Curvature Definition and 872 Threads

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...

In Differential Geometry by Heinrich Guggenheimer (if you have the book, the proof I am asking about is Theorem 2-19), he gives the angle between a chord through points s and s' and the tangent at s, as the integral of the curvature (with respect to arc length) from s' to s. I'm not sure how he...

I just read a sentence in GRAVITATION by MTW (aka the "Princeton Phonebook") that made me realize a confusion wrt the metric, connection, and curvature. In short how are g_{\mu\nu}, \Gamma^{\alpha}_{\mu\nu}, and R^{\alpha}_{\beta\mu\nu} distinguished? They all include the description "how space...

Suppose one had a solid sphere just slightly larger than its Schwarzschild radius. What would the curvature of the surface look like to a local observer? Would it curve downwards, or appear flat, or curve upwards? If my brain was working a bit better today, I'd calculate it myself from the...

May i know how to find the curvature of a point on a wave function? e.g wave function: y=Asin(wx) and i want to find the curvature of a point at for example x=x0 and y= Asin (wx0) thank you very much

If you take a homogenously distributed spherical mass and compress it to a smaller radius while maintaining its overall total energy and momentum, will it change the curvature of space (gravity) outside of the original sphere? According to Newton, it stays the same. However, it seems like...

Homework Statement Show, that a three-dimensional space with constant curvature K is charaterized by the following equation for the Riemann curvature tensor: R_{abcd} = K \cdot \left(g_{ac}g_{bd}-g_{ad}g_{bc}\right) Homework Equations The Attempt at a Solution Hi folks, I would like to...

Hi, I'm new here. I want post a specific question that's been rattling around in my head. Basically, if you consider the curvature of 3 dimensional space into a 4'th dimension due to gravitational field, has anyone considered the 'direction' of that curvature ? If you think about the...

If the Ricci scalar R happens to be zero (everywhere according to our metric), is that the definition of a 'flat' space time? And how are flat space times related to Minkowski space precisely? ARE they the SR space exactly? Thanks. Just trying to understand why 'flat space time' is a...

Assuming gravity is matter curving space as Einstein says, isn't our theory of dark matter just an assumption that because more gravity is required to explain galaxy formation that it must be caused by unseen matter? Why do we assume that the curvature of space required must be caused by matter...

hello i understand that in a flat space the metric is \eta_{uv}dx^udx^v...i know that this means that the light follows straight geodesic in this space time... but ¿what would means that metric is f(t)\eta_{uv}dx^udx^v where f(t)=infinite in t=0 and f(t)=0 in t=infinite...obvious i...

Do Einstein's field equations explicitly show that energy alone can curve the metric of spacetime? True, energy is included in the stress-energy tensor, but is it assumed that energy in of itself curves spacetime? Or, is it possible that only energy "embedded" in mass contributes to...

How can one work out what terms like: (g^{cd}R^{ab}R_{ab})_{;d} are in terms of the divergence of the Ricci curvature or Ricci scalar? One student noted that since: G^{ab} = R^{ab} - \frac12 g^{ab}R {G^{ab}}_{;b} = 0 that we could maybe use the fact that G^{ab}G_{ab} = R^{ab}R_{ab} - \frac12...

Hi, From what I know, science is the study of the observable world, its theories are supported by evidence. Now GR is a theory, and it informs that mass curves the 'fabric' of space and time. The thing I don't understand is that there is no evidence of mass curving spacetime, then how is...

Homework Statement I have a graph of y=lg(x) which is supposed to mimic the curvature of a beam, or I can use y =√x to be more precise. But in essence between two points x2 and x1, I need to find the radius of curvature R so as to find the bending stress on it. Homework Equations...

Hello, I am trying to find an interpolating curve between a few points that has minimal curvature. That means, as close to a straight line as possible. Reading a document about cubic splines, they say that \kappa \left ( x \right )=\frac{|f''\left ( x \right )|}{\left ( 1+\left [ f'\left...

Homework Statement This is not exactly a question, but I am trying to understand the derivation of radius of curvature from a boof I'm reading. I would be extremely grateful if someone is able to help me. Homework Equations Let u and n be the tangent and normal unit vectors respectively...

