Curvature Definition and 872 Threads

Find Curvature of Ellipse given x=3*cos(t) and y=4*sin(t) at the points (3,0) and (0,4) Relevant equations: curvature at r(s) is k(s)=||dT/ds|| when r(s) is arc length parametrization and T is the unit tangent vector I usually use the formula k(t)= (||r'(t) x r''(t)||)/||r'(t)||^3 So...

In a nearby thread: I believe it was correctly concluded "yes" because an electromagnetic (EM) field has energy and pressure so it curves spacetime; it does have a gravitational effect. Although it was not discussed, this effect is much, much smaller than the electromagnetic effects because...

Homework Statement I'm currently self-studying Carroll's GR book and get stuck by proving the following identity: K^\lambda \nabla _\lambda R = 0 where K is Killing vector and R is the Ricci ScalarHomework Equations Mr.Carroll said that it is suffice to show this by knowing: \nabla _\mu...

The Kretschmann curvature scalar is defined to be K = RabcdRabcd where Rabcd is the Riemann tensor. I believe I heard in class that this scalar can be used to demonstrate the existence of a curvature singularity. Can somebody tell me why this is so? Also, I heard that it is better (easier I...

Homework Statement The cornea, a boundary between the air and the aqueous humor, has a 3.0cm focal length when acting alone. What is its radius of curvature? Homework Equations n(1)/s + n(2)/s' = (n(2)-n(1))/R where, n is the index of refraction (unitless), s' is the img dist (cm)...

Homework Statement (Only need help for b) I think but I'll post the whole problem) (All values are SI units) (O; i, j, k) orthonormal basis. A particle moves following this law: r = 8t2 j A disk with radius 2 in the plane XOY rotates around Z with constant angular speed: ω = 3 k. At the time...

Why do things orbiting, i.e. free-falling, around Earth float away from each other? Why don't they both free fall toward Earth together? I remeber hearing once that if you let go of 2 objects while 'floating' in space they both go away from you and away from each other. Is this due to curving...

I've been using various methods of finding curvature, and using the forumla K=\frac{||\vec{r}'X \vec{r}''||}{||\vec{r}'||^3} I took the cross product of my two vectors and came up with (just assume this is correct, if my question isn't answered i'll post all related work)...

i need to calculate radius of curvature of 1 MeV KE electron in 1 Tesla magnetic field. r = mv / eB what is the (relativistic) speed, v, of the electron? (ans: 0.941 c ??) then i think you use p(rho) = mo. V / sqrt (1 - v^2 /c^2) = mv (ans: 7.59E-22 kg m/s) e = 1.6E-19 C B = 1...

Folks, I'm in the process of trying to understand spacetime curvature in general relativity. My question might sound odd, but I'm wondering how to best conceptualize spacetime distortions due to a moving mass. If there is a large mass, e.g. a planet, moving through spacetime, the curvature...

Homework Statement How do we argue that gravity is not a force due to curvature of space-time? Homework Equations I'm new.. I don't even understand the eqn of tensor calculus. The Attempt at a Solution No force is needed for as massive objects follows the curvature of spacetime...

Hi, I've been reading through Yau's proof of the Calabi conjecture (1) and I was quite intrigued by the relation R_{i\bar{j}} = - \frac{\partial^2}{\partial z^i \partial \bar{z}^j } [\log \det (g_{s\bar{t}}) ] derived therein. g_{s \bar{t} } is a Kahler metric on a Kahler manifold (I'm...

Parallel transport, as one means of quantifying the curvature of a coordinate space, enables changes in a vector's components, when it is carried around variously oriented loops in that space, to be properly measured, i.e. by comparisons made at the same location. Those changes which are...

