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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Shape of the universe [duplicate]

I have just started learning GR (but have some rudimentary knowledge on differential geometry) and came across this statement: "the universe is flat with only a 0.4% margin of error". I have read ...
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3answers
124 views

General relativity without curvature?

Is there a reformulation of general relativity without curved space time, just with fields (like classical E&M)? Edit: removed the part about E&M with curvature (multiple posts).
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1answer
61 views

How are tidal gravity and curvature related?

I see tidal gravity mentioned in the literature sometimes, and sometimes people even say something like that is the “real gravity”. I am confused about the significance of tidal gravity, and what ...
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1answer
373 views

2D space-time curvature

Actually, why is the space-time curvature considered 2D plane. As 2-D dimensional space-time curve is used to explain why moon revolves around the earth stating because the massive objects wraps the ...
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0answers
18 views

Rotating object influence on space time [closed]

We know that an object with a big mass curls the space time, but if an object starts rotating, so the experienced gravity reduces, will this make the space time curl less?
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1answer
73 views

Why $C_{abcd}C^{abcd}$ in de Sitter–Schwarzschild metric doesn't depend on $\Lambda$

With the de Sitter–Schwarzschild metric: $$ds^2=-\left(1-\frac{2M}{r}-\frac{\Lambda r^2}{3}\right)dt^2+\left(1-\frac{2M}{r}-\frac{\Lambda r^2}{3}\right)^{-1}dr^2+r^2d\theta+r^2\sin^2\theta d\phi$$ I ...
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1answer
35 views

Can you compare curvatures in space, spacetime and hidden space?

Spacetime curvature is given by the cosmological constant, that produces a De-Sitter spacetime. It is non-zero. But space curvature is nearly zero (how close to zero, compared to the cosmological ...
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3answers
110 views

GR time dilation cares only about local curvature, right?

Regarding this remark on Worldbuilding SE and the discussion leading up to it: Can someone properly knowledgeable on the subject please explain whether the time dilation due to being in a gravitation ...
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1answer
66 views

Does $R_{\mu \nu \sigma \rho} R^{\mu \nu \sigma \rho} \propto R$ hold?

For $R_{\mu \nu \sigma \rho}$ the Riemann-tensor and $R$ the Ricci-scalar: Does $R_{\mu \nu \sigma \rho} R^{\mu \nu \sigma \rho} \propto R$ hold? Or is there any way to relate $R$ approximately ...
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2answers
69 views

Does acceleration warp space?

I know that mass warps spacetime and gravity and acceleration are equivalent so does acceleration warp spacetime too?
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1answer
53 views

Can everything be described without anything needing to actually “bend”? [closed]

Is space bending because gravity actually causes small particles to move differently? If large source of gravity is somewhere are particles extending towards it, creating a "bend" in space? So "bend" ...
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1answer
62 views

What force causes massive objects to bend space? [duplicate]

The visualization of gravity as shown by this video is pretty good at explaining how massive objects bend space, and such bending causes objects around it to fall towards it (a.k.a: gravity). ...
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1answer
41 views

Covariant derivative commutator on spinors [closed]

What is this object $[\nabla_{\mu},\nabla_{\nu}]\epsilon$ in terms of curvature tensor $R_{\mu\nu}$? Where $\nabla_{\mu}$ is the covariant derivative on a four sphere and $\epsilon$ is spinor. PS: I ...
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1answer
98 views

Curvature of spacetime and the equivalence principle

Assuming the Einstein equivalence principle, formulated as following: The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the ...
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2answers
83 views

Telling if geometry is curved without Riemann tensor

If you're only interested in telling whether a certain geometry is flat or curved, and you do not need to know in which way it is curved, do you still need the Riemann tensor? When I try to visualize ...
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2answers
51 views

Do Heavy objects like planets and stars create Curve in spactime fabric?

According to Newton moon revolves around the earth because of gravitational pull! But Einsteins quoted that its not the pull but Earth's mass creates a curve in spacetime and Moon revolves on the edge ...
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1answer
70 views

Does charge bend spacetime like mass? [duplicate]

Does charge bend spacetime like mass? I'm not asking if electromagnetic forces can be described geometrically, but if EM fields could correspond to particular curvatures of spacetime, like gravity ...
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2answers
155 views

Can anyone explain me how time can bend according to Einstein in simple way? [duplicate]

I am just 16 and curious to learn about Theory of Relativity. Can any one explain it simple enough for me to understand? I read that it is bending of time-space or space-time that causes gravity. How ...
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1answer
82 views

Symmetry of extrinsic curvature tensor

I am trying to solve following problem: In a spacetime of signature (+, −, −, −), let $$ u^au_a = 1, \quad A_{ab} = \nabla_cu_dh^c_{\; a}h^d_{\; b}, \quad h_{ab} = g_{ab} - u_au_b $$ Show that ...
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2answers
52 views

Human max speed in open space

Suppose you are an astronaut forgotten in the middle of nowhere, between our solar system and proxima centauri's. Now, you are out of fuel. I heard that with some kind of movements, someone in free ...
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3answers
1k 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
116 views

Calculation of Einstein tensor for weak gravitational field

I am studying A First Course in General Relativity (2nd Ed.) by Bernard Schutz. I have some difficulty in deriving Eq.(8.32) on P.193, the form of Einstein tensor for weak gravitational field, which ...
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4answers
831 views

When space bends, what are the lines that are being bent?

In an electric field diagram, the lines represent the electrostatic force vector at the position. These lines are bent when you place a charge into the system. What is the equivalent description ...
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0answers
31 views

Zero mean curvature and maximal volume

I have a 1+1 asympotically flat spacetime system (spherical symmetry) for which I'm trying to find the hypersurface which maximizes the volume enclosed in a sphere of a given radius $r=R$. **It seems ...
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2answers
212 views

How can the universe be flat?

