Curvature Definition and 872 Threads

Dear all, I was wondering how the radii of curvature can be calculated of a flexible chain (polymer chain). I have the x,y and z values of the polymer chain. For a 2D chain, I can calculate the curvature radii (http://www.intmath.com/applications-differentiation/8-radius-curvature.php). I am...

Hey guys, I'm new here. I got a problem from my professor that is different from any other problems we have done. I'm stuck and need a little help.Homework Statement r(t) = <cos(t), t, 2sin(t)> Find parametric equations for the circle of curvature at (0, pi/2, 2)The Attempt at a Solution I...

I am struggling to make sense out some things. Hopefully someone can help or at least offer some different point of view. Let's examine a differential curve parameterized by arc length that maps some interval into an oriented surface (lets call it N(s)). The surface has a unit normal field...

Hello All, As far as the Newtonian mechanics and Einstein's GR is concerned, I am a little bit confused in the following things: (a) Concerning the bending of light due to gravity: Some lectures and opinions show that light bends due to the force of gravity as shown in the event of a solar...

I'm writing a sci-fi story and I'd like to make it, at the very least, scientifically plausible (in the way that alcubirre warp drives are possible assuming we could get our hands on something with negative mass which, as far as we know, doesn't exist). The basic assumption for these questions...

I am just wondering - is space-time curvature in the presence of energy-momentum ( i.e. in interior solutions to the EFEs ) always pure Ricci in nature ? I had a discussion recently with someone who claimed that, but personally I would suspect that not to be the case in general, since I see no...

Homework Statement The speed of a car increases uniformly with time from 50km/hr at A to 100km/hr at B during 10 seconds. The radius of curvature of the bump at A is 40m. if the magnitude of the total acceleration of the car’s mass center is the same at B as at A, compute the radius of...

Curvature of Space-time: What is it? General relativity talks about curvature of space-time due to mass. What does it actually mean by 'curve'. Is the space made of something that we can say is curving? If space is purely 'empty', then what is getting curve? Or is it that curving is just an...

This is something I have pondered for some while... it is so obvious that there must be an answer and is probably a silly question, but I haven't found an answer yet... so... Gravity is a consequence of the localised curvature of space. According to Relativity, space (space-time) is...

I have a basic question regarding the invariants that can be formed from the Riemann curvature tensor, specifically the Kretschmann scalar. Does this quantity have any physical significance, in the sense that it is connected to anything physically measurable or observable ? My current...

Homework Statement Let ##\Gamma ## be trajectory which we got from ##z=xy## and ##x^2+y^2=4##. Calculate the curvature ##\kappa ## and vectors T, N and B (B is perpendicular to T and N). Homework Equations The Attempt at a Solution Well, the hardest part here is of curse to...

Is there a single equation that can model both spatial and temporal metric contraction simultaneously? And also what's that equation that can model the actual degree of curvature n space-time that uses trig functions and how do you use that in combination with the two previously mentioned...

Prior to expansion the inflaton field had a large potential energy. I wonder whether there are any considerations or calculations to evaluate how to this energy curves the space created by the big bang. Does it make sense at all to talk about critical vs. actual energy density, the value of...

Sorry if this question seems too trivial for this forum. A grad student at my university told me that a compact Riemannian manifold always has lower and upper curvature bounds. Is this really true? The problem seems to be that I don't fully understand the curvature tensor's continuity etc...

Well, I'm totally in a mess now

Does the curvature of spacetime have a unit?

1. If the current mass density in the Universe was about 10 protons/m3 what would be the current radius of its curvature? What would be the maximum distance between the two points in the Universe? I got the first part but not the 2nd. If I solve the Friedman equation I get the max scale factor...

i am trying to understand the relationship between the two on a local and global scale and how these two concepts are related to the Ricci scalar. Is it correct to say that as far as we know on a global scale, spacetime is flat so that the Ricci scalar is zero. If so, what can be said about...

Hi, I have a problem which you guys probably could help me solve or at least advise how to approach. I am building a mechanical system that consists of 2 steel rods acting as rails and a platform that travels along. I need to find radius of curvature of a steel rod under stress to see by...

