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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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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0answers
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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
125 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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4answers
485 views

If gravitation is due to space-time curvature, how can a body free-fall in a straight line?

According to general relativity, Gravity is due to space-time curvature. Then all paths must be curved. If so, how can there be any straight line motion? The body must follow a curved path. So, there ...
2
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1answer
63 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 ...
0
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1answer
377 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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1answer
223 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 ...
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7answers
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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 ...
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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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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
37 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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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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3answers
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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 ...
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2answers
72 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
43 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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4answers
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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?
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2answers
317 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 ...
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1answer
138 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 ...
0
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1answer
63 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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0answers
119 views

Curvature based derivation of Schwarzschild Metric

I'm a third year maths undergrad and I'm trying to find (and follow) a curvature based derivation of the Schwarzschild metric, if there exists such a proof?
4
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1answer
100 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
84 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
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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 ...
3
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1answer
71 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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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
158 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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3answers
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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 ...
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1answer
1k 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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3answers
215 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 ...
3
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2answers
117 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 ...
0
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2answers
54 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 ...
2
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4answers
833 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
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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
221 views

About Christoffel symbols in Riemann normal coordinates

According to the answer to this post, the Christoffel symbols in Riemann normal coordinates are approximated by $$\Gamma^{k}_{ij}(x)~\sim~\frac{1}{2} R^k{}_{ilj}(x_0) \xi^l \tag{5.10}$$ which came ...
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6answers
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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 ...
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2answers
216 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
82 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
739 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
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1answer
63 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} - ...
0
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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 ...
2
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1answer
188 views

Derivation of Weyl tensor

I want to derive the Weyl tensor along the lines of this derivation, but I am unable to complete it. (I am only interested in $4$ dimension for now.) Every contraction I perform gives either $0=R + 3 ...
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2answers
77 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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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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5answers
147 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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10answers
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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 ...
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1answer
148 views

Is there any relationship between the $E=mc^2$ equation and the $a_n=\kappa v^2$ formula for the normal component to acceleration?

To clarify, I know very little about physics and don't pretend to have any insight whatsoever into relativity beyond what has entered the popular imagination; my knowledge is more or less at the level ...
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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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2answers
1k 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 ...