Curvature Definition and 872 Threads

Homework Statement A cylinder rolls without slippage on a horizontal plane. The radius of the cylinder is equal to r. Find the radious of curvature of the trajectory of points A and B. Homework Equations Ciruclar motion equations. ##R=\frac{1}{C}## The Attempt at a Solution First I drew the...

In reviewing one of Einstein's thought experiments, the accelerating elevator in space, and the resulting bending of light passing through the elevator, Einstein's predicted that light will bend in gravity. Now Einstein's original prediction was off by a factor of 2 because he hadn't yet...

I don't know GR so while answering the question if you prefer not to use that, I would be happy. In the Friedmann Equations, is energy density has an effect on curvature or vice versa? Or they are separate things and they don't affect each other? For example can we have an energy density...

Its stated that empty universe should have a hyperbolic geometry (Milne Universe) but I don't understand how its possible. $$H^2=\frac {8\pi G\epsilon} {3c^2}-\frac {\kappa c^2} {R^2a^2(t)}$$For an empty universe when we set ##\epsilon=0## we get $$H^2=\frac {-\kappa c^2} {a^2(t)}$$...

I have read numerous times that the overall curvature of space in extremely large regions -1000s of megaparsecs say - is zero. I also keep reading that the expansion rate of the universe is increasing, and that the universe is resultantly positively curved. I would be interested in a...

For 2-sphere it is having a curvature of k=1/R ,where R is the radius of the 2-sphere and to make it more generalised we treat the kR as the curvature which is always +1 and is independent of it's radius. My question is how do we treat the curvature term for 3-sphere , And it the curvature term...

If considering a 2D surface (plane) having polar coordinate r,θ (where r is the distance from the origin and θ is the anticlockwise angle from the base line as usual) The metric is now actually ds2=dr2+r2dθ2 If now this 2D surface is given a positive curvature of +1 (equivalent to the surface of...

This is an inquiry about some of the details of the bending of the trajectory if a photon by curved space-time (S-T) from the perspective of general relativity (GR). The environment in this scenario contains a black hole at its center, a photon passing by the black hole a km above the event...

My question is quite simple: what is the fundamental definition of extrinsic curvature of an hypersurface? Let me explain why I have not just copied one definition from the abundant literature. The specific structure on the Lorentzian manifold that I'm considering does not imply that an...

So I'm an Software Engineer, not a physicist, nor a mathematician. So I like to work in the qualitative, not the quantitative. Today I hit on a problem. I've been trying to remove the concept of "down" or "inward" from my thinking of gravity and GR. When people show the concept of space/time...

In an ideal world with uniform gravity field, what does its space-time curvature look like? Is it non-zero? If not, how a free particle would be accelerated with the point view of space-time curvature? By uniform gravity field, I mean a gravity field with same value, same direction everywhere...

As the coordinate singularity at r=2GM doesn't mean a physical singularity as Riemann curvature tensor is smooth although [g][/rr] metric behaves oddly in the case of Schwarzschild solution. Do somebody tell me an authentic reference how is this value is calculated? I have references writing...

Homework Statement Hi everyone! Just wondering if there's a way to prove the symmetry property of the Riemann curvature tensor $$ R_{abcd} = R_{cdab}$$ without using the anti-symmetry property $$ R_{abcd} = -R_{bacd} = -R_{abdc} $$? I'm only able to prove it with the anti-symmetry property and...

I am watching these lecture series by Fredric Schuller. [Curvature and torsion on principal bundles - Lec 24 - Frederic Schuller][1] @minute 34:00 In this part he discusses the Lie algebra valued one and two forms on the principal bundle that are pulled back to the base manifold. He shows...

Riemann tensor is defined mathematically like this: ##∇_k∇_jv_i-∇_j∇_kv_i={R^l}_{ijk}v_l## Using covariant derivative formula for covariant tensors and covariant vectors. which are ##∇_av_b=∂_av_b-{Γ^c}_{ab}v_c## ##∇_aT_{bc}=∂_av_{bc}-{Γ^d}_{ac}v_{db}-{Γ^d}_{ab}v_{dc} ##, I got these...

I'm a complete rookie in this field so please correct me where I go wrong, I just really want a better understanding of this subject. So as far as I am aware, mass causes the space surrounding it to curve or bend. What I want to know is how much does it bend the space? is the bending of space...