Homework Statement I've attached a picture which include the question -- I found it on the net, and it's exactly the same as my homework. Homework Equations I'm not sure which formula to use. It's due pretty soon, so please help! :cry:

Homework Statement Find the curvature of a helix given by the parametric equation r(t)=<acost, asint, bt> where a and b are real numbers Homework Equations I know k=|T'(t)/r'(t)| The Attempt at a Solution and I believe the answer to be k=b/(a2+b2)1/2, I just don't know how to get there

Homework Statement Find the curvature of a helix given by the parametric equation r(t)=<acost, asint, bt> where a and b are real numbers Homework Equations I know k=|T'(t)/r'(t)| The Attempt at a Solution and I believe the answer to be k=b/(a2+b2)1/2, I just don't know how to get there

Homework Statement Proof that, if a particle moves along a space curve with curvature 0, then its motion is a along a line.Homework Equations K=\frac{||r'(t)\times r''(t)||}{(||r'(t)||)^3} (curvature of a space curve)The Attempt at a Solution Assume the curve is smooth, so r'(t) cannot be the...

I'm confused, but when objects travel along the straight lines in curved[/color] space-time, do they undergo acceleration? We know that following geodesics is equivalent to inertial motion (one example is free-fall), but when these inertially moving objects travel in curved spacetime, they...

Hello! In New Scientist this week (actually next week!), there was a question concerning the curvature at the horizon. The formula is as follows for the distance to the horizon: (2*6373*h)^1/2 km; where h is the height of the individual from the ground. Using the exaple it states a towe...

Homework Statement http://steam.cs.ohio.edu/~cmourning/problem1.jpg If the image doesn't load (and it might not, although I'm not sure why), it can be found at: http://steam.cs.ohio.edu/~cmourning/problem1.jpg Homework Equations Part of the problem is I'm not entirely sure what all the...

Homework Statement Given a parametric curve, \alpha(t) = (x(t),y(t) ), not necessarily arc length parameterized show that the curvature is given by: k = \frac{x'y'' - y'x''}{|\alpha'|^{3}} Homework Equations As I understand this the curvature is defined from the point of view of...

It is a standard fact that at any point p in a Riemannian space one can find coordinates such that \left.g_{\mu\nu}\right|_p = \eta_{\mu\nu} and \left.\partial_\lambda g_{\mu\nu}\right|_p. Consider the Taylor expansion of g_{\mu\nu} about p in these coordinates: g_{\mu\nu} = \eta_{\mu\nu}...

Homework Statement (My first post on this forum) Background: I am teaching myself General Relativity using Dirac's (very thin) 'General Theory of Relativity' (Princeton, 1996). Chapter 11 introduces the (Riemann) curvature tensor (page 20 in my edition). Problem: Dirac lists several...

hey folks, as far as I understand, according to Einstein's general theory of relativity, any mass that exists in spacetime causes a curvature in it, right?! now, my question is: does this curve take place in the time dimension (ct) or in spacetime (ct,x,y,z) itself?

Hi, Given a large number of test particles N, it should be possible to determine the Riemann curvature tensor by tracking their motion as they undergo geodesic deviation. Is there a minimum number N that will achieve this in any situation, or does it vary from problem to problem? How...

Say, does anyone happen to know the non-zero components of the Riemann (curvature) tensor for the Schwarzschild metric using r,\phi,\theta and t? Thanks.

Can we draw an infinite equlateral triangle on a negatively curved surface?

Hi all. Is the radius of curvature of a Universe-model constant through-out the Universes' lifetime? Or does this have to be adjusted depending on the redshift we are looking at?

Homework Statement A Curve C is given by the polar equation r=f(theta). Show that the curvature K at the point (r, theta) is K=|2(r')^2 - rr'' + r^2| -------------------- [(r')^2 + r^2]^(3/2) *Represent the curve by r(theta) = r<cos theta, sin theta> Homework Equations I...

what's the maximum area of an equilateral triangle that can be drawn on a positive curvature?

Does exist a system to calculate how mass creates a curvature in relativity, and how it creates redshift?

