Curvature Definition and 872 Threads

Hello, all, the most important results that I know in this topic is the Gauss-Bonnet Theorem (and hence the classification of compact orientable surfaces) and also the Poincare-Hopf index theorem. But there are still some fundamental problems I don't understand. For example, is the...

Is there any properties with the curvature tensors in 3 dimensions? (Maybe between the Ricci tensor and the Ricci scalar, they are proportional to each other? ) I heard about it in a lecture, but I can not remember the details. The 3 dimensional case is not discussed in many reference books...

Hi, I'm new to this forum so maybe this topic has been addressed ad nauseum at some point before, so I apologize if so. But, as the title suggests, do you feel the spacetime curvature is a reality, or is it just a mathematical convenience for making predictions? dm4b

Total "absolute" curvature of a compact surface Hi! Someone could help me resolving the following problem? Let \Sigma \subset \mathbb{R}^3 be a compact surface: show that \int_{\Sigma}{|K|\mathrm{d}\nu} \ge 4\pi where K is the gaussian curvature of \Sigma. The real point is that I want...

The Riemannian curvature tensor has the following symmetries: (a) Rijkl=-Rjikl (b) Rijkl=-Rijlk (c) Rijkl=Rklij (d) Rijkl+Rjkil+Rkijl=0 This is surely trivial, but I do not see how to prove that Rijkl=-Rjilk. :( Thanks.

I have a question... Since solutions to Einstein's field equations are diffeomorphism invariant, does that mean that solutions are metrics of constant curvature?

Homework Statement What curves lying on a sphere have constant geodesic curvature? Homework Equations k^2 = (k_g)^2 + (K_n)^2 The Attempt at a Solution I'm trying to understand the solution given in the back of the book. It says, a curve on a sphere will have constant curvature...

Hi! A google search for my upcomming question lead me to this page, and I am delighted to find an online community of psysics who hopefully will answer my, maybe, very simpel question. In the study of Einstein's General Relativity theory, the picture of a sphere placed in a net representing...

On an oriented surface, the integral of the Gauss curvature over a smooth triangle can be interpreted as the angle of rotation of a vector that is parallel translated once around the three bounding edges. How does one interpret the integral of the Gauss curvature of an arbitrary SO(20 bundle...

Homework Statement Find the curvature K(t) of the curve r(t) = (-4sin(t)) i + (-4sin(t)) j + (5cos(t)) k.Homework Equations K(t) = |r'(t) x r"(t)| / |r'(t)|3 The Attempt at a Solution r'(t) = (-4cos(t))i + (-4cos(t))j + (-5sin(t))k r"(t) = (4sin(t))i + 4sin(t))j + (-5cos(t))k |r'(t)| =...

Homework Statement A golfer hits a golf ball from point A with an initial velocity of 50 m/s at an angle of 25° with the horizontal. Determine the radius of curvature of the trajectory described by the ball (a) at point A, (b) at the highest point of the trajectory.Homework Equations p = V2/an...

I have just created a rainbow artificially and the photo is attached. I didn't expect the bow to come out straight as it has. Does anyone have any idea why it is straight and not curved (or circular)? I know rainbows are normally curved (or spherical) normally because the particles that the...

Is it possible to explain, in one or two paragraphs, what the scalar curvature, R, is as it applies to General Relativity (the Einstein Field Equation, specifically?). This needs to be understandable to a high school AP-C physics student. Signed, Me - the high school AP-C physics student...

I Have a problem understanding that vanishing of the curvature tensor implies that parallel transport is independent of path. With the converse of this assertion I have no problem. The text I'm reading(Lovelock and Rund) explains the converse but treats the direct assertion as trivial. Can...

Hello all, We know that for some well-behaved, smooth/continuous, twice differentiable function of x, f[x] there exists at each point a slope (f ' [x]) and a radius of curvature \rho [x]=\frac{\left(1+f'[x]^2\right)^{\frac{3}{2}}}{f\text{''}[x]} It also seems intuitive to think that at...

what is the radius of curvature of a parabola y^2=4ax at the end of the focal chord ?

Homework Statement how to find the radius of curvature for following curve-: x^2y=a(x^2+y^2) at the point (-2a,2a) Homework Equations radius of curvature= {(1+y1)^3/2}/y2 where y1 and y2 are the first and second order...

I see that general relativity uses tensors to calculate curvature. How exactly does relativity calculate actual curvature. Are the units of curvature m^-1, like regular curvature units? For example, using SI units Ruv - 1/2guvR = (8πG/c^4)Tuv R00 - 1/2g00R = (8πG/c^4)T00 R00 + R/2 = 8πGρ/c^2...

Does anyone know how to even start this problem.. :( I seriously have no idea...spent like few hours to find this out on the library and no luck... Please help a poor guy out...I aint smart enough for this...

If I understand GR correctly, gravity is no real force but only an effect of the curvature of spacetime. Thus, objects subject to no other forces than gravity follow trajectories in spacetime that are geodesics. I find this very hard to understand, because the trajectories of such objects don't...

While posting a reply in another thread, I had an inspiration for a device to measure spacetime curvature. It is well know that we can measure this curvature by measuring the angles of a large triangle or comparing the circumference of a circle to its radius, but his device may may be simpler or...

Can someone explain how a curvature in space-time,can cause motion? (Without telling the pillow example) Thanks

My question concerns the affect of the curvature of space on a stationary object. I understand that the force of gravity is more accurately described as space curvature. Ie, a massive object like the sun or Earth can be visualized as a bowling ball placed on a rubber sheet, creating a curvature...

