Curvature Definition and 920 Threads

hello For the same Friedmann metric, Landau (Classical theory of fields) finds a value for the Riemann curvature scalar which is given in section 107 : R = 6/a3( a + d2(a)/dt2) whereas in MTW , in box 14.5 , equation 6 , its value is : R = 6(a-1 d2(a)/dt2 + a-2 (1 + (d(a)/dt)2 ) ) The...

Homework Statement Metric ansatz: ds^{2} = e^{\tilde{A}(\tilde{\tau})} d\tilde{t} - d\tilde{r} - e^{\tilde{C}(\tilde{\tau})} dΩ where: d\tilde{r} = e^{\frac{B}{2}} dr Homework Equations How to calculate second fundamental form and mean curvature from this metric? The Attempt at a...

if gravity arises from normal accelerations due to the curvature of spacetime...what would the opposite of this "process" represent? to clarify is it possible to describe the opposite of this curvature?? thanks

Homework Statement Hello everyone. :) I'm having trouble simplifying the last little bit of this question that deals with Gaussian curvature. I've taken all the required derivatives, and double checked with my professor to make sure that they're correct. I'm only have trouble with reducing...

I was doing some simple physics with a ball resting on a table and I made this curve (0,0) (25, 6.8) (50, 27.51) (75, 63.4) (100, 112.34) (125, 175.7) (150, 253.3) (175, 345.4) I was wondering if anyone could identify what sort of curve it is? I am just curious. This is not a homework...

I don't really understand the point in Curvature and Torsion, I am wondering if someone could explain them to me. Thank you for your kindness: Why do mathematicians need Curvature and Torsion? What are their main uses??

After some light reading, I'm more confused than ever. Is gravity just a byproduct or effect of the curvature of space? Is it a force that would exist if space didn't curve, even in the presence of mass? (probably a stupid question, sorry!) I've seen various diagrams of the Earth revolving...

Is there always the same "amount" of spacetime curvature in the uni.? Universe is what I meant by uni. Okay, if matter and energy cannot be created or destroyed, and since they are what causes spacetime to curve, does that mean there will always be the same amount of spacetime curvature...

If the curvature is always constant and >0 for a parametrized curve C, does it automatically mean the curve is a circle?

Hi there. I have a dump question for you guys. I really wonder about curvature of spacetime. I read that due to Omega_tot=1 the Universe is assumed to be flat. But on the other hand something like the curvature of the universe is mentioned... I also thought that the energy stress tensor...

This maybe a simple question, but if Earth orbits the Sun due to the Sun's mass 'curving' spacetime, wouldn't we be moving closer to the sun? like if you spun a marble around within a bowl, it ends up in the center. What am I missing here?

The basic equation of GR has a curvature constant Λ on the lefthand (geometric) side. The Friedman equation is derived from the Einstein Field Equation by making a simplifying assumption of uniformity. As a spacetime curvature Λ can be written either in units of reciprocal area or reciprocal...

Use the gauss bonnet theorem to show that the gauss curvature of a closed orientable surface of genus 2 cannot be identically zero euler characteristic is 2-2(2)=-2 so total gauss curvature is equal to -4pi. The integral of zero is zero and not -4pi so gauss curvature is not identically zero...

Einstein discovered that general covariance allows his GR equation to have just TWO gravitational/geometric constants: Newton G and a curvature constant he called Lambda. So the symmetry of the theory requires us to put both constants into the equation and investigate empirically whether or not...

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...

Homework Statement I have a given Metric: ds^{2}=A(u,v)^{2}du^{2}+B(u,v)^{2}dv^{2} And I'm asked to compute its curvature, and use this result to compute the curvature of the poincare metric: Set A=B=\frac{1}{v^{2}} The Attempt at a Solution I'm using Cartan's method. So first I change to an...

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 Prove the following statement K = (a(t) * N(t)) / (llv(t)ll)2 To clear things up it is the dot product of a(t), and N(t). Divided by the magnitude of velocity squared. Homework Equations llV X All / llV(t)ll3 The Attempt at a Solution I used the cross...

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...

Homework Statement Consider the hyperboloid x2+y2-z2 = 1 at the point (1,0,0). Take the normal direction i to the surface. a) Compute the curvature of the circle x2+y2=1 on the hyperboloid (z=0) at the point (1,0). b) Compute the curvature of the hyperbola x2-z2=1 on the hyperboloid...

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...