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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118 views

Is my diagram of spacetime curvature valid (relatively)?

I've been wracking my brain trying to understand what "curved spacetime" really is, and I think replacing one dimension with the time dimension then drawing the world-lines through time was the "aha!" ...
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1answer
113 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
66 views

Why is space (almost) flat? Is it because masses are approximately homogeneously distributed?

The question I have is: Why is space (almost perfectly) flat in our neighbourhood? (I am disregarding the deviations due to the sun and the planets.) Is it correct to say that space is (almost) flat ...
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3answers
73 views

What is the meaning of Einstein's field equation in terms of source and its effects on curvature?

The Einstein's Field Equation is $$R_{\mu\nu}-(1/2)g_{\mu\nu}R=-8\pi T_{\mu\nu},$$ where the left hand side is the curvature term and the right hand side is the source term (see, Hartle). Now, in the ...
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2answers
34 views

Does the value of the Ricci scalar determine the strength of the gravitational field?

If I was solving an equation that contains the Ricci Scalar, and I want to solve the equation in the strong and weak gravity regimes, is right to assume that $R>>1$ for first case and ...
4
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1answer
56 views

Fastest way to find the curvature terms from a given metric [closed]

I want to find the spherically symmetric, static solutions to Einstein's equations $$ R_{\mu \nu} - \frac{1}{2}Rg_{\mu \nu} = 0 $$ in four dimensions using the metric $$ g_{\mu \nu}dx^{\mu}dx^{\nu} ...
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0answers
32 views

Did spacetime curve infinitely about 13.7 billion years ago? [duplicate]

GR/Big Bang Model implies that there was a singularity about 13 billion years ago, in which all the matter and energy along with the observable universe (or perhaps, the entire universe) was ...
2
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1answer
246 views

Why does the Ricci tensor vanishes in Schwarzschild metric? [duplicate]

If the Schwarzschild metric is suppose to describe the behaviour of a spherical object in flat space, so the Schwarzschild is different from the flat metric because it describes curved space so why ...
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0answers
55 views

Quadratic order perturbation terms in the expansion of Ricci tensor [closed]

I want to expand Einstein-Hilbert action for the metric $$ g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu} $$ up to quadratic order in $h_{\mu \nu}$. For this purpose I need to calculate the Ricci tensor ...
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3answers
201 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 ...
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0answers
52 views

Eddington-Finkelstein coordinates: Why $\ln(r-2m)$ instead of $\ln|r-2m|$?

If one considers the Schwarzschild metric $$ \text d s^2 = -V(r)\text d t^2 + \frac{1}{V(r)}\text d r^2 + r^2 \text d \Omega^2\;,\qquad V(r) = 1-\frac{2m}{r}\;, $$ and introduces the ...
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1answer
134 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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0answers
37 views

Does the curvature of spacetime by gravity affect homogeneity and isotropy of the space of the universe?

The FLRW metric starts with the assumption of homogeneity and isotropy of space.(Wikipedia) FLRW metrics of the universe have no or only very weak curvature - It is curved space. In contrast, ...
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1answer
94 views

Physical visualisation of curvature

I was wondering-how do you visualise curvature in the context of general relativity. The gravity well and trampoline analogies are quite wrong, so I want a more realistic approach to it (say, the way ...
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5answers
1k 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 ...
4
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1answer
144 views

Geodesic Deviation between Test Particles from Gravitational Wave

I'm having trouble understanding how Carroll (Spacetime and Geometry, p.296) explains the effect of a passing gravitational wave on test particles. If we have two geodesics with tangents $\vec{U}$, ...
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2answers
35 views

Particle motion characteristic

I'm making a particle motion raffling normal numbers. The normal random numbers raffled are the angles of the directions that the particle is going. The particle speed is constant. Look how this is ...
0
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1answer
49 views

Gravitational time dilation in changing curved space time

Imagine a portion of spacetime which is changing its spacetime curvature because of an object with great mass travelling nearby. For instance, before it was flatter, and after the object passes it ...
4
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3answers
700 views

Where do I start with Non-Euclidean Geometry?

I've been trying to grok General Relativity for a while now, and I've been having some trouble. Many physics textbooks gloss over the subject with an "it's too advanced for this medium", and many ...
5
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1answer
82 views

How is $\Omega_0 = 1$ when the characteristic “teardrop” past light cone seems to admit curvature?

Introduction: The top graphic is just one I pulled from a page describing the process of detecting cosmic curvature. The second graphic is one I drew up to illustrate my misunderstanding. My ...
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2answers
86 views

Curvature gravity and a falling apple? [duplicate]

I know very little of physics after Einstein. I am aware of that Einstein's gravity theory says that the existence of matters creates curvature of a space-time, so that our Earth orbits our Sun. I ...
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2answers
2k 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 ...
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0answers
52 views

Are Laplace Operator and mean curvature exactly the same thing for 2D function?