I'm am new to the forums and have a quick question. When I see diagrams of objects in the universe on the spacetime fabric, the bottom of the object seems to be touching the fabric. My question is, does the fabric touch Earth for example on the south pole like presented in the pictures, or on...

How Relativity say about redshift in terms of curvature?

1. Homework Statement Is the gaussian curvature at a point on the surface \frac{1}{(x^2+y^2+1)^2}? 2. Homework Equations shape operator: S(\textbf{x})=-D_\textbf{x}\hat{\textbf{n}}=\frac{\partial (n_x, n_y)}{\partial (x,y)} Gaussian Curvature = |S(\textbf{x})|...

What is the meaning of the curvature scalar (R) in GR? More precisely, what is the meaning of it's evolution? Why when we are concerning the solar system we take R to be small and when we are concerning the cosmological scales the we assume R to be large? Thanks in advance.

According to contemporary ideas the spin of elementary particle is a certain mysterious inner moment of impulse for which it is impossible a somewhat real physical picture to create. The absence of spin visual picture, in opinion of a number of authors leaves the regrettable gap in quantum...

Under what circumstances do we know whether a given tensor of 4th rank could be the curvature tensor of a manifold? For instance, if I specify some arbitrary functions R_{ijkl} (with the necessary symmetries under interchange i<->j, k<->l, and ij<->kl), is it necessarily the case that there is a...

Hi. I am reading "An Introduction to Modern Astrophysics" by Carroll and Ostlie, for a summer project. In section 27.3 (Relativistic Cosmology) the curvature of a sphere is given by 6\pi \frac{C_{exp}-C_{meas}}{C_{exp}A_{exp}} The situation is as follows: Consider a sphere of radius...

I don't know why this question puzzles me... I believe I can understand the general idea that a lightwave moving in the vicinity of a source of gravity would be deflected by it, as in the "rubber sheet" model, and would curve slightly towards the mass. But is this also valid when the light...

Does relativistic mass curve space-time, i.e., does relativistic mass affect the gravitational field of an object?

A particle is fired into the air with an initial velocity of 60m/s at an angle of 54 degrees from the ground. At time t=5.443, what is the radius of curvature of the path traveled by the particle? I started by coming up with a vector equation for the path traveled by the particle, using the...

http://camoo.freeshell.org/25.8.pdf" Laura Latex source below. I won't be changing this if I edit the file, it's just for convenience if you want to grab latex code. n sec. 25.8, he says "recall the dual $^\ast F$ of the Maxwell tensor F. We could imagine a 'dual' U(1) gauge...

please see http://camoo.freeshell.org/dual_gauge.pdf" Thanks Laura

What exactly are the tangent and the normal accelerations of a projectile motion and how are they expressed mathematically? What is curvature radius? What is its expression? How is it derived ?

Homework Statement What is the minimum radius of curvature for an alpha particle, He, moving at 2.0x10^-6m/s in a magnetic field of 2.9x10^-5T? Homework Equations F=qvBsin(theta)...then F=BILsin(theta) The Attempt at a Solution F=qvBsin(theta)...

I have a question about spatial curvature. Before asking the question, I will summarize my understanding of the distinction between spatial curvature and spacetime curvature. If I have this wrong, I would appreciate corrections. Spacetime curvature occurs locally in the presence of gravity...

What is radius of curvature and why is it equal to 2f where f is the focal length of a lens or a mirror ?

We can all see what curvature of space looks like, just by throwing a ball and watching it follow the natural geodesic. But what does curvature of time look like? How do we experience it? We typically experience the passage of time in what seems to be a forward linear manner. The...

Inside a spherical cavity centered at the Earth's center, is the space-time curvature is 0 or not 0? Would the clock run more slowly?

I'm trying to find the equation of a curve in R3 where k=t=a/(s^2+b) where k is the curvature, t is the torsion and a,b are constants contained in R. I've spent weeks on this problem and at the moment it's driving me nuts since I always seem to end up with an impossible integral. Any help or...