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 do we look at non-flat geometries in Cosmology?

In Cosmology we use the Robertson-Walker-Metric which follows from the cosmological principle & mathematics. This metric leaves three cases for a possible curvature (or geometry) of space (not ...
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99 views

If gravity is due to curvature, how does gravity work in situations with no curvature?

The strength of the gravitational field falls off as the inverse square of the distance from a spherical source. It only falls off as the inverse of the distance from an extended cylindrical or line ...
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Help with the Ricci tensor and Chistoffel Symbols [closed]

I really am confused with certain notations of the Ricci tensor and the Christoffel symbols. I'm looking to evaluate $R_{00}$ from my lecture notes, but I'm a little stuck at one point. The Ricci ...
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1answer
47 views

Is there inflationary solution in $R^2$ theory in Jordan frame?

In the Starobinsky $R^2$ inflation model, one usually uses a conformal transformation from Jordan frame to Einstein frame in which the action can be written just like Einstein action + scalar field ...
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32 views

Why is spacetime curved by mass but not charge [duplicate]

According to general relativity theory, the deformation of spacetime is proportional to energy tensor $$T_{\mu\nu}.$$ $$ R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi G T_{\mu\nu}. $$ Does it mean that ...
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Direction of Gravity [duplicate]

I ran into a pde that has a parameter in it that can be either $+1$ or $-1$. They say the $+1$ case corresponds to gravity pointing upwards, whereas the $-1$ case applies to gravity pointing ...
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1answer
50 views

See behind the black hole

Why in this video does the 2nd black hole appears to change size and appear larger the farther away it gets? How can you see behind it? http://www.youtube.com/watch?v=ENd8Sz0AFOk
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1answer
67 views

What exactly does the Kretschmann scalar implies and how does it work?

From the General Relativity class lectures I understood that this particular invariant, the Kretschmann scalar namely $$R_{\mu\nu\lambda\rho} R^{\mu\nu\lambda\rho}$$ is really important because, ...
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59 views

How can I prove that for a Killing vector $\nabla^a \nabla_a \xi^\mu = -R^b_a \xi^a$? [closed]

I'm taking a course on General Relativity and I'm trying to prove that for a Killing vector field $\xi^\mu$ the following equation holds: $$\nabla^a \nabla_a \xi^\mu = -R^\mu_a \xi^a$$ Where $R_ab$ ...
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37 views

Proper time and asymptotic flatness

I'm trying to understand the concept of asymptotic flatness in general relativity, and came up with the following question: If the proper time $\tau$ is infinite for a timelike geodesic, does it mean ...
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2answers
66 views

How does bending in space-time caused by mass energy translate into acceleration of object? [duplicate]

Let say we have 2 similar apples separated by a distance apart, just their mass energy alone is sufficient to cause bending in the space-time. I think it is this bending in the space-time that cause ...
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4answers
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What allows us to assume spacetime is flat when no normal matter is present?

Dark matter causes a bend in spacetime. We see this through gravitational lensing. But what allows us to assume spacetime is flat when no normal matter is present? Why can't we say that dark matter ...
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3answers
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Why does the curvature of space decrease during inflation but increase after it?

According to the Friedmann equation, curvature of space will increase with time/expansion of space, but I've also read that during Inflation, the expansion caused the Universe to flatten out. What is ...
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31 views

Stress-Energy Tensor of Riemann's Curvature Field

Einstein, in "The Foundation of the General Theory of Relativity," as well as most modern lecturers in the subject, use a pseudo-tensor for the stress-energy of the gravitational field. By ...
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1answer
63 views

Gravity: Why Do things fall to Earth? [duplicate]

If gravity is in reality spacetime geometry why when I drop an object on the surface of the Earth does it fall to the ground? Does spacetime push it?
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5answers
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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 ...
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4answers
136 views

Which tensor describes curvature in 4D spacetime?

I heard these two statements which don't work together (in my mind): In 4D spacetime the curvature is encoded within the Riemann tensor. He holds all the information about curvature in spacetime. ...
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1answer
95 views

Spacetime curvature effect on chemistry

Do current chemistry / astrophysics / stellar chemistry calculations include the effects of the curvature of spacetime on chemical reactions? For example, the heat transfer from a point closer to the ...
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1answer
210 views

Riemann tensor with 2nd and last indice the same will vanish?

I calculated that Riemann tensors are antisymmetric with respect to 2nd and last indice,as the symmetry properities of $R_{\rho\nu\sigma\mu}$ goes. $$R^{\omega}_{\ \ \ ...
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1answer
72 views

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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How does curved spacetime cause motion revisited

There was a previous question titled "Why would spacetime curvature cause gravity?" asked March 10, 2014. The answer given was essentially that since the time component of an object in curved space ...
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2answers
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Is the popular explanation given for gravity in General Relativity misleading? [duplicate]

In most popular explanations of General Relativity, both in print and film/television, gravity is demonstrated using an example of a 2 dimensional plane being flat, then when putting a heavy object in ...
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Show that $R_{\mu\nu}=C g_{\mu\nu}$ from the vacuum Einstein equation with a nonzero $\Lambda$ [closed]

If I begin with the vacuum field equation with a nonzero cosmological constant: $$R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}R+g_{\mu\nu}\Lambda=0$$ How can I show that $$R_{\mu\nu}= ...
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1answer
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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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5answers
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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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1answer
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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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1answer
39 views

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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1answer
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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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2answers
181 views

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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0answers
92 views

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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44 views

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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50 views

Motion without matter

Suppose you have a vacuum spacetime with non zero cosmological constant then you can show that two test particles will move toward each other or apart depending on whether it is negative or positive ...
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53 views

Covariant Derivative commutator on a Spinor [closed]

I am trying to prove 8.14 of Supergravity - Freedman. The equation that I am trying to show is $$\gamma^\mu \nabla_\mu \gamma^\nu \nabla_\nu \psi = (g^{\mu\nu}\nabla_\mu \nabla_\nu - ...
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21 views

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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1answer
39 views

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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61 views

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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1answer
118 views

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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2answers
123 views

Falling with same acceleration and meaning of gravity

My question is what does falling with same acceleration has to do with what Einstein concluded concerning the gravity in terms of the curvature?
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119 views

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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1answer
60 views

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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1answer
154 views

Ricci tensor as relativistic Hamiltonian

I am little bit dissapointment with action integral in General relativity. The action integral is: $$ \int Rd^{4}x=\int R_{ij}g^{ij}d^{4}x\tag{1} $$ Where $$ ...
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1answer
98 views

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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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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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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2answers
161 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
103 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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2answers
553 views

2D space-time curvature [closed]

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
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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 ...