Tagged Questions

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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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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What is the sum of the angles of a triangle on Earth orbit?

Gauss went out and measured triangles made up of mountain peaks to show that the angles sum up to 180 degree. However, general relativity leads to non-Euclidian space and I would like to get a better ...
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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?
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Local inertial frame

In general relativity we introduce local inertial frames to be such frames where the laws of special relativity holds. Let $\xi^{\alpha}$ the coordinates in the local inertial frame, so we get ...
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Thought experiment on space curvature due to gravity

Let's say you built a huge, straight rigid beam out in space, far from the sun (outside the orbit of Pluto). It is very long, say 200,000 miles long, but can be very narrow. Then you move it to the ...
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Is there proof gravity bends space or is it just the most convenient explanation?

I have read this sentence in an article: The theory [of general relativity] holds that gravity is geometry: particles are deflected when they pass near a massive object not because they feel a ...
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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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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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Vanishing of the Ricci tensor in higher spacetime dimensions

I understand how, if the Riemann tensor is 0 in all its components, since we construct the Ricci tensor by contracting the Riemann, Ricci tensor would be 0 in all components as well. I've read that ...
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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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Expanding the Ricci tensor by summing over indices

I had an attempt at deriving the Schwarzschild metric. This is a 4-dimensional problem where the indices are being summed from 0 to 3. I got up to the part where I calculate the Ricci tensor which is ...
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Dissipation in hyperbolic and spherical space

Say that we have a point that emits point particles that all travel at the same velocity in a random direction and neither their velocity nor their direction changes and neither does the position of ...
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Why does the FLRW metric assume constant curvature?

So the FLRW metric takes the following form in reduced-circumference polar coordinates. $$ds^2 = -c^2 dt^2 + a^2(t) \left(\frac{dr^2}{1 - k\, r^2} + r^2 (d\theta^2+sin^2\theta\, d\phi^2)\right)$$ ...
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What altitude you need to see the earth's curvature? [duplicate]

i have seen several videos and articles that nobody has seen earth's curvature except nasa. if someone has tried to go up to see earth curvature it seemed flat to them which makes some people to think ...
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Confusion in Special Relativity: Rotating frame of reference

Suppose we are observing a rotating frame from an inertial frame, free from gravity, and try to measure the circumference of a circle drawn in the rotating frame. Since our measuring rod would be ...
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How are mass and density treated in general relativity? [duplicate]

Background: I am confused by how mass relates to the equations in general relativity. For example, given a certain mass density distribution, I am unsure how to express a system in terms of GR. ...
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Covariant derivative acting on $\delta R$

In this question I am seeking a nice methodology. Maybe the experts on the site can help me on this. I am trying to find the GHY term for a higher derivative theory of gravity. Partly, I have figured ...
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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 ...
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How can a finite amount of matter be uniformly distributed in a flat, infinite space?

There are some properties of the Universe I find in the (mostly popular) literature which are often described as "the most probable in case of our Universe". I can't put them together in a way that ...
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3D-representation of space-time

When I read something about GR, I nearly always see some pictures that look like trampolines, like this one. I know that the curvature of space-time is described by the Riemann-Tensor $R$. I was ...
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Finding the Riemann tensor for the surface of a sphere with sympy.diffgeom

I have implemented a SymPy program that can calculate the Riemann curvature tensor for a given curve element. However, I am encountering problems solving for the case when the curve element is the ...
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Falling with same acceleration led to geometric meaning of gravity?

In Hartle's book, he says: Einstein saw that the experiment fact that all bodies fall with the same acceleration in a gravitational field led naturally to an understanding of gravity in terms of ...
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Can space and time separately be curved?

How can I imagine curved time, if it is not a part of four dimensional spacetime? Similarly for space. What are the measurable, observable consequences of these two phenomena in a laboratory or in ...
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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 ...
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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 ...
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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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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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The Ricci tensor and its relation to volume

From Wikipedia's entry on Ricci tensor, In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a geodesic ball in ...
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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 ...
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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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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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Can the curvature of space be measured using a swimming pool?

I read somewhere that if the center of mass of two twelve ton elephants was one meter apart then the space they curve would be one nanometer longer. (Though I doubt elephants come that dense) This ...
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How many tons of lead is needed to curve space 1 nanometer? [closed]

How many tons of lead is needed to curve space 1 nanometer?
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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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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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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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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 ...
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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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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 ...
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 ...
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 ...