The curvature tag has no wiki summary.
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1answer
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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
71 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
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3answers
208 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, ...
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1answer
46 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):
$$
\Delta A_{k} ...
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2answers
91 views
Ricci tensor for a 3-sphere without Math packets
Let's have the metric for a 3-sphere:
$$
dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right).
$$
I tried to calculate Riemann or Ricci tensor's ...
1
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0answers
103 views
How to calculate Riemann and Ricci tensors for a sphere? [closed]
Let's have the metric for a sphere:
$$
dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right).
$$
I tried to calculate Riemann or Ricci tensor's ...
1
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0answers
25 views
How to prove the derive the expression for space part of Riemann tensor for homogeneous and isotropic space-time?
It's not a homework!!
For spheric, hyperbolic and flat case
$$
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 ...
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3answers
113 views
How scalar curvature of following spacetime can be equal to zero?
For an interval of this spacetime,
$$
ds^{2} = c^{2}dt^{2} - c^{2}t^{2}(d \psi^{2} + sh^{2}(\psi )(d \theta^{2} + sin^{2}(\theta )d \varphi^{2})),
$$
scalar curvature is equal to zero. Also, Ricci ...
-2
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0answers
84 views
How to calculate a scalar curvature fast? [closed]
Let's have a metric tensor
$$ g^{\alpha \beta} = \frac{1}{\left( 1 + \frac{ct}{R} \right)^{2}}\begin{bmatrix} \frac{1 - \frac{r^{2}}{R^{2}}}{\left(1 + \frac{ct}{R}\right)^{2}} & ...
0
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1answer
90 views
What is the curvature of an empty universe?
My calculations tell me an empty universe has hyperbolic curvature. Is this correct? If it is, can anyone help me understand why this is intuitively?
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2answers
56 views
Changing the scalar curvature (k = 0,+1,-1) with coordinate transformations?
I would like to prove that I can (or can't) change curvature of space, k = 0,+1,-1, via general coordinate transformations, which in principle can mix space and time coordinates together.
1
vote
1answer
81 views
Ricci scalars for space and spacetime, local and global curvature
If Ricci scalar describes the full spacetime curvature, then what do we mean by $k=0,+1,-1$ being flat, positive and negative curved space?
Is $k$ special version of a constant "3d-Ricci" scalar?
...
0
votes
1answer
83 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 ...
4
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3answers
138 views
How do you tell if a metric is curved?
I was reading up on the Kerr metric (from Sean Carroll's book) and something that he said confused me.
To start with, the Kerr metric is pretty messy, but importantly, it contains two constants - ...
0
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2answers
123 views
If the universe is 3D, how is space-time like a “fabric”? [duplicate]
I have been taught that space-time should be viewed as a fabric and that objects with a large gravitational influence indent that fabric. My question is, if the singularity of a black-hole punctures ...
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0answers
158 views
Spacetime around a Black Hole
If we consider the sun, then space-time is curve around it. My question is that what is the kind of curvature of space and time around the black hole. Is that space and time more curved around the ...
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0answers
37 views
Curved space to flat space calculation
When changing the curved space co-ordinate into a flat space co-ordinate if a cone. I got the result transformation that i cannot get a transformation at the vertex(apex) why?
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0answers
31 views
In which direction does space “bend”? [duplicate]
Gravitation is often depicted as a ball on a cloth that curves a hole into space. But in what direction does this hole form? Into the direction the object is moving?
4
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1answer
141 views
Does the curvature of space-time cause objects to look smaller than they really are?
What's the difference between looking at a star from a black hole and looking at it from empty space?
My guess is that the curvature of space-time distorts the wavelength of light thus changing the ...
3
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2answers
100 views
How/why can the cosmic background radiation measurements tell us anything about the curvature of the universe?
So I've read the Wikipedia articles on WMAP and CMB in an attempt to try to understand how scientists are able to deduce the curvature of the universe from the measurements of the CMB.
The Wiki ...
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2answers
200 views
Is the curvature of spacetime invariant? Could it be characterized as the ether?
I'm writing a paper for a Philosophy of Science course about GR/SR and I'm wondering if I can (1) characterize the curvature of spacetime as invariant and (2) argue that this is what Einstein referred ...
3
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1answer
117 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 ...
0
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1answer
31 views
What is the Willmore energy of the Earth (or the geoid)?