Let's assume we study 2D function/surface f(x,y). Then Laplace Operator is defined as: $$\nabla^2 f=\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}$$ And the mean curvature: let ...
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1answer
96 views

General relativity: is curvature of spacetime really required or just a convenient representation?

I'm not really far into the general theory of relativity but already have an important question: are there formulations that can do without spacetime curvature and describe the general theory of ...
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2answers
168 views

Curvature of spacetime as a real thing?

I get the curvature tensor in General Relativity, it is “just” math. Does space-time REALLY curves as a tangible thing, or is Einstein proposing a mathematical abstraction? More naively, please ...
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1answer
33 views

Flatness and Kinetic Energy

Why the curvature parameter can be interpreted as the difference between the average potential energy and the average kinetic energy of a region of space? Curvature ...
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1answer
335 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)
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2answers
92 views

Is the surface of a heavy sphere bigger than $4 \pi r^2$ due to general relativity?

I am unfortunately not familiar with the mathematics behind general relativity. However, on a heavy planet (say a sphere) gravity will bend space-time in a way that an object initially in rest, will ...
2
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2answers
89 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 ...
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7answers
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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 ...
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9answers
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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 ...
2
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2answers
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Is the curvature of space around mass independent of gravity?

Is the curvature of space caused by the local density of the energy in that area?Could gravity be a separate phenomenon only arising from the curvature of space? For instance if the density of energy ...
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5answers
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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 which are ...
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4answers
248 views

Space-time curvature creates gravity or is it (could it be) vice-versa too?

Mass (Energy) creates space-time curvature and thus it forms the reason for gravity. Can it be vice-versa too? Like, mass created gravitational field, gravitational field created space-time curvature? ...
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2answers
83 views

Conservation in space-time curvature

Pardon this possibly naive question. I'm starting to poke around in the topic of General Relativity (as soon as I can pull myself back up out of the vortex of underlying mathematics that I've gotten ...
0
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1answer
42 views

Relationship between mass and the radius of curvature of space and time

What is the relationship between mass and the radius of curvature of space and time created due to the presence of the mass? please give the mathematical relation if there is any?
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1answer
144 views

If space warps distort moving objects' trajectories, does it mean that static objects are immune to gravity? [closed]

If gravity is just space distortion, which affects trajectories of moving objects, then a static object (not moving, thus no trajectory) will not suffer any type of accelerating force from gravity? ...
14
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3answers
346 views

Is the flatness of space a measure of entropy?

This is a bit quirky: For a very long time I've found Stephen Hawking's evaporating small black holes a lot more reasonable and intuitive than large black holes. The main reason is that gravity is ...
2
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3answers
287 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 ...
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2answers
223 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, ...
4
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1answer
42 views

How can we see that the Riemann curvature tensor is covariant?

The Riemann curvature tensor, using the conventions of wikipedia, is written in terms of Christoffel symbols as: $$ \tag{1} R^\lambda_{\,\,\mu \nu \rho} = \partial_\nu \Gamma^\lambda_{\,\,\rho \mu} - ...
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3answers
2k 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 ...
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3answers
59 views

Space time curvature bends back

If our perception of space-time curvature is gravitation and Reduced Gravity Plane can reach weightlessness on some point of its trajectory, doesn't that mean that when Reduced Gravity plane reaches ...
2
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1answer
65 views

Curvature of a particle move

I'm simulating a particle movement following a normal distribution. How this is done: My particle has a constant speed v and every step the particle move, I ...
4
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2answers
141 views

Physical meaning of harmonic function?

In complex numbers, we define a harmonic function as a twice continuously differentiable function such that the Laplace operator acting on it gives zero. Can anybody explain me the physical ...
0
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1answer
58 views

Do the concepts of intrinsic and extrinsic curvatures imply that all spaces are embedded in a higher dimensional space?

The concepts of intrinsic and extrinsic curvature seem to imply that all spaces must be embedded in a higher dimensional space? What does this imply for physical reality?
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2answers
62 views

How can the universe be flat and have no center if universal mass-energy content is finite?

WMAP measurements confirm that the universe is flat within a 0.4% margin of error. If we assume the universe is flat and there is no 'center' then how could the mass energy content be finite since ...
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0answers
58 views

Coordinate Symbol confusion in general relativity

In a previous post (Finding the metric tensor from the Einstein field equation?), the equation used lambda, rho mu and nu (not sure of the names of the letters!) for the Ricci tensor and swapped to a, ...
5
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1answer
618 views

Gravity is curved geometry: A fact of nature or model-dependent interpretation?

We are regularly taught in high-schools and universities that, according to General Relativity (GR), gravity is nothing but a manifestation of space-time curvature (which, in its turn, is caused by ...
5
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0answers
141 views

Euclidean black hole extrinsic curvature

I have read that the extrinsic curvature at the horizon of a euclidean black hole is zero? Does anybody know how this can be shown?