Homework Statement I had my college math courses in 1955-1957, so I'm rusty. Lately interested in Radius of Circle of Curvature. I don't have a math typing program, so I'll try to describe the equation that I found recently, but it's complexity [though so far, I can handle any common...

If we parameterize the arc length of a vector valued function, say, r(s) and r(s) has constant curvature (not equal to zero), then r(s) is a circle. Thus, |T'(s)| = K but to prove it we would need to show |T'(s)| = K => <-Kcos(s), -Ksin(s)> and integrate component-wise two times, right?

Homework Statement Find the unit tangent, normal and binormal vectors T,N,B , and the curvature of the curve x=−4t y=−t2 z=−2t3 at t=1. Homework Equations The Attempt at a Solution I found T=(-4/sqrt(56),-2/sqrt(56),-6/sqrt(56)) which is correct. But I keep getting N wrong...

"Visualizing" Ricci curvature Can someone help me visualize the Ricci curvature? Since it is easier to visualize a surface bending in 3-D, let's try to view this as a sheet with one spatial dimension and one time dimension and embedding into euclidean 3-D. Since the metric can always be...

A (3-d or higher) metric which is flat except for one non-trivial metric function of a different coordinate - eg changing dx2 to f(y)dx2 in Euclidean or Minkowski metric [but not f(x)dx2] - is curved if f(y) has a non-zero second derivative; there is no way to make the f(y) 'disappear', ie to...

Homework Statement I am using the boundary element method to solve unknowns to the Laplace equation from classic potential flow theory for the time evolution of a fluid air interface. At each time step, I need to solve a material derivative equation numerically at every node along an interface...

Is there any particular reason that scalar curvature is defined R = g^{ab}R_{ab} instead of R = R^{ab}R_{ab} ? Do both scalars share the property that they are zero if and only if every component of R^{a}_{bcd} is also zero?

hello! I need to find curvature tensor of sphere of R radius. How can I start? thanks!

After a while of trying to understand this, it's still kind of confusing. I get how the planets orbit around the sun because of the sun's spacetime curve, but if one were to drop something, how does spacetime curvature cause it to fall? How can everything in the universe have a gravitational...

A proton is deflected by a magnetic field, and curves away. It states that just before it curves, the force acts towards the centre of curvature. Is this the point where it turns, i.e. the turning point of the parabola, or the centre being like the centre of a circle?

General Relativity explains that every single piece of mass alters the space-time fabric, creating a curve. But since we live on a three-dimensional universe shouldn't the curvature alter the fabric in every dimension, creating some kind of weird closed curve that might look as a bubble around...

Ok, I have been wondering. I heard that Space/time is Curved when it comes into contact with mass correct? If this is so, then would it be possible to calculate the volume(or some equation) for the curvature of space/time around my own mass or a mass larger then mine? I have already...

Say we have a real field that satisfies: E^2 = P^2 + m^2 Assume spacetime is 4D. Assume the field is at rest and grab a single point of this field and slowly displace it a distance x. Just as an anchored string (string with an additional sideways restoring force) with fixed end points will...

I got this question on an Assigment, and ..man, am having really hard time with it Here is how it start : "A student used the apparatus shown below to measure the radius of the curvature of the path of electrons as they pass through a magnetic field that is perpendicular to their path. This...

Hi, I am having some problems understanding this concept, I hope you can help. I studied on Hobson, Efstathiou and Lasenby, in chapter 16 on Inflationary cosmology that in cosmological perturbation theory we need to express quantities in a gauge invariant way, very clear so far. The problem...

in this article http://en.wikipedia.org/wiki/General_relativity it says in the first paragraph that the curvature of spacetime is related to the stress energy tensor i.e. the matter in the universe. i.e. the curvature at a particular point depends on the matter present at that point...

Homework Statement How do I show that the curve makes a constant angle with the vertical (k) direction? Homework Equations The Attempt at a Solution I know the curvature. What else do I need to know?

i am trying to make a piece of stainless steel of varying thickness bend in such a way that it creates a cam profile in much the same way a fishing rod works. for example if the thick end has a 5mm by 10mm cross section and the thin end is 1mm by 10mm crosssection and the change in thickness...

