Curvature Definition and 872 Threads

On the surface of a sphere, we can find the radius of cuvature of the sphere by: angle excess / area = 1/ r_s^2 http://en.wikipedia.org/w/index.php?title=Angle_excess&oldid=543583039 If we use triangles, for instance, the angle excess is the sum of the angles of the triangle minus 180...

Hey. so you have two formulas for curvature: The ordinary: |dT/ds| = |a|/|v|2 And the advanced: |v x a|/|v|3 = |a|*sin(α)/|v|2 = |aN|/|v|2 But the problem is, those two formulas aren't the same? The top one has acceleration divided by speed squared, while the bottom one has normal component...

When does the arc of the space curvature is large and when is it small?

There's something very fundamental about the curved structure of spacetime that is confusing me. Einstein is saying that gravity can bend starlight. In other words, if I have this right, a star's light will follow the curvatures of spacetime created by a large body of mass, like the sun. Here's...

I can't see how to get the following result. Help would be appreciated! This question has to do with the Riemann curvature tensor in inertial coordinates. Such that, if I'm not wrong, (in inertial coordinates) R_{abcd}=\frac{1}{2} (g_{ad,bc}+g_{bc,ad}-g_{bd,ac}-g_{ac,bd}) where ",_i"...

Hi All, Just wanted to know, is there any experimental or observational evidence today, that electromagnetic fields can cause spacetime curvature? Either direct or indirect?

I am trying to understand Gaussian curvature. This led me into looking at principle curvature. Now If one takes a look at the picture of the "Saddle Surface" on Wikipedia here: http://en.wikipedia.org/wiki/Principal_curvature I see that at the point p on the saddle where curvature goes both...

Peter Donis and Nugatory taught me a lot about spacetime curvature yesterday, but it has left me with so many questions. It sounds like mass slows down time as it warps spacetime. So, I suppose this means: more mass = more spacetime curvature = less time elapsing. Okay, in addition to...

Question: does the physical curvature of spacetime ever "move"? Something isn't adding up with Einstein's theory--or, more likely, I'm just not understanding it correctly! How can we say that the curvatures of spacetime created by the presence of stress-energy is giving us a continuum? When I...

Where should I start from to show that curvature scalar (RiemannScalar) is 2\frac{R_1212}{det (g_μ√)} ?

Hi these are questions from my test review that i am unsure of, i posted question and my answer if you can tell me if I've gotten right answer that would be much appreciated! Let C be the curve with the equations x = 2 - t^3, y = 2t - 6, z = \ln(t) Find the point where C intersects the...

If gravity rises from the fact that mass bends space-time and stuff falls in because it actually follows a straight line in a curved space as it moves by a gravitating object - doesn't that mean that a relatively stationary particle would not fall in the the claws of gravity as it would NOT be...

Homework Statement For my high school physics coursework I must investigate factors affecting the focal length of a lens. I have focused on radii of curvature and completed my data collection and verified the accuracy using the lens makers equation. However, in the conclusion I am really...

Mass curves spacetime. The relative acceleration of nearby geodesics of free test particles indicates the sign of the spacetime curvature. Convergent geodesics mean positive, divergent negative curvature. But also the metric expansion of space curves spacetime. The geodesics may be convergent...

Homework Statement Find the curvature of the polar function r = 5sin(2θ). Homework Equations All of the usual curvature equations. The Attempt at a Solution I want to turn this into a vector value function, so I can use the normal curvature equations, but that seems worse. I am...

Homework Statement r(t)={(4+cos20t) cost,+(4+cos20t) sint,+0.4sin20t} Calculate the curvature of r[t] for 0≤t≤4pi Homework Equations k = | r' x r'' | / | r' |^3 The Attempt at a Solution r[t_]:={4+Cos[20t]*Cos[t],4+Cos[20t]*Sin[t],0.4Sin[20t]} k[t_]:=Norm[Cross[r',r'']]/Norm[r']^3...

So I ran into a question; Show that there is no metric on S^2 having curvature bounded above by 0 and no metric on surface of genus g which is bounded below by 0. honestly I have no idea what is going on here. I know that a Genus is the number of holes in some manifold or the number of...

