Questions tagged [curvature]

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 Calabi-Yau manifold.

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78 votes
15 answers
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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 ...
Zac's user avatar
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20 votes
5 answers
6k views

How does "curved space" explain gravitational attraction? [duplicate]

They say that gravity is technically not a real force and that it's caused by objects traveling a straight path through curved space, and that space becomes curved by mass, giving the illusion of a ...
Mason Wheeler's user avatar
21 votes
6 answers
4k views

Better explanation of the common general relativity illustration (stretched sheet of fabric)

I've seen many science popularisation documentaries and read few books (obviously not being scientist myself). I am able to process and understand basic ideas behind most of these. However for general ...
Pavel Horal's user avatar
159 votes
6 answers
54k 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" ...
user1648764's user avatar
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36 votes
8 answers
6k views

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 ...
Dale's user avatar
  • 6,004
68 votes
1 answer
9k 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 ...
Leos Ondra's user avatar
  • 2,123
38 votes
3 answers
6k 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 ...
Rijul Gupta's user avatar
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11 votes
5 answers
11k views

Does a moving object curve space-time as its velocity increases?

We always hear how gravity bends space-time; why shouldn't velocity? Consider a spaceship traveling through space at a reasonable fraction of the speed of light. If this spaceship, according to ...
Armend Veseli's user avatar
46 votes
4 answers
16k views

What is the physical meaning of the connection and the curvature tensor?

Regarding general relativity: What is the physical meaning of the Christoffel symbol ($\Gamma^i_{\ jk}$)? What are the (preferably physical) differences between the Riemann curvature tensor ($R^i_{\ ...
Sklivvz's user avatar
  • 13.4k
26 votes
3 answers
8k views

Nature of gravity: gravitons, curvature of space-time or both?

General relativity tells us that what we perceive as gravity is curvature of space-time. On the other hand (as I understand it) gravity can be understood as a force between objects which are ...
dF_'s user avatar
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70 votes
17 answers
138k views

What is the simplest way to prove the Earth is round?

Assume you've come in contact with a tribe of people cut off from the rest of the world, or you've gone back in time several thousand years, or (more likely) you've got a numbskull cousin. How would ...
user avatar
45 votes
14 answers
7k views

Why does the speed of an object affect its path if gravity is warped spacetime?

I think I understand the idea of thinking about gravity not as a force pulling an object towards another object but instead a warping of space so that an object moving in a straight line ends up ...
Cormac Mulhall's user avatar
23 votes
8 answers
5k views

Physical meaning of non-trivial solutions of vacuum Einstein's field equations

According to Einstein, 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 ...
Dubious's user avatar
  • 523
13 votes
9 answers
15k 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 ...
german's user avatar
  • 131
1 vote
2 answers
3k views

Bowling ball on a rubber sheet analogy - what pulls the ball down [duplicate]

Possible Duplicate: Does the curvature of spacetime theory assume gravity? Since I read Cosmos long ago, I see the same analogy about the balls rolling on a rubber sheet used to explain how ...
Ariel Popovsky's user avatar
2 votes
2 answers
3k 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)
Jitter's user avatar
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29 votes
5 answers
5k 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 ...
Erick Weil's user avatar
28 votes
2 answers
4k views

Why does dark energy produce positive space-time curvature?

My understanding is that dark energy, or equivalently a positive cosmological constant, is accelerating the expansion of the universe and I have read that this gives empty space-time positive ...
Daniel Mahler's user avatar
22 votes
4 answers
16k views

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 ...
PunkZebra's user avatar
  • 989
5 votes
3 answers
2k views

Clarifying what metric counts as flat space

In (2D) Cartesian coordinates, the Euclidean metric... $$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ ...is flat space. If the diagonal elements are exchanged for other real numbers ...
ben's user avatar
  • 1,517
68 votes
6 answers
46k 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 ...
Džuris's user avatar
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15 votes
4 answers
28k 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?
Hakim's user avatar
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12 votes
6 answers
5k views

The Fabric of Space-time?

I am not an academic in anyway, just someone interested in the story that is our universe. So my apologies if this isn't a well thought out inquiry. I've been struggling with a concept for some ...
Kaplan's user avatar
  • 137
9 votes
2 answers
14k 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, ...
ta3920's user avatar
  • 415
2 votes
4 answers
13k views

Non-zero components of the Riemann tensor for the Schwarzschild metric

Can anyone tell me which are the non-zero components of the Riemann tensor for the Schwarzschild metric? I've been searching for these components for about 2 weeks, and I've found a few sites, but ...
Omar M.'s user avatar
  • 31
12 votes
5 answers
7k views

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 ...
oz1cz's user avatar
  • 223
9 votes
2 answers
4k 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 ...
Paul Manta's user avatar
5 votes
2 answers
719 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 ...
Rez's user avatar
  • 51
17 votes
4 answers
4k views

ALL "forces" as manifestations of properties of space-time

I apologize if this seems like a quack question, but I need some insights by those who know much more than me in Physics. Anyway, the gravitational "force" (not really a force) is a manifestation of ...
1989189198's user avatar
11 votes
2 answers
1k views

How do spatial curvature and temporal curvature differ?

