Curvature Definition and 872 Threads

Homework Statement verify that |T ' (s)| = |T ' (t)| / |r ' (t)| Homework Equations K = |dT / ds| K = |r'(t) x r''(t)| / |r'(t)|^3 the x is a cross product The Attempt at a Solution I don't know how to start this problem because one side of the verification is in terms of arc length...

Is the curvature of GR accumulative? Can you integrate the curvature immediately around various points to find the curvature around a larger area? Thanks.

Consider the curve which is graph of a smooth function $$ f : (a,b) → R$$. Show that at any $$ {x}_{0}\:s.t\:{x}_{0} ∈ (a,b)$$ the curvature is $$\frac{{f}^{''}({x}_{0})}{{(1+{{f}^{'}({x}_{0})}^{2})}^{3/2}}$$.

Let k(x) be the curvature of y=ln(x) at x. Find the limit as x approaches to the positive infinity of k(x). At what point does the curve have maximum curvature? You're supposed to parametrize the graph of ln(x), which I found to be x(t)=(t,ln(t)). And you're not allowed to use the formula with...

A maximally symmetric is a Riemannian n-dimensional manifold for which there is n/2 (n+1) linearly independent (as solutions) killing vectors. It is well known that in such a space $$R_{abcd} \propto (g_{ab}g_{cd} - g_{ac}g_{bd}) .$$ How is this formula derived for a general maximally...

Why curvature of a plane curve is k = \frac{d\phi }{ds} ? I know that curvature of a plane is \frac{\left | r'(t) \times r''(t) \right |}{\left | r'(t) \right |^3} , and that led to this. k = \frac{\left | \frac{d^2s}{dt^2} \right |sin\phi }{\left | \frac{ds}{dt}\right |^2} But I can't go...

The Friedmann equation states that $$(\frac{\dot a}{a}) = \frac{8\pi G}{3} \dot \rho + \frac{1}3 \Lambda - \frac{K}{a^2},$$ where ##a, \rho, \Lambda, K## respectively denotes the scale factor, matter density, cosmological constant and curvature. Now, I'm trying to get at an intuition on...

I am trying to expand $$\varepsilon^{{abcd}} R_{{abcd}}$$ by using four identities of the Riemann curvature tensor: Symmetry $$R_{{abcd}} = R_{{cdab}}$$ Antisymmetry first pair of indicies $$R_{{abcd}} = - R_{{bacd}}$$ Antisymmetry last pair of indicies $$R_{{abcd}} = - R_{{abdc}}$$...

If I understand everything correctly, space near (but outside) a mass is curved negatively, so that if I create a triangle with, for example, rigid rods and the mass in its center, the angles would sum up to less than 180°. (If I am mistaken, please correct me.) On the other hand, the typical...

I’ve just read Schutz: “Gravitation from ground up”. He says that there are 4 (not all independent) sources of gravity: 1. density of active gravitational mass = (rho + 3 P) 2. active curvature mass (generating spatial curvature) = (rho - P/c^2) 3. ordinary momentum...

So the universe starts with an amount of matter, radiation, a fixed spatial curvature constant and a cosmic constant. Due to the expansion of the universe matter and radiation dillute and the spatial curtvature decreases whereas the cosmic constant remains fixed. Radiation dillutes faster...

1. Compute the curvature. α(t) = (cos^3t, sin^3t) This is not a unit-speed curve. I want to use κ(t) = \frac{||T'(t)||}{||σ'(t)||} When I find α'(t) and then its norm, I run into an impasse. Am I supposed to use a trig identity?

I was recently trying to test something out with the Riemann tensor. I used only 2 dimensions for simplicity sake. As I was deriving the Riemann tensor, I noticed that it looked as if all of the elements were going to come out to be 0 (which they all did). Therefore, this coordinate system is...

String effective action: S=-\frac{1}{2\lambda_{\text{s}}^{d-1}}\int d^{d+1}x\sqrt{|g|}e^{-\phi}\left[R+(\nabla\phi)^2+2\lambda_{\text{s}}^{d-1}V(\phi)-\frac{1}{12}H^2\right]+S_m where H^2=H_{\mu\nu\alpha}H^{\mu\nu\alpha}\\ H_{\mu\nu\alpha}=\partial_\mu B_{\nu\alpha}+\partial_\nu...

Homework Statement If two lenses have the same radii of curvature but different indexes of refraction their focal lengths won't be equal because the one with the greater index of refraction will undergo greater refraction and will have a smaller f. Doesn't this contradict the definition of...