The question is: What is the minimum radius of curvature of a jet, pulling out of a vertical dive at a speed of v, if the force on the pilot's seat is 7 times his weight? The way I thought to answer this is just to say that, 7 mg, the net force on the seat will be equal to the...

Here is the question: Here is a link to the question: Find the curvature of the curve? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

So Newton says that gravity is an attractive force and some people believe in gravitons to transmit that attractive force, but Einstein says the attraction is actually due to moving along the curvature of spacetime (caused by the bodies' mass). I'm not asking which is correct, but my question is...

Homework Statement I have to prove two of the curvature formulas. The first one is (V X A) / l V3l The other one is a(t) * N(t) / l V(t) I2 Homework Equations I have a hint from my professor, but it is all confusing. I need a youtube video or something to get started on these...

The so called f (R)-gravity could be, in principle, able to explain the accelerated expansion of the Universe without adding unknown forms of dark energy/dark matter but, more simply, extending the General Relativity by generic functions of the Ricci scalar. However, a part several...

On occasion I notice there is some talk about "graviton" particles, I would have thought astro/ quantum sciences were past that idea. I am quite aware of a basic rule "Don't fall in love with your theories" so a gravity particle might exist, more on that later. In my understanding of...

Hi! Can someone please explain how does the limit in the attachment equals the curvature of a trajectory? I do not understand it. Why is it defined this way? ζ=dx/dl and it is in the direction of T. Thank you!

I am trying to improve my understand of the basic elements of GR. I have read that the Earth orbits the sun because spacetime between the Earth and the sun is warped, mainly due to the sun’s mass. The Earth follows a geodesic, which is the equivalent of a straight line in curved space...

Author: John Lee Title: Riemannian Manifolds: An Introduction to Curvature Amazon link https://www.amazon.com/dp/0387983228/?tag=pfamazon01-20 Prerequisities: "Introduction to Smooth Manifolds" by Lee seems like a prereq. Level: Grad Table of Contents: Preface What Is Curvature? The...

What happens to the Reimann tensor at the event horizon of a black hole? Do some of the 24 components become zero or infinite? What happens to parallel transport of a vector on the surface of an event horizon that is different than on a surface outside the event horizon? I'm newly educated...

Hi, having a bit of trouble with this question "In an argon ion laser ( λ = 514nm) the minimum beam waist is 1.0mm and is close to the plane mirror. Calculate the radius of curvature of the beam at the output mirror. 1.15m away" Attempt at a solution: θ = 2λ/pi W02 R = Z + ZR2/Z ZR =...

If I take it by literally meaning: Mass causes space time to curve. A rubber sheet where the mass is there, it causes the dent, the curvature. So it means the greater the momentum, the greater the curve or the dent. Now if we have a very big mass, I mean to say big in terms of size, the...

Homework Statement r(t)=<t^2,lnt,tlnt> Homework Equations k= |T '(t)| / |r '(t)| The Attempt at a Solution My professor's answer sheet solved the problem using the other method, k(t)=|r '(t) x r ''(t)| / |r '(t)|^3 and that answer ends up being 0.3, while mine is 0.4. I...

Space-Time Curvature Question! Hi Guys, A question about the curvture of space-time by mass. Where is the point of maximum curvature?? Is it at the centre of mass (i.e.. the middle of the body) The reason I ask, is that when space-time curvature is shown visually it makes out like it is a...

R_{a}_{b}_{c}^{d}ω_{d}=((-2)\partial_{[a}\Gamma^{d}_{b] }_{c}+2\Gamma^{e}_{[a]}_{c}\Gamma^{d}_{[b]}_{e})ω_{d} good, me question is about of: 1.- as appear the coefficient (-2) und the (2)? 2.- it is assumed that...

I was wondering if there were any mechanical engineers that can answer a few questions I have regarding an assignment that I have been set. We have to choose a suitable beam to support a monorail. we are looking for a moment of deflection of around 10mm. Using the universal beams table bs 4 1993...

