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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If massive gravitons fill 'empty' space and are displaced by matter then is curved spacetime the state of displacement of the gravitons? [on hold]

APS Physical Review D: Massive gravitons as dark matter and gravitational waves Precursor: Bigravitons as dark matter and gravitational waves The massive graviton is a viable candidate of dark ...
6
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2answers
334 views

Interpreting the Kretschmann scalar

How do you interpret the Kretschmann scalar (in general relativity)? What can you tell from it? The Kretschmann scalar is defined as $$K = R_{abcd} R^{abcd} $$ where $R_{abcd}$ is the Riemann ...
1
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0answers
47 views

What is the difference between intrinsic and extrinsic curvature? [migrated]

In general relativity, energy bends spacetime. However, this doesn't mean that a fifth dimension for spacetime to "bend into" exists." That is, spacetime isn't embedded in a higher dimensional space, ...
12
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4answers
597 views

Curvature of Hilbert space

That may appear as a dumb question, but: Does Hilbert space have curvature, or is it a flat space? How and why?
19
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6answers
2k views

Does curved spacetime change the volume of the space?

Mass (which can here be considered equivalent to energy) curves spacetime, so a body with mass makes the spacetime around it curved. But we live in 3 spatial dimensions, so this curving could only be ...
2
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1answer
107 views

Mathematical expression of energy storage

I'm trying to develop an idea which is as follows. Put simply, imagine a flat sheet of material which, when distorted (I.e. curved in the third dimension) stores energy. Now, by calculating the ...
0
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1answer
29 views

What is the fate of a hyper-sphere universe?

This PBS video postulates a positively curved universe which is in the shape of something the narrator calls a "hypersphere". http://youtu.be/AwwIFcdUFrE?t=6m33s Such a universe would have no edge, ...
3
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0answers
85 views

What is the geometric explanation for why the interior angles of a triangle sum to 180 degrees in both Euclidean space and Minkowski spacetime?

Four-dimensional Euclidean space has the same topology and affine structure as Minkowski spacetime, though the two have different metric structures. Given that the interior angles of a triangle ...
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2answers
81 views

General relativity applications other than gravity

Do the Einstein field equations successfully predict/describe physical processes other than gravitational ones?
3
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5answers
562 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 ...
3
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1answer
150 views

Does mass compress space-time?

My understanding of relativity explains that the presence of mass warps space-time so that light travelling through the warp follows at straight line but the warp itself is curved and therefore the ...
4
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0answers
46 views

Is the Weitzenböck connection the only connection with Torsion but without Curvature?

In teleparallel gravity, the (local) connection coefficients of the Weitzenböck connection are given by $$ \Pi^{\beta}{}_{\mu\nu}= h^{\beta}_{i} \partial_{\nu}h^{i}_{\mu} - \Gamma^{\beta}{}_{\mu\nu}...
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1answer
84 views

How is Riemann tensor related to the curvature in the coordinates?

I came across statements such as "the acceleration observed in a weak gravitational field is mainly due to curvature in the time coordinate. " I want to know how we can explicitly find the curvature ...
21
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2answers
2k 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 ...
3
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2answers
4k views

Radius of curvature and focal length

Is the radius of curvature of a convex or concave lens longer than the focal length of the lens? Does the center or curvature affect the focal point in a lens?
3
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2answers
112 views

Proving constant curvature

I'm currently on section 5.1 in Wald's book. He is trying to prove that the cosmological principle implies that space has constant curvature. Given a spacelike hypersurface $\Sigma_t$ for some fixed ...
0
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0answers
30 views

How do we measure curvature of space? [duplicate]

I know according to general relativity, space-time is curved near mass. But I have also read that at large distances space can be curved too, and that Gauss was the first one to measure it. My ...
0
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3answers
922 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?
0
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2answers
108 views

How does gravity truly work in the bend of spacetime?

If gravity is caused from the bend in space time from a large mass, why do all objects fall towards earths center and not strait down to below earth? Sorry i am not an expert in any fields just trying ...
2
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3answers
3k 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?
2
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2answers
241 views

Different signatures

I was working out the christoffel symbols, once where the metric that I am using has (+---) signature and another time where it has (-+++) signature because two books had different signatures and I ...
8
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8answers
2k 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 ...
2
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1answer
122 views

Are all maximally symmetric spacetimes constant curvature spacetimes?

A $d$ dimensional maximally symmetric spacetime is a spacetime with the maximum allowed number of Killing vectors. This number is $\frac{d(d+1)}{2}$. Constant curvature spacetimes are spacetimes ...
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1answer
38 views

Second covariant derivative, computation problem

I am having a question on the wikipedia article http://en.wikipedia.org/wiki/Second_covariant_derivative Using the notation therein I don't get why $(\nabla_{u}\nabla_{v}w )^a=u^c\nabla_{c}v^b\nabla_{...
3
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0answers
86 views

Scalar Curvature of a Conformally Flat Metric

Suppose that you have a metric $g_{\mu\nu}=\phi^2\eta_{\mu\nu}$ for some function $\phi$. There is a standard formula for what the scalar curvature $R$ looks like in terms of $\phi$, which is given by ...
5
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1answer
260 views

Killing tensor and Riemann tensor identity

I know that if we have a Killing vector then it's straightforward to show the identity: $$\nabla_a \nabla_b K_c = R_{cba}^k K_d$$ I'm now trying to show the following identity for a $(0,2)$ Killing ...
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0answers
9 views

Coordinate Equation of a curve which bends all the parallel incoming rays from infinity towards a single point

How should i proceed on to find the coordinate equation of a curve such that it bends all the parallel rays coming from infinity towards a single point. Yes I know that it would be a 2nd degree ...
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2answers
103 views

Flat space Solution of Einstein Field Equation

Does a trace-free energy-momentum tensor $T_{\mu}^{\mu} = 0$ ensure that the Einstein's field equations have a flat space solution?
4
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1answer
61 views

How can you tell if spherical-like coordinates are locally flat across the origin?

