Tagged Questions
7
votes
1answer
124 views
Uniqueness of Riemann curvature tensor
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 ...
2
votes
1answer
37 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 ...
1
vote
1answer
56 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
votes
1answer
81 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
$$ds^2=R^2(d\phi_1^2 + \sin{\phi_1}^2 ...
0
votes
1answer
47 views
Does the actual curvature of spacetime hold energy?
My understanding of GR is that curvature of spacetime reflects the density of energy-matter. Does the curvature itself have energy? Or if energy is assigned to curvature it simply reflects the energy ...
3
votes
0answers
76 views
Curvature and spacetime
Suppose that it is given that the Riemann curvature tensor in a special kind of spacetime of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a ...
6
votes
3answers
261 views
Why Can We Observe Space Curvature / Warping At All?
I don't understand why we are able to see and measure curvature / warping of space at all.
Space as I understand it determines distances between objects, so if space were "compressed" or warped, ...
1
vote
0answers
131 views
How to calculate Riemann and Ricci tensors for a sphere? [closed]
Let's have the metric for a sphere:
$$
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 ...
0
votes
1answer
97 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?
0
votes
2answers
65 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
98 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
91 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
votes
3answers
144 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
votes
2answers
133 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 ...
0
votes
0answers
164 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 ...
0
votes
0answers
39 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?
1
vote
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
votes
1answer
145 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 ...
5
votes
2answers
205 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
votes
1answer
123 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 ...
1
vote
1answer
202 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
vote
2answers
173 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
$R_{\mu\nu}=g^{\sigma\rho}R_{\sigma\mu\rho\nu}$.
0
votes
1answer
122 views
Homogeneous gravitational field and the geodesic deviation
In General Relativity (GR), we have the geodesic deviation equation (GDE)
...
2
votes
1answer
194 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
votes
2answers
131 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 ...
4
votes
2answers
215 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 ...
7
votes
1answer
165 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
votes
1answer
155 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
votes
1answer
221 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 ...
1
vote
1answer
67 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?
-9
votes
1answer
213 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
vote
1answer
106 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
votes
2answers
203 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?
4
votes
0answers
58 views
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 ...
3
votes
4answers
326 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 ...
6
votes
1answer
556 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 ...
3
votes
2answers
393 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
votes
1answer
345 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 ...
0
votes
2answers
195 views
How can we model intrinsic curvature?
Can it only be done in Euclidean space? Doesn't Euclidean space only model extrinsic curvature?
2
votes
1answer
282 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 ...
1
vote
1answer
441 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 ...
5
votes
1answer
126 views
Source term of the Einstein field equation
My copy of Feynman's "Six Not-So-Easy Pieces" has an interesting introduction by Roger Penrose. In that introduction (copyright 1997 according to the copyright page), Penrose complains that Feynman's ...
3
votes
2answers
126 views
How can I vizualize and understand curved spaces in general relativity?
I'm taking a basic physics class and the teacher described space with a special table that has curves and black holes etc. He would throw a metal ball down onto it and the class would watch it circle ...
6
votes
2answers
360 views
How can a point-like particle “feel” gravity, if locally the curvature of spacetime is always flat?
I imagine a point-like particle can only experience the local properties of spacetime. But locally there is no curvature and no gravity, as it is often stated that
Locally, as expressed in the ...
3
votes
2answers
228 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 ...