I have always read that a wormhole will quickly collapse in on itself due to its own gravity, forming a black hole, unless it is held open by some exotic matter that has a negative energy density. But couldn't there exist a wormhole with zero spacetime curvature? It would therefore have no...

i asked this question before, but i didn't ask it quite right so i didn't get a satisfactory answer.. curvature is define as how quickly/ abruptly a curve changes with respect to its arc length.okay so the normal vecor (N = T ') is the change in the tangent vector of a curve with respect to...

Homework Statement Find the curve whose curvature is 2, passes through the point (1,0) and whose tangent vector at (1,0) is [1/2 , (√3)/2 ]. The Attempt at a Solution I know I must use the Fundamental theorem of plane curves but I don't know how to apply it correctly here.

General Relativity says that every object that has mass make a curvation to space-time. Ι want to ask from what depends the curvation. Only from his mass? It depends from the size of object? For example let's say that we have one object with 10 meter size,and an other with 1000,but they have...

Consider the case of a right circular helical curve with parameterization \(x(t) = R\cos(\omega t)\), \(y(t) = R\sin(\omega t)\), and \(z(t) = v_0t\). Find the curvature and torsion curve. http://img30.imageshack.us/img30/7828/gwi.png We can then parameterize the helix \begin{align*}...

What is the radius of curvature formula for an ellipse at slope = 1? I have found b^2/a, and a^2/b for the major and minor axis, but nothing for slope = 1. Thanks.

Does anyone know if it is possible to construct a compact 3-manifold with no boundary and negative curvature? I ask this question in the Cosmology sub-forum because I see in various writings of cosmologists that it is often taken for granted that a negatively curved Universe must be infinite...

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hi In a local inertial frame with g_{ij}=\eta_{ij} and \Gamma^i_{jk}=0. why in such a frame, curvature tensor isn't zero? curvature tensor is made of metric,first and second derivative of metric.

Homework Statement Find the equation of scalar curvature for homogenous and isotropic space with FLRV metric. Homework Equations ## R=6(\frac{\ddot{a}}{a}+\left( \frac{\dot{a}}{a}\right )^2+\frac{k}{a^2}) ## The Attempt at a Solution ##G_{AB}=R_{AB}-\frac{1}{2}Rg_{AB}##

Homework Statement Show that all components of Riemann curvarue tensor are equal to zero for flat and Minkowski space. Homework Equations The Attempt at a Solution ## (ds)^2=(dx^1)^2+(dx^2)^2+...+(dx^n)^2 \\ R_{MNB}^A=\partial _{N}\Gamma ^{A}_{MB}-\partial _{M}\Gamma ^{A}_{NB}+\Gamma...

if space curve C=<f(t),g(t),h(t)>, and v=\frac{dC}{dt}=<\frac{df(t)}{dt},\frac{dg(t)}{dt},\frac{dh(t)}{dt}> Why is curvature defined this way? κ\equiv\frac{d\widehat{T}}{dS} \hat{T}=unit tangent vector S=arc length to elaborate, for a space curve, i understand what \frac{dT}{dt} is, but what...

Hello, would someone know what is the smallest radius of curvature achievable with current gradient index optics (GRIN) technology? I mean, how much could one "curve" a ray of light? Many thanks! :smile:

Gravitation is described on one hand as curvature of space in the presence of matter. It is also described as a field acting through gravitons on matter. How can the two views be reconciled?

Homework Statement Let C be a curve given by y = f(x). Let K be the curvature (K \ne 0) and let z = \frac{1+ f'(x_0)^2}{f''(x_0)}. Show that the coordinates ( \alpha , \beta ) of the center of curvature at P are ( \alpha , \beta ) = (x_0 -f'(x_0)z , y_0 + z) Homework Equations The...

Just wondering, if the way to describe the movement of objects through spacetime is to say that they fall through the curves created in 4D spacetime, then is it a stupid question to ask why objects don't rise through spacetime? Or is this the same thing and rising and falling are one of the same...

hello Can you perhaps explain what does the Riemann curvature scalar R measure? or is just an abstract entity ? What does the Ricci tensor measure ? I just want to grasp this and understand what they do. cheers, typo: What DO they measure in the title.

Homework Statement Find the curvature K of the curve, where s is the arc length parameter: \vec{r}(t) = \langle 2 \cos t , 2 \sin t, t \rangle Homework Equations s(t) = \int_a ^t ||\vec{r}'(u)||du The Attempt at a Solution I know I need to find the arc length function, in order to find the...

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

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...