically correct to say curved space ...
2
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1answer
120 views

Flat poster on a wall gaining curvature over time

Assuming you have a flat poster with no curvature, why is it that when you pin it to the wall (with thumbtacks) it gains curvature as seen in the picture below. When I put the poster up it was ...
2
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6answers
251 views

Curvature of Spacetime

I have been exploring for some time both the Special and General Relativity, hoping to glean at least a conceptual grasp of their basic tenets. In reading the book "Gravitation" by Misner, Thorne and ...
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1answer
56 views

Riemann normal chart and special relativity

When you pick Riemann normal coordinates at a point, then the Christoffel symbols vanish and the metric is flat, but the Riemann curvature tensor does not necessarily vanish. Since Einstein said that ...
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1answer
150 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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1answer
769 views

$\pi$ and the Curvature of Space

If one draws a circle on a sphere and measures the ratio of the diameter to the circumference, that value varies depending on the diameter of the circle compared to the diameter of the sphere it is ...
2
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2answers
106 views

Frames, Tetrads and GR

Given a general metric, $g_{ab}$ I can select an orthonormal basis $\omega^{a}$ such that, $$g_{ab} = \eta_{ab}\omega^a \otimes \omega^b$$ where $\eta_{ab}$ = $\mathrm{diag}(1,-1,-1,-1).$ We may ...
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1answer
105 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 ...
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1answer
771 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 ...
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1answer
208 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
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1answer
56 views

Finding the components of the Riemannian tensor given the components of a metric

I am looking at a manifold of dimension $n$ (And I am considering a local co-ordinates system $x_1,x_2,\ldots x_n$) and the metric defined by the components $g_{ij} = \frac{\delta_{ij}}{x_1^2}$. I'm ...
2
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1answer
109 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 ...
2
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2answers
332 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 ...
2
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1answer
207 views

What's the difference between the equivalence principle and curvature of spacetime?

Calculating using the equivalence principle only accounts for half the deflection of light, whereas the other half is from curvature of space-time. But isn't the equivalence principle the same thing ...
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2answers
173 views

Galilean transformations and Frenet Frame

How I can prove that the curvature and torsion of a curve are invariant under the Galilean transformations? In my physics book a hint is the isometries of Galilean transformations, but it's still ...
2
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1answer
210 views

Curved lines in a picture (Photography)

My problem is when I take a picture (a close one) the straight edge looks a little curved. In a standard camera, like a CyberShot. I would like to know if there is some relationship between the ...
2
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1answer
49 views

Detail of deriving Berry Curvature From Berry Connection

The Berry curvature of the $n^{\mathrm{th}}$ eigenstate of Hamiltonian $H$ for the vector of external parameters $\vec{R}$ can be derived in part by writing the following two lines: $B^n(\vec{R}) ...
2
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1answer
195 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 ...
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0answers
430 views

de Sitter and anti de Sitter metric

Is the following correct for the distance $d$ from the origin $(0,0)$ to point $(t,x)$ in the 2-dimensional de-Sitter and anti de-Sitter spaces? Here, $t$ is time and the distance may be called the ...
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1answer
298 views

Superposition of Ricci scalars [closed]

Suppose I have two point/line singularities in spacetime (what is important to me is that they are localized). Also suppose I have some fields in spacetime and that the two singularities interact with ...
2
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1answer
127 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 ...
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3answers
274 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 ...
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1answer
153 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)
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2answers
178 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
464 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?
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1answer
92 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?
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1answer
82 views

The source of gravitation in a spacetime without matter

In a discussion concerning: Physical meaning of non-trivial solutions of vacuum Einstein's field equations there were a number of answers claiming that the flatness of the Ricci space (Rµv=0) ...
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2answers
44 views

Killing vector contractions along isometric curves

Imagine $\xi_{\nu}$ is a Killing vector field on a manifold. Does $\xi_{\nu}\xi^{\nu}$ remain constant along any isometric curve defined by the Killing vector field? My guess is that yes since as ...
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1answer
109 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: ...
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1answer
186 views

Weyl & Riemann curvature tensors and gravitational “physical” quantities in Einstein vacuum equations

If we look at the Einstein vacuum equations, that is without matter (there is the possibility or curvature without matter), for instance we may consider gravitational waves. The question is: Is there ...
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1answer
160 views

Space time curvature real or theoretical (mathematical)?

Assuming one were in a capsule of some kind, with no window or instruments, and you swung into the gravitational field of a massive object (planet). Assuming no atmosphere to provide friction, could ...
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1answer
251 views

In what way is the Riemann curvature tensor related to 'radius of curvature'?

In Misner, Thorne & Wheeler, they say, in their delightful 'word equations' that $$\left(\frac{\mathrm{radius\,\, of \,\,curvature}}{\mathrm{of\,\, spacetime}}\right) = ...