Hello all, I'm new to GR and trying to understand everything in general now... I was looking at pictures like this one http://upload.wikimedia.org/wikipedia/commons/2/22/Spacetime_curvature.png for a very long time. I made two major "experimental" conclusions here: 1. A meter near planet...

Hi, I am working on the Laughlin model of Quantum Hall Effect, which relates to the concept ' adiabatic curvature'. The paper didn't include much details, and I knew little about berry phase. Could some one please give me some idea that how is the local adiabatic curvature deduced? ( the...

How would you prove that if the curvature of a 'curve' in R3 is zero that the line is straight? All I have learned about is the Serret Frenet equations which I thought only apply when the curvature is non-zero? How do you define normals/binormals in this case? I'm not sure if this is...

Hello- would someone mind clarifying the distinction between curvature of a function and the concavity? I would prefer if you could keep it to the one or two dimensional case, since my math background is just multivariable. Thank you very much.

hi i need a affirm of curvature in polar coordinates. i need now please

General relativity has it that the spacetime continuum is curved. The physics of continuum is dealt with [stress] tensors. My questions: (1) The presence of a mass creates the curvature in spacetime. By how? (2) If the curvature due to matter is positive, is the curvature due to antimatter...

I was just gong to learn general relativity(not with maths) but with some very basic tutorials given over internet. I also watched the animated series of general realtivity. Everywhere i see,matter bends spacetime( a fabric of spac and time woven ). And when there is matter than this...

Hi, I know that you can determine that the Gaussian curvature of a cone tends to infinity at the vertex, but seeing as the curvature anywhere else on the cone is zero, how is this possible?

We have described the distortion in spacetime which Einstein derived in GR as a "curvature" of spacetime. This is barely more descriptive than "warping" spacetime. I understand that what this means is that spacetime varies from being Euclidean, having distortion caused around objects of mass...

hi, how does general relativity work INSIDE stars and planets, since the mass is no longer concentrated within a point, so there are necessarily gravitationnal effects outwards and not only inwards?

Can we say that each subatomic particle affects space time such that collectively as big as a planet it explains why there is gravity? Thank you very much.

Homework Statement An alpha particle and beta particle, each with kinetic energy 35keV , are sent through a 1.1T magnetic field. The particles move perpendicular to the field. Homework Equations no idea. The Attempt at a Solution theres nothing on curvature radius in my...

Homework Statement I've got this problem on polar coordinates which says: A particle moves along a plane trajectory on such a way that its polar coordinates are the next given functions of time: r=0.833t^3+5t \theta=0.3t^2 Determine the module of the speed and acceleration vectors for this...

I'm a bit confused about this and would like for someone to help me get this straight. I read in wikipedia that a manifold with more than three dimensions, like spacetime, is conformally flat if its Weyl tensor vanishes. I think all FRW metrics are conformally flat, so I guess our universe is...

Homework Statement A civil engineer is asked to design a curved section of roadway that meets the following conditions: With ice on the road, when the coefficient of static friction between the road and rubber is 0.1, a car at rest must not slide into the ditch and a car traveling less...

How would we express the curvature of the 2-dimensional surface of a sphere without referring to a radius of curvature or any other extra-dimensional description?

Homework Statement If a plane is flying level at 950 km/h and the banking angle is not to exceed 40 degrees what is the minimum curvature radius for the turn?Homework Equations possibly F = ma = mv2 / r ? The Attempt at a Solution no idea where to start on this one, not sure where the angle...

Hi I need to calculate the radius of curvature of a reflector.I have a sound source (Ultrasonic transducer of 40 mm operating at 50 Khz) in air . I am trying to generate a standing wave using this sound source .As curved reflectors can help to amplify the sound pressure (I actually don't...

In the previous section he derived the components, with respect to the coordinate bases associated with a polar coordinate system, of the Riemannian metric tensor field on S2, the unit 2-sphere: g = \begin{pmatrix}1 & 0 \\ 0 & \sin^2(\theta) \end{pmatrix} where \theta is the zenith angle...

Homework Statement A train enters a curved horizontal section of track at 100 km/hr and slows down with constant deceleration to 15 km/hr in 12 seconds. An accelerometer mounted inside the train measures a horizontal acceleration of 2 m/s^2 when the train is 6 seconds into the curve. Calculate...

Homework Statement If you paint a dot on the rim of a rolling wheel, the coordinates of the dot may be written as (x,y)=(R\theta+Rsin(\theta), R+Rcos(\theta) where \theta is measured clockwise from the top. Assume that the wheel is rolling at constant speed, which implies \theta = \omegat...

Hi everyone, I have a question related about the relation between potentials, connections and curvature in gauge theories. In Newtonian physics, the common starting point is Newton's law, which determines the motion in terms of the derivative of the potential, i.e. sth. like \ddot...

What would the units be on the curvature of spacetime? G(curvature)=8πGT/c^4

How can I calculate the curvature of a 3D hyperboloid? I mean, what parameters do I need to calculate the intrinsic curvature? I guess to calculate the extrinsic curvature as seen from a 4D space I would just need a curvature radius, right? Thanks

Question: How do I use this formula to find the Radius of curvature? Formula: M/I= σ/Y = E/R (M = bending moment, I = second moment of aria, σ = stress, y = distance from nutral axia, E = modulus of elasticity & R = radius of curvature) Attempt: In this question, I have all of the...

In the definition of the extrinsic curvature, there is the normal vector. It depends on the sign of the normal vector? Because a normal vector can be directed in two ways. For example the curvature of a circle on the plane has different curvature from inside and outside! But this is...

Please read and critique this argument for me please, any help is appreciated. Imagine a geodesic, and a matter wave that traverses this geodesic. The action of this matter wave determines the motion of the matter wave along this geodesic over a given space-time interval, and is specified...