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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Can a curvature in time (and not space) cause acceleration?

I realize that the curvature of space-time causes acceleration (gravity). Is it possible to have a curvature only of space, or a curvature only of time? If so, would a curvature only of space, or a ...
3
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2answers
260 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 ...
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3answers
365 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 ...
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0answers
64 views

Feynman Path integrals in space with holes in it [closed]

Feynman Path Integrals are a way of calculating the wave function of quantum mechanics. It usually integrates every possible path through all of space. I wonder if there is any study of Feynman path ...
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0answers
70 views

How to derive the cigar soliton solution to the Ricci flow equation? [closed]

I am trying to derive the cigar soliton solution to the Ricci flow equation. Such solution has the form $$ {\frac {{{\it dx}}^{2}+{{\it dy}}^{2}}{{{\rm e}^{4\,t}}+{x}^{2}+{y}^{2 }}} $$ I am ...
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93 views

What Would Negative Mass Do To Spacetime?

It's known that positive mass bends space-time to create a curvature. But if something had negative mass what would it do? Make it flat or like a crest?
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50 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
147 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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315 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}) ...
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334 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 ...
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0answers
145 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 ...
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2answers
107 views

Curvature gravity and a falling apple? [duplicate]

I know very little of physics after Einstein. I am aware of that Einstein's gravity theory says that the existence of matters creates curvature of a space-time, so that our Earth orbits our Sun. I ...
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3answers
354 views

If gravitation is due to space-time curvature, how can a body free-fall in a straight line?

According to general relativity, Gravity is due to space-time curvature. Then all paths must be curved. If so, how can there be any straight line motion? The body must follow a curved path. So, there ...
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1answer
222 views

Two definitions of Riemann curvature tensor

I am relatively used to the coordinate free expression of the Riemann tensor: $$ R(X, Y)Z=\nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z - \nabla_{[X, Y]} Z, $$ where $\nabla$ is the Levi-Civita connection ...
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1answer
128 views

Riemann Curvature Tensor Symmetries Proof

I am trying to expand $$\varepsilon^{{abcd}} R_{{abcd}}$$ by using four identities of the Riemann curvature tensor: Symmetry $$R_{{abcd}} = R_{{cdab}}$$ Antisymmetry first pair of indicies ...
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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?
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1answer
1k 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 ...
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3answers
2k views

Ricci scalar for a diagonal metric tensor

I was wondering if there is a general formula for calculating Ricci scalar for any diagonal $n\times n$ metric tensor?
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4answers
104 views

Any tips on evaluating Riemann tensor?

I am calculating the Riemann tensor for the Schwarzschild solution. I've calculated all 9 non-vanishing Christoffel symbols already. Now I need to evaluate the Riemann tensor and I find no easy way to ...
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1answer
200 views

If space warps distort moving objects' trajectories, does it mean that static objects are immune to gravity? [closed]

If gravity is just space distortion, which affects trajectories of moving objects, then a static object (not moving, thus no trajectory) will not suffer any type of accelerating force from gravity? ...
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1answer
76 views

Curvature of a particle move

I'm simulating a particle movement following a normal distribution. How this is done: My particle has a constant speed v and every step the particle move, I ...
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7answers
475 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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2answers
212 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
85 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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2answers
381 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
244 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 ...
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1answer
141 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 ...
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2answers
130 views

Curved space-time VS change of coordinates in Minkowski space

I'm looking for a rather intuitive explanation (or some references) of the difference between the metric of a curved space-time and the metric of non-inertial frames. Consider an inertial reference ...
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2answers
92 views

How is the Ricci scalar $R=0$ here?

Given the metric in the form: $$ds^2 =-A(r)dt^2 +B(r) dr^2 dr^2 +r^2(d\theta ^2 +\sin^2\theta d\phi^2)$$ Papapetrou in his book said that $R=0$ But when I performed it I didn't get zero. For ...
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4answers
345 views

Where does the idea gravity=curvature of spacetime really come from?

I have been searching for quite a while but mostly found the answer: Einstein's genius. Quite unsatisfactory. I know and understand that the idea gravity=curvature of spacetime works. Furthermore I ...
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2answers
565 views

What is the meaning of space-time curvature?

What is the difference between the Space-time curvature and Space curvature?
2
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1answer
78 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 ...
2
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1answer
394 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
97 views

Can energy bend space? [duplicate]

I know mass bends the space around it and I also remember matter can be converted into energy and vice versa, so my question is: can energy interact with space in a similar fashion as matter does?
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1answer
77 views

Can a lone black hole in a closed Universe evaporate?

If there is a closed Universe which only has a black hole in it, can that black hole evaporate? As the black hole evaporates, it gives off energy, which will eventually come back and be re-absorbed ...
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1answer
146 views

How to understand the curvature of this metric?

Suppose we have the metric $ ds^2 = dr^2 + \alpha^2 d\phi^2$, where $\alpha$ is a constant, $0 \leq r \leq \infty$, $ 0 \leq \phi \leq 2 \pi$ and we identify points $\phi = 0$ with points $\phi = ...
2
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1answer
646 views

proper distance and proper length

I am wondering if I mix up the notion of proper distance and proper length. I have two cuves in Schwarzschild space-time describing the flight of two photons (think of it as photons guided in by ...
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1answer
153 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
259 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} ...
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2answers
1k views

Is the curvature of space around mass independent of gravity?

Is the curvature of space caused by the local density of the energy in that area?Could gravity be a separate phenomenon only arising from the curvature of space? For instance if the density of energy ...
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1answer
94 views

How can I use Einstein's field equations to find the metric tensor? [duplicate]

I have watched and read a lot on the topic of General Relativity and the geometry behind it. I am confident that I can derive an approximation of the the stress-energy-momentum tensor with just the ...
2
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1answer
148 views

Is my diagram of spacetime curvature valid (relatively)?

I've been wracking my brain trying to understand what "curved spacetime" really is, and I think replacing one dimension with the time dimension then drawing the world-lines through time was the "aha!" ...
2
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1answer
83 views

Fastest way to find the curvature terms from a given metric [closed]

I want to find the spherically symmetric, static solutions to Einstein's equations $$ R_{\mu \nu} - \frac{1}{2}Rg_{\mu \nu} = 0 $$ in four dimensions using the metric $$ g_{\mu \nu}dx^{\mu}dx^{\nu} ...
2
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1answer
133 views

Size of the Universe: Curved vs flat? Finite vs infinite?

I have recently heard the theory that the Universe may be smaller than observed but may be curved to the extent that light rays may have looped past us once already and hence appear to have originated ...
2
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1answer
120 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
178 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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2answers
535 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
275 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
188 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
265 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 ...