An automobile moves at a constant speed over the crest of a hill traveling at a speed of 85.7km.h. At the top of the hill a package on a seat in the rear of the car barely remains in contact with the seat. What is the radius of curvature (m) of the hill? 85.7km/h=2.38m.s v=2.38m/s...

I have some questions related to bivector space, the curvature tensor and Cartan geometry. 1) Because of its antisymmetric properties R_{\mu\nu\alpha\beta}=-R_{\nu\mu\alpha\beta}, R_{\mu\nu\alpha\beta}=-R_{\mu\nu\beta\alpha}, the Riemann curvature tensor can be regarded as a second-rank...

Homework Statement Oh boy... Here we go... At what minimum speed must a roller coaster be traveling when upside down at the top of a circle so that the passengers will not fall out? Assume a radius of curvature of 7.4 m. I honestly have nothing. To begin with, I drew a free-body diagram...

(i) show that R_{abcd}+R_{cdab} (ii) In n dimensions the Riemann tensor has n^4 components. However, on account of the symmetries R_{abc}^d=-R_{bac}^d R_{[abc]}^d=0 R_{abcd}+-R_{abdc} not all of these components are independent. Show that the number of independent components is...

Homework Statement A factory has a machine which bends wire at a rate of 4 unit(s) of curvature per second. How long does it take to bend a straight wire into a circle of radius 8? My professor likes to give us things we've never seen before on our weekend quizzes. The above is one such...

Homework Statement The motion of a particle is defined as: x=[(t-4)3/6]+t2 y=t3/6-(t-1)2/4 Find the acceleration and radius of curvature at t=2 Homework Equations a=(dv/dt)(et)+(v2/\rho)(en) where et and en are the tangential and normal unit vecotrs to the curve and \rho is...

A curved space and be flattened out locally, giving a flat "subspace" while increasing the curvature around that "subspace", right? If so, wouldn't it be possible to make any curved space flat everywhere and concentrate all the curvature at a single point and move that point infinately far away...

Homework Statement Find the curvature \kappa(t) of the curve \\r(t)=(2sint)i +(2sint)j +(3cost)k Homework Equations \\\k(t)= (\left|T'(t)\right|) / (\left|r'(t)\right|) The Attempt at a Solution I found \\\\r'(t)= (2cost)i + (2cost)j + (-3sint)k \\\\\...

Kip Thorne says (Lecture in 1993 Warping Spacetime, at Stephan Hawking's 60th birthday celebration, Cambridge, England,) Comments, interpretations, appreciated. I thought classical time was always symmetric ...apparently not. Is this same description applicable to a "big crunch" as...

Hi, i read a few books (pop ones) and one thing that left me thoroughly confused was the relation between graviton and space curvature, is there any relation between the two... if we find a graviton then does it mean that space is not curved? regards Monty

Can anyone prove the following formula: R_{abf}^{\phantom{abf}e} \Gamma_{cd}^f = R_{abc}^{\phantom{abc}f} \Gamma_{fd}^e + R_{abd}^{\phantom{abd}f} \Gamma_{cf}^e I found it in "General Relativity" by Wald (in slightly different notation).

In the BBC film Einstein and Eddington, Eddington describes the theory of spacetime using a table cloth (space), a loaf of bread (sun) and a piece of fruit (a planet). The Bread is placed in the middle of the table cloth, this forms curves in the cloth. He then takes a piece of fruit and...

Please help me!A Problem about radius of curvature Please forgive my poor English! I have been thinking about a problem in Courant's <Introduction to Calculus and Analysis> for 2 days.After trying all possible methods, I am now exhausted ,almost give up and lose hope.really need someone's...

Homework Statement The hyperbola y = 1/x in the first quadrant can be given the parametric definition (x, y) = (t, 1/t), t>0. Find the corresponding parametric form of its evolute, and sketch both curves in the region 0<x<10, 0<y<10 Homework Equations Curvature formula...

My understanding is that a physical object moving through a spatial curvature gradient (as distinguished from spacetime curvature gradient) will not automatically experience an internally stress-free change in its physical dimensions consistent with the changing background spatial geometry. But...