While looking at the metrics of different spacetimes, i came across the "Ellis wormhole", with the following metric: $$c^2d\tau^2=c^2dt^2-d\sigma^2$$ where $$d\sigma^2=d\rho^2+(\rho^2+n^2)d\Omega^2$...
Gotbread's user avatar
  • 151
8 votes
5 answers
1k 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 ...
user avatar
7 votes
8 answers
3k views

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 ...
charles stewart's user avatar
7 votes
2 answers
2k 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 ...
XYZT's user avatar
  • 769
6 votes
2 answers
2k 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 ...
Worldsheep's user avatar
17 votes
4 answers
4k views

Why do the Einstein field equations (EFE) involve the Ricci curvature tensor instead of Riemann curvature tensor?

I am just starting to learn general relativity. I don't understand why we use the Ricci curvature tensor. I thought the Riemann curvature tensor contains "more information" about the curvature. Why is ...
Stan Shunpike's user avatar
12 votes
10 answers
7k views

Why doesn't planet Earth expand if I accelerate upwards when standing on its surface?

According to General Relativity I am being accelerated upwards by planet earth while writing this question. But a curious person on the the other side of the planet relative to me would have the same ...
Megahyttel's user avatar
6 votes
7 answers
791 views

What are the analogues of $F_{\mu\nu}$ in General Relativity?

In electromagnetism, the measurable gauge-invariant quantities are the electric and magnetic fields or the six independent components of the field strength tensor $F_{\mu\nu}$. What are the analogues ...
Solidification's user avatar
6 votes
2 answers
550 views

How can locally Euclidean space of zero curvature accumulate to non-zero global curvature?

I'm having trouble understanding how a space can be locally Euclidean, with zero curvature, and globally non-Euclidean, with curvature. If the space had locally approximately zero curvature, I see how ...
userManyNumbers's user avatar
5 votes
5 answers
2k views

How does the curvature of spacetime induce gravitational attraction?

I don't know how to ask this more clearly than in the title.
Dale's user avatar
  • 6,004
5 votes
2 answers
2k 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 ...
user avatar
4 votes
3 answers
661 views

Why does GTR not need a higher dimension to describe the bending of spacetime?

I am a bit confused on how GTR uses intrinsic curvature instead of extrinsic curvature. Maybe it is just a misunderstanding, but I will do my best to describe my question: If we take an object of $n$ ...
jng224's user avatar
  • 3,687
42 votes
6 answers
9k 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.
Nathan BeDell's user avatar
23 votes
3 answers
5k views

If black holes are just empty vacuum of space inside, then what causes the curvature?

I have read this question: The fundamental confusion many have about black holes is thinking that they are discrete "things" surrounded by horizons and other phenomena. But they are ...
Árpád Szendrei's user avatar
18 votes
2 answers
2k views

Can matter be described as the result of the curvature of space, instead of vice versa?

Can matter be described as the result of the curvature of space, rather than the curvature of space being the result of matter, and energy being the cause of the curvature of space?
Kane's user avatar
  • 181
12 votes
2 answers
1k views

How much choice did Einstein have in choosing his GR equations?

General relativity was summarised by Wheeler as "Spacetime tells matter how to move; Matter tells spacetime how to curve". I have a fairly good mental picture of how the first part works. However, I ...
N. Virgo's user avatar
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9 votes
4 answers
2k views

How to prove that a flat spacetime admits Minkowski coordinates?

How should I prove the following in general relativity? A flat spacetime can be covered by Minkowski coordinate neighborhoods. A flat spacetime with the trivial topology can be covered by a global ...
Zwiebach Friobie's user avatar
9 votes
1 answer
9k views

Is the scalar curvature of the Schwarzschild solution 0?

The Schwarzschild solution is meant to be a solution of the vacuum Einstein equations. That is $$R_{\mu\nu}=0.$$ So, the Ricci tensor must be null for $r>0$. Now, if the scalar curvature is ...
Yossarian's user avatar
  • 6,007
8 votes
4 answers
1k views

What is the geometric interpretation of the Einstein tensor $R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R$?

The Riemann curvature tensor $R_{\mu \nu \rho \sigma}$ has the geometric interpretation of giving how much parallel transport fails to close around tiny loops. The Ricci tensor $R_{\mu \nu}$ is the ...
user1379857's user avatar
  • 11.5k
8 votes
4 answers
583 views

How could we know that the relativistic curvature of universe is intrinsic?

General relativity tells us that the universe is bent by gravity, but this curvature is intrinsic to the universe (the universe bends, but not in a fourth spatial dimension, the universe having only ...
Jorge Arturo Juarez's user avatar
7 votes
5 answers
870 views

Intuition for why mass and energy curve the space-time fabric and for why this relationship is linear?

The force of gravity does not exist I understand(-ish) that, following general relativity, an apple falls onto earth, not because there is a force pulling earth and the apple toward each other but ...
Remi.b's user avatar
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