In general relativity, with spherical-like coordinates in a radial gauge, I have a metric that looks like: $$-g_{tt}\mathrm{d}t^2 + g_{rr}\mathrm{d}r^2 + r^2(\mathrm{d}\theta^2 + \sin^2\theta\ \...
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1answer
60 views

How energy would be consumed for bending spacetime?

If we could assume that relativity theory is correct about spacetime bending. Can we calculate energy used for moving 1 kg of object in 1 meter by changing the shape of spacetime (simulate gravity)? ...
0
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0answers
54 views

Why Newton's gravitational constant remains unchanged in relativity though gravity is not a force?

I know that Einstein described gravity as a curvature of spacetime. So, It is not a "force" but why Einstein had to accept Newton's gravitational force constant?
0
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1answer
33 views

About relativity [duplicate]

We know that the curvature of spacetime is gravity itself and it is not a force.so,why do we feel our weight in a curve spacetime but not in a straight(I mean not curve) space time like zero gravity ...
0
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1answer
63 views

Why does two masses with energy attract each other?

I have heard that every mass attract another mass with a force directly proportional to the multiplication of their masses and inversely proportional to the square of distance between them, but newton ...
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2answers
73 views

Gravity and circular logic [closed]

My research into gravity indicates that warped spacetime, with time as the major influence, is gravity. It also indicates gravity causes time dilation. Why is this not a circular argument. It is ...
8
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3answers
6k views

Radius of curvature

I have come across a question that asked me to find the radius of curvature of a projectile. As far as I know, the path of a projectile is a parabola and I have found mention of the radius of ...
3
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2answers
113 views

Is this video's notion of general relativity correct? [duplicate]

In this video it explains the path of the apple in the general relativity version of gravity as being a straight line on a curved surface. Is this valid? Edit: this isn't a duplicate of the supposed ...
0
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0answers
36 views

How does Einstein's curved space time produce acceleration in a free falling object? [duplicate]

I never really thought much about all of this before so I'm definitely a newbie. Please excuse my ignorance. If I understand what I have read so far: if spacetime is curved time would click by at a ...
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2answers
83 views

What is more fundamental: Geometry and Topology or physical matter? [closed]

Since, there is always an interplay between gravity and the fabric of spacetime. I wonder which is more fundamental: Geometry and Topology or physical matter?
2
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2answers
193 views

Hawking Radiation and Curvature

From a simple point of view in GR/QM we take a virtual pair creation and presumable they reunite shortly in flat space-time probably representable by a space-time warpage that generates geodesic ...
3
votes
1answer
70 views

How could the universe be hyperbolic if hyperbolic space isn't symmetrical?

In the 2-D projections of the shape of the universe shown here, we see that the flat universe and the spherical universe are perfectly symmetrical, so any triangle drawn anywhere on them will be the ...
3
votes
3answers
191 views

Why does matter curve space time? [duplicate]

I am under the impression that Einstein never explains in his General Theory of Relativity, why matter curves spacetime; could explanations please be given?
3
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1answer
40 views

$f^{\prime}(R)=0$ in $f(R)$ gravity

Suppose in a certain $f(R)$ gravity theory, $f^{\prime}(R)=0$ for some finite value of $R$. (e.g. let $f(R)=R+\alpha R^2$ with $\alpha<0$. $f^{\prime}(R)=0$ at $R=-\frac{1}{2\alpha}$.) Also ...
0
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3answers
67 views

Space time curvature and gravity [duplicate]

Is Space time curvature responsible for gravity or Gravity responsible for the curvature in space-time.
1
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1answer
46 views

Can the curvature of the Universe be a function of time?

Apologies for the repetition here, but can our Universe experience (either physically or mathematically) non-constant curvature that is a function of time?
3
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3answers
147 views

Do gravitational waves affect light?

Gravity "bends" light, predicted with theory of relativity and subsequently observed: how does gravity and gravitational waves achieve this effect, and shouldn't this effect be present wherever there'...
0
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0answers
39 views

What´s the physical foundation of the assumption that the curvature of spacetime can be quantised? [duplicate]

At the moment different paths (by percentual very few people in the world) are taken to arrive (that is, if an arrival exists) at a theory that can quantise the curvature of spacetime. Considering the ...
5
votes
1answer
54 views

Did our Universe experience a curvature dominated phase?

So, my question is simple: did our Universe experience a curvature dominated phase? Or, rather, could our Universe have experienced a curvature dominated phase? This seems quite shruggish, at first ...
2
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1answer
372 views

Higher-Dimensional Metrics in (Hyper)-Spherical Coordinates

I want to compute the components of the Riemann curvature tensor (for a case similar to the Schwarzschild solution) in 4 + 1 dimensions, but I want to use a higher-dimensional analogue of spherical ...
1
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2answers
84 views

Pendulum on a train

I've seen multiple questions about a pendulum on a train and most say to use $T = 2 \pi (L/F)^{1/2}$ and I have done this to compare the pendulum's periods before being on a train and then once its on ...