If the universe is very large relative to the observable universe and it is spherical, and the observable universe is well away from the outside region of the sphere, more towards the center, is spacetime approaching flat for the observable universe? I always assumed the answer is yes, but then...

Homework Statement A particle of mass ##m## moves without slipping inside a bowl generated by the paraboloid of revolution ##z=b\rho^2,## where ##b## is a positive constant. Write the Lagrangian and Euler-Lagrange equation for this system. Homework Equations...

(i.e the height you need to be at to look forward) (This isn't in physics subforum because it's more of a geometry question, this is NOT a homework question and it's more of a random thought) Basically, I've been thinking what height would you have to be at, to see 1° of Earth's curvature. So...

I want to know if there is some simple metric form for Ricci curvature in dimensions generally. In this paper https://arxiv.org/abs/1402.6334 , formula (5.21), the authers seem had a simple formula for Ricci curvature like this ##R= -\frac{1}{\sqrt{-g}} \partial^\mu \big[\sqrt{-g}(g_{\mu\rho}...

Let's say you can bend a paper...how about bending it upward. a slope I'm saying as we saw spactime in 3d...we all know how it looks..the lines are attracted toward Earth but why doesn't it deflects them and maybe negative mass is linked with it. In other words, someone under the trampoline...

I have some questions about the curvature of space (NB not of spacetime) near a planet like Earth. Unambiguously defining space curvature requires choice of a coordinate system, so I choose the Swarzschild system. Here are my questions: Would constant-time hypersurfaces under the Swarzschild...

Regarding curvature of spacetime/space: At some given point in a gravitational field, spacetime is curved at that point and this is a constant. (I'm assuming this is true). Although we can talk about the curvature of spacetime, I never hear anyone talking about the curvature of space. Can...

Many textbooks use the space (spacetime, actually, but for now only space is good enough) around a spherically symmetrical Schwarzschild object to demonstrate curvature of space due to gravity. Let’s consider two shells around such a Schwarzschild object (say a neutron star of 1 solar mass)...

Hi everyone. Could you help me to find the way to prove some things? 1)Changing of body velocity or reference frame don't contribute to spacetime curvature 2)On the contrary the change of body mass causes the change of curvature in local spacetime I use the assumption that if we have the same...

Einstein's theory states that curvature of space (created by a celestial body around itself) determines the orbital path of other celestial bodies around it within that curved space by a constant lateral force acting towards the centre upon that revolving body. Then why is that a similar force...

Friedmann Equation without the cosmological constant can be written as; $$H^2=\frac {8πρG} {3}-\frac {1} {a^2(t)}$$ for ##Ω>1## (which in this case we get a positive curvature universe), and for a matter dominant universe; $$ρ=\frac {1} {a^3(t)}$$ so in the simplest form the Friedmann equation...

Firstly, I am asking for your patience and understanding because my maths formalism is not going to be rigorous. In another thread here in this forum, I set an example for which now I am asking further instructions. I am going to ask about time-like surfaces immersed in Minkowskian space-time...

Hi all, I understand the mathematics behind special relativity pretty well, but I only have a bare conceptual understanding of general relativity. My understanding is that energy, momentum and stress (as described in the energy-stress tensor) are what contribute to space-time curvature and...

Homework Statement Determine the point on a plane curve f(x) = ln x where the curvature is maximum.[/B]Homework Equations k(x) = || T ' (x) || / || r ' (x ) || k (x) = f '' (x) / [ 1 + ( f '' (x))2 ] 3/2[/B]The Attempt at a Solution f ' (x) = 1/x f " (x) = -1/x2 k(x) = 1/x2 / { [1 +...

So General Relativity explains the force of gravity as mass/energy induced curvature of spacetime. This correctly predicts gravitational time distortion, nonlinear geodesics and gravitational lensing, the anomalous precession of planetary orbits, the schwarzchild metric, and so on. Could the...

I wonder if someone would field a beginner's muse I had: If gravity is just an illusion of the curvature of space caused by mass, does not the matter within that space follow the curve? and what is the granularity of that curvature? Does the curvature exist in the space between the nucleus and...

