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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Does the curvature of spacetime theory assume gravity?

Whenever I read about the curvature of spacetime as an explanation for gravity, I see pictures of a sheet (spacetime) with various masses indenting the sheet to form "gravity wells." Objects which are ...
32
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10answers
17k views

How exactly does curved space-time describe the force of gravity?

I understand that people explain (in layman's terms at least) that the presence of mass "warps" space-time geometry, and this causes gravity. I have also of course heard the analogy of a blanket or ...
15
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2answers
929 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 ...
25
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3answers
3k views

Why is spacetime curved by mass but not charge?

It is written everywhere that gravity is curvature of spacetime caused by the mass of the objects or something to the same effect. This raises a question with me: why isn't spacetime curved due to ...
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7answers
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Surely space-time Curvature does not explain gravity, it just describe its effects?

In special relativity co-moving objects see the other's 4-velocity as being only temporal. When they move relative to each other they see the other's 4-velocity has rotated so that it points less in ...
69
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5answers
14k views

Why would spacetime curvature cause gravity?

It is fine to say that for an object flying past a massive object, the spacetime is curved by the massive object, and so the object flying past follows the curved path of the geodesic, so it "appears" ...
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3answers
266 views

How do we know dark matter isn't curved spacetime?

Could Dark Matter actually just be spacetime with curvature "leftovers" from the big bang or the inflationary epoch?
20
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7answers
10k views

Laplace operator's interpretation

What is your interpretation of Laplace operator? When evaluating Laplacian of some scalar field at a given point one can get a value. What does this value tell us about the field or it's behaviour in ...
6
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6answers
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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 ...
4
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2answers
172 views

5D Ricci Curvature

As part of a hw problem for a class, we're supposed to be deriving the equivalence given in equation 2.3 of this paper http://arxiv.org/abs/1107.5563. I was wondering if there is some special ...
3
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2answers
258 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 ...
2
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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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1answer
414 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)
4
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2answers
429 views

Visualizing gravity in 3D

We've all seen the depiction of gravity bending space downwards, and so attracting objects into the dent it creates, cf. e.g. this and this Phys.SE posts. That's intuitive and makes a lot of sense, ...
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5answers
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How to measure the curvature of the space-time?

I know G.R. change our vision of space and time as a unique surface than can bend. We can associate the curvature of the space-time as the gravity created by the mass of planets, stars... But how can ...
4
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1answer
271 views

Computing Curvature via Cartan Formalism

Given a metric $g_{\mu \nu}$, one can select an orthonormal basis $\omega^{\hat{a}}$ such that, $$ds^2= \omega^{\hat{t}}\otimes\omega^{\hat{t}} - \omega^{\hat{x}} \otimes \omega^{\hat{x}} - ...$$ By ...
7
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2answers
510 views

What is the geometrical interpretation of Ricci tensor?

In differential geometry and general relativity space is said to be flat if the Riemann tensor $R=0$. If the Ricci tensor on manifold $M$ is zero, it doesn't mean that the manifold itself is flat. So ...
5
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3answers
300 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 ...
2
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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...)
4
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2answers
202 views

How is the shape of the universe measured by scientists?

I would like to learn how scientists go about measuring the large-scale curvature of the universe to determine if the universe is closed 'i.e. spherical', flat, or open 'i.e. saddle shaped'. My ...
2
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3answers
2k 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?
13
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1answer
504 views

What are the local covariant tensors one can form from the metric?

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 ...
30
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4answers
3k views

Why does a flat universe imply an infinite universe?

This article claims that because the universe appears to be flat, it must be infinite. I've heard this idea mentioned in a few other places, but they never explain the reasoning at all.
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5answers
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Naive visualization of space-time curvature

With only a limited knowledge of general relativity, I usually explain space-time curvature (to myself and others) thus: "If you throw a ball, it will move along a parabola. Initially its vertical ...
6
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5answers
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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
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2answers
2k views

What does it mean for objects to follow the curvature of space?

In science documentaries that touch on general relativity, it is often said that gravitational pull isn't an actual a pull (as described by classical physics), but rather one body travelling in a ...
2
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2answers
106 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 ...
0
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3answers
523 views

Calculating the Riemann tensor for a 3-Sphere

I have worked out all the connection symbols for the 3-sphere using calculus of variations, cf. this Phys.SE post. So to find the Riemann tensor I am trying to find all the nonzero components of: ...
7
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3answers
432 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 ...
3
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1answer
51 views

Spacial curvature and expanding space

If we take the analogy that in an empty space the space is just a flat sheet then if there is a single planet or a star then the flat sheet will curve below the planet leaving a curvature shaped like ...
3
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2answers
187 views

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 ...
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3answers
197 views

Is the universe flat?

There are more than one way to view the description of the universe as flat. There is the description of an open, flat or closed universe in terms of it's fate, expansion forever away from gravity, or ...
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2answers
119 views

What are the factors affecting the spacetime curvature?

Large masses in space as stars and planets cause a curvature in the spacetime fabric. What are the factors that affect this curvature? Is it only mass? And can we conclude these factors using Tensors? ...
14
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3answers
365 views

Is the flatness of space a measure of entropy?

This is a bit quirky: For a very long time I've found Stephen Hawking's evaporating small black holes a lot more reasonable and intuitive than large black holes. The main reason is that gravity is ...
8
votes
4answers
545 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 - ...
4
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2answers
210 views

Physical meaning of harmonic function?

In complex numbers, we define a harmonic function as a twice continuously differentiable function such that the Laplace operator acting on it gives zero. Can anybody explain me the physical ...
4
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1answer
158 views

Geodesic Deviation between Test Particles from Gravitational Wave

I'm having trouble understanding how Carroll (Spacetime and Geometry, p.296) explains the effect of a passing gravitational wave on test particles. If we have two geodesics with tangents $\vec{U}$, ...
4
votes
1answer
588 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 ...
2
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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 ...
1
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1answer
132 views

Has a metric formulation of electromagnetism ever been attempted? [duplicate]

I understand that electromagnetic fields carry energy, and this energy curves spacetime gravitationally. That's not my question. I'm asking if anyone has tried to formulate electromagnetism in such ...
8
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1answer
443 views

Maxwell's equations in curved spacetime

I know that we can write Maxwell's equations in the covariant form, and this covariant form can be considered as a generalization of these equations in curved spacetime if we replace ordinary ...
7
votes
1answer
438 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 ...
5
votes
3answers
278 views

A thought experiment on vision and curved spacetime

What follows is a long self-made example to deal with my conceptual issues of visualizing curved spacetime. Imagine an observer floating somewhere in space. He feels no strain on his body, ...
5
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1answer
429 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 ...
5
votes
4answers
845 views

The Sun as a gravitational lens

Since the Sun is a gravitational lens with as focal length of 550 AU for visible light, with an immense amplification factor, shouldn't it light up objects hanging out there? We should get solar ...
4
votes
4answers
207 views

What makes a coordinate curved?

Bear with me while I try to explain exactly what the question is. The question Can a curvature in time (and not space) cause acceleration? is imagining a coordinate system in which the curvature is ...
3
votes
1answer
154 views

The relationship between spin and spinor curvature

The identity, $$ -\gamma^b{\mathcal{R}}_{ab} = {\mathcal{R}}_{ab}\gamma^b = \frac{1}{2}\gamma^b R_{ab}$$ is presented in the answer to the question Dirac Equation in General Relativity. How does ...
2
votes
1answer
91 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
votes
1answer
632 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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2answers
1k 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 ...