This post in influenced by 3 new threads in our cosmology forum. Recent observational data favors positive curvature of our Universe. The question I have, however, is why positive curvature implies spatially finite Universe? Yes, it might look quite obvious if we embed curved space into higher...

Hello everyone, I am self teaching some elementary notions of differential geometry. Rather, I should say I am concentrating on mean and gaussian curvature concepts related to a physics application I am interested in. I see one has to evaluate an integral that goes as...

Hi all, I was just interested in verification of a concept. If we are given the full Riemann tensor in the form which implies constant curvature (i.e. lambda multiplying metric components) does this imply that the Ricci tensor vanishes? The question stems from why the vacuum equations cannot be...

Double contraction of curvature tensor --> Ricci scalar times metric I'm trying to follow the derivation of the Einstein tensor through double contraction of the covariant derivative of the Bianchi identity. (Carroll presentation.) Only one step in this derivation still puzzles me. What I...

Mass curves space. And speed near the the speed of light increases mass. So for someone traveling near c and is passing a partice at rest, the traveling observe feels like he's at rest and the other particle is moving. So if the other particle is moving wrt his rest frame, does he see an...

Hi All, I was wondering if it is correct to say that a vanishing metric determinant is a necessary (but probably not sufficient) condition for a curvature singularity to exist at some point(s), or is one forced to construct the full Kretschmann scalar? Cheers! FD

Homework Statement Let \vec{X(t)}: I \rightarrow ℝ3 be a parametrized curve, and let I \ni t be a fixed point where k(t) \neq 0. Define π: ℝ3 \rightarrow ℝ3 as the orthogonal projection of ℝ3 onto the osculating plane to \vec{X(t)} at t. Define γ=π\circ\vec{X(t)} as the orthogonal projection...

Homework Statement 1) The magnetic field everywhere is tangential to the magnetic field lines, \vec{B}=B\hat{e}t, where \hat{e}t is the tangential unit vector. We know \frac{d\hat{e}t}{ds}=(1/ρ)\hat{e}n , where ρ is the radius of curvature, s is the distance measured along a field line and...

Homework Statement If you look at yourself in a shiny Christmas tree ball with a diameter of 8.1 cm when your face is 35. 0cm away from it, where is your image? Homework Equations 1/do + 1/di = 1/f The Attempt at a Solution 1/di = 1/4.05 cm - 1/35.0 cm di = 4.6 cm I...

If I start out with a flat beam of length a and then I fix one side and then bend the other side up to form an arc with height h, is that enough information to determine the radius of curvature of the bent beam? If so, how would I do it? Thanks...

Homework Statement Prove that the only surfaces with zero mean curvature are either planes or hyperbolic curves with the equation: y = \frac{\cosh (ax+b)}{a} rotating alone the x axis.Homework Equations The Attempt at a Solution I made an attempt by devoting the equation of the surface as r =...

Homework Statement A mass spectrometer is an important tool in the study of air pollution. However, one of the difficulties faced by scientists is that carbon monoxide molecules (CO), which are major contributors to air pollution, have very nearly the same mass as harmless nitrogen molecules...

The simple question is whether it is possible to have curvature in a beam with no bending moment (similar to how there can be strain without stress)?The main example I have to discuss what lead me to this question is a beam which has been prestressed concentrically and so is undergoing only...

Hi, I am trying to understand a theoretical problem involving the contact between two surfaces. I have uploaded a screen shot of the mathematical formulations of the solution. I understand most of the solution, except the principal curvatures. I have tried to look up principal curvature...

I've posted a bunch of analysis questions as of late. I'm going to change things up a little bit and ask something that involves manifold theory. Here's this week's problem: ----- Problem: (i) Let $\omega$ be a 1-form. Use the structure equations \[\begin{aligned}d\theta^1 &=...

"Lawn Mower Curvature" Given a curve C in the plane, if you pick a consistent perpendicular direction, you can construct a new curve by moving out a fixed distance ε from the curve in that direction. For small values of ε, the area between these two curves will be approximately equal to ε times...

So in GR spacetime is curved in the x, y, z, and t axis but suppose you are given a surface in x, y, z could you find the curvature of time by simply measuring the rate in change in curvature of the surface in the x, y, z axis?