I was working out the components of the Riemann curvature tensor using the Schwarzschild metric a while back just as an exercise (I’m not a student, and Mathematica is expensive, so I don’t have access to any computing programs that can do it for me, and now that I’m thinking about it, does...

Homework Statement Homework Equations The Attempt at a Solution The normal acceleration of the particle at any instant is given by an = v2/r . v is the speed at any time and r is the radius of curvature . Minimum radius will occur when ratio v2/an is minimum . I think this will occur when...

TL;DR Why does the Einstein equivalence principle imply that all forms of (non-gravitational) energy source curvature? Now, as understand it, the Einstein equivalence principle (EEP) implies (or at least suggests) that gravity is the manifestation of spacetime curvature, the reason being that...

Hello! I was thinking about the Riemann curvature tensor(and the torsion tensor) and the way they are defined and it seems to me that they just need a connection(not Levi-Civita) to be defined. They don't need a metric. So, in reality, we can talk about the Riemann curvature tensor of smooth...

$\textit{$13.4.7$ Find the curvature of the curve $r(t).$}$ \begin{align*}\displaystyle r(t)&=(5+9 \cos 8t) i - (4 + 9 \sin 8t)j + 6k\\ v&= -72\sin(8t)+72\cos(8t)\\ |v|&=\sqrt{(-72\sin(8t))^2 +(72\cos(8t))^2}\\ &=5184\\ \frac{v}{|v|}&=\frac{-72\sin(8t)+72\cos(8t)}{5184}\\...

Why is it that the EM force can't be modeled on spatio-temporal curvature the way gravity can be?

I recently was able to view a 193 foot building from 24 miles away. The base of the building is approximately 15 feet above sea level and my eye level was approximately 9 feet above sea level. I was viewing the building across a Lake. I could see a substantial amount of the building, which...

General relativity suggests that path of light is curved around sun. This curvature is not dependent upon frequency of the photon. What is the physical difference between 'curvature of space' and 'curvature of space-time' ? We can make measurements at two points in space at same time. But there...

For example, the curvature due to a mass; does that curvature continue passing from within to outside the mass's light cone? If so, is the mass subject to the external curvature? If not, does the curvature have a discontinuity at the light cone surface?

Does the amount by which an object changes the spacetime curvature depend on relativistic mass or the rest mass? Through this question I just want to answer whether momentum equals [relativistic mass * velocity] or is it [rest mass * gamma * velocity]. Both the formulas might be the same but I...

When I read explanations about the early Universe and the oscillations of the photon-baryon fluid before recombination, effects of the cosmological constant and of the curvature of the Universe on the fluid are never discussed. Only dark matter, baryons, and photons are mentionned. Dark energy...

Can someone please explain to me why time drastically slows for anyone near an extremely curves spacetime. I see it as the flow of time almost becomes slowed due to extreme curvature, what can explain this? What do physicists "see" time as? Also, I'm not entirely educated on this topic but I...

This is probably a bad question, but can it be transformed away? Say Alice is on Earth and Bob is far away in outer space. Bob would think that Alice's clock is running slow. Alice would think Bob's clock is running fast. A third observer, say Carl, anywhere in spacetime would have to observe...

Hi there. I was wondering that if mini or micro black holes could theoretically exist, and if not all black holes need to "devour' matter, then could it be possible that all things we perceive to have gravity could possibly be caused by a mini or micro black hole at the center of massive objects...

I've been trying to understand how we know that the observable universe is flat, and I'm having difficulty finding any sources that explain exactly how the calculations were done. On this WMAP website (https://map.gsfc.nasa.gov/mission/sgoals_parameters_geom.html), it says: "A central feature of...

I know two kinds formulas to calculate extrinsic curvature. But I found they do not match. One is from "Calculus: An Intuitive and Physical Approach"##K=\frac{d\phi}{ds}## where ##Δ\phi## is the change in direction and ##Δs## is the change in length. For parametric form curve ##(x(t),y(t))##...

So let's say that there are several different treatments for being underweight. One way to deal with this is to make repeated measurements of each patient's weight across time. So I can fit a model to the panel data, regressing weight on treatment as an indicator variable, and then plot the...

A body is moving in deep space substantially away from others influences of gravitational force ( just assume). Suddenly, a big planet appears in It's vicinity. As per relativity it's not some force issuing forth from the planet that attracts the body towards itself. The body just keeps on...