We know how the curvature of a vector V or a manifold is depicted by the following formula: dx\mudx\nu[∇\nu , ∇\mu]V Now we know that the commutator is simply the Riemann tensor. My question here is: How do you actually apply that vector V to the Riemann tensor? Here is an example of what I...

Homework Statement A lens of power of -5.0D has a surface which is convex of radius of curvature of 15.0cm. The lens is made of material of refractive index of 1.50. What's the radius of the other surface of lens? Homework Equations The Attempt at a Solution since power = 1/f...

Homework Statement by taking the lower curvature as r1 , and the upper curvature as r2 , i don't know whether r1 is 20cm , r2 is 10cm or vice versa. But according to the ans r1= 10 cm , r2= 20cm . why is it so? Homework Equations The Attempt at a Solution

Homework Statement when the glass is partially cut( as shown in the photo ) , the centre of curvature is inside the denser medium (glass), so the centre of curvature should be lower than point Q in the diagram . am i correct? by saying that the centre of curvature is inside the denser medium...

How can we define SPACE TIME CURVATURE with respect to dark energy and dark matter ?

1. Gravity is the geometric curvature of space-time caused by massive objects. 2. Dark Matter surrounds galaxies. 3. Dark Matter is thought to be critical in galaxy formation. 4. The mass of Dark Matter creates curvatures in space-time around baryonic matter which forms galaxies. What roles...

The Riemann curvature of a unit sphere is sine-squared theta, where theta is the usual azimuthal angle in spherical co-ordinates, and this is shown in many textbooks. But since a sphere is completely specified by its radius, then as far as I can see its curvature should be a function of its...

In the formulation of connections on principal bundles, one derives an expression for the covariant exterior derivative of lie-algebra valued forms which is given by $$D\alpha = d \alpha + \rho(\omega) \wedge \alpha,$$ where ##\rho: \mathfrak g \to \mathfrak{gl}(\mathfrak g)## is a...

I wonder if curvature necessarily means space has been removed. The typical example is forming a "curved" surface by cutting out a triangle from a flat surface, and then gluing the remaining side back together. This forms of a cone which is a type of curved surface. What is the generalization of...

What does spacetime curvature means?

Hey guys, I'm working on a summer research project right now in diff. geo. I'm at the point where I have to define the spin coefficients for my spacetime. I'm following an appendix in another paper related to my problem (the equivalence problem for 3D Lorentzian spacetimes). In the appendix I...

I am working on a paper that provides the following formula for computing radius of curvature at a point on a surface. \frac{1}{\rho_c}=\frac{\partial G/\partial S}{2\sqrt{E}G} where E,G are first fundamental coefficients and S is the arc length parameter. Can anyone please tell me the...

After my studies of metric tensors and Cristoffel symbols, I decided to move on to the Riemann tensor and the Ricci curvature tensor. Now I noticed that the Einstein Field Equations contain the Ricci curvature tensor (R\mu\nu). Some sources say that you can derive this tensor by simply...

A second SR question that has been on my mind lately is that of hyperbolic nature of Minkowski space. The fact that the invariant interval, or lines of constant delta S trace out a hyperbola according to the equation, ##x^2-(ct)^2=S^2##, is fascinating to me and seems to imply that space-time...

When we talk about space-time curvature or the curvature of space, how many different "types" of curvature are there according to GR? For example, the rounded surface of a cylinder is curved in only 1 dimension, while the other is flat. For a sphere, both dimensions of the surface are curved...

I am trying to compute the geodesic (or tangent) radius of curvature of the geodesic circle by using the below formula. \frac{1}{\rho_c}=\frac{\partial G/\partial S}{2\sqrt{E} G} where s is the arc length parameter and E, G are the coefficents of the first fundamental form. Can you...

Does anyone have a good reference or references that go into detail on rigorous/formal developments of 2-point correlation functions for curvature perturbations (and related perturbations) in the cosmological context? I'm using the TASI lectures in inflation, Mukhanov, and Dodelson but none of...

Hey there, This isn't a homework question, it's for deeper understanding. So I'm learning about unit normal/tangent vectors and the curvature of a curve. I have a few questions/points. 1) So my book states that we can express acceleration as a linear combination of the acceleration in the...