Okay, so I just want to clarify a few things. According to what I have read, we have measured the universe to be flat, and the shape of the universe is directly related to the mass-energy density. ...
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0answers
54 views

Is the Big Bang notion compatible with the flat space? (A problem with the **moment** of Big Bang, not with the place of it) [duplicate]

A passage from a paper: "If one imagines running the clock backward in time, any given region of the universe shrinks and all galaxies in it get closer and closer until they smash together in ...
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2answers
80 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 ...
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1answer
66 views

How are these two Riemann tensor equations equivalent?

Poisson in A Relativist's Toolkit defines the Riemann tensor as$$A_{\,;\alpha\beta}^{\mu}-A_{\,;\beta\alpha}^{\mu}=-R_{\phantom{\mu}\nu\alpha\beta}^{\mu}A^{\nu}.$$ Foster and Nightingale's A Short ...
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1answer
62 views

How can the D'Alembertian of a field be interpreted intuitively?

The D'Alembertian operator is defined as $$ \Box = g^{\nu\mu}\nabla_\nu\nabla_\mu $$ For the Minkowski metric in Cartesian coordinates that is $$ \Box=\frac{1}{c^2}\frac{\partial^2}{\partial t^2} - ...
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2answers
75 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 ...
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5answers
144 views

How to explain space-time curvature in two minutes? [closed]

How would you explain what is the curvature of space-time in a short period of time like 2 minutes to non experience people?
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1answer
31 views

Calculating curvature of spacetime when energy is present

I am only about half-way of studying SR and GR and I am not yet familiar with a formula to calculate the curvature of spacetime when energy is present. To be more specific, I want to calculate ...
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3answers
213 views

How does warped space actually look (visually)?

Recently, I was reading about space warping due to extreme gravity and at speeds approaching c, but in books, they always show space in 2D and depth to show space distortion. I was wondering how ...
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1answer
44 views

Light trajectory

We have observed stars where "we should not" Some people say that gravity can alter light trajectory. Some people say that gravity actually alter the space on which light travels. Which one is ...
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1answer
35 views

If there are any discernible effects of travelling through curved space..?

If one is on a spaceship traveling through a 'very' curved section of space-time, are there experiments one could perform on the ship that would reveal measurable differences between the very curved ...
4
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1answer
137 views

How warped spacetime bends trajectories of light and moving objects?

I fail to see why the light follows something like the blue line and not the green line on the attached image. Figure 1 - light bends around warped spacetime Afaik. something similar happens ...
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1answer
82 views

Commutator of Gauge Covariant derivatives

What is the physical meaning of $$ [D_{\mu}, D_{\nu}] ~\propto~ F_{\mu, \nu}, $$ where $D_{\mu}$ is the gauge covariant derivative and $F_{\mu,\nu}$ is the field strength? Is it just a definition? ...
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4answers
162 views

Any tips on evaluating Riemann tensor?

I am calculating the Riemann tensor for the Schwarzschild solution. I've calculated all 9 non-vanishing Christoffel symbols already. Now I need to evaluate the Riemann tensor and I find no easy way to ...
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4answers
121 views

How is gravity proportional to space-time curvature in the rubber-sheet analogy?

In General Relativity, Einstein established that gravity is due to the curvature produced by objects in space. We all know that gravity is proportional to mass. The picture Einstein painted looks ...
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0answers
33 views

FRW Metric maximally symmetric, derivation, $R=3K$ or $R=6K$ confusion, two different texts

I'm looking at Tod and Hughston Introduction to GR and writing the metric in the two forms; [1]$$ ds^{2}=dt^{2}-R^{2}(t)(\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2})) $$ [2] $$ ...
2
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1answer
974 views

Earth curvature refraction for dummies

I keep being presented with 'earth curvature experiment' videos recently, by flat/concave earth advocates. It seems to be their favorite "evidence" that Earth is not spherical. Debunking this gets ...
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0answers
177 views

Can the universe be round but still infinite?

Can the universe still be infinite in space if its curvature is > 1? Is a manifold of positive curvature necessarily compact? Does the Tarski paradox have any bearing on the finite or infinite ...
3
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1answer
74 views

The actual space curvature

What is the curvature of our physical space, according to the latest experimental data? I've found it somewhat difficult to find a definitive answer to the question, because the spacetime curvature ...
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3answers
334 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?
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2answers
114 views

Does mass curve space?

Just to be sure, according to the theory of General Relativity, my understanding is that mass curves space-time. My question is, can mass also curve space?
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2answers
128 views

Once I calculate the Riemann curvature tensor, what do I do with it?

I am considering the Schwarzschild metric. I have calculated my Christoffel symbols and am able to calculate the Riemann tensor (I think). In short, I have done a bunch of work to find this thing ...
4
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2answers
84 views

Schwarzschild: Proof that $\{r<2m\}$ is a black hole

I saw the following proof to show that $\{r<2m\}$ is a black hole in the Schwarzschild metric. Consider the Schwarzschild metric: $$ g=-V(r)\text d t^2 + \frac{1}{V(r)}\text d r^2 + r^2 \text d ...
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1answer
134 views

Can energy bend space? [duplicate]

I know mass bends the space around it and I also remember matter can be converted into energy and vice versa, so my question is: can energy interact with space in a similar fashion as matter does?
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1answer
70 views

Bianchi Identity using null tetrad

I'm currently looking at the Newman-Penrose Formalism, and trying to understand where there sets of equations come from. For that, I need to know how I can write the second Bianchi identity for the ...
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1answer
59 views

Weyl scalar calculation

I'm trying to compute Weyl scalars, but don't really understand the formulae for them, in the sense I don't understand how to compute them. Let's take ...