Wikipedia defines the Willmore energy as:
$$e[{\mathcal{M}}]=\frac{1}{2} \int_{\mathcal{M}} H^2\, \mathrm{d}A,$$
where $H$ stands for the mean curvature of the manifold $\mathcal{M}$.
What is the ...
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1answer
190 views
In what way is the Riemann curvature tensor related to 'radius of curvature'?
In Misner, Thorne & Wheeler, they say, in their delightful 'word equations' that
$$\left(\frac{\mathrm{radius\,\, of \,\,curvature}}{\mathrm{of\,\, spacetime}}\right) = ...
1
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2answers
159 views
What is the Riemann curvature tensor contracted with the metric tensor?
Can the Ricci curvature tensor be obtained by a 'double contraction' of the Riemann curvature tensor? For example
$R_{\mu\nu}=g^{\sigma\rho}R_{\sigma\mu\rho\nu}$.
2
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2answers
114 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...)
0
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1answer
116 views
Homogeneous gravitational field and the geodesic deviation
In General Relativity (GR), we have the geodesic deviation equation (GDE)
...
2
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1answer
176 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
123 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
votes
3answers
156 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
87 views
Is there any Calculator capable of calculating and displaying differential geometry? [closed]
Is there any Calculator capable of calculating and displaying differential geometry (display curvature of spacetime)?
$$ds^2~=~g_{ab}dx^adx^b.$$
4
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2answers
208 views
Space-time geometry and metric
I am confused in one question in general relativity, why we can always express a space-time geometry only by metric. It means a metric, which is just about distance in tangent space, can tell us all ...
2
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1answer
78 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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1answer
155 views
Is spacetime flat inside a spherical shell?
In a perfectly symmetrical spherical hollow shell, there is a null net gravitational force according to Newton, since in his theory the force is exactly inversely proportional to the square of the ...
2
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1answer
148 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 ...
2
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1answer
205 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 ...
2
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2answers
258 views
Equation of the saddle-like surface with constant negative curvature?
What is the equation for the saddle-like 2d surface (embeded in 3d Euclidean space with cartesian coordinates x, y and z) with constant negative curvature frequently used to illustrate open universe ...
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1answer
66 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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2answers
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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 ...
5
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1answer
157 views
Curvature and edge state
If the boundary of quantum hall fluid has non-constant curvature, how will it affect the edge state which is usually described in chiral Luttinger fluid?
-9
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1answer
201 views
Why are we talking about space curvature as if we know what space is? [closed]
1) Why are we talking about space curvature as if we know what space is?
Every question about gravity seems to evoke an answer involving "space curvature" which seems like an undefined placeholder ...
1
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1answer
104 views
What is the curvature scalar $\Psi_{4}$?
What is the curvature scalar $\Psi_{4}$?
Is it related to the scalar curvature $R$?
What does its real and imaginary parts represent?
4
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2answers
187 views
What is the variation of Gauss-Bonnet term a total derivative of?
What is the variation of Gauss-Bonnet term total derivative of?
i.e. Variation of Gauss-Bonnet combination $= \nabla_{\mu} C^{\mu}$.
What's $C^{\mu}$ in 4-dimensions?
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0answers
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gravitational convergence of light
light has a non-zero energy-stress tensor, so a flux of radiation will slightly affect curvature of spacetime
Question: assume a flux of radiation in the $z$ direction, in flat Minkowski space it ...
0
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0answers
55 views
Space-time & solar mass
Does the space-time curvature described by Einstein have any affect on the accuracy of our determination in the age of a star or globular cluster? How does this affect our interpretion of how old we ...
3
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4answers
317 views
Gravitation is not force?
Einstein said that gravity can be looked at as curvature in space- time and not as a force that is acting between bodies. (Actually what Einstein said was that gravity was curvature in space-time and ...
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0answers
277 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 ...
6
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1answer
531 views
What is the stress energy tensor?
I'm trying to understand the Einstein Field equation equipped only with training in Riemannian geometry. My question is very simple although I cant extract the answer from the wikipedia page:
Is the ...
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2answers
372 views
Where do I start with Non-Euclidean Geometry?
I've been trying to grok General Relativity for a while now, and I've been having some trouble. Many physics textbooks gloss over the subject with an "it's too advanced for this medium", and many ...
2
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1answer
332 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 ...