In the Euclidean plane, assume a differentiable function $$y=f(x)$$ exists. At any given point, say $$(x_0,y_0)$$, the line tangential to $$y=f(x)$$ at this point intersects the x-axis at an angle $$\phi$$. The curvature of this curve, $$\kappa$$, is the rate of change of $$\phi$$ with respect...

Hi All: I am curious about the definition of mean curvature and its apparent lack of invariance under changes of coordinates: AFAIK, mean curvature is defined as the trace of the second fundamental form II(a,b). II(a,b) is a quadratic/bilinear form, and I do not see how its trace is invariant...

Homework Statement Find the curvature of ##r(t) = \frac 4 9 (1+t)^ \frac 3 2 i + \frac 4 9 (1-t)^ \frac 3 2 j + \frac 1 3 t \hat k## at t=0 Homework Equations K=1/|v| * |dT/dt| The Attempt at a Solution Found v. ##v= \frac 2 3 (1+t)^ \frac 1 2 i - \frac 2 3 (1-t)^ \frac 1 2 j + 1/3 \hat...

Is it possible for a riemann manifold to change its curvature? In practice could the universe in general change its curvature by time? (let's say in the past it was negative and today it's almost flat tending to positive); If not which theorem disproves it?

I am intrigued to see what spacetime curvature is like in reality. Most images or ways to imagine it tend to look at spacetime as a fabric which it is not precisely. So how would be best to imagine it... Do any of the picture demonstrate this? What is the best way to imagine it?

I was listening to one of Leonard Susskind's cosmology lectures. He talks about the factor K having values of +1/0/-1 corresponding to positive/flat/negative curvature. We don't know what the real value of K is. But then as he discussed K at the big bang and K now, it seemed that he was...

Why can't EM attraction/repulsion be modeled as spatial curvature the way gravity can be? And for that matter, why can't the strong and weak nuclear forces be modeled that way either? Or can they?

Homework Statement Homework Equations r=mv/qb mv=sqrt(2*KE*m(alpha)) m(alpha)=6.64e-27 kg The Attempt at a Solution i was just wondering how to get the answer (7.6e-4 m). i get path curving down, and do r=sqrt(2*1e3eV*6.64e-27kg*1.6e-19J/eV)/(q*B) =15.18e-4m so to get the...

Light bending in the "hot plate" model of curvature In the Feynman lectures, feynman describes the hot plate model of space curvature and shows that light is bent around the center of the plate, see Fig. 42-6 http://www.feynmanlectures.caltech.edu/II_42.html#Ch42-S1 However, the hot plate...

Hello, this isn't a homework problem, so I'm hoping it's okay to post here. I would like to know the correct way to mathematically express the idea in my title. It is intuitively obvious that as the radius of a circle increases, it's curvature decreases. I looked it up and found that...

Hi Bear with my possible ignorant. I am puzzled over this dilemma. If General Relativity states that gravity is the curvature of spacetime, that is, no spacetime no gravity, and the cause of curvature is matter (mass), it means that if no matter, there is no gravity. I understand that...

The problem is stated in the attachment. I would include my attempt at the question if I got anywhere. I'm really only looking for a hint as to how I set up the solution. PS, I understand how to work out the angle if the car wasn't moving. Thanks Homework Statement...

Urgent!How to find the point where Maximum Curvature occurs on graph? Homework Statement The graph(2D) is plotted from experimental data, not from a given equation and within certain limit. I want to know the x&y coordinate where the maximum curvature occur via software means, not manually...

Hi everybody, As you know, the Einstein field equation R_{μ\nu} - 1/2Rg_{μ\nu} =κT_{μ\nu} implies that at any point with vanishing energy-momentum tensor the Ricci curvature also vanishes: T_{μ\nu} = 0 \Rightarrow R_{μ\nu} = 0 hence a Ricci-flat space-time (the vacuum...

I've been studying the Einstein field equations. I learned that the Ricci curvature tensor was expressed as the following commutator: [∇\nu , ∇\mu] I know that these covariant derivatives are being applied to some vector(s). What I don't know however, is whether or not both covariant...

All over the news we see the results of the recent detection of gravity waves from the early universe. Which got me wondering: The early universe was much more dense than at the present. It therefore seems that spacetime was much more curved than it is, on average, today. Is this...

Most of us are familiar with the fact that in Loop gravity the area and volume observables have discrete spectrum. The discrete spectrum of the area operator, leading to a smallest positive measurable area, has lots of mathematical consequences that have been derived in the theory. It helps...